You may read it now or later:
A Case Study on the Paradox of
Free Choice Permission
— Master Thesis —
Master of Logic Program/ ILLC
Katrin Schulz
supervised by Prof. Dr. F. Veltman
1
Contents
1 Introduction 3
2 Classical Approaches
to the Free Choice Paradox 6
2.1 Performative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Constative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 The Proposal 18
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Free Choice as Clausal Implicature:
The Proposal of Gazdar . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.3 The Data in Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Developing the Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 A Gricean Notion of Entailment . . . . . . . . . . . . . . . . . . . . 26
3.2.2 Information Orders. Introduction . . . . . . . . . . . . . . . . . . . . 27
3.2.3 How to establish Properties of Minimal States . . . . . . . . . . . . 28
3.2.4 The Basic Information Order . . . . . . . . . . . . . . . . . . . . . . 29
3.2.5 Information Orders with Modal Information . . . . . . . . . . . . . . 33
3.2.6 The Objective Information Order . . . . . . . . . . . . . . . . . . . . 34
3.2.7 Competence and Multi-Modal Belief . . . . . . . . . . . . . . . . . . 38
3.3 Applying the Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1 Solution 1: Weakening the Order . . . . . . . . . . . . . . . . . . . . 40
3.3.2 Solution 2: Weakening the Logic . . . . . . . . . . . . . . . . . . . . 43
3.3.3 Solution 3: The Combination . . . . . . . . . . . . . . . . . . . . . . 45
4 Discussion of the Proposal 46
4.1 Comparing the three proposed Approaches . . . . . . . . . . . . . . . . . . 46
4.1.1 Strong Competence and Weak Order . . . . . . . . . . . . . . . . . . 46
4.1.2 Weak Competence and Weak Order. . . . . . . . . . . . . . . . . . . 48
4.1.3 Weak Competence and Strong Order . . . . . . . . . . . . . . . . . . 49
4.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Evaluating the Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 Conclusions 58
2
1 Introduction
Certainly one the most influential pragmatic theories of the 20th century is the theory
of conversational implicatures proposed by Grice [1989]. It has been applied to various
semantical problems, inspired many developments in pragmatics, and also received attention
in other areas like philosophy and social sciences.
Grice developed the theory as a reaction to the growing bulk of criticism brought forward
against the fundamental idea of formal semantics: to describe natural language semantics
by mapping natural language to some formal language for which a precise and well-studied
semantics is given. The wide range of problems which classical truth-conditional semantics
had to face led an increasing number of scholars to question this logicistic program.1 Grice
proposed to rescue the logical stance towards semantics by explaining the critical data as
due to additional, pragmatic interpretation mechanisms (this enterprise will be called in
the following the Gricean Program).
Making pragmatics responsible for the problems truth-conditional semantics had to
face asks, of course, for a powerful theory of pragmatics that can account for the critical
observations. This role Grice intended his theory of conversational implicatures to play. He
claimed that besides semantical conventions there are also rationally motivated rules that
govern our use of language. According to him these rules, the maxims of conversation, are
not arbitrary but determined by the common interest of the participants to maximize the
gains of communication. The inferences that can be derived on the assumption that the
speaker behaves according to the maxims are called conversational implicatures.
Irrespective of its popularity, the theory of conversational implicatures has also been under
constant attack. A central objection is that the account lacks precision. The resulting
vagueness in its predictions makes it impossible to evaluate the question whether Grice’s
proposal does rescue truth conditional semantics from its threads. One of the main goals of
pragmatics has been to overcome this defect and to propose a more detailed description of
the class of conversational implicatures (e.g. Horn [1972], Gazdar [1979], Hirschberg [1985]).
However, a completely satisfying proposal in this direction is still missing. This raises the
question whether a precise formulation of the theory is possible at all2. Perhaps Grice’s
attempt to save classical truth-conditional semantics may have only shifted the problem to
the realm of pragmatics. Now, it is this part of interpretation that lacks a logic.
However, there are good reasons to believe that the mentioned attempts to improve on
the clarity of Grice’s theory did not exhaust their possibilities. When looking at the proposals
made it catches the eye that a rather limited set of technical tools are used to
formalize Grice’s theory. The main role is still played by classical deductive logic; the logic
of Frege and Tarski. But also logic has had its revolutions since their times, among them
the development of non-monotonic reasoning.
Non-monotonic logics were invented in particular to describe common sense reasoning,
1A similar development can be observed in cognitive science half a century later. There a lot of evidence
seems to show that not only language but human reasoning in general is not driven by exact logic; see, for
instance, the argumentation of van Lambalgen & Stenning [2001] contra such a conclusion.
2See the very sophisticated discussion by Davis [1998] who indeed argues that Grice’s theory of conversational
implicatures cannot be made precise.
3
especially to account for the strange fact (logical speaking) that in every day life people
jump to conclusions even if they are not absolutely reliable. This has the consequence that
when later on evidence occurs that shows the conclusions drawn to be false, then, of course,
these ‘rash’ conclusions have to be abandoned. This property, that one loses inferences if
they stand in conflict with more reliable information, also gave those logics their name: nonmonotonic.
In this respect they clearly differ from classical deductive logic where inferences
once obtained will always be valid no matter how much information is added.
Non-monotonicity has always been considered to be a central feature of conversational
implicatures.3 This suggests that techniques developed in non-monotonic logic may be of
use to formalize the theory of Grice. However, this possibility has been widely ignored until
now in the research on conversational implicatures.4 This is even more surprising given that
there are development in this area that bear a close connection to the theory of Grice. For
instance, the notion of only knowing developed by Halpern & Moses (1984) and recently
generalized by van der Hoek et al. (2000) shows a strong relation to one of the maxims
postulated by Grice: the first submaxim of Quantity.
The goal of the research presented in this thesis was to investigate in how far tools of
non-monotonic logic can help to make the theory of conversational implicature precise and
the Gricean Program successful. However, to pursue this aim with respect to Grice’s theory
and its adequateness in general would not have been achievable within the limits of a master
thesis. Therefore, we will concentrate on one particular phenomenon that challenged
classical truth-conditional semantics and study the questions whether the notion of conversational
implicature can be made precise insofar as to account for this concrete observation.
The phenomenon investigated is free choice permission. As has often been observed (e.g
von Wright [1969] and Ross [1941]) that an utterance of (1a) intuitively entails (1b) and
(1c): on hearing (1a) we will conclude that the addressee is allowed to take an apple and
she is allowed to take a pear (although not both at the same time). She has free choice
between both fruits.5
(1) a. You may take an apple or a pear.
b. You may take an apple.
c. You may take a pear.
The discussion that arose around the treatment of this observation mirrors to a great
extend the general problems truth-conditional semantics got into and from which Grice
intended to rescue it: It appeared to be impossible to give formally precise meanings to the
involved expressions in a way that can account for the free choice reading - at least, not
without making unacceptable predictions in other respects. Therefore, von Wright decided
3However, in the linguistic literature they do not refer to this property as non-monotonicity but call it
instead the cancellability of conversational implicatures. This terminus has been also used by Grice himself.
4An exception is the dissertation of J. Wainer [1991], who tries to formalize scalar implicatures using
McCarthy’s [1986] notion of circumscription. In Levinson [2000], it is pointed out that using such techniques
may be very useful for the problem of formalizing these inferences but this idea is not worked out further.
5Even though free choice permission is a phenomenon that does apply to a wide range of constructions
we will concentrate here on English sentences containing the connective ‘or’.
4
to speak of the paradox of free choice permission, and the question was raised whether one
should not conclude that the way English speakers (and the same observation is made in
many other languages too) understand (1a) simply has no logic.
In this thesis we will take a Gricean stance towards the problem. Hence, we will not try
to unmask the paradox as illusion by proposing an adequate semantical description of the
sentence, but instead look for a Gricean, pragmatic explanation for free choice permission.
We will additionally follow Grice insofar as we try to keep the semantics of (1a) simple and
close to classical truth-conditional semantics.
In sum, the thesis pursuits two different but strongly connected aims. First, it tries to
account for a notorious problematic observation concerning the interpretation of certain
sentences of English: the paradox of free choice permission. More particularly, the intention
is to account for this observation in terms of Grice’s theory of conversational implicatures.
Secondly, before this theory can be used first it has to be made precise. Here, the intention
is to provide a (part-wise) formalization using results from non-monotonic logic.
The thesis is organized as follows. In the next section we will start with a more extended
discussion of the phenomenon of free choice permission. The aim is to get a good
understanding of the phenomenon that has to be described. Also two classical accounts to
the problem will be discussed that do not only constitute central pillars in the literature
on this topic but also build the fundament of the approach developed here: the extensive
discussion of free choice permission by Hans Kamp and the proposal of Ede Zimmermann.
Afterwards, in section 3, a part-wise formalization of Grice’s theory of conversational implicatures
is proposed. Building on the work of Halpern & Moses [1984] a pragmatic notion
of entailment is introduced that is intended to describe the conversational implicatures due
to the first submaxim of Quantity and parts of the maxim of Quality out of Grice’s inventory
of conversational maxims. The inferences obtained this way essentially come down
to Gazdar’s [1979] clausal implicatures. We will show that such a notion of entailment
together with an assumption of competence of the speaker - and here we build strongly on
Zimmermann [2000] - allows the interpreter to derive the free choice permission. In section
4 we will discuss the proposal of the previous section. As it will turn out, the approach
has to fight with one central shortcoming: it over-generates. Some ideas where to look for
a solution of this problem will be introduced but a detailed investigation has to wait for
another location. We finishes with a section on conclusions and outlines of further research.
5
2 Classical Approaches
to the Free Choice Paradox
As stated in the introduction we will concentrate in this thesis on the question how to
account for the phenomenon of free choice permission. This section is dedicated to a closer
investigation of the problem we have to solve. Furthermore, a selection of approaches to
the paradox of free choice permission will be discussed that were of great influence to the
account presented here.
The reader might remember from the introduction the example used to clarify the notion
of free choice permission. The critical observation was that an utterance of (2a) intuitively
entails (2b) and (2c): on hearing (2a) we will conclude that the addressee has the permission
to take an apple and she has the permission to take a pear.
(2) a. You may take an apple or a pear.
b. You may take an apple.
c. You may take a pear.
The challenge for a semanticist who faces this observation is now to assign meanings
to the expressions involved in (2a), describe how these meanings are to be combined and
perhaps other processes involved in interpreting this utterance such that finally she can
correctly predict the way English speakers understand sentences involving these expressions.
In particular, a notion of entailment | between sentences of English has to be definable
in the theory of interpretation (probably depending on the context) such that the following
relation holds:
(FCP) (2a) | (2b) ^ (2c)
However, this seems to be not at all a simple task. Traditionally, ‘or’ receives the same
semantics (the same truth conditions) as the corresponding logical operator _ in classical
logic: for two sentences ‘A’ and ‘B’, ‘A or B’ is true in case at least one of the both sentences
holds. This truth function is called an interpretation as inclusive disjunction.6 But if one
assumes ‘or’ to take wide scope over ‘may’ (and, hence, that the surface structure of (2a) is
due to ellipsis), then the inference (FCP) will not be valid. It would hold, however, if in this
wide scope analysis ‘or’ were interpreted as conjunction. That led some scholars to suggest
that in this context ‘or’ has indeed a conjunctive interpretation (e.g. Jennings [1994]). But
notice, that there is a difference between the interpretation of (2a) and the one of (3).
(3) You may take an apple and a pear.
While the latter allows the addressee to take both (at the same time), (2a) certainly excludes
this possibility. Hence, ‘or’ has to make its own contribution to the interpretation
6This is, thus, the semantic analysis Grice would like to defend with his theory of conversational
implicatures.
6
of (2a) and cannot be simply taken to be semantically the conjunction in this context (as
Jennings [1994] does propose).
Let us do a step backward. We observed that assuming ‘or’ to take wide scope over ‘may’
in (2a) and interpreting it as inclusive disjunction does not allow us to account for the inference
pattern (FCP). Still, a classical approach to ‘or’ could be rescued by claiming that
‘may’ has wide scope over ‘or’ in (2a) and then proposing a logic of ‘may’ that is operating
on the disjunction in its scope in a way that the free choice inferences becomes valid.
Unfortunately, there are some problems with such a strategy. First, it has been observed
by different authors (e.g. Zimmermann [2000]) that a parallel sentence giving ‘or’ explicitly
wide scope (see (4)) does also allow a free choice permitting reading.
(4) You may take an apple or you may take a pear.
Second, as we will see in this section, several attempts to make the meaning of ‘may’ precise
were not able to give an explanation of the free choice effect in a semantical way and, even
more, the question has been raised whether such a way of proceeding can be successful
at all. A very cautious approach to the meaning of ‘may’ is to try to axiomatize it by a
list of intuitively valid statements and rules of derivation and then look for a sound and
complete semantic. However, it seems to be very difficult to find a system of logic that
captures the observed inference. The system either has no theorem may(p _ q) ! may(p)
or, if we add the inference as an axiom, produces unintuitive consequences. For instance,
it seems natural to assume a theorem may(p) $ ¬ must (¬p), but together with the free
choice axiom we can now derive by some simple assumptions about the behavior of ! and
the axiom ¬¬p $ p that must(p) ! must(p ^ q), which is absurd. Because of apparently
fundamental misfit between how the logic of ‘may’ is supposed to look like and the way we
understand and reason with sentences such as (2a) von Wright [1969] spoke of a paradox of
free choice permission.
Although we will restrict us in this work to free choice readings of utterances containing
the sentence connector ‘or’, it is important to notice that ‘or’ does not seem to stand
alone with many of its strange interpretation properties, among them the one discussed
here. It shares a pattern of variance in interpretation with a whole class of expressions
of English and this holds even cross-linguistically. This class can be roughly described as
expressions standardly analyzed as existence quantifiers. In (5) some examples are given.
(5) Du darfst dir einen meiner Stifte nehmen.
Du darfst dir irgendeinen Stift aus der blauen Tasche nehmen.
Du kannst irgendwann morgen vorbeischauen.
The conclusion we have to draw from this observation is that, on the one hand, and
fortunately, if we have a solution for the puzzle of ‘or’ this answer may be extended immediately
to a whole class of problems. On the other hand, it also means that if our solution
cannot be generalized, we have to motivate why we need an inhomogen analysis.
7
2.1 Performative Approaches
An extensive discussion of the problem of free choice permission can be found in two papers
([1973] and [1979]) of Hans Kamp. In his first paper on this topic, Kamp claimed that
the paradox of free choice is due to the illegitimate step of reasoning with a descriptive
logic on performative utterances. Thus, the mistake made by earlier attempts to find a
semantics for ‘may’ that can account for the problematic inference was to use the wrong
kind of semantic theory to describe the meaning of permissions as (2a). In turn, the thesis
is that if one takes the right speech act - sensitive semantics the free choice inference will
fall out as semantic consequence. Given this position, the program of Kamp was to develop
an extension of recursive truth-conditional semantics to a semantics of permissions and to
show that the latter accounts for the free choice reading.
Kamp starts with reviewing the work of Lewis [1970] who has given a model-theoretic
analysis of the illocutionary force of permission and obligation sentences. The utterance of
such sentences in a context w is interpreted as an action changing the deontic options left
to the hearer (slave) by the speaker (master) in w. The deontic options are modeled by the
set of worlds in which the hearer fulfills all the restrictions previously imposed on him by
the speaker. Obligations, obl , are now interpreted as imposing additional restrictions on
this set: the options shrink to those worlds where holds (i.e. to those where its truthconditions
are fulfilled).7, 8 Permissions, on the other hand, extend the deontic options of
the hearer. By uttering a permission per some of the worlds where holds become deontic
options of the hearer. This adds a non-monotonic element to the logic of permission: earlier
valid obligations will be lifted and become invalid in turn.
In this setting Kamp sketches the following solution of the choice paradox. Sentences of
the form ‘NP may VP’ are treated as the sentence per([NP V P]) in the formal language of
Lewis.9 Kamp proposes that the notion of inference also changes with the speech act. He
defines a new notion of entailment !P for permissions: a permission sentence per entails
a permission sentence per if in all contexts the latter is redundant after executing the
former. per is redundant after per if all options that per would add are enclosed in the
options added by per . Let [per ]p
w be the set of worlds added to the options by uttering the
permission per in context w, then per permission-entails per (per !P per ) iffdef
[per ]p
w [per ]p
w. This notion of entailment is now proposed by Kamp to be responsible
for the free choice inferences. Hence, what he has to show is that according to the proposed
account for the meaning of permissions the options added by the permission of a disjunction
per( _ ) have to enclose the options added by the permission of per and per .
It is clear that whether the idea works depends on how the set of options contributed
by a new permission is defined. But, as the title of his article signals, Lewis observes that
this is no trivial question to solve. Suppose the options are restricted by the orders to clean
the house and not to drink any beer from the fridge. Now the speaker permits the hearer to
drink a beer from the fridge. We can in reaction not add all worlds where the hearer drinks
7... in some nearby future. We slightly simplify he sophisticated model theory used by Lewis and Kamp.
They take deontic options to be variants of the actual world (they have the same history) where all obligations
are fulfilled in some nearby future.
8This is the same kind of operation that an assertion executes in dynamic semantics on the common
belief state in the utterance context w.
9However, no connection is made by Kamp between ‘may’ and per.
8
a beer from the fridge to the option sets. This is so, because they also include worlds where
she does not clean the house, for instance. So, how is the extension operation restricted?
The idea on which Lewis and Kamp base some of the answers they consider is that
we have to add only worlds that are maximally close to the previous option set. The
induced change on the deontic alternatives has to be in some sense minimal. Different
approaches were discussed by Lewis, Stalnaker and Kamp to make this idea more concrete
but they generally fit in one formal pattern: the update is modeled as contraction
with respect to an absolute, transitive and irreflexive order
This order is intended to closeness to the deontic options of the hearer. The deontic options,
in consequence, constitute the minimal elements of the order. A permission per
is interpreted as adding the closest worlds making true to the set of deontic options:
[per ]p
w = {w1 2 [ ]w|¬9w2 : w2
in consequence, constitute the minimal elements of the order. A permission per
is interpreted as adding the closest worlds making true to the set of deontic options:
[per ]p
w = {w1 2 [ ]w|¬9w2 : w2
The central question such an approach raises is how to specify the order, which, in turn,
determines the properties of the function [·]p
w. Stalnaker suggested to define it as reprehensibility:
how much the hearer deviates from the optimal conformistic behavior. This would
mean that a permission per adds the set of least reprehensible -worlds to the deontic
options. Unfortunately, the notion of ‘reprehensibility’ is still too vague to make concrete
predictions. One may interpret it as counting how many previously imposed obligations the
hearer breaks in a world - an idea behind database accounts independently developed by
Lewis [1970] and Kamp [1973] and generalized in the work of Harper [1976].12 Kamp gives
a different utility oriented interpretation of reprehensibility: according to him, permissions
can have different weights depending on their costs for the speaker. But no matter which
elaboration is chosen, the question is: can it account for the free choice inferences?
It is easy to see that a minimal contraction account based on any transitive, irreflexive
and total order per( _ ) !P per( ) ^ per( ) gives us exactly in case the minimal worlds
of each disjunct are all of equal distance from the optimum.13,14 Hence, the free choice permission
is not generally valid for minimal contraction of the option set. However, it may be
the case that the specific order chosen to model change on deontic options allows to derive
the choice principle in those cases where it actually occurs. But neither of the ideas on
what reprehensibility may mean we discussed above will have this property. For instance,
Kamp claims that there are contexts where it is intuitively more reprehensible to take a
pear than to take an apple but where utterances of (2a) would still allow for free choice
10The terminus contraction is taken over from the literature of belief revision. A contraction is an operation
defined on a preference structure: a class of models M together with an order. Applying the operation to a
sentence gives you the minimal elements among the subclass of M where is true.
11That this idea was so present in the discussion is not surprising: just a few years before Lewis had
published his very influential book on the semantics of counterfactuals in which he developed exactly this
kind of preferential interpretation mechanism.
12Harper himself did not apply his system to permission sentences. Only many years after the publication
of Harper’s work van Rooy [2000] made an attempt in this direction.
13That means, no worlds where is true is more minimal than any world where is true and the other
way around.
14We restrict our considerations here to the narrow scope analysis, which is considered by Kamp. Of
course, one could also think about the behavior of a wide scope disjunctions per( ) _ per( ) in such a
setting, which would come with its own problems. For instance, what should it principally mean to utter a
disjunction of two performatives?
9
permission. Hence, his utility based perspective on the order will make false predictions.
And, as van Rooy [2000] points out, neither the order introduced by Harper [1976] makes
the right predictions. We have to conclude that these theories about the meaning of (2a)
cannot account for the free choice inference.
Facing the problems of his proposal, Kamp introduces a new kind of explanation for free
choice permission: maybe it is not due to language conventions of how to interpret permissions,
maybe it is due to reasoning on rational behavior in conversational contexts - maybe
it is a Gricean conversational implicature! Hence, Kamp adopts the strategy of the Gricean
program: rescue semantics by letting pragmatics solve the problems. The difference is that
it is Kamp’s performative permission semantics that has to be saved now.
Kamp discusses a derivation via the maxim of Brevity, one of the rules that, according
to Grice, is obeyed by rational and cooperative speakers. It asks the speaker, roughly,
to keep his contributions short. The derivation of free choice permission Kamp proposes
based on Brevity runs as follows: if it was the intention of the speaker to allow only one of
the disjuncts (as Kamp’s semantics would predict in case the wrong results are obtained)
there would have been a shorter expression the speaker could and, following brevity, should
have used, namely (2b) or (2c). However, she did not. Hence, this cannot be the intended
interpretation of the utterance. But so far, the reasoning brings us only half the way to
the free choice inferences. Now, Kamp continues, we have to rationally motivate why a free
choice permitting reading should be chosen as the intended interpretation, instead.15 Kamp
does not provide an answer to this question, but one may think of the following continuation
of the argument:16 the hearer considers the possibility that she based her interpretation
on the wrong ordering of worlds and tries to figure out the right one. But she cannot take
the order to prefer the other disjunct instead because then the same argument would apply
again. Hence, she concludes that the speaker based her utterance on an order where both
disjuncts are equally reprehensible.
But besides the objection that it is difficult to motivate why the interpreter should
think that the mistake sits precisely in her idea of the order, there is something strange
with Kamp’s explanation in general. As Kamp [1973] notices himself and as was also pointed
out in the introduction, the surprising properties of the interpretation of ‘or’ are displayed
by other quantificational expressions as well. A sentence like “You may pick any flower.”
also receives a free choice interpretation. But here Kamp’s complexity argument will obviously
fail: listing the kinds of flowers that may be picked will raise the length of utterance.
We conclude that also this Gricean attempt to derive the free choice inferences based on a
performative semantics is not very convincing.
If this were not bad enough already, the performative approach to sentences containing
‘may’ has to face other problems as well. Merin [1992] discusses extensively the perspectives
of an account based on a contraction function and presents a whole list of objections.
Among other things he raises a question that often goes unnoticed in discussing the behav-
15This is the big problem any derivation of Brevity implicatures has to face: they easily exclude certain
interpretations but have problems in singling out the interpretation obtained instead. Already Grice notices
that and suggests that brevity implicatures are actually more conventional than conversational.
16An alternative has been suggested and criticized by Merin [1992].
10
ior of ‘or’: do the approaches brought forward also cope with other coordinators, especially
with ‘and’ ? As he points out, there is a general problem with the minimal contraction
approach when it comes to interpret conjunctive permissions. According to it, a permission
per(A _ B) adds to the options of the addressee the closest worlds where A ^ B is true.
Hence, all the options added this way will be such that both conjuncts are true. The package
deal reading is obtained: no matter what the order is the hearer is not permitted to do one
thing without the other. While there is a reading of (6) that can be paraphrased as ‘You
may take an apple provided that you also take a pear, and vice versa.’ this interpretation
only seldom occurs.
(6) You may take an apple and a pear.
The dominating interpretation of sentences as (6) seems to be that the speaker allows the
addressee to choose between four options (e.g. van Rooy [2000]): to take both, an apple
and a pear, to take either or to take neither of them. No account using contraction will get
rid of the package deal prediction.17
To the difficulties for the contraction approach to permission sentences we discussed above,
others have to be added that concern performative approaches in general.
First, permission sentences, especially the examples under consideration, are in the indicative
mood. Given that this mood normally corresponds to assertion one should give the
constative approach a serious try before the dominant speech act - sentence mood parallelism
is broken. Furthermore, Kamp observes that the sentence (2a) has also a reportive,
and thereby assertive use, and even then a free choice reading can occur. Thus, we have to
come up with an additional explanation for free choice readings of assertive utterances.
Another problem is that the free choice reading is not restricted to sentences containing
‘may’ but seems to generalize to other types of disjunctions as well. Kamp’s examples are
(Kamp [1979], p. 281)
(7) a. We may go to France or stay put next summer. (with the epistemic reading of
‘may’)
b. I can drop you at the next corner or drive you to the bus stop.
c. If John had come to the party with Alice or with Joan we might have had some
fun.
17Notice, that this is above all a problem for a narrow scope analysis: when the performative update works
on the permission per( ^ ). One may think about solving the problem by proposing (i) that in case the
reading without package deal occurs the conjunction has wide scope over the permission (but remember that
Kamp does not make any connection between per and ‘may’), and (ii) that a conjunction of performatives
is interpreted as successive execution of the subordinated permission sentences. Of course, this can only be
implemented in a system where permission/obligation sentences do not only update the deontic options but
the whole order on which the contraction operation for permission sentences is defined (as, for instance, the
approach of Harper [1976]). However, while we would get rid of the general prediction of the package deal
reading this way, whether we get the dominant four - choices - reading described above would still depend
on the specific order we chose.
11
hence, the free choice inference is not bound to a specific speech act. It may even occur with
sentences that seem to have primary assertive use. This is no principal argument against
a performative approach, but it teaches us that the explanation given for choice inferences
has to fit in a more general frame.
A stronger point against performative accounts is the following. Reportive uses of
permission sentences do not always allow for free choice inferences, especially not when the
utterance is continued as in (8) or if for other reasons it is known that the knowledge of the
speaker on the topic of interest is limited.
(8) You may take an apple or a pear - but I don’t know which.
In consequence, if we try to describe free choice in reportive sentences we have to make
sure that the inference is not always derived and its occurrence is bound in some sense to
what is known about the competence of the speaker. This could be ignored by an account
that only tries to deal with performative permission sentences. However, Kamp claims that
there are also permissions that have no free choice inferences, for instance (9), used by a
father addressing his son.18 (Kamp [1979], p. 271)
(9) You may go to Shoal Creek or go to Shingle Creek. But stay away from the dangerous
one.
Kamp suggests the following explanation for the difference in reading. In case we observe
free choice permission the act in each disjunct is permitted. In (9), on the contrary, the
disjunction as a whole is permitted and extends the deontic options. He considers the possibility
that a scope ambiguity serves as a basis for this difference in interpretation. Again,
Kamp is not satisfied with this analysis, among other points for the following reason. Utterances
of (4) repeated here as (10), for which an underlying narrow scope logical form would
be difficult to motivate, prominently get the incompetence and not the free choice reading
while Kamp’s proposal takes this logical form to underly the free choice interpretation.
(10) You may take an apple or you may take a pear.
Finally, also a serious gap in every performative proposal Kamp discusses speaks against
his way of approaching the problem: there is no interpretational contribution of ‘may’. One
may think that ‘may’ is the part of the sentence that makes it a permission, but Kamp never
brings the performative back to the occurrence of this verb, nor does he derive the use of the
performative interpretation function compositionally from this part of the structure.19 So,
we are faced with the very strange situation that a systematic ingredient of the utterance
form of permissions does not contribute to the interpretation.
18Merin [1994] supports the analysis of (9) as performative by pointing out that (9) can be put in the imperative
mood, while ‘may’ sentences continued with ‘... but I don’t know which.’ resist such a transformation.
19Perhaps this is so for the following reason. If ‘may’ would contribute compositionally the imperative
mood then the performative interpretation function would only play a role in the scope of the verb. But
what is then the interpretation function applied on top of the whole sentence?
12
2.2 Constative Approaches
Facing all these difficulties, Kamp finally gives up his initial position to account for the
paradox of free choice permission by a performative interpretation of sentences as (2a).
Instead, he tries to arrive at the choice inferences by taking the involved utterances to be
assertions coming with truth-conditional meaning. ‘May’ is now interpreted as an existential
modal operator claiming that among the deontic options of the hearer in a world w there
is an option where holds.20
Also for this constative and modal approach Kamp has to observe that the free choice
principle does not hold simply by the semantics outlined above: based on the classical
notion of entailment for assertions it is still not valid that (2a) entails (2b), for instance,
no matter whether a wide or narrow scope analysis of the disjunction with respect to the
modal operator is taken. Again, Kamp considers the possibility to get the inference as a
conversational implicature using the maxim of Brevity. The situation is slightly different
from the one we considered in the last section when he tried to rescue the performative
account by a Gricean argument - simply because different meanings are assigned to (2a).
With respect to the performative proposal, the interpreter had to face a maxim clash: the
meaning predicted by semantics was not compatible with the maxim of Brevity. Considering
the constative interpretation, there is no obvious shorter expression to express the assigned
truth conditions. This time Kamp claims, that, in a first step, by brevity all possibilities
can be excluded where the speaker knows what is the single disjunct that is permitted.
The reason Kamp gives is that if the speaker had this knowledge she should have used a
shorter expression. Hence, Kamp here uses brevity not as demanding the speaker to code
the meaning she wants to convey the shortest way possible, but to do so with her knowledge.
The interpreter learns that way that the actual world has to be such that the speaker takes
every disjunct to be possibly permitted. Kamp now claims there are only two situations
where this can be the case: (i) the speaker is incompetent, or (ii) the speaker is indifferent.21
If it is known by the interpreter that the former is not the case, then she infers that the
speaker is indifferent and hence that every disjunct permitted.
The free choice inference here becomes a context dependent phenomenon bound to
what is known by the interpreter on the competence of the speaker. This picture fits the
observations with which we finished the last section much better. But again, the proposal
lacks details. Kamp notices this but does not give any elaboration of his ideas, instead
he immediately presents arguments that make the correctness of such a Gricean account
questionable. He observes that the choice inference can also occur if the permission sentence
is embedded in a complex construction as in (11) and can even contribute to the semantic
meaning of this construction.
20We learn not much about the character of the accessibility relation Kamp assumes besides that it is
reflexive: the actual world is among the deontic options. Hence, he only allows consistent sets of obligations
to be imposed on the hearer.
21Given Kamp’s view on assertive permissions (they report that the performative act has occurred) one
may question his decision to contrast incompetence of the speaker with indifference instead of with competence.
It is here much less clear why the indifference of the speaker will warrant that the permission has
actually been given. This is more obvious in case reporting permissions is not about previously executed
acts of permitting but for instance, about reporting preferences of the speaker. The force of permissions may
then be due to a sharing of preferences between speaker and hearer that determines an authority relation.
13
(11) Usually you may only take an apple. So if you may take an apple or take a pear, you
should bloody well be pleased.
How can a pragmatic process as the concept of conversational implicature be that deeply
involved in the calculation of the semantic meaning of (11)? Other counter arguments can
be brought forward. Remember the observation made in the last section: the free choice
inference also occurs with utterances as ‘You may take any apple you want’. Again, Kamp’s
argumentation using Brevity would not generalize to these cases. Furthermore, Brevity cannot
do the job Kamp assigns to it. According to Kamp’s derivation the utterance has to be
a maximally short expression of the knowledge of the speaker. But Brevity, as formulated
by Grice, only asks the speaker to efficiently code the meaning she wants to convey. This
maxim can, hence, not be the reason why the speaker should intend to communicate all she
knows.22
To summarize, Kamp left us with a broad and sophisticated discussion of what does not
work when one tries to account for the paradox of free choice permission and provides a
much more general picture of the phenomenon we have to describe. However, he also left
us the problem.
Several years later another very interesting constative approach to the problem has been
brought forward by Ede Zimmermann [2000]. A central feature of his account is a more
general perspective on the phenomenon. Zimmermann aimed at an account of the free
choice inferences that can also deal with (7a). He even added to the list of phenomena that
have to been explained the observation that a simple factive sentence involving ‘or’ as in
(12) allows to infer (in an honest conversation) that the speaker takes both conjuncts as
epistemic possibilities.
(12) Peter or Marie took the beer from the fridge.
Zimmermann’s approach to the free choice inferences is based on two fundamental
claims.
• Semantically, ‘or’ denotes a conjunction of epistemic possibilities which he speaker
sees23, (hence (12) is interpreted as stating that the speaker thinks it possible that
Peter took the beer from the fridge and the speaker thinks it possible that Marie took
the beer from the fridge).
• The choice principle is a pragmatic inference based on the assumption that the speaker
is competent.
These two assumptions together allow him to account for the free choice inferences based
on a logical form where ‘or’ has wide scope over the modal operators. Let’s see how this
works.
22But for this demand other conversational maxims can be made responsible. Actually, this will be the
fundament of the approach presented in the third section of this thesis.
23Zimmermann does not make the restriction to the belief state of the speaker, but we can ignore this
aspect for our purposes.
14
‘May’, shortened as 4, and ‘might’, shortened as 3, are analyzed, as in Kamp [1979], as
unary modal operators. Hence, the formal system in which the proposal is situated is modal
(predicate) logic. The operators are interpreted with respect to accessibility relations24 R4
and R3, respectively. R3 connects a world w of a model M with all words the speaker takes
as epistemically possible in w. Zimmermann does not model belief but knowledge, hence,
in every model M = hW,R4,R3, V i every world is among its epistemic alternatives: 8w 2
W : hw,wi 2 R3. R4 connects w will all deontic alternative states of affair. Remember,
that truth is defined in modal logic with respect to a state: a tuple of a model M and a
world w in the set of worlds WM in M. For a sentence the truth conditions of 4 are
defined as follows: M,w |= 4 iff 9v 2 WM : hw, vi 2 R4 ^ M, v |= . Truth of 3 is
defined analogously: M,w |= 3 iff 9v 2 WM : hw, vi 2 R3 ^M, v |= .
Now, we can formalize Zimmermann’s semantic analysis of ‘or’ in the following way:
M,w |= or ' ,def M,w |= 3 ^ 3'.
Zimmermann formalizes competence using Groenendijk & Stokhof’s [1984] definition of
exhaustive knowledge:
A speaker is competent with respect to a predicate P (with domain D) in a state
hM,wi iffdef 8v 2 W : wR3v ! (8x 2 D : v 2 P(x) $ w 2 P(x)).
From the definition of competence Zimmermann derives the following Authority Principle:
If a speaker is competent with respect to predicate P in state hM,wi then it holds:
8x 2 D : (9v 2 W : wR3v ^ v 2 P(x)) ! (8v 2 W : wR3v ! v 2 P(x)). That means, if
a speaker competent with respect to P takes it as possible that some object has property
P then she knows that the object has the property.25 For the derivation of the choice principle
Zimmermann instantiates P with the set of deontic options of the hearer in w, the
set {v 2 W|hw, vi 2 R4}. For a speaker competent with respect to this set, the authority
principle says that if she takes a certain world to be possibly a deontic option then she
knows that it is a deontic option.
Now, we have everything in place to face the choice inferences again. We start with the
plain disjunction A or B. By the semantic interpretation rule the sentence is equivalent to
3A^3B - what, given the introduced semantics, immediately accounts for the interpretation
of (12). Next, we apply the account to the example (2a) containing a deontic modality.
4A or 4B is semantically interpreted as 34A ^ 34B. If the speaker is competent with
respect to the deontic alternatives of the hearer in state hM,wi, and, hence, the authority
principle holds we can infer M,w |= ¬3¬4A ^ ¬3¬4B. Now, Zimmermann uses his
assumption that the beliefs of the speaker are correct (reflexivity of R3) to make the step
to M,w |= 4A ^ 4B, the free choice inference.26 Finally the epistemic case: (7a). 3A or
24Binary relations on the set of worlds WM of a model M.
25The derivation of the authority principle uses both directions of the equivalence in the definition of
competence.
26The author does not understand why Zimmermann takes the detour via knowledge. The inference
34 ! 4 is simply the “!” direction of the definition of competence that he already uses in deriving
the authority principle. Hence, reference to reflexivity is not necessary to account for the free choice
interpretation.
15
3B is semantically equivalent to 33A ^ 33B. To allow a derivation from w |= 33 to
w |= 3 Zimmermann introduces the Self-Reflection Principle: Any world w satisfies
8v 2 W : wR3v ! (8v0 : vR3v0 $ wR3v0). This frame condition legitimates the desired
derivation and therefore we can state that also for these disjunctions Zimmermann is able
to derive the choice inference: w |= 3A ^ 3B.
There have to be didactic reasons why Zimmermann introduces the self-reflection principle
additionally to his notion of competence, because the former principle is nothing more
than an instantiation of his concept of competence: now, P is taken to be the set of epistemic
options of the speaker. Thus, Zimmermann’s proposal is more uniform than it may look at
first sight. Furthermore, assuming a speaker to be competent with respect to R4, her own
epistemic state, is equivalent with taking the accessibility relation R3 to be transitive and
euclidic. These conditions are very popular if it comes to characterize the way we reason
about our own beliefs. They assign an agent positive27 and negative28 introspective power
with respect to her belief state. This connection adds to the plausibility of the proposed
formalization of competence.
The derivation of the free choice inference via a competence axiom has another attractive
consequence. As Zimmermann noticed himself, he can straightforwardly account for
the observation that free choice permission is not always observed and that it is omitted
especially in contexts where the speaker explicitly states his incompetence: the derivation
of free choice permission simply relies on taking the speaker to be competent. On the other
hand, if the disjunctive sentence is about the epistemic state of the speaker, one observes
that the choice principle seems to be always present (if the speaker is taken to be honest).
This can now be explained by pointing out that we take every healthy speaker to be a priori
competent on his own belief state.
As these results show: Zimmermann’s proposal is much more successful in accounting for
the free choice inference than any previously discussed alternative. But it has to face its
limitations. We have seen above how the approach generates the correct predictions in
case ‘or’ has wide scope over the modal operators. However, the interpretation mechanism
can also be run on the syntactic analysis giving ‘or’ narrow scope - unfortunately producing
very unsatisfying results. Zimmermann noticed this himself and proposes to exclude narrow
scope readings for independent reasons, but this part of his proposal seems a bit ad hoc.
Zimmermann makes also wrong predictions in case ‘must’ occurs instead of ‘may’. It
seems straightforward to analyze deontic ‘must’ as the dual of ‘may’: r =def ¬4¬, and
epistemic ‘must’ as the dual of ‘might’: 2 =def ¬3¬. But then, his semantic analysis of a
sentence like ‘You must or '’ is: 3r ^ 3r'. Under the assumption, that the speaker
is competent you can now derive that r ^ r', hence, You must and you must ' - a
result that is obviously too strong.
Such sentences, which contain necessity modalities ((13) gives another example) are
rarely discussed in the free choice literature. The simple reason is that they seem to be
unproblematic.
(13) Mr. X must take a taxi or a boat.
27If the agent knows then he knows that he knows .
28If an agent does not know then he knows that he does not know .
16
But, actually, they play their tricks with us, too. As Zimmermann noticed, also for imperatives
there exists an incompetence reading made especially strong by continuation with
‘... but I don’t know which’. But this is not the only possible interpretation. Zimmermann
observes ‘... there appears to be a more straightforward construal according to which Mr.
X’s obligation are unspecific as to the exact means of transport.’ (Zimmermann [2000], p.
283). Hence, there is a reading of (13) from which you can conclude (14a) and (14b), the
obligation (13) still allows Mr. X to chose which disjunct of the obligatory disjunction he
is going to fulfill.
(14) a. Mr. X may take a taxi.
b. Mr. X may take a boat.
The similarity between these inferences of (13) and the choice inference of (2a) plus the
parallel absence of the inferences if the speaker is taken to be incompetent strongly suggest
to speak also for (13) of a free choice inference and to look for one general explanation.29
Further support for this point comes from the fact that the observation extends to epistemic
‘must’:
(15) That must be Tom or Bill.
Inference: That must be Tom. & That must be Bill.
But, much stronger than for ‘may’, people have difficulties to get the free choice inference
for (16), where ‘or’ has explicitly wide scope over ‘must’, and even found it questionable
whether it can get such a reading at all.
(16) You must clean the kitchen or you must go shopping.
All this cannot be explained by the proposal of Zimmermann.
In our opinion, these mispredictions are mainly due to his semantic analysis of ‘or’. This
part of his proposal can even be criticized independently from the free choice topic. For
instance, if his analysis would be correct, how could it be that sentences like (17) are so
useful to communicate the speaker’s strong belief in the first disjunct - something obvious
if one assume inclusive truth-conditions for ‘or’?
(17) Peter is in love or I’m a monkey’s uncle.
And one should be clear about the fact that it is the idea to get the epistemic possibilities
for each disjunct from the semantic meaning of ‘or’ that bound his approach to a wide scope
analysis for this coordination in free choice sentences. This critic does not affect the role
competence plays in Zimmermann’s approach. On the contrary, this aspect of his theory is
very taking.30
29In view of this it becomes clear that the heading conjunctive reading or conjunctive inference, that is
often found in the literature, is clearly not the right peg for the free choice reading. The parallel sentence of
(13) with wide scope conjunction: ‘Mr. X must take a taxi and Mr. X must take a boat.’ does not describe
the observations. Furthermore, Kamp suggests in his papers that the free choice inference has to be seen as
essentially connected with constructions that bring possibilities to the attention of the hearer - this is what
all the sentences around (7a) have in common. The observation above makes that questionable or trivial.
30Notice the parallelism between Zimmermann’s approach and the involvement of competence in the
Gricean argument of Kamp’s constative proposal.
17
3 The Proposal
3.1 Introduction
We come now to the main part of the thesis. In the following, a new constative approach
to the free choice problem is developed. Because it is a constative approach, we will only
discuss the reportive use of permission and obligation sentences.31 And given the general
aim of the paper to defend the Gricean program, a central feature of the approach to be
presented will be that it fulfills the restrictions imposed by Grice on a theory on interpretation.
In particular we will assume a simple and classical semantics: ‘or’ will be interpreted
as inclusive disjunction and the modalities are analyzed as unary modal operators. We will
see that we can, nevertheless, account for the free choice inferences - if we treat them as a
pragmatical phenomenon.
The new proposal will display the same general structure that characterized the account
of Zimmermann [2000] and that is also present in the sketched Gricean and constative approach
of Kamp [1979]. Both establish the free choice inferences on two premisses. The first
ingredient was that the speaker considers certain states of affairs as epistemic possible.32
For our core-example (18a) it is the assumption that the speaker takes both, (18b) and
(18c) as possibly true. Sometimes, this information already accounts for the free choice
observation, for instance, for examples like (12): ‘Mary or Peter took the beer from the
fridge’. But for the free choice reading of (18a) a second premise was necessary and this
was the assumption that the speaker is competent on the subject she is talking about.
(18) a. You may take an apple or a pear.
b. You may take an apple.
c. You may take a pear.
It is obvious that a semantic theory as sketched above will not be able to provide us
with these requisites: given an analysis of ‘may’ as a modal operator on deontic possibilities,
(18a) does neither convey that (18b) and (18c) are possibly permitted according to
the speaker, nor that the latter is competent on the deontic options. And because neither
Zimmermann’s idea to put the first premise in the semantics of ‘or’ nor Kamp’s suggestion
to derive it as conversational implicature via Brevity is convincing we have to come up with
something new with respect to this part. However, we will follow Zimmermann in taking
free choice permission to depend on the information that the speaker is competent on what
is permitted.
The core problem of the new approach is thus to give a account of the first ingredient:
the derivation of information about the epistemic alternatives the speaker considers. We
will describe them as conversational implicatures derived from the assumption that the
31We will not argue for any position with respect to the questions whether there is a performative use and
how it should be treated.
32Remember, however, that both approaches differ in which level of interpretation they take to be responsible
for this information.
18
speaker obeys the maxim of Quality (T ) and the first submaxim of Quantity (Q1) as introduced
by Grice [1989]. We will give a precise description of a class of implicatures that
can be derived from these maxims and show that the needed inferences on the epistemic
state of the speaker are among them. The formal part is an application of work of Halpern
& Moses [1984], that has been recently generalized by van der Hoek et al. [1999, 2000].
This formalization serves at the same time as an argument against critics on the Gricean
program we started with: the notorious and justifiable complaint of Kamp and many other
scholars that all Gricean approaches lack (a convincing) formally precise implementation.
3.1.1 Free Choice as Clausal Implicature:
The Proposal of Gazdar
Actually, this is not the first time that the relevant information about the epistemic possibilities
of the speaker has been analyzed as due to the Gricean maxims T and Q1. At least
part-wise already Gazdar, in his dissertation from [1979], took them to be such conversational
implicatures.33 The account presented here is strongly related to his proposal. To
study Gazdar’s approach is therefore a good starting point to introduce some general ideas.
Gazdar distinguishes two classes of inferences which he analyzed as generalized Q1 implicatures.
The first class can be described as inferences that some kind of strengthening of
the claim made does not hold (as far as the speaker knows). These are the so-called scalar
implicatures. The second class consists of claims that the speaker does not know for some
strengthening of the given utterance whether it holds or not. Gazdar called this group of
inferences clausal implicatures. The following examples of the latter notion are given by
Gazdar (Gazdar [1979], p.50).
(19) a. If John sees me, then he will tell Margaret.
Implicature: I don’t know that John will see her.
b. My sister is either in the bathroom or in the kitchen.
Implicature: I don’t know that my sister is in the bathroom and I don’t know
that my sister is in the kitchen.
The reader will have noticed that the clausal implicatures of the second sentence give
exactly the kind of information that Zimmermann [2000] analyzed as the semantic contribution
of ‘or’, the information on the epistemic state of the speaker we are looking for. Here,
they appear as pragmatic inferences, as clausal implicatures of a simple34 disjunction. Of
course, this observation raises the question whether we can account for the free choice inferences
simply by combining Gazdar’s concept of clausal implicatures with Zimmermann’s
formalization of competence. This will now be investigated.
Gazdar describes the following procedure to calculate clausal implicatures. First, he
defines the set of potential clausal implicatures (pcis) of a compound sentence (Gazdar
33However, he did not consider an application to the paradox of free choice permission.
34A sentence is called a simple disjunction, if the only sentential operator it contains is ‘or’ and this
operator occurs only once.
19
[1979], p.59, 3 is defined as the dual of 2, and 2 stands for the speaker knows that ).35
|
2 {3 ,3¬ } and is a subsentence of
such that neither entails nor its negation¬
As the reader can see, clausal implicatures are essential inferences about the limitations
of the knowledge of the speaker. For instance, for example (19b) the definition predicts the
following set of pcis: { the speaker does not know that her sister is in the bathroom, the
speaker does not know that her sister is not in the bathroom, the speaker does not know that
her sister is in the kitchen, the speaker does not know that her sister is not in the kitchen}.
But not all potential clausal implicatures are predicted by Gazdar to become part of the
interpretation of an utterance. First, they have to pass a strict consistency check: A pci becomes
part of the interpretation of an utterance if it is a consistent extension of the common
ground together with the assumption that the speaker knows her utterance to be true36 and
all other potential clausal implicatures. Because of this filter-condition, clausal implicatures
become context dependent and, for instance, the misprediction of Zimmermann concerning
example (17) is avoided. However, if nothing special is commonly known, uttering ‘ or
’ where and are logically independent propositions will clausal-implicate 3 and 3 ,
Zimmermann’s semantic meaning of ‘ or ’ . Hence, Gazdar can account for the free
choice inference which a simple disjunction as (12) or (19b) normally comes with.
But Gazdar cannot only account for these observations. If we take sentences containing
epistemic operators, his approach also predicts the free choice inferences we observed.
Let’s see how. For a sentence 3( _ ) Gazdar predicts the following set of pcis:
{3 ,3 ,3¬ ,3¬ ,3( _ ),3¬( _ )} - among them the free choice inferences 3 and
3 . Whether the pcis are actually generated depends on the context. If the logic of the
knowledge operator is S4, as Gazdar assumes, and nothing else is commonly known on
and , then all pcis in this set are predicted to be actually derivable from the utterance.
Hence, Gazdar can also account for a free choice reading of 3( _ ).37 Let us consider
another case: 2( _ ). We arrive at exactly the same set of pci’s. The element 3¬( _ )
is not consistent with the assumption that the speaker believed what she said and will
therefore be not predicted as actual implicature. However, the free choice implicatures 3
and 3 survive the consistency check and, hence, Gazdar can account for the free choice
interpretation of examples like (15)!38
Unfortunately, Gazdar’s proposal makes generally incorrect predictions if applied to
disjunctions containing other modal operators 4 with 4 6= 3. In consequence, he cannot
account for free choice in deontic contexts. With respect to a narrow scope disjunction
4( _ ) there is no hope at all to obtain the needed implicatures, because in this case
there will not be any information about the modality 4 in the set of pcis of this formula:
35Gazdar gives one further condition, but this one can be ignored for our purposes. Furthermore, he
assumes S4 to be the logic of the modal operator 3.
36This is Gazdar’s formalization of the T -implicatures an utterance comes with.
37The wide scope analysis gets also a free choice reading in S4. However, this is not the case, if the logic
of 2 is assumed to be KD45. In this case the relevant potential clausal implicatures are generated but no
potential implicature will survive the consistency check (see the discussion below for 4 _4 ). Thus, even
semantically equivalent sentences may generate different sets of implicatures under Gazdar’s treatment.
38The pcis of a sentence 2 _ 2 are different, but still the free choice interpretation is derived. For the
logic KD45 we obtain the same results.
20
{3 ,3¬ ,3 ,3¬ ,3( _ ),3¬( _ )}. For the sentence giving the disjunction wide
scope, 4 _ 4 , the situation looks a little bit better. Gazdar predicts the following set
of pcis: {3 ,3 ,34 ,34 and the respective negations}. A first thing to notice is that
the first two members of this list together with their negative counterparts should not be
predicted as conversational implicatures. Though in case 4 in 4 _ 4 is interpreted as
deontic ‘may’ one (normally) infers from an utterance of this sentence that the speaker
takes the asserted deontic options also to be epistemically possible, this inference should
rather be analyzed as part of the appropriateness conditions (presuppositions) of permissions
(and obligations). To support this standpoint, notice that this inference accompanies also
sentences like (20), and, thus, projects trough negation.
(20) You must not take an apple or a pear.
That 3 and 3 should not be predicted to be conversational implicatures of an utterance
of 4( _ ) is even more obvious if one takes 4 to stand for other modalities like, for
instance, the belief state of other agents. While also in this case an utterance of (21) does
have a free choice reading (Peter thinks it possible that Mr. X is in Berlin and he thinks it
possible that Mr. X is in Amsterdam), one does not infer, of course, that also the speaker
thinks it possible that Mr. X is in Berlin.
(21) Peter believes that Mr. X is in Berlin or in Amsterdam.
But let us consider the other pcis Gazdar predicts for the sentence 4( _ ). Again,
Zimmermann’s semantic interpretation of the sentence is part of the pcis. Therefore, one
may think, if we add an assumption of competence to the context with respect to which
4( ^ ) is interpreted, then we will be as successful in deriving the free choice inferences
as was Zimmermann. But his formalization of competence will allow us not only to derive
from the potential clausal implicature 34 the fact 4 but also to conclude from its
negative counterpart 3¬4 that ¬4 holds.39 Hence, we cannot add the pcis we need
to such a context without generating contradiction with other elements of this set. They,
therefore, do not pass Gazdar’s consistency test and are predicted not to occur. The results
for sentences containing deontic ‘must’ are equally bad.40
After such a successful beginning the question arises: what went wrong? And can we
repair Gazdar’s approach without major changes? The discussion above revealed two different
problems of his approach. For one thing, there is something essentially wrong with
the generation process of the pcis, as, for instance, the unintuitive pci 3p for a sentence
4(p_q) shows. This part of his theory demands serious revision. In particular, the syntactic
aspect of Gazdar’s notion that allows you to take subsentences out of their modal contexts
has to be removed. The problem concerning the incompetence that prevents Gazdar from
predicting the free choice inference for 4 _4 , on the other hand, seems to be less severe.
To avoid it one could change the generation process so as to not include the negative pcis
or weaken the notion of competence, for instance.
39To see this, the axiomatization of Zimmermann’s definition of competence in section 3.2.7 may be of
use.
40There is, of course, no difference in the results if we would assume as epistemic logic KD45.
21
An objection against the proposal of Gazdar of a totally different character is that it
lacks a precise motivation as a formalization of the theory of Grice. And again, it is above
all the syntactic aspect of his description of pcis that does not seem to fit. Gazdar sketches
a derivation of the implicatures from the maxims Q and T that is in its structure as old
as the theory of Grice itself: if the (rational and cooperative) speaker had known the truthvalue
of a subsentence of her utterance, then she should, in order to obey Q1 and T , have
used a sentence that semantically conveys this information. Hence, the speaker cannot know
whether the sub-sentence holds or not. Given the formulation of the maxims it is absolutely
unclear, why this reasoning should apply to sub-sentences of the utterance, and to those
only.
To develop our approach and overcome the deficits discussed above we will again start
from Grice and motivate a new description of the conversational implicatures due to T and
Q1. In the end we will see that the inferences we describe can be best understood as, again,
Gazdar’s clausal implicatures. In particular, all the appealing inferences he predicts are
maintained.
3.1.2 Technical Preliminaries
But before we can really start introducing the new approach, the basics of the formalization
of natural language we will use have to be laid out. This will be done here: We will make
our model-language precise, define its semantics and introduce some technical machinery
we will need later on.
Language Our language L is generated from a finite set of propositional atoms P =
{>,?, p, q, r, ...}, the logical connectives ¬,^,_, and ! and finitely many unary modal
operators {41,42...,4n}. We will use ri to shorten ¬4i¬ for all i 2 {1, ...n}. 3 is
the special modal operator 4i for some i 2 {1, ..., n} that refers to the belief state of the
speaker and 2 ¬3¬ expresses in turn that the speaker believes . 4 and r are used
as meta-variables over the set of modalities of L.
| · | : L −! N denotes the modal depth of formulas, defined recursively: for p 2 P and
, 2 L, |p| = |>| = |?| = 0, |¬ | = | _ | = | ^ | = max{| |, | |}, |4 | = |r | =
| | + 1.
We will frequently make use of sublanguages L0 L. For instance, the languages
L0
(n) = { 2 L0| | | n, n 2 N}, the set of L0-sentences with modal depth smaller or equal
to n, will play a central role. We call L(0), the language that contains the modal-free part
of L, the basic language. Another important construct we will use is 2L0 = {2 | 2 L0},
the language that shifts L0 into the belief state of the speaker.
We introduce the following abbreviations for certain L formulas:
(K) r( ! ) ! (r ! r ), (4) 2 ! 22 ,
(Dual) r $ ¬4¬ , (5) ¬2 ! 2¬2 ,
(D) 2 ! 3 .
22
Semantics A frame for L is an (n+1)-tuple of a set of worlds W and for every modality
4 of L a binary relation R4 over W. A model for L is a tuple consisting of a frame for
L and an interpretation function V for the non-logical vocabulary of L; a function from
p 2 P to characteristic functions over W. Let F = hW,R41 , ...,R4ni be a frame for L and
M = hF, V i a model. For w 2 W, R4i [w] denotes the set {v 2 W|hw, vi 2 R4i}. We call
the tuple s = hM,wi, where w 2 W, a state. Truth of a sentence of L with respect to a
state is defined along standard lines. We will give here only the definition of truth for a
modal formula 4 : M,w |= 4 iffdef there is a v 2 W such that v 2 R4[w] and M, v |= .
A sentence is valid on a class of states S (|=S ) iffdef for all s 2 S: s |= . A set of
formulas entails a formula relative to a class of states S ( |=S ) iffdef for all s 2 S:
s |= implies s |= . If = { }, we write |=S . F,w |= holds iffdef for all valuations
V : hF, V i,w |= . F |= iffdef for all w 2 W: F,w |= .
Without going into much details here, models for L can also be treated as models for the
first-order language L1 corresponding to L (see Blackburn et al. (2001), definition 2.44).
For 2 L1 we write Mk − iffdef is true in the model M = hF, V i. Fk − iffdef for all
valuations V : hF, V ik − .
For A1, ...An 2 L, StateKA1...An is the set of states s = hM,wi with M = hF, V i such
that for all i 2 {1, ..., n}: F,w |= Ai.41 We shorten |=StateKA1...An
with |=KA1...An. In this
thesis we will work with (subsets of) StateKD45. The sentences (D), (4), and (5) locally
correspond to well known first order formulas, what allows us to get a good grasp of the
structure of the states in StateKD45.
Fact 1 42 Let F = hW,R41 , ...,R4ni be a frame for L and w 2 W. Then the following
correspondences hold:
F,w |= (D) iff Fk − 9x : R3(w, x)
F,w |= (4) iff Fk − 8x, y : R3(w, x) ^ R3(x, y) ! R3(w, y)
F,w |= (5) iff Fk − 8x, y : R3(w, x) ^ R3(w, y) ! R3(x, y)
We know, hence, that the underlying frames of the states in StateKD45 are with respect
to R3 locally (in w) transitive, euclidical and non-blind43. Conceptually, that means that
we adopt a particular perspective on how the belief state of a speaker looks like: the speaker
has positive and negative introspective power and the absurd belief state is excluded. This
restriction has been necessary because of the limited resources of the study at hand.
The reader may be surprised by the choice to ask only for the local validity of the
formulas (D), (4), and (5). In consequence, we allow models where, for instance, the speaker
thinks it possible that some other agent takes him to be in a non-transitive, non-euclidian,
or absurd belief-state. To realize the idea that KD45 is a general restriction of what belief
states can possibly look like, one should impose the axioms globally on the underlying
frames. Then one should, of course, also single out the class of modal operators that
describe the belief state of other agents and impose parallel conditions on them. One
41Hence, Ai is valid in w on the frame underlying M.
42Proof: Because (D), (4), and (5) are Sahlqvist-formulas we know immediately that they have firstorder
correspondents (see Blackburn et al. (2001), theorem 3.54). Applying the Sahlqvist - van Benthem
algorithm gives us the claimed equivalencies.
43A state shM,wi is non-blind in w with respect to R3 of M iffdef R3[w] 6= ;.
23
reason why this is not done here is that in this paper we will never come in a situation
where we will talk about such deeply embedded belief. Hence, the restriction would never
show up. Furthermore, later on we will consider restrictions on states that are only plausible
when imposed locally.44
Another concept that will - in different disguises - play an important role in the theory
we are going to develop is n-bisimulation. Let hM,wi, hM0,w0i 2 S be two states in S. We
say that hM,wi, hM0,w0i are n-bisimilar (hM,wi =n hM0,w0i) if there exists a sequence of
binary relations Zn · · · Z0 with the following properties (for i + 1 n):
(i) wZnw0
(ii) If vZ0v0 then 8p 2 P : V (p)(v) = V 0(p)(v0)
(v and v0 agree on all proposition letters)
(iii) If vZi+1v0 and u 2 R4[v]
then there exists u0 with u0 2 R0
4 [v] and uZiu0 (forth)
(iv) If vZi+1v0 and u0 2 R0
4[v0]
then there exists u with u 2 R4[v] and uZiu0 (back).
We will frequently make use of the following well-known theorem:45
Theorem 1 Let L be a modal language with a finite set of atomic propositions: |P| = m 2
N. For two states s, s0 2 S where S is a set of states for L the following are equivalent.
(i) s =n s0
(ii) s and s0 agree on all ' 2 L with modal depth at most n.
Finally, we define for A1, ...,An 2 L the KA1....An-closure (CLKA1....An(M)) of a model
M = hW,R41 , ...,R4n, V i as a model M0 with M0 = hW,R0
41
, ...,R0
4n
, V i 2 StateKA1...An,
for all 4i, R4i R0
4i
, and there is no M00 = hW,R00
41
, ...,R00
4n
, V i 2 StateKA1...An such
that for some 4i R4i R00
4i
R0
4i
.
3.1.3 The Data in Overview
Our aim is to describe the diverse free choice inferences we met in section 2. Because we have
to account for inferences, the core of the proposal should be to define a notion of entailment,
lets call it | , such that the relevant inferences become valid with respect to it. Given
our formal model-language L as described above this means that the following entailment
statements have to be valid. Let p, q 2 L(0) be two logically independent propositions
({A|B} has to be read as ‘A is the premise or B is the premise’):
(D1) p _ q | 3p ^ 3q
44The choice to work with local restrictions has some consequences for the logic of |=S: the normal modal
logic with additional axioms (D), (4), and (5) is not sound with respect to StateKD45. However, it is not
difficult to find a sound and strongly complete proof system for this class of states: the only thing one has to
do is to restrict in the normal modal logic with additional axioms (D), (4), and (5) the rule of generalization
to formulas that are also derivable in the minimal normal modal logic K. Because syntactic considerations
play no role in the rest of the paper we abstain from discussing this topic in more detail.
45For a discussion see Blackburn et al. [2001], p. 74.
24
(D2) {3(p _ q)|3p _ 3q} | 3p ^ 3q
(D3) {4(p _ q)|4p _4q} | 4p ^4q
with competence of the speaker.
(D4) {4(p _ q)|4p _4q} | 3(4p ^ ¬4q) ^ 3(4q ^ ¬4p)
without competence of the speaker, 4 6= 3
(D5) {2(p _ q)|2p _ 2q?} | 3p ^ 3q
(D6) {r(p _ q)|rp _ rq?} | 4p ^4q
with competence of the speaker.
(D7) {r(p _ q)|rp _ rq} | 3(rp ^ ¬4q) ^ 3(rq ^ ¬4p)
without competence of the speaker, r 6= 2
(D8) 3(p ^ q) | 3(p ^ ¬q) ^ 3(¬p ^ q) ^ 3(¬p ^ ¬q)
(D9) 4(p ^ q) | 4(p ^ ¬q) ^4(¬p ^ q) ^4(¬p ^ ¬q)
with competence of the speaker.
The restriction to logically independent propositions reflects the observation that we
want the inferences only to be generated in case they are consistent with the context. For
instance, we don’t want (D1) to be valid if the antecedent is the logical form of (17), here
repeated as (22).
(22) Peter is in love or I’m a monkey’s uncle.
(D2), (D3), (D5) and (D6) are the core observations: the free choice inferences for
epistemic and deontic modalities. Zimmermann has added (D1) to the free choice inferences:
the intuition that simple occurrences of ‘or’ (no additional sentence operator is involved)
allow the inference that both disjuncts are considered to be possible by the speaker. (D4)
and (D7) describe the alternative free choice free interpretation for the deontic examples.
We observed that these readings are particularly forced if the speaker states explicitly her
incompetence and that they are not possible for epistemic modalities. (D8) and (D9),
finally, refer to one of the points of criticism Merin brought forward against the minimal
contraction approach: it cannot handle conjunction properly. As we will show, our approach
will predict exactly the reading van Rooy [2000] described as the normal interpretation of
such sentences, namely (D8) and (D9).
No claim will be made in advance with respect to the choice of scope on which each
inference is based. As we have noticed above, free choice as well as the incompetence
inference can come with the surface narrow scope form - but there may be a deletion
process working on deep structure, hence, this does not allow us to conclude that both
inferences have to be derivable from a narrow scope logical form. For surface wide scope
we noticed that free choice readings are more difficult to get and are reported to be absent
for all-quantifying modalities. However, because this was not tested properly, we will not
exclude this analysis for free choice. We will just see what our approach predicts.
25
3.2 Developing the Framework
3.2.1 A Gricean Notion of Entailment
Our starting point is Grice’s theory of conversational implicatures, especially the first submaxim
of Quantity (Q1): Make you contribution as informative as required (for the current
purpose of exchange).46 Because of the contrast with the second submaxim and the
applications Grice discusses, this maxim is generally understood as an imperative to maximize
the quantity of relevant information that is transmitted.47 But the speaker is not
allowed to provide any bit of relevant information she can think of. The maxim of Quality
(T ) restricts her contributions to facts she believes to hold and for whose truth she has
sufficient evidence. Both maxims can be brought together in the following interpretation
rule for a hearer:48
The contribution of a rational and cooperative speaker encodes
all of relevant information the speaker has; she knows only .
We now introduce a pragmatic notion of entailment, | , building on this rule of interpretation.
| is defined to hold in case is true on all models where represents all
the speaker knows. In other words, | describes the inferences an interpreter can draw from
an utterance if she assumes the speaker to behave rational and cooperative; in particular,
to obey T and Q1.
Fortunately, the question how to formalize that a certain formula expresses all an agent
knows has gotten some attention from logicians. Our account for the free choice inferences
highly relies on this work. More specifically we are going to use an approach introduced by
Halpern & Moses [1984] and generalized by van der Hoek et al. [1999, 2000].
The fundamental idea behind their formalization is strikingly simple. The states where
a sentence is all a speaker knows are taken to be those states where she is in the minimal
belief-state, given that she believes . Hence, we impose an order on all the states s
where the speaker believes her utterance , i.e. s |= 2 , that compares how much the
speaker believes in these states. And then we define as the class of pragmatic inferences
those sentences that hold on the minimal elements of this order. It is obvious that such a
formulation of a notion of entailment constitutes nothing more or less than an instance of
interpretation in preference structures. We summarize our considerations in the following
46We will not discuss why a rational cooperative speaker should obey the maxims of conversation. We
simply try to capture what an interpreter can infer if she assumes that the speaker does so.
47Although the restriction to relevant information has often been neglected in the literature.
48This paraphrase does not exactly capture what Grice says in his maxim Q2. According to his formulation
it may be the case that two possible utterances of the speaker are both maximally informative and relevant
but nevertheless convey different information. Our formulation implicitly assumes that there exists exactly
one maximum. We can make this claim because we will order sentences only with respect to the semantic
notion of entailment |=. Then it is the case that if two sentences are logical independent their conjunction
is more informative than both of them and therefore we can assume the existence of a unique maximum.
However, |= may not be an appropriate description of the order (how does it deal with the modification of
Grice: ‘as informative as required’, for instance). Nevertheless, if adaptations are necessary they can take
the work presented here as a starting point.
26
definition of a notion of pragmatic entailment.
Definition 1 (The Inference Relation | )
Let be a partial order on some class of states S. We define for sentences , 2 L:
| S iffdef 8s 2 S : [s |= 2 ^ 8s0 2 S : s0 |= 2 ) s s0] ) s |= .
This relation of entailment is the hart of the approach developed here.49
3.2.2 Information Orders. Introduction
The crucial question for an approach along these lines is whether we can find a convincing
definition of the order , hence, of what it should mean that in one state the speaker knows
more than in another. Until now we have said nothing on this point. This will be our next
topic.
Van der Hoek et al. [1999] introduce an appealing general characterization of such
orders. The underlying idea is very simple: we use our language to define the order. A
speaker is said to believe less in a state hM,wi than in a state hM0,w0i with respect to a
certain sublanguage L0 L, if the set of sentences of L0 she believes in the first state is a
subset of the set of sentences of L0 she believes in the second. To make this more precise
we define the notion of an information order.
Definition 2 (Information Orders)
A sublanguage L0 L defines with respect to a class of states S the relation iffdef
8s1, s2 2 S : s1 s2 , 8' 2 L0 : s1 |= ' ) s2 |= '. An order that is defined by a
sublanguage L0 L is called an information order.
The restriction to orders that have defining languages has some consequences for the
notions of entailment that can be defined in terms of them. For instance, these orders are
partial but not necessarily total orders. Hence, it may be the case that for a sentence
there exists more that one minimal state. Given the application we have in mind the orders
we are interested in compare states with respect to the beliefs of the speaker. Hence, we
will use information orders defined by languages 2L for L L. If for a sentence there
exist with respect to the order defined by such a belief-language more than one minimal
state this means that for the interpreter a speaker of can be in (with respect to 2L )
different minimal belief states - that she believes in the minimal belief states for different
things. But in such a situation the speaker has to have L -beliefs she did not communicate.
Thus, it is obvious for the interpreter that she did not obey the maxims Q1 and T (as
they are interpreted by this account). Such sentences should then be pragmatically not
well-formed. The notion of entailment that can be defined on basis of an information
order using definition 1 reflects this consideration: if there are different minimal states,
49Some reader may miss semantics in this definition: the truth conditions of the antecedent play no role.
This is true and the notion of pragmatic entailment defined above is not conservative with respect to the
semantic entailments of an utterance - hence, does not truly strengthen the semantic meaning of an utterance
as one might expect. However, this can be easily achieved by defining | s by 8s 2 S : s |= ( ^ 2 ) &
... . Also with respect to this notion, the results we are going to discuss will hold. (This is not trivial: by
extending the selection conditions, the set of minimal models may change.)
27
then | ?, because there will be no state that fulfills the antecedence in definition 1.
Halpern & Moses called such sentences, sentences that allow for different with respect to
2L incomparable minimal belief state, dishonest.
Definition 3 (Honesty)
A sentence 2 L is called honest with respect to a language 2L L and a set of states
S, iffdef {s 2 S|s |= 2 ^ 8s0 2 S : s0 |= 2 ) s s0} 6= ;, where is the order defined
by 2L .
There is an interesting and useful connection between being honest and another property
of formulas. Let’s say that a formula 2 L has the disjunction property with respect to a
language 2L L and a class of states S iffdef is satisfiable in S and for every finite set of
sentences 1, 2, ..., n 2 2L : if |=S 1 _ ... _ n then for some i 2 {1, ..., n}: |=S i.50
Now, assume that is honest with respect to 2L and S. This means that there exists a
state s 2 S such that s |= 2 and 8s0 2 S : s0 |= 2 ) s s0. Assume at the same time
that 2 does not have the disjunction property with respect to 2L and S. Then, there
is some sequence of sentences 1, ..., n 2 2L , such that 2 |=S 1 _ ... _ n but for no
i 2 {1, ..., n}: 2 |=S i. Of course, if 2 |=S 1 _ ... _ n then also s |= 1 _ ... _ n.
Because of the way the truth function of _ is defined, from this it follows that there is
some k 2 {1, ..., n} such that s |= k. However, by the choice of the 1, ..., n it cannot
be the case that 2 |= k. Hence, there has to be an s0 2 S such that s0 |= 2 and
s 6|= . But this impossible: we have chosen s such that 8s0 2 S : s s0 and, hence,
8 2 2L : s |= ) s0 |= ! This proves the following fact:
Fact 2 If is honest with respect to 2L (L L) and a class of states S, then 2 has
the disjunction property with respect to 2L and S.
This result will prove to be quite useful for establishing that certain formulas are not honest.
3.2.3 How to establish Properties of Minimal States
In the last two sections we introduced the central concepts of the new approach. We defined
a pragmatic notion of entailment - the notion by means of which we want to account for
the free choice inferences - by strengthening the inferences one can get from the premise
that the speaker believes what she said with taking the speaker to be in the minimal belief
state possible given that she believes what she said. And we characterized a class of orders
that can be used to compare the beliefs of the speaker. What else do we have to do before
we can see whether the approach gives us the free choice inferences?
Of course, we still have to choose a particular information order. A general characterization
is not enough. But before we are going to discuss this question, we will address
another problem: given a class of states and an information order, how can we establish
that certain inferences are valid with respect to the notion of entailment defined by this
order when applied to this class of states? How can we establish properties of minimal
states? This question is the topic of the present section.
50The definition can be easily extended to arbitrary languages.
28
We will use a state-construction technique to show that certain sentences hold on the minimal
states for a sentence . It will turn out that the same technique can also be used to
prove that a sentence is honest. Let S be a set of states and an information order on
S with defining language 2L . Assume that we could define a mapping
W
from the set of
subsets of S to S such that for arbitrary T S (i)
W
(T) 2 S and (ii) 8s 2 T :
W
(T) s,
hence
W
would map subsets T of S on a state that is -smaller that every s 2 T. Then, we
could in terms of this mapping characterize a class of honest formulas for the order-defining
language 2L .
Fact 3 If 2 is
W
-persistent, i.e. 8T S : (8s 2 T : s |=S 2 ) )
W
(T) |= 2 ), then is
honest with respect to S and 2L .
To see this take T = {s 2 S|s |= 2 }. W According to the assumptions we have made about
, we have
W
(T) 2 S and 8s 2 T :
W
(T) s. Furthermore, because 2 is
W
W -persistent:
(T) |= 2 . But this means that with
W
(T) we have found some state s 2 S such that
s |= 2 and 8s0 2 S : s0 |= 2 ) s s0. Hence, is honest with respect to S and 2L .
More relevant for our applications is the following fact.
Fact 4 If 8s 2 S : s |= 2 ^ ¬ ) [9s0 2 S : s0 |= 2 & s0 s], then | S .
Assume that |6 S . By definition, it follows that 9s 2 S : s |= 2 & [8s0 2 S : s0 |=
2 ) s s0] & s |= ¬ . But by the antecedent of fact 4, we know that 9s00 2 S : s00 |=
2 ^ s00 s - hence, contradiction.
We will see later on that an operation
W
that has the properties we have stated above
will also allow us to easily show the validity of the antecedent of fact 4. In sum, the main
work for establishing the free choice inferences will lay in finding such a mapping
W
. But it
will turn out that this is not a particularly difficult enterprise either. An adapted version
of the disjoint union of modal models will do the job.
3.2.4 The Basic Information Order
Now, it’s time to let all the techniques run on an example. We will, therefore, use this
section to study a simple but nevertheless for the free choice inferences very useful instance
of an information order: the basic information order.
The basic information order is the order defined by the language 2L(0).51
Definition 4 (The Basic Information Order 0)
For s, s0 2 StateKD45
52 we define
s 0 s0 iffdef 8' 2 2L(0) : s |=S ' ) s0 |=S '.
51This is one of the descriptions Halpern & Moses [1984] use for the definition of their notion of minimal
knowledge.
52As stated earlier, when discussing applications we will restrict our considerations to StateKD45.
29
0 compares how much the speaker knows on sentences that contain no modal operators.
Of course, this is a strong limitation of order-relevant information - and we will have to
extend the defining language to get all the intended inferences. But for the moment we will
stick to this simplification.
It is easy to see that the following simple fact holds.
Fact 5 For hM,wi, hM0w0i 2 StateKD45: hM,wi 0 hM0,w0i ,
8v0 2 R0
3[w0]9v 2 R3[w]8p 2 P : V (p)(v) = V 0(p)(v0).
Thus, we could have equivalently used the stated model-condition to define the order. To
give a rough reformulation of the statement, it says that a speaker knows in a state hM,wi
less than in a state hM0,w0i if for every epistemic possibility she sees in hM0,w0i there will
be an epistemic possibility of the speaker in hM,wi that gives exactly the same interpretation
to propositional atoms.
With the use of the basic information order we already can account for the inferences
(D1), (D2), (D5) and (D8) from section 3.1.3.
By inserting 0 in Definition 1 we get the first concrete instance of our pragmatic
entailment relation: | 0
KD45 holds iffdef on the 0-minimal set of StateKD45 where
the speaker believes , is valid.
How can we now, for instance for (D1), establish that the inference is valid for | 0
S?
Given the considerations of the last section, we want to use fact 4 for this purpose. But to
be able to do so we first have to show that its antecedent holds: we have to show that a state
s1 where the speaker believes the antecedent of (D1) but where the consequence 3p^3q is
not true is for 2(p_q) not minimal with respect to 0. Assume, without loss of generality,
that the consequence fails because s1 |= ¬3p. An idea how to find for s1 a strictly smaller s
could be to define it by adding to the epistemic possibilities the speaker distinguishes in s1
one where p holds. In the resulting s, ¬3p would be false and, hence, because ¬3p 2¬p
and 2¬p is in our order defining language 2L(0) it follows that ¬(s1 0 s). On the other
hand, because we only added an epistemic possibility where p holds, 2(p _q) is still true in
s, and in general the speaker should only have less L(0)-beliefs in s than in s1. In sum, s
would be a state such that s |= 2(p _ q) and s 0 s1, and the antecedent of fact 4 for (D1)
would be established.
In the following the argumentation sketched above is made more precise. For this
purpose we define the s used above as the result of a merge of s1 with a KD45-state where
2p holds. This operation will be a function with the properties described in the last section.
For the basic information order the merge comes down to a simple variation of the disjoint
union of modal models.
Definition 5 (The Merge Function
W
1)
Let T be a subset of StateKD45 indexed with the set I. Assume that the set of worlds of all
states in T are disjoint and w is a world not occurring in any domain.
W
1(T) is the state
hCLKD45(M0),wi, where M0 is defined as follows.
W0 =
S
i2I Wi [ {w},
R0
3 =
S
i2I Ri,3 [ {hw, vi|v 2 Ri,3[wi], i 2 I},
30
R0
4 =
S
i2I Ri,4 for 4 6= 3,
V 0(p) =
S
i2I Vi(p).
The picture below shows how the operator
W
1 merges two states s1 = hM1,w1i and s2 =
hM2,w2i to a state s = hM,wi =def
W
1({s1, s2}). As we can see, there are two ways in
which arrows are added to the epistemic accessibility relation of s: arrows from the new
point w to epistemically accessible points of s1 and s2 come by the definition of M0. Arrows
connecting the epistemic possibilities of different states with each other are added by the
closure operation that is applied to the model M0 in the definition.
w
w1
w2
by
construction
by
closure
R
"
"
"
"
"
"
"
b
b
b
b
b
b
b
```````
•
@@
•
@@
•
Figure 1:
The presentation of the states is highly
simplified. Only some essential R3-
connections are depicted. Thick lines represent
R3-arrows of the merged states s1
and s2, thin lines stand for R3- arrows inserted
by the merge operator. The arrowdirection
is always from left to right (connections
represented by curved lines hold
in both directions).
By definition,
W
1(T) 2 StateKD45. Its set of R3-accessible worlds is the union of
the epistemic alternatives of the speaker in every s 2 T with their respective valuation for
atomic propositions. But that means for hM,wi =def
W
1(T) that for every hMi,wii 2 T and
for every vi 2 Ri,3[wi] there is an epistemic alternative v 2 R3[w] that assigns in hM,wi
exactly the same valuation to atomic propositions as vi in hMi,wii: 8vi 2 Ri,3[wi]9v 2
R3[w]8p 2 P : V (p)(v) = Vi(p)(vi). But because this is exactly what defines the order 0
(see definition 2), we obtain: 8s 2 T :
W
1(T) 0 s.
Fact 6 For all T StateKD45 the following holds:
(i)
W
1(T) 2 StateKD45,
(ii) 8s 2 T :
W
1(T) 0 s.
Hence,
W
1 has all the properties to apply for the role of
W
in fact 3. Thus, we can use
the function to characterize a class of honest formulas for StateKD45 and 2L(0). And a
short consideration reveals that the antecedents of the inferences in (D1), (D2), (D5) using
the first antecedent, and (D8) are persistent under
W
1. Hence, they are all honest with
respect to 2L(0) and StateKD45. Take, for instance, the sentence =def 2(p _ q) and
assume hM1,w1i, hM2,w2i 2 StateKD45 make this sentence true. Hence, 8v1 2 R3[w1] :
M, v1 |= p _ q and the same for hM2,w2i.
W
1 maps those states on a new state hM,wi such
that R3[w] is simply the union of R1,3[w1] and R2,3[w2] - whose members all make p _ q
true. Of course, it will also hold that 8v 2 R3[w] : M, v |= p _ q. Hence, is persistent
under
W
1.
We will now again show that (D1) holds with respect to StateKD45 and | 0 but now
formally more precisely, using
W
1. By assumption, p and q are logical independent propositions.
Assume furthermore that there is a state s1 2 StateKD45, such that s1 |= 2(p _ q)
31
but s1 6|= 3p ^ 3q. Without loss of generality, we assume s1 |= 2¬p. Notice again that
2¬p 2 2L(0), whereby 2L(0) is the defining language of the information order 0. Because
p 6 ?, we can find a state s2 2 StateKD45 such that s2 |= 2p. It follows that s2 |= 2(p_q).
Let s =def
W
1({s1, s2}). Because 2(p _ q) is
W
1-persistent, s |= 2(p _ q). By construction,
we additionally have s |= 3p and hence s 6|= 2¬p. Thus, we found a formula in the defining
language that holds at s1 but not at s. Together with fact (6.ii), this allows us to conclude
s 0 s1. To summarize, we have found for every KD45-state verifying 2(p _ q) that does
not make the free choice inference true a strictly smaller KD45 state. Hence, we verified
the antecedent of fact 4 and can conclude that (D1) is valid with respect to StateKD45 and
| 0.
Notice, that (D1) is not necessarily valid for logical dependent p, q 2 L(0).53 Gazdar’s
inconsistency check on pcis now is a natural consequence of the non-monotonicity of the
notion of entailment we defined.
Let’s take as second example the inference (D5) with the first antecedent 2(p _ q). In
this case a state s1 that verifies the antecedent 22(p_q) ( KD45 2(p_q)) but falsifies the
conclusion 3p ^ 3q of (D5) will again (without loss of generality) imply the order-relevant
sentence 2¬p. To show that such a state s1 cannot be minimal with respect to the order 0,
this time, we have to merge s1 with a witness for 3p such that the stronger claim 22(p_q)
still holds in the merge. But the s2 we used above does already fulfill this requirement:
s2 |= 22(p _ q) and hence s =def
W
1({s1, s2}) |= 22(p _ q) because the sentence is
W
1
persistent. And again we can conclude from s 6|= 2¬p and fact (6.ii) that s 0 s1. Hence,
s1 is for 22(p _ q) not minimal with respect to 0 and (D5) follows by fact 4.
The validity of the inferences (D2) and (D8) can be proven by the same means. However,
the argument does not extend to (D5) using the second antecedent 2p _ 2q. The reason
is simply that 2p _ 2q is not
W
1-persistent. But the inference (D5) does hold! Actually
we obtain 2p _ 2q | 0
KD45 ?, because the formula 2p _ 2q is not honest with respect to
2L(0) and StateKD45, i.e. there is no state s 2 StateKD45 such that s |= 2(2p _2q) and
8s0 2StateKD45 : s0 |= 2(2p _ 2q) ) s 0 s0. How to see that 2p _ 2q is dishonest? If a
speaker believes 2p _ 2q she may, for instance, believe that p but not believe that q. Let
s1 be a state where this is the case. On the other hand, she may also believe that q but not
believe that p. Assume that this holds in s2. Because 2p,2¬p,2q,2¬q are in the orderdefining
language 2L(0) we have s1 6 0 s2 and s2 6 0 s1. Hence, if 2p_2q were honest, then
there would exists an s 2StateKD45 such that s |= 2(2p _ 2q) and s is 0-smaller than
both, s1 and s2. However, if s is smaller than s1, then s |=KD45 ¬2p because the speaker
can believe only less in s that she believes in s1. For the same reason s |=KD45 ¬2q. But
then s 6|=KD45 2(2p _ 2q) what contradicts our first assumption about s. Hence, 2p _ 2q
has to be dishonest with respect to 2L(0) and StateKD45. Or, in other words, a speaker
who utters 2p _ 2q cannot at the same time obey the maxims T and Q1: she cannot at
the same time believe in her utterance and say all she believes. This sentence is, given our
formalization of Grice’s maxims, pragmatically not well-formed.54
53If, for instance, p = ?, then we would not be able to find such an s2 as we used in the proof.
54The same point can be made much easier but less intuitively using the disjunction property. In
StateKD45 we have obviously: 2(2p _ 2q) |=KD45 2p _ 2q. Both, 2p and 2q, are in the order-defining
language 2L(0). In order to have the disjunction property, 2(2p _ 2q) should entail one of these sentences.
32
But this is not such a bad result! Sentences like (23a) and (23b) are indeed reported to
be odd. In particular, they do not allow a free choice reading.
(23) a. Mr. X must be in Amsterdam or Mr. X must be in Frankfurt.
b. I believe that A or I believe that B.
The approach at hand can explain these intuitions by referring to the notion of pragmatic
well-formedness that it allows to define.
This argument nicely illustrates the power of the notion of honesty. Normally, we are
not able to recognize whether a speaker is really obeying the maxims of Grice: whether
she does tell us the truth and gives us all the information she has. The interpreter has to
assume that the speaker is trustable. Dishonest sentences, however, directly compromise
the speaker. They can never be uttered in accordance with the Gricean maxims. They are
therefore also a very powerful testing device for the account presented here.
The information order 0 does not allow us to derive the other inferences in section 3.1.3 we
want to account for. In particular, we cannot account for free choice permission. The reason
is easily found: the basic information language does not contain any sentence that refers to
belief the speaker has about deontic modalities (or any other modality). Thus, if asked to
compare two states, the order is not able to see differences in belief about modalities. So
how can we expect any minimization effect on such kind of information? The strategy to
overcome the limitation we face here should be to extend the order defining language. This
will be done in the following sections.
However, a small modification of what we have said above is necessary. Perhaps the
reader has recognized that in using the basic information order we do have a certain minimization
effect on what the speaker beliefs on at least one modality: her own belief state.
But this is due to the particular logic of belief used here: KD45. This logic demands a close
relation between what the speaker believes and her beliefs on her beliefs: the speaker is assumed
to be competent on her beliefs. Therefore, minimizing the L(0)-beliefs of the speaker
will also minimize her beliefs on these beliefs. To make this a little bit more precise, consider
the language L defined by the following BNF: ' ::= p(p 2 L(0))|' ^' |' _' |2' .
This language allows to express beliefs of various depths but prohibits the occurrence of 2
under negation. Now, it can be shown that 8' 2 2L 9'0 2 2L(0) : ' KD45 '0. Hence, we
could have defined 0 equivalently using 2L . However, while | o also minimizes beliefs
the speaker has on what she does believe, it is easy to see that her beliefs on what she does
not believe do not count for the order.55
3.2.5 Information Orders with Modal Information
Of course, extending the defining language will in turn widen the class of properties of the
states that the information order is sensible to - this is actually our intention. But, to
Of course, this is not true. Hence, the formula cannot have the disjunction property and, therefore, using
fact 2, 2p _ 2q cannot be honest with respect to 2L(0) and StateKD45.
55Because of the introspective power of the agent minimizing beliefs on beliefs will actually result in
maximizing beliefs on what the speaker does not believe.
33
prove validity on minimal states with respect to such more complex orders we need to know
how the minimal states look like. We need a model-theoretic definition of the order. But
how to establish such results? Van der Hoek et al. [1999] provide an interesting technique
that simplifies this problem for certain order defining languages. They use the notion of
Ehrenfeucht - Fra¨ıss´e - Orders to this purpose:
Definition 6 (Ehrenfeucht - Fra¨ıss´e - Orders)
Let n, n 2 N, be an enumerable set of partial orders. We define the Ehrenfeucht-Fra¨ıss´e-order
as follows.
hM,wi hM0,w0i ,def 8n 2 N8v0 2 R0
3[w0]9v 2 R3[w] : hM, vi n hM0, v0i.
For this notion they prove the following result.56
Theorem 2 (Collecting)
Let L and {L
(n)|n 2 N} be a system of sublanguages such that for all n 2 N: L
(n) =
L \ L(n). If for all n 2 N the language L
(n) defines the order n and L is closed under
_, then 2L defines , i.e.
s s0 , 8 2 2L : s |= ) s0 |= .
Given this result, if we know the model conditions of the orders defined by the finite
sublanguages {L
(n)|n 2 N}, then we also know that the order defined by the language 2L
is the Ehrenfeucht- Fra¨ıss´e order of these ‘finite’ orders. And to establish model-conditions
for a finite language with finite modal depth is much less demanding.
3.2.6 The Objective Information Order
In this section we will study an appealing candidate for an information order that respects
the knowledge an agent has on other modalities. However, at the end we will see that the
notion of entailment defined by this order will not be able to give us in StateKD45 all the
inferences we are looking for.
To find a modality-sensible defining language we use the simple idea to extend the basic
language L(0) with all sentences of L that say something on modalities other than the
belief state of the speaker.
Definition 7 (The Objective Information Order o)
Let Lo L be the language defined by the BNF o ::= p(p 2 L(0))|¬ o| o ^ o|r ,r 6=
2( 2 L). We call Lo, following Halpern57, the objective information language and define
the objective information order, o, for s, s0 2 StateKD45 as follows:
s o s0 , 8' 2 2Lo : s |= ' ) s0 |= '.
56For a proof see van der Hoek et al. [1999]. The result relies on the finiteness property of the set of
atomic propositions.
57According to van der Hoek et al. [2000] the objective information order has been introduced by Halpern
in a paper where he extends the approach in Halpern & Moses [1984] to the multi-agent case.
34
Building on the result of the last section, van der Hoek et al. [1999] show that the
following equivalent model-theoretic definition of o can be given.
Fact 7 We define on KD45 for n 2 N the relations =−3
n as follows:
hM,wi =−3
0 hM0,w0i ,def 8p 2 P : V (p)(w) = V 0(p)(w0)
hM,wi =−3
n+1 hM0,w0i ,def
8<
:
hM,wi =−3
n hM0,w0i &
84 6= 38v 2 R4[w]9v0 2 R0
4 [w0] : hM, vi =n hM0, v0i (forth) &
84 6= 38v0 2 R0
4 [w]9v 2 R4[w] : hM, vi =n hM0, v0i (back)
Then o is the Ehrenfeucht- Fra¨ıss´e order defined by the orders =−3
n for n 2 N:
hM,wi o hM0w0i , 8n 2 N8v0 2 R0
3[w0]9v 2 R3[w] : hM, vi =−3
n hM0, v0i
The proof works by induction on the maximal modal depth n of the sublanguages
Lo(
n) = Lo \ L(n). For every n 2 N one has to show that Lo(
n) define the respective relation
=−3
n .58 Then one applies theorem 3 from section 3.2.5. Remember from section 3.1.2 that
=n stands for n-bisimulation between two states. The relation =−3
n is a weakening of =n:
R3/ R0
3 -branches starting at the root of two states that are compared are not considered.
In other words, the relation =−3
n ignores, the beliefs of the speaker on her own beliefs.
The extended order o defines, based on definition 1, a new notion of pragmatic entailment:
| o
KD45. The question we now have to address is: can this entailment relation account for
the inferences (D1) - (D9)? To investigate this problem we would like to use the same
techniques as in section 3.2.4 when we studied this question for the basic information order.
In particular, we would like to make use of fact 4. However, to establish the antecedent of
this fact, we cannot rely on the merge function
W
1 any longer. Remember that we need
our operation
W
to fulfill two conditions: (i) 8T StateKD45 :
W
(T) 2 StateKD45, and
(ii) 8s 2 T :
W
(T) o s. But for
W
1 we only showed the claim (ii) for the basic information
order 0. In this case the only thing we needed was that for every s0 = hM0,w0i 2 T and
every v0 2 R0
3 [w0] there exists for hM,wi =def
W
(T) a v 2 R3[w] that agrees with v0 in
the valuation of propositional atoms. For the objective information order, as fact 7 tells us,
we need much more. v and v0 have to agree on a great part of their generated submodels:
hM0, v0i =−3
n hM, vi for every n 2 N!
W
1 cannot warrant this. For suppose that in the
generated submodel of v0 in M0 there is a kind of circle leading back to a world v00 2 R0
3[w],
hence, there exists a sequence of accessibility relations hR0
1,4i
, ...,R0
n,4l
i of M0, 41 6= 3
and a sequence of worlds hv0
1, ..., v0
n+1i, such that 8i(1 i n) : v0
i+1 2 R4i [v0
i ] and v0
1 = v0
and v0
n+1 = v00. If you now merge hM0,w0i with other states using
W
1, then in the resulting
state hM,wi R3[v00] will be much bigger than R0
3 [v00]. Hence, in M v00 generates a different
submodel than in M0. But therefore, a relevant part of the submodel generated by v0 in the
new model M will change too and, even though =−3 ignores 3-branches starting from v0,
hM, vi 6 =−3 hM0, v0i. To make sure that we do not get in any trouble of this kind, we need
a new merge operation that does not pool the set of epistemical possible worlds directly.
Instead, we let it work on copies of them. We define:
58Essentially for the proof is the well-known result of modal logic that to decide whether a sentence of a
modal language with modal depth n 2 N is true it is sufficient to look n-deep into the model.
35
Definition 8 (The Merge Function
W
2)
Let T be a subset of StateKD45 indexed with I. Assume that the domains W of all states
s 2 T are disjunct and w is a world not occurring in any domain. Let ˜· be a copy-function, an
injection from
S
i2I Wi to a distinct set of worlds D.
W
2(T) is the state hCLKD45(M0),wi,
where M0 is defined as follows.
W0 =
S
i2I Wi [ { ˜ vi|vi 2 Ri,3[wi], i 2 I} [ {w},
R0
3 =
S
i2I Ri,3 [ {hw, ˜vi|v 2 Ri,3[wi], i 2 I},
R0
4 =
S
i2I Ri,4 [ {h ˜ vi, uii|vi 2 Ri,3[wi] ^ ui 2 Ri,4[vi] ^ i 2 I} for 4 6= 3,
V 0(p) =
S
i2I Vi(p) [ { ˜ vi|vi 2 Vi(p), i 2 I}.
The picture below demonstrates the working of the operator
W
2 for the merge of two states
s1 = hM1,w1i and s2 = hM2,w2i to a state s = hM,wi.
w
w1
w2
v1
˜v1
41
41
42
copy ?
copy 6
"
"
"
"
""
b
b
b
b
bb
``````
•
@
@
@
•
•
•
@
•
-
Figure 2:
Now, modalities different from 3 do occur in
the picture. They are represented by arrows annotated
with the modality to which they refer.
Otherwise the same conventions as for the last
picture are in force. The graphic illustrates how
the use of copies for the merged states breaks
the circles that caused the problem we discussed
above. If we had merged s1 and s2 directly
using
W
1, then hM1, v1i 6 =−3
3 hM, v1i, because
wWe would have obtained that in hM,wi =def
1({s1, s2}) from v1 one can reach by 41, 42
and 3 all elements of R2,3[w2]. However, this is
not possible in M1.
For ˜ v1 this problem does not occur. Notice that even though hM, v1i =−3
n hM, ˜ v1i for n 2 N
- hence, they make exactly the same sentences in Lo true, ˜ v1 differs from v1 by not seeing
in M under any modality 4 6= 3 any other point of R3[ ˜ v1].
Fact 8 For all T StateKD45 the following holds:
(i)
W
2(T) 2 StateKD45,
(ii) 8s 2 T :
W
2(T) o s.
The proof is quite simple given what we have said above. Take an arbitrary T
StateKD45, hMi,wii 2 T and let hM,wi =def
W
2(T). We have to show that 8n 2 N8vi 2
Ri,3[wi]9v 2 R3[w] : hM, vi =−3
n hMi, vii. We know that by construction 8vi 2 Ri,3[wi] :
˜ vi 2 R3[w]. But this copy ˜ vi of vi is a v 2 R3[w] such that hM, vi −3
n hMi, vii for all
n 2 N by the way the construction is set up.
Together with fact 3 this result allows us to establish a general class of 2Lo-honest formulas:
If 2s is
W
2-persistent, then is 2Lo-honest. Actually, all antecedents of the
36
inferences in the data-section, beside the part of (D5) we already disqualified in the last
section, are honest with respect to 2Lo .
Parallel to the argumentation in section 3.2.4, we can establish the validity of the free
choice inferences using fact 4. However, not all inferences from section 3.1.3 will come out
as valid. While we obtain (D1), (D2), (D4), (D5), (D7), and (D8), the inferences (D3),
(D6), and (D9) do not hold.
Let’s start by discussing one inference we already had as valid given the basic information
order: (D2). Let p, q 2 L(0) be two logically independent propositions. Let s1 2 StateKD45
be a model of the formula that verifies 23(p _ q) but fails the conclusion of (D2), hence,
without loss of generality, s1 |= 2¬p, where 2¬p 2 2Lo, our order defining language.
Because p and q are logically independent, p is a contingent fact and, hence, there will be a
state s2 2 StatesKD45 such that s2 |= 3p. It follows that s2 |= 3(p_q) and, by the validity
of axiom (5) in StateKD45, s2 |= 23(p _ q). Let s =def
W
2({s1, s2}). Because 23(p _ q)
is
W
2-persistent, we obtain that s |= 23(p _ q). Additionally, s |= 3p by construction, and
hence, s o s1. By fact 4, 3(p _ q) | o
KD45 3p ^ 3q.
However, we will not be able to conclude by a parallel argument that in the set of ominimal
states for 24(p _ q), 4p holds and hence to account for (D3). The reasoning fails
the moment we want to conclude that on a state s2 where 4p is valid, 24p is valid too. In
the derivation above we could do the step from 3p to 23p because of axiom (5): because
we assumed the speaker to have negative introspective power. We have nothing similar for
4.
Instead of (D3) we do get (D4), the incompetence inference. Let p, q 2 L(0) be two logical
independent propositions and let s1 be a model of the formula that verifies 24(p _ q)
but fails the conclusion of (D4). Without loss of generality, s1 |= 2(¬4p _ 4q). The
formula 2(¬4p _ 4q) is an element of the order defining language 2Lo. p ^ ¬q is consistent
and hence, there will be a state s2 2StatesKD45 such that s2 |= 2r(p ^ ¬q). It
follows that s2 |= 24(p _ q). Let s =
W
2({s1, s2}). Then, by persistence of the antecedent
we have s |= 24(p _ q). Additionally, s |= 3(4p ^ ¬4q) by construction and hence,
s 6|= 2(¬4p_4q). This allows us to conclude that s o s1. Hence, by fact 4, (D4) is valid.
To summarize, on StateKD45 the incompetence inference (D4) is valid with respect to
| o while the free choice permission (D3) is not. Actually this is not such a bad result.
As observed in the literature, the incompetence reading does exist. Additionally, the free
choice reading seem to be bound to the assumption that the speaker is competent. We did
not assume any connection between the deontic facts and the knowledge of the speaker here
- hence, we should not obtain free choice permission. On the other hand, for the epistemic
modality we build competence into the system - by assuming the axioms (4) and (5) to be
valid. As Zimmermann did, we can explain the observation that epistemic free choice seems
always valid by claiming that interpreters always take a speaker to have full introspective
power.
The aim of the next section has to be to generalize the concept of competence behind
introspection to other modalities - here we will rely on the work of Zimmermann [2000] -
and then investigate whether we can make the inferences (D3) and (D6) valid this way.
37
3.2.7 Competence and Multi-Modal Belief
As we have observed in the last section we will be only able to derive the intended inferences
in the given setting if we assume additional restrictions on the class of models on which
utterances are interpreted. There has to be some kind of connection between what is valid
on a certain modality and the speaker’s beliefs with respect to this modality: the speaker
has to be assumed to be in some sense competent on the relevant modality.
In how we make this idea precise we rely on Zimmermann’s [2000] formalization of
competence. As stated in section 2.2, Zimmermann, building on a proposal of Groenendijk
& Stokhof [1984], defines competence by the following first-order model condition.59
Definition 9 (Competence)
A speaker is strongly competent in a state hM,wi 2 StateKD45 (where M = hF, V i and
F = hW,R4i , ...,R4ni) with respect to a modality 4 iffdef
8v 2 W : v 2 R3[w] ) (R4[v] = R4[w]).
It is easy to prove that this condition is characterized in modal propositional logic by
the two axioms (C14) and (C24) described below, i.e. a speaker is strongly competent
in some state s = hM,wi if the underlying frame locally (hence, in w) satisfies (C14) and
(C24).
(C14) r ! 2r
(C14) will be called the axiom of positive competence. Intuitively it expresses that if a
sentence holds in the R4-admissible worlds then the speaker knows this. If one substitutes
2 for r one obtains the positive introspection axiom (4). Using the same strategy as in the
proof of fact 1 one can show that the following holds for any frame F = hM,R41 , ...,R4ni
for the language L and any point w in this frame.
F,w |= C14 iff Fk − 8v 2 W : v 2 R3[w] ) (R4[v] R4[w])
We turn to the second axiom.
(C24) ¬r ! 2¬r
(C24) will be called the axiom of negative competence. Intuitively it expresses that the
speaker knows also for every sentence that is not valid on the R4 accessibility relation that
this is the case. It comes as no surprise that this is a multi-modality generalization of
negative introspection or euclidicity:
F,w |= C24 iff 8v 2 W : v 2 R3[w] ) (R4[v] R4[w])
59The (intensional) predicate w x.P (w)(x) in his definition is instantiated here with the characteristic
function of the R4-accessibility relation: w v.wR4v.
38
As these considerations show, the definition of competence used by Zimmermann [2000]
is a generalization of the concept of full introspective power to arbitrary modalities. Now
we have to investigate whether imposing this generalized model condition on the class of
states that is considered (hence, to work on StateKD45C14C24 instead of on StateKD45)
allows us to account for the free choice permission as much as assuming introspective power
of speaker allowed us to account for the epistemic free choice inferences. Thus, the question
is whether (D3) and (D4) are valid conditional on assuming the speaker to be competent
in the sense defined above.
Unfortunately, if we use the notion of entailment defined by the objective information
order o on the set StateKD45C14C24 we obtain a system that is much too strong: if the
speaker is assumed to be competent on 4, then every consistent sentence is dishonest
with respect to StateKD45C14C24 and o. Or, in other words, given the way | o interprets
the maxims T and Q1 a speaker competent on 4 as formalized in C14 and C24 cannot
utter any consistent sentence and be obeying these maxims. The reason for this is that
| o demands a strongly competent speaker to provide so much information that there is
no finite sentence to encode it all. In some more detail: for every ' 2 L both, 2r' and
2¬r' are in the order defining language 2Lo. Recall that this means that the order o
prefers states where these sentences are false. But if a speaker is strongly competent with
respect to 4 then for every ' 2 L: 2r' and 2¬r', cannot be false at the same time
(given the formalization we have chosen a speaker strongly competent on 4 knows for every
' 2 L which of r' and ¬r' is true; hence, either 2¬r' or 2r' holds). That means
that there will be at least two incompatible classes of minimal states: one where 2r' is
true, and one where 2¬r' holds. But this would unmask the speaker to be dishonest.
Hence, if a competent speaker does not want to be trapped by not obeying the maxims T
and Q1 (as formalized in | o) then she already has to exclude one of the possibilities with
her utterance of : 2 |=KD45C14C24 2r' or 2 |=KD45C14C24 2¬r'. But the same
argument applies for every ' 2 L! Thus, for every sentence ' her utterance has to convey
whether she believes r' or ¬r'. However, there can be no finite and consistent sentence
that is that strong: that decides every sentence valid on the R4 accessible worlds. Hence,
the general dishonesty result.60
Given these results we have to accept that the combination of strong (positive and
negative) competence with respect to some modality 4 with a defining language that takes
every fact on 4 to be relevant for the order does not result in an appropriate notion of
pragmatic entailment. Two ways to improve the situation can be considered: (1) we can
restrict the order defining language 2Lo and, hence, weaken the pragmatic entailment, or
(2) we can weaken our notion of competence. As we will see, both choices can be turned
into a successful approach to the free choice inferences.
60Exactly the same point can also be made by using the disjunction property of honest formulas.
Parallel to the situation in StateKD45, StateKD45C14C24 has theorems (T1) 2r $ r (Proof:
())2r )D 3r ,Def ¬2¬r )C24 r ,(() by C14) and (T2) 2¬r $ ¬r (Proof: ())
2¬r $Def ¬3r !D ¬2r !C14 ¬r , (() by C24) for the modality 4. Hence, we have
|=KD45C14C24 2r _2¬r (Proof: |=KD45C14C24 _¬ . Instantiate : |=KD45C14C24 2r'_¬2r'.
Using T1 and T2 : |=KD45C14C24 2r' _ 2¬r'), without the disjuncts being theorems, too. Because formulas
that are honest with respect to StateKD45C14C24 and o have to fulfill the disjunction property
it follows that if is o-honest in StateKD45C14C24 then for all 2 L: |=KD45C14C24 2r or
|=KD45C14C24 2¬r , which is impossible to fulfill with a consistent and finite sentence.
39
3.3 Applying the Framework
3.3.1 Solution 1: Weakening the Order
In this section we will introduce a first combination of a logic and a notion of entailment
that gives us the free choice inferences. It will be derived from the framework of the
previous section by weakening the information order o on which the notion of entailment
| o was based. However, the notion of competence as used by Zimmermann [2000] will be
kept. For the moment, we will assume that the language L has only one modality r 6= 2.
Furthermore, we assume that the speaker is strongly competent with respect to 4. Hence,
we are working in StateKD45C14C24. We will come back to the multi-modal case at the
end of this section.
One way to look at the problem we ended up with in the last section is that we chose a
too strict formalization of the conversational maxims T and Q1. As they are described by
| o a speaker who wants to obey the maxim has to give nearly every information she has.
The only thing we excluded from Lo were the beliefs the speaker has about she does not
believes. But all other sentences counted as relevant for the information order. Perhaps we
can obtain a more natural notion of pragmatic entailment when we allow the speaker to
withhold more information. The problem, then, becomes to find the right restriction that
fits our intuitions about when utterances give sufficient information. A promising approach
is to generalize the restrictions we have already working for 2-sentences in Lo to every
modality the speaker is competent on. Hence, we systematically remove all sentences from
Lo where r is the modality with widest scope and occurs under negation. We call the new
information order that is defined by this language the positive information order: 61
Definition 10 (The Positive Information Order +)
Let L+ L be language defined by the BNF '+ ::= p(p 2 L(0))|'+ ^ '+|'+ _ '+|r',r 6=
2(' 2 L). We call L+, the positive information language and define the positive information
order, +, for s, s0 2 StateKD45C14C24 as follows:
s + s0 , 8' 2 2L+ : s |= ' ) s0 |= '.
This order asks the speaker to give all information she has on what is necessary with respect
to R4, but she may know more about what is possible in R4 than she says. Also for the
positive information order an equivalent model-theoretic definition can be given.
Fact 9 We define the relations .3
n for n 2 N on StateKD45C14C24 as follows:
hM,wi .−3
0 hM0,w0i ,def 8p 2 P : V (p)(w) = V 0(p)(w0)
hM,wi .−3
n+1 hM0,w0i ,def
hM,wi .−3
n hM0,w0i &
8v0 2 R0
4 [w]9v 2 R4[w] : hM, vi =n hM0, v0i (back)
Then + is the Ehrenfeucht- Fra¨ıss´e order defined by the orders .−3
n for n 2 N:
hM,wi + hM0w0i , 8n 2 N8v0 2 R0
3[w0]9v 2 R3[w] : hM, vi .−3
n hM0, v0i
61Notice, that this notion differs from what van der Hoek et al. [2000] call the positive information order.
40
It does not come not as a surprise that the order .−3
n differs from the relation =−3
n of
fact 7 only in the absence of the forth condition - which is responsible for perseverance of
sentences of the form 2¬r , 2 L. Because the order + has not been discussed by van
der Hoek et al. the proof of fact 9 is given here. We show by induction on n 2 N that the
languages L+
(n) = L(n) \ L+ define the relation .−3
n . Then, the claim follows immediately
by theorem 3.
For n = 0, L+
(0) = L(0), s .−3
0 s0 , s =0 s0, and the claim follows from theorem 1.
Assume L+
(n) would define .−3
n . We show that L+
(n+1) defines .−3
n+1. Thus:
hM,wi −3
n+1 hM0,w0i , 8' 2 L+
(n+1) : hM,wi |= ' ) hM0,w0i |= ' (*).
(“)”) Notice first that for all 2 L+
(n+1) we can find a sentence
Vl
i=1
Wk
j=1 'i,j 2
L+
(n+1), where for all i, j either 'i,j 2 L(0) or 'i,j = r i,j for i,j 2 L(n), such that
Vl
i=1
Wk
j=1 'i,j .62 Now, assume that for some 2 L+
(n+1) : M0,w0 6|= . We have
to show that then also M,w 6|= , given that hM,wi .−3
(n+1) hM,w0i. M0,w0 6|= ,
9i(1 i l)8j(1 j k) : M0,w0 6|= 'i,j . Firstly, assume 'i,j 2 L(0). Then, because
hM,wi .−3
(n+1) hM,w0i entails hM,wi .−3
0 hM,w0i, M,w 6|= 'i,j . On the other hand, if
'i,j = r i,j , then 9v0 2 R0
4 [w0] : M0, v0 6|= . But hM,wi .−3
(n+1) hM0,w0i entails also,
that 8v0 2 R0
4 [w0]9v 2 R4[w] : hM, vi =n hM0, v0i. By theorem 1, we can conclude
that 9v 2 R4[w] : M, v 6|= i,j and, hence, M,w 6|= r i,j . In sum we have shown that
9i(1 i l)8j(1 j k) : M,w 6|= 'i,j and, thus, M,w 6|= .
(“(”) Assume, that the left-side of the condition (*) does not hold. Hence, for some
hM,wi, hM0,w0i 2 StateKD45C14C24 either (a) hM,wi 6.−3
n hM0,w0i, or (b) 9v0 2 R0
4[w0]
8v 2 R4[w] : hM, vi 6 =n hM0, v0i. We show that in both cases the right side of (*) does not
hold, either. In case (a) the induction assumption allows us to conclude that there exists
' 2 L+n L+ n+1 such that M,w |= ', while M0,w0 6|= '. Hence, the claim. In case (b) by
theorem 1 it follows that there exists a v0 2 R0
3[w0] such that for every v 2 R4[w] there
exists a ' 2 L(n) such that M, v |= ' but M0, v0 6|= '. Because L(n) is finite, we can find a
finite sequence h'1, ..., 'mi with '1, ..., 'm 2 L(n) such that 8v 2 R4[w] : M, v |=
Wm
i=1 'i
and M0, v0 6|=
Wm
i=1 'i. Hence, M,w |= r
Wm
i=1 'i, but M0,w0 6|= r
Wm
i=1 'i. However,
r
Wm
i=1 'i is an element of L+ n+1. Thus, we conclude that hM,wi 6.−3
n+1 hM0,w0i. q.e.d.
In terms of the information order + again a notion of entailment | + can be defined
using definition 1. Now, we want to study which of the inferences of section 3.1.3 are valid
with respect to this notion. A first thing that catches the eye is that on StateKD45C14C24
the formula rp _ rq is not honest with respect to 2L+. The sentence has incomparable
minimal states: one, where rp but not rq holds and the speaker is informed about this,
and one, where rq but not rp holds - again known by the speaker.63
62This is a straightforward consequence of how L+ is defined.
63Using the disjunction property of honest formulas one argues as follows: the problem is - as one might
expect - that by (T1) and (T2) (see footnote 56) 2(rp _rq) |=KD45C14C24 2rp _ 2rq (Proof: 2(rp _
rq) 2(¬rp ! rq) )K 2¬rp ! 2rq )T2 ¬rp ! 2rq rp _ 2rq )T1 2rp _ 2rq) but none
of the disjuncts is entailed. Hence, the formula fails the disjunction property of the language 2L+ and
StateKD45C14C24 and cannot, given fact 2, be honest.
41
Next, we want to establish the inferences (D3) and (D6) with respect to | +. If we
want to use the same kind of argument we employed earlier we need again a proper mergeoperation
W
3 on StateKD45C14C24. Thus, we need (i) 8T StateKD45C14C24 :
W
(T) 2
StateKD45C14C24, and (ii) 8s 2 T :
W
3(T) + s. Notice that if we would drop the
competence axioms and apply | + to StateKD45 we could again use
W
2 as merge operation.
Condition (ii) would still be fulfilled because + is a weakening of the order o. Hence,
s o s ) s + s0. However, if we take the axioms C14 and C24 to be valid, then we have
to face the fact that
W
2(T) is normally not an element of StateKD45C14C24. Therefore,
we have to define a new merge function
W
3 - but notice that
W
3 differs from
W
2 only in the
stronger closure applied to M0.
Definition 11 (The Merge Function
W
3)
Let T a subset of StateKD45C14C24 indexed with I. Assume that the domains Ws of all
states s 2 T are disjunct and w is a world not occurring in any domain.
W
3(T) is defined
as the state hCLKD45C14C24(M0),wi where M0 is defined as in definition 8.
Now property (i) is immediately warranted for
W
W 3, but we might have lost (ii): 8s 2 T :
3(T) + s. Fortunately, it turns out that we did not.
Fact 10 For all T StateKD45C14C24 the following holds:
(i)
W
3(T) 2 StateKD45C14C24,
(ii) 8s 2 T :
W
3(T) + s.
Intuitively, this can be seen by the observation that the only difference between
W
3 W and
2 is that for
W
3 extra arrows are added to R4[w] and R4[v] for v 2 R3[w] in
W
2(T).
But about these extra arrows + does not care - exactly because we dismissed the forth
condition when going from =−3
n to .−3
n . Hence, we get
W
3(T) + W
2(T) + s for all
s 2 T.
Based on
W
3 we can use fact 3 to establish: If 2 is persistent under taking
W
3 then
is +-honest with respect to StateKD45C14C24. The antecedents of (D3) and (D6)
are persistent under union
W
3 and we can continue to the question central to the whole
discussion: do they | +-entail their consequences? The answer is affirmative. We will only
discuss (D6), (D3) works along the same lines. Let s1 be a state for the honest formula
2r(p_q) that does not satisfy the consequence 4p^4q of (D6). Without loss of generality,
we assume that s1 |= r¬p. By C14, we can conclude that s1 |= 2r¬p, whereby 2r¬p 2
2L+. Because p 6 ? there will be a state s2 2 StateKD45C14C24 such that s2 |= 2rp
and, in consequence, s2 |= 2r(p _ q). Let s =def
W
3({s1, s2}). By the
W
3-persistence
of 2r(p _ q) we have s |= 2r(p _ q) and by construction we know s |= 34p. Hence,
s 6|= 2r¬p, which allows us to conclude, together with fact 10.ii, that s + s1. Now, we
can apply fact 4 and arrive at (D6).
As an aside: in this system (D4) and (D7) will not be valid, of course. According to
KD45C14C24, incompetence of the speaker is not possible and hence the conclusions of
(D4) and (D7) are not consistent with C14 and C24, let alone valid on minimal models.
In this section we restricted our considerations to a language that contained only one additional
modal operator: r. As we have seen, in this case if the hearer thinks the speaker
42
to be competent on 4 in the sense of C14 and C24, then | + predicts the free choice
inferences (D3) and (D6) to be valid. Hence, if we interpret 4 as ‘may’ we do obtain free
choice permission. Of course, it would be preferable if we could extend the account to a
richer language with more modalities - including, for instance, the belief states of other
speakers or different kinds of deontic options. But then, one would also like to consider
the possibility that the speaker is also competent with respect to these modalities. Such
an extension is easily possible. And it is also not difficult to imagine how a language 2L
should be defined that allows, for instance, free choice inferences for a set of modalities 1
on which the speaker is competent and no free choice for the modalities: L should look like
Lo, except that we exclude for r 2 1 sentences with negative occurrences of r not embedded
under another modality. The respective model-conditions are also straightforward, as
is the validity of the free choice inferences (D3) and (D6) for r 2 1 and the incompetence
inferences (D4) and (D7) for r 2 2.
It is even possible to have a notion of entailment that is independent of the choice of 1
and nevertheless provides the free choice inferences for modalities the speaker is competent
on. This is the entailment defined by a language that allows no modality with widest scope
to occur under negation: the BNF is ' ::= p(p 2 P)|' _ '|' ^ '|r ,r 6= 2( 2 L). Let
us call it | ++. One important difference to | + is that with respect to | ++ (D4) and
(D7) are not valid for any modality on which the speaker is not taken to be competent (as
described by the axioms C14,C24).64
3.3.2 Solution 2: Weakening the Logic
Recall the discussion at the end of section 3.2.7: applying | o on StateKD45C14C24 will
predict dishonesty for all consistent sentences of the language L. The problem was that,
on the one hand, a speaker who is competent on a modality r knows everything that can
be expressed on this modality. On the other hand, the information order o also demands
her to say everything she knows on r. Both obligations together can never be met by
uttering a consistent sentence. In the last section we discussed a way out of this unpleasant
situation by weakening the notion of entailment | o. Another possibility is, of course, not
to weaken what the speaker has to communicate, but to take him to be less competent.
Parallel to the last section, where we removed sentences from the language that concerned
knowledge of the speaker on what is not necessary with respect to the modality r, we now
consider the case of an interpreter that does not take the speaker to be competent on these
sentences: we weaken KD45C14C24 to KD45C14. In this case, the speaker knows of
excluded 4-options that they are excluded, but she may take it possible that more options
are excluded than actually are. For instance, applied to deontic options, she may take it as
possible that more obligations exist than actually do.
On StateKD45C14 | o will not disqualify every consistent sentence as dishonest. But
do we obtain the free choice inferences? Once more we run our machinery. Again, we have
64Given the problem that brought us to consider + one may also think about using an order that instead
of skipping the forth condition of o skips the back condition. This would result in minimizing what the
speaker takes to be possible and hence maximizing her belief. This approach does not lead to an appeling
concept of honesty. Because the speaker’s belief can be maximized in many different ways, honesty becomes
a very rare thing again.
43
to adapt our merge-operator65: we define
W
4 simply by weakening the closure conditions
in definition 11 to KD45C14. We can establish again the following fact:
Fact 11 For all T StateKD45C14 the following holds:
(i)
W
4(T) 2 StateKD45C14,
(ii) 8s 2 T :
W
4(T) o s.
In consequence, we have again our result about the Lo-honesty of
W
4-persistent formulas.
It turns out that with the exception of 2p _ 2q all antecedents of the inferences in section
3.1.3 are honest. Particularly, rp _ rq is honest in StateKD45C14 with respect to 2L+.
This may be a bit surprising on first glance, but take a look at the states sketched in figure
3. While there is no state in StateKD45C14C24 that is o-smaller than s1 and s2, s3 2
StateKD45C14 is o- smaller than both. Actually, s3 is the result of merging s1 and s2
using
W
4. s3, however, while in StateKD45C14 is not an element of StateKD45C14C24.
Thus, the sentence rp _rq becomes honest because in StateKD45C14 less restrictions are
imposed on the competence of the speaker.
@ @I
@@R
•
@ @I
@@R
•
@ @I
@@
-
@@R
-
•
s1 s2
s3
w1 w2
w3
p
p,q
p
p,q
p,q
q
p,q
q
p
q
p,q
p
p,q
p,q
q
Figure 3:
In this picture, arrows present
4- admissible worlds, lines 3-
admissible worlds. Again, only
R3-arrows from the root of the
state are drawn. Furthermore,
only information on the deontic
possibility of p and q is specified.
Not mentioning a proposition
letter at a world means that
the proposition is not true there.
Also in this system we can account for the free choice inferences (D3) and (D6) (with
respect to the latter now for both antecedents). To show that a state s1 in StateKD45C14
that entails, for instance, 24(p _ q) but not the conclusion 4p ^ 4q cannot be minimal
with respect to o and StateKD45C14 one can use the
W
4 merge with the same kind of s2
as used in the last section to prove the validity of (D6).
But actually, the inferences (D4) and (D7) are valid too! This could have been expected,
because in both cases the sentences you obtain by negating their conclusions are part of the
order defining language 2Lo. Hence, the interpreter tries to make them false when interpreting
the speaker using | o, and, hence the consequences true. On the other hand, the
consequences are also consistent with the logic and the assumption that the speaker believes
the respective antecedents. Therefore, in sum, the free choice inferences occur together with
the inference that the speaker is not competent on the modality. In particular, it is predicted
the interpreter will learn more about what is valid on R4 than she thinks the speaker knows.
65W
3 is not appropriate because it fails condition (ii) in fact 11.
W
2, on the other hand, fails condition (i).
44
We conclude this section with two final remarks. First, in the discussion above we assumed
that the speaker is taken to be competent on one modality r. The discussed results
all extend to the case that the logic is enriched with C140 for other modalities 40 of the
language.
Furthermore, it is interesting to look at the second possibility to weaken the logic: by
giving up C14 and keeping strong negative competence C24. There seems to be nothing
that speaks generally against such a concept of competence. However, it will not allow us
to account for the data we want to capture. The order + minimizes also what the speaker
knows about what is possible on 4-accessible worlds. Hence, in the minimal states as much
as possible statements of the form 3r will be valid. Following C24 the interpreter thinks
that all the speaker takes possibly to be valid on R4 is actually valid (because if not, the
speaker would know so). Hence, from 3r the interpreter infers r . This leads to results
that are again much too strong. For instance, 4p _ 4q will be dishonest, because the
sentences are consistent with rp ^ r¬q as well as with rq ^ r¬p.
3.3.3 Solution 3: The Combination
Finally, it is interesting to note, that also the combination of | + with StateKD45C14,
hence, the combination of weakening the order and weakening the notion of competence
allows us to derive the free choice inferences. The results from the last sections can be
transfered nearly directly. One point that may need some discussion is the fact that also
with respect to + we can use the merge
W
4 without losing fact (11.ii) for the positive
information order. But this follows immediately from the fact that s o s0 ) s + s0. The
only other point that may be problematic by going from o to + is that the sentences
we used to show that states that do not verify the conclusion of our inferences are not
minimal are no longer in the order defining language. It appears that this is no problem for
the free choice inferences (D1), (D2), (D3), (D5) with the first antecedent, (D6) for both
antecedents and also for the inferences concerning the conjunction (D8) and (D9). The
relevant sentences 2¬p, 2r¬p and 2r¬(p ^ ¬q) are all in 2L+. Again, this combination
of a concept of competence and a notion of entailment differs from the one in section 3.2.6
in not disqualifying the second antecedent of (D6) as dishonest. From the solution studied
in the last section this approach differs by not making the incompetence inferences (D4)
and (D7) valid. Assume, for instance, that (D7) does not hold in some s1 2 StateKD45C14.
This means in particular that s1 6|= 3(rp^¬rq). Hence, s1 |= 2(¬rp_rq). This is not a
sentence of 2L+ and hence a state that simply differs from s1 in not entailing 2(¬rp_rq)
will not be strictly smaller than s1 with respect to +.
Just for the sake of completeness, we will discuss the inference (D9). Assume, there
exists s1 2 StateKD45C14 such that s1 |= 24(p ^ q) and, without loss of generality,
s1 6|= 4(p ^ ¬q). By C14 it follows that s1 |= 2r(¬p _ q). The latter sentence is in the
order defining language 2L+. Because p and q are logically independent from each other,
there are states such that p ^ q as well as p ^ ¬q is true. Take a s2 2 StateKD45C14 such
that s2 |= 2(rp^4q^4¬q). It follows that s2 |= 24(p^q). Hence, for s =def
W
4({s1, s2})
it hold that s |= 24(p ^ q). Furthermore, by construction, s 6|= 2r(¬p _ q) and together
with fact 11(ii) (8s 2 T :
W
4(T) + s) we can conclude that s + s1. In combination with
fact 4 this proves the claim.
45
4 Discussion of the Proposal
The topic of this last chapter is an evaluation of the account developed in the thesis. We
start with comparing the three combinations of a notion of pragmatic entailment and a class
of states on which the entailment is evaluated that turned out to predict the free choice
inferences. After that we will evaluate the approach in general.
4.1 Comparing the three proposed Approaches
At the end of the third section we discussed three combinations of a notion of entailment
(capturing conversational implicatures due to the maxims Q1 and T ) and a class of states
on which the notion of entailment is applied (modeling assumptions about the competence
of the speaker) that all predicted the free choice inferences. In the following we will broaden
our view and evaluate the plausibility of the accounts with respect to other questions. As
we will see they do not always perform equally good. In the end we will come up with a
ranking of the three approaches concerning their general adequacy and choose one of them
as the most convincing account for free choice inferences.
4.1.1 Strong Competence and Weak Order
The first successful system, discussed in section 3.3.1, took a very strong perspective on
competence: both, the positive (C14) as well as the negative (C24) competence axiom
were assumed to be valid in the context of interpretation. This is what we called strong
competence of the speaker with respect to 4. To avoid general dishonesty we had, in
turn, to weaken what counts as the relevant information the speaker has to convey and,
hence, to restrict the language that defined the pragmatic entailment. We discussed two
possible adaptations of the objective information language 2Lo. In variant 1 we removed for
modalities 4 the speaker is assumed to be competent on sentences from Lo where a negative
occurrence of r is not embedded under another modal operator; this led to the notion | +.
In a second variant all sentences that contain a negative occurrence of a modal operator
in wide scope position are removed from Lo - irrespective of competence; the resulting
notion of entailment was | ++. Both system are, despite the weaker information orders,
quite strong in their predictions. For instance, we obtain that an utterance of the sentence
rp _ rq66 made by a competent speaker is pragmatically not well-formed. In particular,
no free choice inferences can be derived. The reason is simple: the sentence is not honest
with respect to the order defining language 2L+/ 2L++ and StateKD45C14C24. Or, to
put it in other words, given the way | + and | ++ formalize the conversational maxims T
and Q1 a competent speaker uttering rp _ rq cannot be obeying these maxims.67
Another pleasant property of the combination of a weak information order with a strong
notion of competence is that it suggests a straightforward way how to account for the
possible cancellation of free choice inferences. As we observed in section 2 free choice
readings do not occur in case the speaker explicitly mentions her incompetence (cf. the
examples (8) and (9) here repeated as (24) and (25)).
66This is the second antecedent in the inference schema (D6), which we flagged with a question mark.
67Recall that all three accounts we discussed predict the sentence 2p _ 2q to be pragmatically not wellformed
for exactly the same reason.
46
(24) You may take an apple or a pear - but I don’t know which.
(25) You may go to Shoal Creek or go to Shingle Creek. But stay away from the dangerous
one.
What does this approach predict in such a situation? Of course, any statement of the
speaker by which she claims to be (partly) incompetent will be inconsistent with taking
her to be strongly competent. Hence, an interpreter that takes the speaker to be strongly
competent on the deontic options will not be able to directly incorporate the information
of the examples (24) and (25) in her belief state. She has, presumably, first to weaken her
belief about the competence of the speaker and to give up strong competence. In this thesis
we cannot discuss the topic of belief change, so we can only suggest that this adaption of
the belief state of the interpreter leads to a new state on which | + and | ++ will not make
the free choice inference valid.68,69
Already in section 3.3.1 we hinted at a difference between the two notions of weak
entailment, | + and | ++. The inference schemas (D4) and (D7), which we wanted to
obtain for modalities the speaker is not taken to be competent on, are not valid for | ++.
To give a concrete example, it is predicted that a possibly incompetent speaker of (25)
takes it as possible that the addressee may go to Shoal creek, but it is not predicted that
the speaker takes it as possible that the addressee may go to Shoal Creek and may not
go to Shingle Creek. This seems contra intuitions, according to which the sentence is
understood as implying that the addressee may go only to one of the creeks and the speaker
does not know which one. The reason why this inference is missing is that sentences like
2(4p^¬4q)) are not elements of the order defining language L++ and, hence, not subject
to minimization. Therefore, we cannot derive that the speaker takes 4p ^ ¬4q to be an
epistemic possibility and, in turn, establish the consequence of (D4). Even though this may
be no serious defect of | ++ we have an alternative notion, | +, that can account for (D4)
and (D7).
One might argue that also | + has its weak point: it depends on what the hearer believes
about the competence of the speaker. One may ask why an interpreter should adapt the
completeness expectations she has for the utterance of the speaker this way. The following
observation hints at an explanation. If the speaker is believed to be strongly competent
68Notice that whether we obtain this result indeed depends on the way the belief change of the interpreter
is modeled. If, for instance, in reaction to (24) or (25) the interpreter adopts a belief state where not C24
but still C14 is valid, then | + and | ++ would still derive the free choice inferences. However, this belief
adaption is arguably to specific. Why should the interpreter choose to give up only C24?
69Of course, this outline of an explanation of the cancellation observation builds on the assumption that
the interpreter first takes the speaker to be strongly competent. Only in these situations can we assume that
belief change is necessary. But if competence is assumed to be something the hearer learns from incoming
information it may be the case that she has only partial knowledge about the competence of the speaker but
still enough to derive the free choice inferences. Then the speaker’s announcement of incompetence may not
be strong enough to contradict the belief of the interpreter and the free choice inferences may still go trough.
An example for such a situation is again the case where the hearer thinks C14 to be valid but does not
know about C24. A sentence like (24) does not stand in conflict with such a belief state. And, as we know
from section 3.3.3, application of | +, | ++ on this belief state would still render the free choice inferences
valid. An interesting option how to exclude such a situation (and they seem to be contra intuition) is to
take also competence to be an assumption the hearer makes about the speaker. We will come back to this
perspective in section 4.2.
47
with respect to some modal operator 4, then | + together with the assumption of strong
competence are already enough to infer all the speaker’s beliefs on Lo − L+, the part left
out of the language. Therefore, in case strong competence of the speaker is commonly
known a speaker does not need to mention these beliefs and a hearer will not expect her
to do so. That the speaker will not give this information seems even more reasonable given
conversational maxims as the second submaxim of Quantity. It asks the speaker only to
communicate what is necessary. There is no time to go into more details on this point. It
has to be left for further research. But so far there seems to be no true objection against
| +.
4.1.2 Weak Competence and Weak Order.
We come now to another approach to the free choice inferences, which was discussed in
section 3.3.3. As we observed there, also the application of the weak notions of entailment
| +/ | ++ to a class of states where the speaker is only taken to be positive competent on
some modality 4 allows to account for the free choice inferences.
Because the notion of entailment used remains the same, what was said above on this
topic applies here as well. But there are some negative consequences of dropping C24 that
have to mentioned. First of all, we lose the suggested account for cancellation in case the
speaker mentions her incompetence. The reason is that the logic KD45C14 leaves the
speaker elbow-room to be in some sense incompetent on 4 and this space is not occupied
by the information order. This has the effect that in +/ ++-minimal states the speaker
does not have to have complete beliefs on what does and does not hold on R4. Therefore,
if the speaker claims to be incompetent on the modality 4 this does not necessarily conflict
with what the hearer has to think of the beliefs of the speaker in order to derive the free
choice inferences. Hence, no inconsistency does occur that forces the interpreter to give up
C14 and, in consequence, the free choice inferences may still go through.70
We will not dwell on possible solutions for this problem but instantly turn to another
suspicious property of this approach that is strongly related. Without C24 we are no longer
able to derive by | +, | ++ from 4(p_q) that 2(4p^4q), hence, that the speaker knows
the free choice inference to be valid. (To illustrate this point, figure 4 sketches two states
for 4(p _ q) that are minimal with respect to + but where C24 does not hold. In s2
2(4p ^ 4q) is not true.) This result does not seem to be correct. Intuitively, we take a
speaker of 4(p _ q) to know that the addressee has free choice.71
Finally, this variant of the approach loses, compared with the one discussed above, the
possibility to disqualify sentences like def rp _ rq as pragmatically not well-formed.
However, there is a straightforward way to repair this defect. In a minimal state for such a
sentence it will be the case that the utterance itself, , is not true.72 This is certainly not a
70Notice, that this is particularly true for the two examples of cancellation discussed here : (24) and (25).
71Notice that the loss of this particular information seems also to be responsible for the difficulties to
account for cancellation. If the speaker is assumed to be aware of the information she implies, then continuation
with “ ... but I don’t know which” would be inconsistent with uttering 4(p _ q) (something along the
same lines can be said about (25)). Hence, in this case one may again be able to account for the cancellation
effects using belief change.
72In the minimal belief state of this sentence the speaker distinguishes an epistemic possibility where
rp ^ ¬rq holds and one where rq ^ ¬rp is true. Hence, ¬2rp ^ ¬2rq. By C14 (which we assume to be
48
s1
H HY
@@R
p *
q
p,q
p
q w1
•
s2
H HY
A
A
A
-
@@R
*
HHj
-
q
p
p,q
p
q w2
p
q
•
Figure 4:
The conventions for this figure are the same as for figure
3: arrows present 4-admissible worlds, lines 3-
admissible worlds. Only 3-connections from the root of
the state are plotted. Furthermore, only information on
the deontic possibility of p and q is given. Absence of a
proposition letter at a world means that the proposition
does not hold there.
plausible model for the utterance. Already at the beginning of section 3.2.1 we discussed the
possibility to strengthen our notion of pragmatic entailment by incorporating the semantic
meaning of the utterance: to take the truth of the utterance to be part of the requirements
minimal states have to fulfill. It turns out that with respect to such an extended notion
of entailment the sentence rp _ rq would again become dishonest. This can be seen as
follows. The first thing to notice is that for two states s1, s2, such that s1, s2 |= rp _ rq
they either entail rp or rq. Assume now that s1 is minimal for rp _ rq with respect to
+. Without loss of generality, s1 |= rp. We obtain s1 + s2 ) s2 |= 2rp (this follows
straightforward from C14 and the fact that rp is in the order defining language). However,
there are states s3 making rp _ rq true that do not entail 2rp: take, for instance, the s3
sketched in figure 5. Hence, there are states s3 such that for a minimal state s1: s1 6 + s3.
In turn, there cannot be an unique minimal belief state. This implies that with respect to
the set of states where 4p _ 4q is true the sentence is dishonest with respect to L+ and
StateKD45C14.
s3
H HY
@@
-
*
HHj
q
p,q
q
q
p,q
w2
•
Figure 5:
4.1.3 Weak Competence and Strong Order
Finally, in section 3.3.2 we discussed the possibility to account for the free choice inferences
by combining positive competence with the notion of entailment | o based on the strong
objective information order. This approach faces similar problems as does the last one
and actually even in increasing strength. Because the order minimizes not only belief on
R4-necessities but now also on R4-possibilities and again positive competence does not lay
restrictions on the speaker’s belief about what is possible on R4-admissible worlds, a hearer
infers using | o that the speaker is as far as possible incompetent on R4-possibilities. For
instance, remember the states s1 and s2 sketched in figure 3. According to the objective
valid in this variant of the approach) it follows that ¬rp ^ ¬rq. Thus, ¬(rp _ rq).
49
information order, s2 o s1 and even s2 is not minimal for 4(p _ q) with respect to o.
Particularly, it turns out that using | o a speaker of 4(p _ q) while implying free choice
permission is inferred not to know that this inference holds. These predictions are clearly
not in accordance with intuitions.
Given this problem it is not surprising that also the difficulties to predict cancellation of
the free choice inferences in case incompetence is stated transfers from the system discussed
in the section above. And, as much as for this approach, the pleasant dishonesty properties
of the first system are not valid.73
Finally, this system predicts that a strongly competent speaker is always dishonestly
withholding information. Again, this goes against intuitions.
4.1.4 Conclusions
Let us summarize the discussion so far. While all three accounts we ended up with in
section 3 predict the free choice inferences, some differences occur in other respects. Only
the first system is able to disqualify the marked inferences in (D5) and (D6) because of
dishonesty. However, we have seen that a straightforward modification of the definition of
the notion of pragmatic entailment may extend the results also to the other two systems.
Furthermore, the intuitions on the well-formedness of sentences like rp _ rq were not
without some doubts. Therefore, the force of this argument in favor of the first account
is not particularly strong. However, both other accounts give unintuitive results for the
hearers picture of the belief state of the speaker after minimization: while intuitive in case
free choice permission obtains the speaker is taken to know this, the system combining
| + with positive competence is not able to make this prediction, and | o even allows to
infer the opposite. These problems lead also to difficulties with the suggested account for
cancellation if incompetence is stated. We conclude that therefore the first approach should
be preferred. This still allows us to choose between | + and | ++. As we have seen, | ++
fails some of the inference schemes we want to be valid. Therefore, | + has to be selected
as the notion of pragmatic entailment that gives the best results. For the rest of the paper
we will always refer to this notion and evaluate it using the strong version of competence.
4.2 Evaluating the Approach
While in the last section we discussed the three successful ways in which the proposed pragmatic
account to the free choice inferences can be spelled out, now we turn to a general
evaluation of the plausibility of this pragmatic approach.
If one considers the list of pragmatic inferences we wanted to obtain (see section 3.1.3)
the approach defended in this thesis is very successful: we were able to come up with a
formally precise Gricean notion of entailment that, indeed, makes all the inferences on this
list valid. But there are also some problems, or at least, open questions connected to the
way free choice inferences are approached here. One point that needs further discussion is
73However, also the proposed solution discussed above extends to the present system. The argumentation
can be copied directly. Because the defining language of the objective information order is an extension of
the order defining language for the positive information order, the latter order is an extension of the former
one.
50
the status of the speaker’s competence. The way it is formalized here (as part of the belief of
an interpreter) has difficulties to account for some further observations. First, when we look
at the data it catches the eye that taking the speaker to be competent seems to be rather
the rule than the exception. Free choice is the normal interpretation disjunctions undergo;
that this reading is not intended has to be signaled explicitly by mentioning incompetence.
Because in the proposed account the free choice inferences depend on competence it is predicted
that the hearer normally believes the speaker to be competent. On the other hand,
one glance on the formalization makes it quite clear that competence is a strong restriction
on the belief of the interpreter. One might question whether in every situation where a free
choice interpretation occurs indeed the hearer has the factual evidence that the speaker is
competent on the matter of discourse. Against this background, it seems more appropriate
to understand also the speaker’s competence as a default assumption of the interpreter.
And remember that also the suggested explanation for cancellation of free choice inferences
works much better with this perspective on competence.
An adaption of the presented interpretation function that reflects such a competence
assumption can be easily given. But this change in the theory provokes some troublesome
questions. For instance, how to motivate such a competence default? None of the Gricean
maxims seems to contain a competence assumption. Is it nevertheless something that is
reasonable to assume in certain situations? Or is it a linguistic convention?
We will leave these questions unanswered and come to another, more serious problem of
the proposed account. As the reader may have recognized, the approach presented here - at
least in the way it is spelled out in section 3 - strongly over-generates. What we mean with
over-generate in this context is that besides free choice much more pragmatical inferences
are predicted and many of them are not in accordance with intuitions. The concrete problem
can be formulated particularly clearly using the order defining languages. The reader
will remember that in case 2L is used to define an information order , everything that
is taken to be in L is subject to belief-minimization and hence, if possible, predicted not to
be decided in the speaker’s belief state. It seems that we took too much of L to be relevant
for the order, hence, we chose an L that is too big. The language fragment L+ contains,
for instance, all atomic propositions P. We obtain, therefore, for arbitrary p, q, r 2 P:
4(p _ q) | + 3r ^ 3¬r ^ 34r ^ 34¬r. And under the additional assumption that the
speaker is competent on 4, 4(p _ q) | + 4r ^ 4¬r.74 Or, to use more natural examples,
given that the sentences Aunt Hetty is making apple pie, Mr. X is in Berlin and You take
a banana can be expressed in our formal language, we obtain, for instance, that (26a) | +-
entails (26b), (26c) | +-entails (26d), and (26e) | +-entails, assuming C1may and C2may
to be valid, (26f). These predictions are certainly wrong.
(26) a. Tomorrow we have better wether or we have to stop.
b. The speaker doesn’t know whether aunt Hetty is making apple pie.
c. Mr. X might be in Amsterdam or in Frankfurt.
d. Mr. X might be in Berlin.
74The same holds for | o and | ++.
51
e. You may take an apple or a pear.
f. You may take a banana.
What we, thus, need to make the approach to the free choice inference defended here
work is a general description of that part of L+ with respect to which the belief of the
speaker really should be minimized.
But even if we had such an description of the order defining language, we still might
be in serious trouble. The point is that because of the specific way in which we motivated
our approach not any description will do. The defined notion of entailment was set out as
formalizing the concept of conversational implicature; as description of what it means to interpret
a speaker as obeying certain Gricean maxims. Hence, whatever language restrictions
we impose, they have to have a Gricean motivation.
In the remainder of this section we will discuss some possible ways to approach this
problem and connected difficulties. However, no final solution will be given. Further investigations
have to show whether the challenge of adequate language restrictions can be
answered or whether the problem of over-generation reveals that, in the end, we chose the
wrong approach.
How could a Gricean description of the order defining languages look like? Let’s start
with a negative example: with a kind of description that can not be motivated this way.
Intuitively, there seems to be a close connection between the inferences we want to obtain
(and, hence, the sentences that should define the order) and what has been actually said by
the speaker. Given this observation, a first idea one might have is to determine the order
defining language using the logical form of the utterance. Gazdar’s calculation of potential
clausal implicatures comes close to such a way of proceeding. But, as already mentioned
when discussing his approach, if it is really the form of the uttered sentence that matters
how should a Gricean motivation for such a dependency look like?75 Is it not the case that
then by definition the generated inferences belong - as form-properties - to the sphere of
semantics? Therefore, we have to conclude that an approach along these lines would not
be in accordance with the general understanding of free choice inferences advanced here.
To find an appropriate description of the order defining language a promising strategy
might be to start directly from Grice and asks oneself what kind of order defining languages
can get a motivation from his theory of conversational implicatures. Or, to reformulate the
question, what part of the belief of the speaker would this theory predict to be exhausted
by her contribution? In the formulation of Q1 Grice explicitly marks that the speaker only
has to be complete with respect to relevant information. In the maxim of Relevance (and
also in the second submaxim of Quantity) he even asks the speaker not to give irrelevant information.
Hence, a restriction of the order defining language to sentences that are relevant
would fit into a Gricean setting.
There can be two different ways of how to incorporate relevance in our description of
the free choice inferences. First, what the speaker thinks to be relevant may be taken by the
interpreter to be a subjective variable that she can access only indirectly - by the utterance
75We would clearly need more information about the form than pure complexity. Hence, a motivation via
Grice’s maxim of Brevity would not be an option.
52
made. In this case the order defining language could be defined as those set of propositions
the utterance made is in some sense relevant to, the facts the speaker is giving information
about. Second, relevance may also be an inter-subjective contextual factor that can be
manipulated by the discourse participants, for instance, by asking questions. In this case
the order defining language is defined as those sentences that serve the commonly known
goal of discourse at the moment when the speaker makes her contribution.
For both perspectives on relevance there exists a wide range of proposals how to make
them precise. To give a simple example how to formalize the first perspective on relevance:
one could, for instance, require that the utterance of the speaker gives information about a
sentence in case the logical form of the utterance and this sentence are logically dependent
on each other. This would, of course be a very coarse-grained notion of relevance. For
the second kind of approach we can make use of the well-developed theories of relevance
determined in terms of contextual given questions or decision problems (e.g. Groenendijk &
Stokhof [1984], van Rooy [2003], van Rooy & Schulz [submitted]). We will use here a rather
simple concept of relevance proposed in this area to illustrate some aspects of relevancebased
approaches to the order defining language. Let us assume that the meaning of a
question is a partition Q of the logical space, hence, a set of sets of states (cf. Groenendijk
& Stokhof [1984]). We define a sentence ' to be about Q ( 2About(Q)) iffdef 8q 2 Q :
(|=q ') or (|=q ¬').76 Our notion of entailment | is then defined for answers to the
question Q by the language 2L where L contains those and only those sentences that are
about Q, (L =defAbout(Q)).
What would such an approach predict, for instance, for an utterance of (26e)? It seems
quite natural to assume in context of the utterance: ‘You may take an apple or a pear’ a
question of the form ‘Which fruits may I take?’. The set of sentences that are about this
question are then L = { you make take an apple (4apple), you may not take an apple
(¬4apple), you make take a banana (4banana), ...}.
A first thing to notice is that this language - and this would also have been the case if
we had used most other notions of relevance - is closed under negation. As we have already
seen when discussing the combination of the objective information language and strong
competence this results in an unintuitively strong notion of dishonesty. To formulate the
problem in some generality:
Fact 12 If (i) the speaker is assumed to be strongly competence with respect to two sentences
1 and 2 (hence, i ! 2 i and ¬ i ! 2¬ i are valid for i = 1, 2), and 1 and 2 and
(ii) their respective negations are elements of L , then the disjunction 1 _ 2 is dishonest
with respect to 2L .
There are different ways to see this.77 Intuitively, the problem is simply that the sentence
2( 1 _ 2) has three different and incomparable minimal models in this setting: one where
1 ^ ¬ 2 holds, one where 2 ^ ¬ 1 is true, and, finally, one where 1 ^ 2 is true. To
give a concrete example and also to illustrate how serious this problem is for our matters,
take 1 = 4p and 2 = 4q. If we assume the speaker to be strongly competent on 4/r
76This notion of aboutness has been independently introduced by Lewis [1998] and Groenendijk [1999].
77You can, for instance, use the disjunction property for 2L . Using strong competence one can derive
2( 1 _ 2) ` 2 1 _2 2 while the antecedent does not entail one of the disjuncts. Hence, it cannot be honest
with respect to 2L .
53
and 4p,4q,¬4p,¬4q 2 L (notice that 4p,4q,¬4p,¬4q 2 About(Q)) then 4p _ 4q
( 4(p _ q)) would turn out to be dishonest.
As with the more general problem of combining strong competence with the objective
information language we have two perspectives now: (i) we can weaken the competence assumption,
or (ii) we can weaken the order defining language. In the first part of section 4 we
have argued that an account that keeps strong competence but describes the order defining
language to be (a subpart of) 2L+ should be preferred. Of course, we can now postulate,
for instance, that the order defining language is the set 2Neg, where Neg= About(Q)\L+
and, hence, for the example we have discussed: Neg = {¬4apple,¬4banana, ...}. Indeed,
in terms of 2Neg we can account for the free choice inferences. But again: what is the
(Gricean) motivation for taking this intersection? Actually, one may argue that Pos =
About(Q)− Neg = {4apple,4banana, ...} is a much more natural candidate in this respect.
For instance, the sentences in Pos are prototypical optimal answers to the question
‘What fruits may I take?’, what leads Hamblin to propose this set to be the meaning of
the question. Of course, minimizing belief with respect to Pos does not give the desired
inferences.78 But a simple adaption brings us back on the right track. We only have to
change how a language defines an information order. First, it is easy to prove that the
following fact holds.
Fact 13 . Let L− be the language defined by the BNF '− ::= p(p 2 L(0))|'− ^ '−|'− _
'−|4',4 6= 3(' 2 L). We define further
s vo s0 iffdef8' 2 Lo : s0 |= 3' ) s |= 3',
s v− s0 iffdef8' 2 L− : s0 |= 3' ) s |= 3'.
Then, it holds 8s 2 StateKD45 : s o s0 , s vo s0 and s + s0 , s v− s0.
In consequence, the notions of entailment that can be defined in terms of the interchangeable
orders (using definition 1) are equivalent too. Hence, minimizing the beliefs of
the speaker with respect to 2Lo/ 2L+ provides exactly the same results as maximizing her
epistemic possibilities with respect to the languages 2Lo/ 2L−. In the same way it can be
shown that vpos defined by Pos is equivalent to neg defined by Neg. And therefore, also
from Pos the free choice inferences can be obtained if we take the perspective of maximizing
the epistemic possibilities of the speaker.79
So far, things are going quite well: we have seen that some problems relevance based
accounts face on the first view can easily be solved. A next question one might want to ask
is which of the two ways to incorporate relevance in the proposed account for the free choice
inferences should actually be adopted: taking relevance to be a contextually given factor
78The order defined by Pos using definition 2 (section 3.2.2) produces inferences that are inconsistent with
a free choice interpretation: having 4apple in the order defining language will lead to an order that prefers
belief states where the speaker does not believe that 4apple is true. In case the speaker is C2may-competent
we even obtain 4(apple _ pear) | pos r¬apple ^ r¬pear.
79As a final observation on this topic, the Gricean argument sketched in section 4.1.1 to motivate the
language restriction from Lo to L+ is independent of the choice of minimizing belief with respect to Neg or
maximizing epistemic possibilities with respect to Pos and, hence, still an option to explain these necessary
language restrictions.
54
or to be determined by the utterance of the speaker. It is important to notice that there
are subtile differences between both perspectives. They make, for instance, different predictions
in case the speaker gives information on contextually irrelevant sentences. In this
case, the latter notion would give rise to implicatures that are not predicted by the former
one. A different situation occurs when the speaker gives no information on sentences that
are contextually relevant. Then, using the first concept of relevance as basis would give rise
to implicatures that do not occur with a language founded on what the speaker is talking
about. We can use these situations where both approaches make different predictions to
test them against each other.
The dialogue fragment (27) is set up as example for a speaker giving contextually irrelevant
information.
(27) A: Was it raining this morning?
B: Yes, and aunt Hetty is making apple pie or muffins.
Even though the response of B is quite strange, especially in those contexts that interests
us here: where aunt Hetty’s activities are absolutely irrelevant for answering A’s question,
intuitively, B is taken not to know what aunt Hetty is preparing for tea. Hence, it seems
as if a formulation of the order defining languages in terms of what the speaker is talking
about is more appropriately describing our intuitions. One may counter with the suggestion
that the speaker’s utterance of the disjunction may change what counts as relevant in the
discourse. Hence, that what the speaker is talking about can be used to repair situations
where the interpreter’s picture of what is relevant and the contribution of the speaker are
not in accordance. One may furthermore suggest that the speaker’s strategy in such a
situation is to add the disjuncts to the set of relevant sentences and, hence, the order
defining language. ‘But why?’ one might ask. In particular, why adding the disjuncts and
not the whole disjunction?
We leave this question unanswered and consider the second test condition. Examples
where the speaker provides (given the semantics assumed here) no information about contextually
relevant sentences can be easily found. Take a situation where (26e) is uttered in
the context of (28).
(28) A: Which fruits may I take?
B: You may take an apple or a pear.
In this situation standard notions of contextual determined relevance (in particular, the
one described above) would predict ‘You may take a banana’ to be a relevant sentence.
However, the speaker B does not provide any information about its truth. What are the
intuitions about this example? Do we infer that the speaker does not know whether it is
true that the addressee A may take a banana or is nothing concluded about the speaker’s
beliefs on the permissibility of taking bananas? The dominant opinion in the literature on
comparable examples (e.g. Groenendijk & Stokhof [1984], van Rooy & Schulz [submitted1]
and [submitted2]) is that while there are contexts in which the answer of B is taken to
represent all the speaker knows (and, hence, the approach using contextually determined
relevance would make the correct predictions), in most situations the interpreter of B’s
utterance will conclude that an apple and a pear is all the addressee may take (hence, taking
55
a banana is prohibited) and this is known to the speaker. Thus, it seems that neither of the
relevance approaches gives in general the right result: while using the utterance to determine
what counts as relevant is unable to account for those cases where indeed the speaker is
inferred to be incompetent about the banana, determining the order defining language with
contextual relevance gives in other contexts a language that is still not restricted enough.
In the latter case the predictions made seem to interfere with the so called exhaustive
interpretation of utterances. The following dialogue fragment gives a standard example for
this notion.
(29) Paul: Who knows the answer?
Paula: John and Mary.
In most contexts Paula’s answer in (29) is interpreted as conveying that John and Mary are
the only people who know the answer. Although the sentence ‘John and Mary know the
answer’ does not bear this information if occurring in isolation (hence, one could argue, it
is not conveyed by its semantic meaning in the strict sense of the word), used as answer it is
understood as giving all those and only those who know the answer. In the same sense (26c)
on page 65 is understood as giving all whereabouts of Mr. X the speaker thinks possible
and (26e) to give a full list of what the addressee may take from the fruit bowl.
When considered more closely, exhaustive interpretation seems to be simply a stronger
variant of the non-monotonic reasoning we modeled in chapter 3. Again, the absence of
information is made meaningful. But instead of inferring from the fact that the speaker
did not claim that some sentence p holds that she does not know whether it does, here
the stronger inference is derived that p does not hold - negation as failure. With respect
to which sentences this inference is made seems to be strongly dependent on what is relevant
in the context of utterance (cf. van Rooy & Schulz [submitted1]).80 Actually, taking
exhaustivity to minimize the set 2Pos described above while at the same time assuming
strong competence on the extension of the question predicate yields in many cases a correct
description. So, let’s adopt for the moment the thesis that exhaustivity can be described
in this way.
We have the following options now. We can first adopt the position that what the speaker
is talking about determines the order defining languages. Then we have to come up with
something else to account for the incompetence readings this approach cannot account for.
Independent of the fact that until now we have said nothing about how to make what the
speaker is talking about precise there are some questions that such a way of proceeding raises.
As may have become clear in section 3 the class of inferences described by our notion of
pragmatic entailment can be seen as generalization of Gazdar’s clausal implicatures. But
there are also strong connections between exhaustive interpretation and Gazdar’s notion
of scalar implicatures (see van Rooy & Schulz [submitted1]). Gazdar took both classes of
conversational implicatures to be due to the first submaxim of Quantity (and the maxim
80Perhaps the context dependence is even stronger here than with free choice inferences. Consider (30).
(30) A: Who is coming for tea?
B: Mary. And aunt Hetty is making [apple pie]F .
For this example an exhaustive interpretation of the second sentence of B seems hardly possible.
56
of Quality). We added in this section a third maxim to the derivation: the maxim of Relevance.
One question that has to be answered now is why we need two different notions of
relevance for both classes of implicatures.
A second option is to take for both: free choice inferences and exhaustification, the
same notion of relevance: contextual determined relevance. But this would mean, given the
preliminary description of exhaustive interpretation we have adopted, that in both cases
2Pos is the order-defining language. It is difficult to see how then the mispredictions we
discussed in connection with example (28) can be excluded. This can, for instance, not
be done simply by executing one inference-generating process after the other (as Gazdar
proposed for his description of clausal and scalar implicatures). As we have described both
processes they work in completely opposite ways on 2Pos: while to obtain the free choice
inferences we have to select the maximal elements of the resulting order, we described exhaustivity
as doing exactly the opposite: looking for its minimal states. It is obvious that,
therefore, no matter which order is adopted, there will be no contribution of the process
running as second.
Of course, this whole discussion depends strongly on the provisional description of the
exhaustive interpretation we have adopted. Therefore, we conclude that in order to find a
solution for the problem we started with: the problem of how to restrict the order defining
languages, first, we have to closely study the phenomenon of exhaustive interpretation and
its interaction with the free choice inferences. But this would go beyond the scope of this
thesis and has, therefore, to be left for another occasion.
57
5 Conclusions
This thesis proposed a formalization for parts of Grice’s theory of conversational implicatures.
The proposal builds on work of Halpern & Moses [1984] recently generalized by van
der Hoek et al. [1999, 2000]. The core of the approach consists of a notion of pragmatic
entailment that is intended to capture the conversational implicatures due to the first submaxim
of Quantity and parts of the maxim of Quality introduced by Grice. Technically, the
defined notion of entailment represents an instance of reasoning in preferential structures:
a sentence is said to pragmatically entail a sentence if in all states where the speaker
is in the unique minimal belief state given that she believes also is valid.
The proposed formalization was then tested by trying to account in terms of it for the
free choice inferences. In particular, we tried to describe free choice permission, a well-known
problem for many semantic and pragmatic theories. It was shown that given contextual
information about the competence of the speaker - and here we build strongly on work of
Zimmermann [2000] - the approach allows us to derive the free choice inferences. Hence, the
introduced formalization of the theory of Grice allows us to describe free choice inferences
as conversational implicatures certain utterances of competent speakers come with.
Actually, three accounts for the free choice inferences were obtained, differing in how
the central notions of competence and pragmatic entailment are spelled out. One of them
appears to be particularly promising. It takes as relation comparing the belief state of
the speaker the positive information order, +, which demands the speaker to give for a
modality 4 she is competent on only the information she has on what according to her
belief is necessary the case on the set of R4-admissible words. She does not have, however,
to convey also her beliefs on what is possible on this set of worlds. The resulting notion of
entailment was then combined with the strong version of the competence assumption. This
system allows us to account for all inferences that have been worked out in section 2 to be
characteristic for the free choice phenomenon.
As has been mentioned earlier the proposed notion of pragmatic entailment is closely related
to Gazdar’s [1979] description of clausal implicatures; not only with respect to the inferences
predicted but also in the way they are thought to be linked to the theory of Grice. A
question that this observation immediately provokes is which account should be preferred.
When discussing Gazdar’s approach we found some of its properties problematic. First, he
was not able to account for free choice inferences of modalities different from the one which
refers to the speaker’s belief state. Furthermore, Gazdar predicts some inferences that are
not reasonable. For instance, he predicts for the sentence 4(p _ q) the potential clausal
implicature 3p. Finally, we complained that the syntactic moment in his description of
potential clausal implicatures (he restricted them to subsentences of the sentence uttered)
has not been motivated with Grice’s theory and that it is also questionable whether such a
motivation can be given at all. While the first type of critics does not apply to the proposal
made here, the second one does. Also the approach developed in section 3 has to fight with
mispredictions. Actually, exactly the same inference we complained about when discussing
Gazdar is also predicted here. And furthermore, in some sense the account is also subject
to the third type of critic on Gazdar: when introducing the approach in section 3 we did
not give a Gricean motivation for the choice of the order defining languages. In section 4.2
58
some attempts for an improvement on this point were made. We discussed the possibility
to determine the order defining languages in terms of relevance. But to be able to see
whether such a way of proceeding is successful it appeared to be necessary first to discuss
the phenomenon of exhaustive interpretation. However, this could not be achieved in the
scope of this thesis and has to be left for another occasion.
To summarize, even though we improve on Gazdar’s approach with respect to some
points the proposal made here is still not fully satisfying. Further research has to reveal
whether it is possible to overcome the remaining shortcomings.
Another topic that needs attention in further work is to what extend the presented formalization
of the theory of Grice (if correct) can and should be extended to inferences due to
other conversational maxims or other phenomena analyzed as conversational implicatures.
Particularly interesting in this respect is an extension to the class of scalar implicatures.
As the reader might remember, Gazdar took these inferences to be consequences of the first
submaxim of Quantity (Q1) too. If he is right in this point one would expect that a correct
formalization of the implicatures due to this maxim should also account for the scalar
implicatures. We already noticed, when discussing the approach of Gazdar, that clausal
and scalar implicatures are clearly distinguished by a different grade of strength: while the
former only convey that certain statements are not believed by the speaker, scalar implicatures
claim that the speaker believes other sentences not to hold.81 How can inferences so
different in character be due to the same maxim?
Interestingly, some linguists working on conversational implicatures (e.g. Soames [1982])
proposed that the strength of Q1 implicatures depends on whether the speaker is taken to be
competent on the subject of conversation. This would suggest that the approach defended
here can also account for scalar implicatures: we already can describe clausal implicatures,
so the only thing we have to do is to apply our pragmatic notion of entailment in contexts
where the speaker is assumed to be strongly competent on the subject of discourse. The approach
to exhaustivity we adopted in section 4.2 realizes this program. We have seen there
that such a way of proceeding leads to certain problems with respect to the free choice
inferences. In van Rooy & Schulz [submitted2], it has been argued that this simple account
for exhaustivity will in general not work. The central problem is that typical scalar implicatures
are also allowed for utterances that are obviously not made by a strongly competent
speaker (if competence is understood as described in section 3.2.7). For more discussion on
the question of how to extend the present approach to scalar implicatures see van Rooy &
Schulz [submitted2].
We close the thesis by going back to the examples that led von Wright [1969] to speak
of a paradox of free choice permission and discuss the explanation given by the account
presented here for von Wright’s paradox.
The problem von Wright faced was that he saw no way to describe a logic of the expressions
involved in permission and obligation sentences that can account for the way we
81The reader may notice the parallels between the description of scalar implicatures given here and the
characterization of the inferences due to exhaustive interpretation, that can be found in section 4.2. One of
the central claims made in van Rooy & Schulz [submitted1] is that both types of inferences are to a wide
extend identical.
59
understand these sentences. In particular, for him there seemed to exist no possibility to
have free choice permission valid together with other necessary inferences without that the
obtained logical system gives rise to absurd predictions. Let us take as an example for such
mispredictions the following chain of reasoning.82
(31) (1) Mr. X may take a taxi.
(2) Mr. X takes a taxi implies that Mr. X takes a taxi or a boat.
(3) Hence, Mr. X may take a taxi or a boat.
(4) Hence, Mr. X may take a boat.
Intuitively, it should not be possible to conclude from (1) that (4) is valid. However,
in a deontic logic (like a logic was supposed to look like at the times of von Wright) that
allows free choice permission as a valid inference (31) can be turned into an argument: (2)
is valid as a tautology of propositional logic. From the premise (1) we conclude by (2) and
a rule of substitution truth conditional weaker sentences in modal contexts that (3) holds.
Then, the free choice inference allows us to conclude from (3) to (4).
Let us first recapitulate the position of Zimmermann [2000] regarding this problem.
First, in contrast to the early position of Kamp [1973], he does not detect the problem in
using a descriptive logic to reason over deontic sentences, but he objects that the wrong
translation of natural language sentences in a descriptive logic is used. More particularly,
Zimmermann claims that the interpretation rule for ‘or’ is wrong. If this is corrected in the
way he proposes (A or B is interpreted as 3A ^ 3B) (3) will no longer be derivable from
premises as (1) and (2).83 On the other hand, Zimmermann predicts that the step from
(3) to (4) is context dependent: it can be made only in case it is known that the speaker is
competent with respect to the deontic options ‘may’ refers to.
What has the account presented in this thesis to say about the paradox? First, together
with Zimmermann, the roots of the paradox are not located in the special features of
the semantics of non-assertive speech acts. But in contrast to him we did not blame the
assumptions von Wright made on the semantic meaning of the utterance involved. In
particular, we adopted an interpretation of ‘or’ as inclusive disjunction. With respect to
semantics, in turn, (2) and the inference to (3) are valid, but the step from (3) to (4) is not
permitted. In this thesis free choice permission is analyzed as a pragmatic inference that
is based on reasoning about the beliefs of a competent speaker that observes the maxims
of conversation in uttering (3). Only on the level of pragmatics an interpreter may reason
from observing an utterance of (3) to the truth of (4). Hence, there is no paradox of deontic
logic but confusion about semantic and pragmatic inferences. Notice finally, that (3) as
an utterance of a speaker who also uttered (1) or concluded (3) from (1) does not allow
us to derive (4). In both cases, given that the speaker of (3) believes (1) to be true, (3)
cannot be all she believes. Therefore the speaker was dishonest with her utterance and an
interpretation according to the maxims of conversation is not possible. Hence, given an
utterance of (31) (1)-(3) from the same speaker (4) is not a valid pragmatic inference.
82This example differs from the one we discussed in section 2. It is a weaker argument for the paradox
than the earlier one because it involves stronger assumptions on the logic of permission/obligation sentences.
Nevertheless, it is more popular in the literature.
83It may even be the case that (2) is not valid anymore. This depends on the modal concept of a belief
state that is used. In S5 and KD45, at least, p ! (3p ^ 3q) does not hold.
60
Acknowledgments
I would like to thank my supervisor Frank Veltman, Paul Dekker and Robert van Rooy
for comments and support during the preparation of the thesis.
References
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[2] Davis, W.A. (1998), Implicature. Intention, Convention, and Principle in the Failure
of Gricean Theory, Cambridge: Cambridge University Press.
[3] Gazdar, G. (1979), Pragmatics, London: Academic Press.
[4] Grice, P. (1989), Studies in the Way of Words, Cambridge: Harvard University Press.
[5] Groenendijk, J. (1999), ‘The Logic of Interrogation’, In: T. Matthews & D.L.
Strolovitch (eds.), The Proceedings of the Ninth Conference on Semantics and Linguistic
Theory, Santa Cruz: CLC Publications.
[6] Harper, W.L. (1976), ‘Ramsey test conditionals and iterated belief change’, In: W.L.
Harper, & C.A. Hooker (eds.), Foundations of Probability Theory, Statistical Inference,
and Statistical Theories of Science, Vol. I, Dordrecht: Reidel.
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1984 Non-monotonic reasoning workshop, New Paltz, NY: American Association
for Artificial Intelligence, pp. 165-193.
[8] Hirschberg, J. (1985), A theory of scalar implicature, Ph.D. thesis, University of Pennsylvania.
[9] Hoek, W. van der, et al. (1999), ‘Persistence and minimality in epistemic logic’, Annals
of Mathematics and Artificial Intelligence, 27, pp. 25-47.
[10] Hoek, W. van der, et al. (2000), ‘A General Approach to Multi-Agent Minimal Knowledge’,
In M. Ojeda-Aciego, I.P. Guzman, G. Brewka, L.M. Pereira, (eds.) Proceedings
JELLIA 2000, LNAI 1919, Heidelberg: Springer Verlag, pp. 254-268.
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[12] Horn, L. (1989), A Natural History of Negation, Chicago: University of Chicago Press.
[13] Jennings, R.E., (1994) The Genealogy of Disjunction, New York: Oxford University
Press.
[14] Kamp, H. (1973), ‘Free choice permission’, In: Proceedings of the Aristotelian Society,
N.S., 74, pp. 57-74.
61
[15] Kamp, H. (1979), ‘Semantics versus Pragmatics’, In: F. Guenther & S.J. Schmidt
(eds.) Formal Semantics and Pragmatics of Natural Languages, Dordrecht: Reidel, pp.
255-287.
[16] Levinson, S.C. (2000), Presumptive Meaning, Cambridge, Massachusetts: MIT Press.
[17] Lewis, D. (1970) ‘A problem about permission’, In: Saarinen et al (eds.) Essays in
Honor of Jaakko Hintikka, Dordrecht: Reidel, pp. 163-175.
[18] Lewis, D. (1973), Counterfactuals, Oxford: Blackwell.
[19] Lewis, D. (1998), Papers in Philosophical Logic, Cambridge: Cambridge University
Press.
[20] McCarthy, J. (1986), ‘Applications of circumscription to formalizing common sense
knowledge’, Artificial Intelligence, 28, pp. 89-116.
[21] Merin, A. (1992), ‘Permission Sentences stand in the Way of Boolean and other Lattice
Theoretic Semantics’, Journal of Semantics, 9, pp. 95-152.
[22] Rooy, R. van (2000), ‘Permission to Change’, Journal of Semantics.
[23] Rooy, R. van, K. Schulz (submitted1), Pragmatic Meaning and Non-monotonic Reasoning:
The Case of Exhaustive Interpretation, unpublished paper.
[24] Rooy, R. van, & K. Schulz (submitted2), Exhaustive Interpretation of Complex Sentences,
unpublished paper.
[25] Soames, S. (1982), ‘How presuppositions are inherited: A solution to the projection
problem’, Linguistic Inquiry, 13: 483-545.
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A Case Study of Wason’s Selection Task’, Journal of Logic, Language, and Computation,
110, pp. 273-317.
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generalized implicatures, Ph.D. thesis, Pennsylvania State University.
[28] Wright, G. H. von (1969), An Essay on Deontic Logic and the Theory of Action,
Amsterdam.
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Language Semantics, 8, pp. 255-290
determines the properties of the function [·]p
w. Stalnaker suggested to define it as reprehensibility:
how much the hearer deviates from the optimal conformistic behavior. This would
mean that a permission per adds the set of least reprehensible -worlds to the deontic
options. Unfortunately, the notion of ‘reprehensibility’ is still too vague to make concrete
predictions. One may interpret it as counting how many previously imposed obligations the
hearer breaks in a world - an idea behind database accounts independently developed by
Lewis [1970] and Kamp [1973] and generalized in the work of Harper [1976].12 Kamp gives
a different utility oriented interpretation of reprehensibility: according to him, permissions
can have different weights depending on their costs for the speaker. But no matter which
elaboration is chosen, the question is: can it account for the free choice inferences?
It is easy to see that a minimal contraction account based on any transitive, irreflexive
and total order per( _ ) !P per( ) ^ per( ) gives us exactly in case the minimal worlds
of each disjunct are all of equal distance from the optimum.13,14 Hence, the free choice permission
is not generally valid for minimal contraction of the option set. However, it may be
the case that the specific order chosen to model change on deontic options allows to derive
the choice principle in those cases where it actually occurs. But neither of the ideas on
what reprehensibility may mean we discussed above will have this property. For instance,
Kamp claims that there are contexts where it is intuitively more reprehensible to take a
pear than to take an apple but where utterances of (2a) would still allow for free choice
10The terminus contraction is taken over from the literature of belief revision. A contraction is an operation
defined on a preference structure: a class of models M together with an order. Applying the operation to a
sentence gives you the minimal elements among the subclass of M where is true.
11That this idea was so present in the discussion is not surprising: just a few years before Lewis had
published his very influential book on the semantics of counterfactuals in which he developed exactly this
kind of preferential interpretation mechanism.
12Harper himself did not apply his system to permission sentences. Only many years after the publication
of Harper’s work van Rooy [2000] made an attempt in this direction.
13That means, no worlds where is true is more minimal than any world where is true and the other
way around.
14We restrict our considerations here to the narrow scope analysis, which is considered by Kamp. Of
course, one could also think about the behavior of a wide scope disjunctions per( ) _ per( ) in such a
setting, which would come with its own problems. For instance, what should it principally mean to utter a
disjunction of two performatives?
9
permission. Hence, his utility based perspective on the order will make false predictions.
And, as van Rooy [2000] points out, neither the order introduced by Harper [1976] makes
the right predictions. We have to conclude that these theories about the meaning of (2a)
cannot account for the free choice inference.
Facing the problems of his proposal, Kamp introduces a new kind of explanation for free
choice permission: maybe it is not due to language conventions of how to interpret permissions,
maybe it is due to reasoning on rational behavior in conversational contexts - maybe
it is a Gricean conversational implicature! Hence, Kamp adopts the strategy of the Gricean
program: rescue semantics by letting pragmatics solve the problems. The difference is that
it is Kamp’s performative permission semantics that has to be saved now.
Kamp discusses a derivation via the maxim of Brevity, one of the rules that, according
to Grice, is obeyed by rational and cooperative speakers. It asks the speaker, roughly,
to keep his contributions short. The derivation of free choice permission Kamp proposes
based on Brevity runs as follows: if it was the intention of the speaker to allow only one of
the disjuncts (as Kamp’s semantics would predict in case the wrong results are obtained)
there would have been a shorter expression the speaker could and, following brevity, should
have used, namely (2b) or (2c). However, she did not. Hence, this cannot be the intended
interpretation of the utterance. But so far, the reasoning brings us only half the way to
the free choice inferences. Now, Kamp continues, we have to rationally motivate why a free
choice permitting reading should be chosen as the intended interpretation, instead.15 Kamp
does not provide an answer to this question, but one may think of the following continuation
of the argument:16 the hearer considers the possibility that she based her interpretation
on the wrong ordering of worlds and tries to figure out the right one. But she cannot take
the order to prefer the other disjunct instead because then the same argument would apply
again. Hence, she concludes that the speaker based her utterance on an order where both
disjuncts are equally reprehensible.
But besides the objection that it is difficult to motivate why the interpreter should
think that the mistake sits precisely in her idea of the order, there is something strange
with Kamp’s explanation in general. As Kamp [1973] notices himself and as was also pointed
out in the introduction, the surprising properties of the interpretation of ‘or’ are displayed
by other quantificational expressions as well. A sentence like “You may pick any flower.”
also receives a free choice interpretation. But here Kamp’s complexity argument will obviously
fail: listing the kinds of flowers that may be picked will raise the length of utterance.
We conclude that also this Gricean attempt to derive the free choice inferences based on a
performative semantics is not very convincing.
If this were not bad enough already, the performative approach to sentences containing
‘may’ has to face other problems as well. Merin [1992] discusses extensively the perspectives
of an account based on a contraction function and presents a whole list of objections.
Among other things he raises a question that often goes unnoticed in discussing the behav-
15This is the big problem any derivation of Brevity implicatures has to face: they easily exclude certain
interpretations but have problems in singling out the interpretation obtained instead. Already Grice notices
that and suggests that brevity implicatures are actually more conventional than conversational.
16An alternative has been suggested and criticized by Merin [1992].
10
ior of ‘or’: do the approaches brought forward also cope with other coordinators, especially
with ‘and’ ? As he points out, there is a general problem with the minimal contraction
approach when it comes to interpret conjunctive permissions. According to it, a permission
per(A _ B) adds to the options of the addressee the closest worlds where A ^ B is true.
Hence, all the options added this way will be such that both conjuncts are true. The package
deal reading is obtained: no matter what the order is the hearer is not permitted to do one
thing without the other. While there is a reading of (6) that can be paraphrased as ‘You
may take an apple provided that you also take a pear, and vice versa.’ this interpretation
only seldom occurs.
(6) You may take an apple and a pear.
The dominating interpretation of sentences as (6) seems to be that the speaker allows the
addressee to choose between four options (e.g. van Rooy [2000]): to take both, an apple
and a pear, to take either or to take neither of them. No account using contraction will get
rid of the package deal prediction.17
To the difficulties for the contraction approach to permission sentences we discussed above,
others have to be added that concern performative approaches in general.
First, permission sentences, especially the examples under consideration, are in the indicative
mood. Given that this mood normally corresponds to assertion one should give the
constative approach a serious try before the dominant speech act - sentence mood parallelism
is broken. Furthermore, Kamp observes that the sentence (2a) has also a reportive,
and thereby assertive use, and even then a free choice reading can occur. Thus, we have to
come up with an additional explanation for free choice readings of assertive utterances.
Another problem is that the free choice reading is not restricted to sentences containing
‘may’ but seems to generalize to other types of disjunctions as well. Kamp’s examples are
(Kamp [1979], p. 281)
(7) a. We may go to France or stay put next summer. (with the epistemic reading of
‘may’)
b. I can drop you at the next corner or drive you to the bus stop.
c. If John had come to the party with Alice or with Joan we might have had some
fun.
17Notice, that this is above all a problem for a narrow scope analysis: when the performative update works
on the permission per( ^ ). One may think about solving the problem by proposing (i) that in case the
reading without package deal occurs the conjunction has wide scope over the permission (but remember that
Kamp does not make any connection between per and ‘may’), and (ii) that a conjunction of performatives
is interpreted as successive execution of the subordinated permission sentences. Of course, this can only be
implemented in a system where permission/obligation sentences do not only update the deontic options but
the whole order on which the contraction operation for permission sentences is defined (as, for instance, the
approach of Harper [1976]). However, while we would get rid of the general prediction of the package deal
reading this way, whether we get the dominant four - choices - reading described above would still depend
on the specific order we chose.
11
hence, the free choice inference is not bound to a specific speech act. It may even occur with
sentences that seem to have primary assertive use. This is no principal argument against
a performative approach, but it teaches us that the explanation given for choice inferences
has to fit in a more general frame.
A stronger point against performative accounts is the following. Reportive uses of
permission sentences do not always allow for free choice inferences, especially not when the
utterance is continued as in (8) or if for other reasons it is known that the knowledge of the
speaker on the topic of interest is limited.
(8) You may take an apple or a pear - but I don’t know which.
In consequence, if we try to describe free choice in reportive sentences we have to make
sure that the inference is not always derived and its occurrence is bound in some sense to
what is known about the competence of the speaker. This could be ignored by an account
that only tries to deal with performative permission sentences. However, Kamp claims that
there are also permissions that have no free choice inferences, for instance (9), used by a
father addressing his son.18 (Kamp [1979], p. 271)
(9) You may go to Shoal Creek or go to Shingle Creek. But stay away from the dangerous
one.
Kamp suggests the following explanation for the difference in reading. In case we observe
free choice permission the act in each disjunct is permitted. In (9), on the contrary, the
disjunction as a whole is permitted and extends the deontic options. He considers the possibility
that a scope ambiguity serves as a basis for this difference in interpretation. Again,
Kamp is not satisfied with this analysis, among other points for the following reason. Utterances
of (4) repeated here as (10), for which an underlying narrow scope logical form would
be difficult to motivate, prominently get the incompetence and not the free choice reading
while Kamp’s proposal takes this logical form to underly the free choice interpretation.
(10) You may take an apple or you may take a pear.
Finally, also a serious gap in every performative proposal Kamp discusses speaks against
his way of approaching the problem: there is no interpretational contribution of ‘may’. One
may think that ‘may’ is the part of the sentence that makes it a permission, but Kamp never
brings the performative back to the occurrence of this verb, nor does he derive the use of the
performative interpretation function compositionally from this part of the structure.19 So,
we are faced with the very strange situation that a systematic ingredient of the utterance
form of permissions does not contribute to the interpretation.
18Merin [1994] supports the analysis of (9) as performative by pointing out that (9) can be put in the imperative
mood, while ‘may’ sentences continued with ‘... but I don’t know which.’ resist such a transformation.
19Perhaps this is so for the following reason. If ‘may’ would contribute compositionally the imperative
mood then the performative interpretation function would only play a role in the scope of the verb. But
what is then the interpretation function applied on top of the whole sentence?
12
2.2 Constative Approaches
Facing all these difficulties, Kamp finally gives up his initial position to account for the
paradox of free choice permission by a performative interpretation of sentences as (2a).
Instead, he tries to arrive at the choice inferences by taking the involved utterances to be
assertions coming with truth-conditional meaning. ‘May’ is now interpreted as an existential
modal operator claiming that among the deontic options of the hearer in a world w there
is an option where holds.20
Also for this constative and modal approach Kamp has to observe that the free choice
principle does not hold simply by the semantics outlined above: based on the classical
notion of entailment for assertions it is still not valid that (2a) entails (2b), for instance,
no matter whether a wide or narrow scope analysis of the disjunction with respect to the
modal operator is taken. Again, Kamp considers the possibility to get the inference as a
conversational implicature using the maxim of Brevity. The situation is slightly different
from the one we considered in the last section when he tried to rescue the performative
account by a Gricean argument - simply because different meanings are assigned to (2a).
With respect to the performative proposal, the interpreter had to face a maxim clash: the
meaning predicted by semantics was not compatible with the maxim of Brevity. Considering
the constative interpretation, there is no obvious shorter expression to express the assigned
truth conditions. This time Kamp claims, that, in a first step, by brevity all possibilities
can be excluded where the speaker knows what is the single disjunct that is permitted.
The reason Kamp gives is that if the speaker had this knowledge she should have used a
shorter expression. Hence, Kamp here uses brevity not as demanding the speaker to code
the meaning she wants to convey the shortest way possible, but to do so with her knowledge.
The interpreter learns that way that the actual world has to be such that the speaker takes
every disjunct to be possibly permitted. Kamp now claims there are only two situations
where this can be the case: (i) the speaker is incompetent, or (ii) the speaker is indifferent.21
If it is known by the interpreter that the former is not the case, then she infers that the
speaker is indifferent and hence that every disjunct permitted.
The free choice inference here becomes a context dependent phenomenon bound to
what is known by the interpreter on the competence of the speaker. This picture fits the
observations with which we finished the last section much better. But again, the proposal
lacks details. Kamp notices this but does not give any elaboration of his ideas, instead
he immediately presents arguments that make the correctness of such a Gricean account
questionable. He observes that the choice inference can also occur if the permission sentence
is embedded in a complex construction as in (11) and can even contribute to the semantic
meaning of this construction.
20We learn not much about the character of the accessibility relation Kamp assumes besides that it is
reflexive: the actual world is among the deontic options. Hence, he only allows consistent sets of obligations
to be imposed on the hearer.
21Given Kamp’s view on assertive permissions (they report that the performative act has occurred) one
may question his decision to contrast incompetence of the speaker with indifference instead of with competence.
It is here much less clear why the indifference of the speaker will warrant that the permission has
actually been given. This is more obvious in case reporting permissions is not about previously executed
acts of permitting but for instance, about reporting preferences of the speaker. The force of permissions may
then be due to a sharing of preferences between speaker and hearer that determines an authority relation.
13
(11) Usually you may only take an apple. So if you may take an apple or take a pear, you
should bloody well be pleased.
How can a pragmatic process as the concept of conversational implicature be that deeply
involved in the calculation of the semantic meaning of (11)? Other counter arguments can
be brought forward. Remember the observation made in the last section: the free choice
inference also occurs with utterances as ‘You may take any apple you want’. Again, Kamp’s
argumentation using Brevity would not generalize to these cases. Furthermore, Brevity cannot
do the job Kamp assigns to it. According to Kamp’s derivation the utterance has to be
a maximally short expression of the knowledge of the speaker. But Brevity, as formulated
by Grice, only asks the speaker to efficiently code the meaning she wants to convey. This
maxim can, hence, not be the reason why the speaker should intend to communicate all she
knows.22
To summarize, Kamp left us with a broad and sophisticated discussion of what does not
work when one tries to account for the paradox of free choice permission and provides a
much more general picture of the phenomenon we have to describe. However, he also left
us the problem.
Several years later another very interesting constative approach to the problem has been
brought forward by Ede Zimmermann [2000]. A central feature of his account is a more
general perspective on the phenomenon. Zimmermann aimed at an account of the free
choice inferences that can also deal with (7a). He even added to the list of phenomena that
have to been explained the observation that a simple factive sentence involving ‘or’ as in
(12) allows to infer (in an honest conversation) that the speaker takes both conjuncts as
epistemic possibilities.
(12) Peter or Marie took the beer from the fridge.
Zimmermann’s approach to the free choice inferences is based on two fundamental
claims.
• Semantically, ‘or’ denotes a conjunction of epistemic possibilities which he speaker
sees23, (hence (12) is interpreted as stating that the speaker thinks it possible that
Peter took the beer from the fridge and the speaker thinks it possible that Marie took
the beer from the fridge).
• The choice principle is a pragmatic inference based on the assumption that the speaker
is competent.
These two assumptions together allow him to account for the free choice inferences based
on a logical form where ‘or’ has wide scope over the modal operators. Let’s see how this
works.
22But for this demand other conversational maxims can be made responsible. Actually, this will be the
fundament of the approach presented in the third section of this thesis.
23Zimmermann does not make the restriction to the belief state of the speaker, but we can ignore this
aspect for our purposes.
14
‘May’, shortened as 4, and ‘might’, shortened as 3, are analyzed, as in Kamp [1979], as
unary modal operators. Hence, the formal system in which the proposal is situated is modal
(predicate) logic. The operators are interpreted with respect to accessibility relations24 R4
and R3, respectively. R3 connects a world w of a model M with all words the speaker takes
as epistemically possible in w. Zimmermann does not model belief but knowledge, hence,
in every model M = hW,R4,R3, V i every world is among its epistemic alternatives: 8w 2
W : hw,wi 2 R3. R4 connects w will all deontic alternative states of affair. Remember,
that truth is defined in modal logic with respect to a state: a tuple of a model M and a
world w in the set of worlds WM in M. For a sentence the truth conditions of 4 are
defined as follows: M,w |= 4 iff 9v 2 WM : hw, vi 2 R4 ^ M, v |= . Truth of 3 is
defined analogously: M,w |= 3 iff 9v 2 WM : hw, vi 2 R3 ^M, v |= .
Now, we can formalize Zimmermann’s semantic analysis of ‘or’ in the following way:
M,w |= or ' ,def M,w |= 3 ^ 3'.
Zimmermann formalizes competence using Groenendijk & Stokhof’s [1984] definition of
exhaustive knowledge:
A speaker is competent with respect to a predicate P (with domain D) in a state
hM,wi iffdef 8v 2 W : wR3v ! (8x 2 D : v 2 P(x) $ w 2 P(x)).
From the definition of competence Zimmermann derives the following Authority Principle:
If a speaker is competent with respect to predicate P in state hM,wi then it holds:
8x 2 D : (9v 2 W : wR3v ^ v 2 P(x)) ! (8v 2 W : wR3v ! v 2 P(x)). That means, if
a speaker competent with respect to P takes it as possible that some object has property
P then she knows that the object has the property.25 For the derivation of the choice principle
Zimmermann instantiates P with the set of deontic options of the hearer in w, the
set {v 2 W|hw, vi 2 R4}. For a speaker competent with respect to this set, the authority
principle says that if she takes a certain world to be possibly a deontic option then she
knows that it is a deontic option.
Now, we have everything in place to face the choice inferences again. We start with the
plain disjunction A or B. By the semantic interpretation rule the sentence is equivalent to
3A^3B - what, given the introduced semantics, immediately accounts for the interpretation
of (12). Next, we apply the account to the example (2a) containing a deontic modality.
4A or 4B is semantically interpreted as 34A ^ 34B. If the speaker is competent with
respect to the deontic alternatives of the hearer in state hM,wi, and, hence, the authority
principle holds we can infer M,w |= ¬3¬4A ^ ¬3¬4B. Now, Zimmermann uses his
assumption that the beliefs of the speaker are correct (reflexivity of R3) to make the step
to M,w |= 4A ^ 4B, the free choice inference.26 Finally the epistemic case: (7a). 3A or
24Binary relations on the set of worlds WM of a model M.
25The derivation of the authority principle uses both directions of the equivalence in the definition of
competence.
26The author does not understand why Zimmermann takes the detour via knowledge. The inference
34 ! 4 is simply the “!” direction of the definition of competence that he already uses in deriving
the authority principle. Hence, reference to reflexivity is not necessary to account for the free choice
interpretation.
15
3B is semantically equivalent to 33A ^ 33B. To allow a derivation from w |= 33 to
w |= 3 Zimmermann introduces the Self-Reflection Principle: Any world w satisfies
8v 2 W : wR3v ! (8v0 : vR3v0 $ wR3v0). This frame condition legitimates the desired
derivation and therefore we can state that also for these disjunctions Zimmermann is able
to derive the choice inference: w |= 3A ^ 3B.
There have to be didactic reasons why Zimmermann introduces the self-reflection principle
additionally to his notion of competence, because the former principle is nothing more
than an instantiation of his concept of competence: now, P is taken to be the set of epistemic
options of the speaker. Thus, Zimmermann’s proposal is more uniform than it may look at
first sight. Furthermore, assuming a speaker to be competent with respect to R4, her own
epistemic state, is equivalent with taking the accessibility relation R3 to be transitive and
euclidic. These conditions are very popular if it comes to characterize the way we reason
about our own beliefs. They assign an agent positive27 and negative28 introspective power
with respect to her belief state. This connection adds to the plausibility of the proposed
formalization of competence.
The derivation of the free choice inference via a competence axiom has another attractive
consequence. As Zimmermann noticed himself, he can straightforwardly account for
the observation that free choice permission is not always observed and that it is omitted
especially in contexts where the speaker explicitly states his incompetence: the derivation
of free choice permission simply relies on taking the speaker to be competent. On the other
hand, if the disjunctive sentence is about the epistemic state of the speaker, one observes
that the choice principle seems to be always present (if the speaker is taken to be honest).
This can now be explained by pointing out that we take every healthy speaker to be a priori
competent on his own belief state.
As these results show: Zimmermann’s proposal is much more successful in accounting for
the free choice inference than any previously discussed alternative. But it has to face its
limitations. We have seen above how the approach generates the correct predictions in
case ‘or’ has wide scope over the modal operators. However, the interpretation mechanism
can also be run on the syntactic analysis giving ‘or’ narrow scope - unfortunately producing
very unsatisfying results. Zimmermann noticed this himself and proposes to exclude narrow
scope readings for independent reasons, but this part of his proposal seems a bit ad hoc.
Zimmermann makes also wrong predictions in case ‘must’ occurs instead of ‘may’. It
seems straightforward to analyze deontic ‘must’ as the dual of ‘may’: r =def ¬4¬, and
epistemic ‘must’ as the dual of ‘might’: 2 =def ¬3¬. But then, his semantic analysis of a
sentence like ‘You must or '’ is: 3r ^ 3r'. Under the assumption, that the speaker
is competent you can now derive that r ^ r', hence, You must and you must ' - a
result that is obviously too strong.
Such sentences, which contain necessity modalities ((13) gives another example) are
rarely discussed in the free choice literature. The simple reason is that they seem to be
unproblematic.
(13) Mr. X must take a taxi or a boat.
27If the agent knows then he knows that he knows .
28If an agent does not know then he knows that he does not know .
16
But, actually, they play their tricks with us, too. As Zimmermann noticed, also for imperatives
there exists an incompetence reading made especially strong by continuation with
‘... but I don’t know which’. But this is not the only possible interpretation. Zimmermann
observes ‘... there appears to be a more straightforward construal according to which Mr.
X’s obligation are unspecific as to the exact means of transport.’ (Zimmermann [2000], p.
283). Hence, there is a reading of (13) from which you can conclude (14a) and (14b), the
obligation (13) still allows Mr. X to chose which disjunct of the obligatory disjunction he
is going to fulfill.
(14) a. Mr. X may take a taxi.
b. Mr. X may take a boat.
The similarity between these inferences of (13) and the choice inference of (2a) plus the
parallel absence of the inferences if the speaker is taken to be incompetent strongly suggest
to speak also for (13) of a free choice inference and to look for one general explanation.29
Further support for this point comes from the fact that the observation extends to epistemic
‘must’:
(15) That must be Tom or Bill.
Inference: That must be Tom. & That must be Bill.
But, much stronger than for ‘may’, people have difficulties to get the free choice inference
for (16), where ‘or’ has explicitly wide scope over ‘must’, and even found it questionable
whether it can get such a reading at all.
(16) You must clean the kitchen or you must go shopping.
All this cannot be explained by the proposal of Zimmermann.
In our opinion, these mispredictions are mainly due to his semantic analysis of ‘or’. This
part of his proposal can even be criticized independently from the free choice topic. For
instance, if his analysis would be correct, how could it be that sentences like (17) are so
useful to communicate the speaker’s strong belief in the first disjunct - something obvious
if one assume inclusive truth-conditions for ‘or’?
(17) Peter is in love or I’m a monkey’s uncle.
And one should be clear about the fact that it is the idea to get the epistemic possibilities
for each disjunct from the semantic meaning of ‘or’ that bound his approach to a wide scope
analysis for this coordination in free choice sentences. This critic does not affect the role
competence plays in Zimmermann’s approach. On the contrary, this aspect of his theory is
very taking.30
29In view of this it becomes clear that the heading conjunctive reading or conjunctive inference, that is
often found in the literature, is clearly not the right peg for the free choice reading. The parallel sentence of
(13) with wide scope conjunction: ‘Mr. X must take a taxi and Mr. X must take a boat.’ does not describe
the observations. Furthermore, Kamp suggests in his papers that the free choice inference has to be seen as
essentially connected with constructions that bring possibilities to the attention of the hearer - this is what
all the sentences around (7a) have in common. The observation above makes that questionable or trivial.
30Notice the parallelism between Zimmermann’s approach and the involvement of competence in the
Gricean argument of Kamp’s constative proposal.
17
3 The Proposal
3.1 Introduction
We come now to the main part of the thesis. In the following, a new constative approach
to the free choice problem is developed. Because it is a constative approach, we will only
discuss the reportive use of permission and obligation sentences.31 And given the general
aim of the paper to defend the Gricean program, a central feature of the approach to be
presented will be that it fulfills the restrictions imposed by Grice on a theory on interpretation.
In particular we will assume a simple and classical semantics: ‘or’ will be interpreted
as inclusive disjunction and the modalities are analyzed as unary modal operators. We will
see that we can, nevertheless, account for the free choice inferences - if we treat them as a
pragmatical phenomenon.
The new proposal will display the same general structure that characterized the account
of Zimmermann [2000] and that is also present in the sketched Gricean and constative approach
of Kamp [1979]. Both establish the free choice inferences on two premisses. The first
ingredient was that the speaker considers certain states of affairs as epistemic possible.32
For our core-example (18a) it is the assumption that the speaker takes both, (18b) and
(18c) as possibly true. Sometimes, this information already accounts for the free choice
observation, for instance, for examples like (12): ‘Mary or Peter took the beer from the
fridge’. But for the free choice reading of (18a) a second premise was necessary and this
was the assumption that the speaker is competent on the subject she is talking about.
(18) a. You may take an apple or a pear.
b. You may take an apple.
c. You may take a pear.
It is obvious that a semantic theory as sketched above will not be able to provide us
with these requisites: given an analysis of ‘may’ as a modal operator on deontic possibilities,
(18a) does neither convey that (18b) and (18c) are possibly permitted according to
the speaker, nor that the latter is competent on the deontic options. And because neither
Zimmermann’s idea to put the first premise in the semantics of ‘or’ nor Kamp’s suggestion
to derive it as conversational implicature via Brevity is convincing we have to come up with
something new with respect to this part. However, we will follow Zimmermann in taking
free choice permission to depend on the information that the speaker is competent on what
is permitted.
The core problem of the new approach is thus to give a account of the first ingredient:
the derivation of information about the epistemic alternatives the speaker considers. We
will describe them as conversational implicatures derived from the assumption that the
31We will not argue for any position with respect to the questions whether there is a performative use and
how it should be treated.
32Remember, however, that both approaches differ in which level of interpretation they take to be responsible
for this information.
18
speaker obeys the maxim of Quality (T ) and the first submaxim of Quantity (Q1) as introduced
by Grice [1989]. We will give a precise description of a class of implicatures that
can be derived from these maxims and show that the needed inferences on the epistemic
state of the speaker are among them. The formal part is an application of work of Halpern
& Moses [1984], that has been recently generalized by van der Hoek et al. [1999, 2000].
This formalization serves at the same time as an argument against critics on the Gricean
program we started with: the notorious and justifiable complaint of Kamp and many other
scholars that all Gricean approaches lack (a convincing) formally precise implementation.
3.1.1 Free Choice as Clausal Implicature:
The Proposal of Gazdar
Actually, this is not the first time that the relevant information about the epistemic possibilities
of the speaker has been analyzed as due to the Gricean maxims T and Q1. At least
part-wise already Gazdar, in his dissertation from [1979], took them to be such conversational
implicatures.33 The account presented here is strongly related to his proposal. To
study Gazdar’s approach is therefore a good starting point to introduce some general ideas.
Gazdar distinguishes two classes of inferences which he analyzed as generalized Q1 implicatures.
The first class can be described as inferences that some kind of strengthening of
the claim made does not hold (as far as the speaker knows). These are the so-called scalar
implicatures. The second class consists of claims that the speaker does not know for some
strengthening of the given utterance whether it holds or not. Gazdar called this group of
inferences clausal implicatures. The following examples of the latter notion are given by
Gazdar (Gazdar [1979], p.50).
(19) a. If John sees me, then he will tell Margaret.
Implicature: I don’t know that John will see her.
b. My sister is either in the bathroom or in the kitchen.
Implicature: I don’t know that my sister is in the bathroom and I don’t know
that my sister is in the kitchen.
The reader will have noticed that the clausal implicatures of the second sentence give
exactly the kind of information that Zimmermann [2000] analyzed as the semantic contribution
of ‘or’, the information on the epistemic state of the speaker we are looking for. Here,
they appear as pragmatic inferences, as clausal implicatures of a simple34 disjunction. Of
course, this observation raises the question whether we can account for the free choice inferences
simply by combining Gazdar’s concept of clausal implicatures with Zimmermann’s
formalization of competence. This will now be investigated.
Gazdar describes the following procedure to calculate clausal implicatures. First, he
defines the set of potential clausal implicatures (pcis) of a compound sentence (Gazdar
33However, he did not consider an application to the paradox of free choice permission.
34A sentence is called a simple disjunction, if the only sentential operator it contains is ‘or’ and this
operator occurs only once.
19
[1979], p.59, 3 is defined as the dual of 2, and 2 stands for the speaker knows that ).35
|
2 {3 ,3¬ } and is a subsentence of
such that neither entails nor its negation¬
As the reader can see, clausal implicatures are essential inferences about the limitations
of the knowledge of the speaker. For instance, for example (19b) the definition predicts the
following set of pcis: { the speaker does not know that her sister is in the bathroom, the
speaker does not know that her sister is not in the bathroom, the speaker does not know that
her sister is in the kitchen, the speaker does not know that her sister is not in the kitchen}.
But not all potential clausal implicatures are predicted by Gazdar to become part of the
interpretation of an utterance. First, they have to pass a strict consistency check: A pci becomes
part of the interpretation of an utterance if it is a consistent extension of the common
ground together with the assumption that the speaker knows her utterance to be true36 and
all other potential clausal implicatures. Because of this filter-condition, clausal implicatures
become context dependent and, for instance, the misprediction of Zimmermann concerning
example (17) is avoided. However, if nothing special is commonly known, uttering ‘ or
’ where and are logically independent propositions will clausal-implicate 3 and 3 ,
Zimmermann’s semantic meaning of ‘ or ’ . Hence, Gazdar can account for the free
choice inference which a simple disjunction as (12) or (19b) normally comes with.
But Gazdar cannot only account for these observations. If we take sentences containing
epistemic operators, his approach also predicts the free choice inferences we observed.
Let’s see how. For a sentence 3( _ ) Gazdar predicts the following set of pcis:
{3 ,3 ,3¬ ,3¬ ,3( _ ),3¬( _ )} - among them the free choice inferences 3 and
3 . Whether the pcis are actually generated depends on the context. If the logic of the
knowledge operator is S4, as Gazdar assumes, and nothing else is commonly known on
and , then all pcis in this set are predicted to be actually derivable from the utterance.
Hence, Gazdar can also account for a free choice reading of 3( _ ).37 Let us consider
another case: 2( _ ). We arrive at exactly the same set of pci’s. The element 3¬( _ )
is not consistent with the assumption that the speaker believed what she said and will
therefore be not predicted as actual implicature. However, the free choice implicatures 3
and 3 survive the consistency check and, hence, Gazdar can account for the free choice
interpretation of examples like (15)!38
Unfortunately, Gazdar’s proposal makes generally incorrect predictions if applied to
disjunctions containing other modal operators 4 with 4 6= 3. In consequence, he cannot
account for free choice in deontic contexts. With respect to a narrow scope disjunction
4( _ ) there is no hope at all to obtain the needed implicatures, because in this case
there will not be any information about the modality 4 in the set of pcis of this formula:
35Gazdar gives one further condition, but this one can be ignored for our purposes. Furthermore, he
assumes S4 to be the logic of the modal operator 3.
36This is Gazdar’s formalization of the T -implicatures an utterance comes with.
37The wide scope analysis gets also a free choice reading in S4. However, this is not the case, if the logic
of 2 is assumed to be KD45. In this case the relevant potential clausal implicatures are generated but no
potential implicature will survive the consistency check (see the discussion below for 4 _4 ). Thus, even
semantically equivalent sentences may generate different sets of implicatures under Gazdar’s treatment.
38The pcis of a sentence 2 _ 2 are different, but still the free choice interpretation is derived. For the
logic KD45 we obtain the same results.
20
{3 ,3¬ ,3 ,3¬ ,3( _ ),3¬( _ )}. For the sentence giving the disjunction wide
scope, 4 _ 4 , the situation looks a little bit better. Gazdar predicts the following set
of pcis: {3 ,3 ,34 ,34 and the respective negations}. A first thing to notice is that
the first two members of this list together with their negative counterparts should not be
predicted as conversational implicatures. Though in case 4 in 4 _ 4 is interpreted as
deontic ‘may’ one (normally) infers from an utterance of this sentence that the speaker
takes the asserted deontic options also to be epistemically possible, this inference should
rather be analyzed as part of the appropriateness conditions (presuppositions) of permissions
(and obligations). To support this standpoint, notice that this inference accompanies also
sentences like (20), and, thus, projects trough negation.
(20) You must not take an apple or a pear.
That 3 and 3 should not be predicted to be conversational implicatures of an utterance
of 4( _ ) is even more obvious if one takes 4 to stand for other modalities like, for
instance, the belief state of other agents. While also in this case an utterance of (21) does
have a free choice reading (Peter thinks it possible that Mr. X is in Berlin and he thinks it
possible that Mr. X is in Amsterdam), one does not infer, of course, that also the speaker
thinks it possible that Mr. X is in Berlin.
(21) Peter believes that Mr. X is in Berlin or in Amsterdam.
But let us consider the other pcis Gazdar predicts for the sentence 4( _ ). Again,
Zimmermann’s semantic interpretation of the sentence is part of the pcis. Therefore, one
may think, if we add an assumption of competence to the context with respect to which
4( ^ ) is interpreted, then we will be as successful in deriving the free choice inferences
as was Zimmermann. But his formalization of competence will allow us not only to derive
from the potential clausal implicature 34 the fact 4 but also to conclude from its
negative counterpart 3¬4 that ¬4 holds.39 Hence, we cannot add the pcis we need
to such a context without generating contradiction with other elements of this set. They,
therefore, do not pass Gazdar’s consistency test and are predicted not to occur. The results
for sentences containing deontic ‘must’ are equally bad.40
After such a successful beginning the question arises: what went wrong? And can we
repair Gazdar’s approach without major changes? The discussion above revealed two different
problems of his approach. For one thing, there is something essentially wrong with
the generation process of the pcis, as, for instance, the unintuitive pci 3p for a sentence
4(p_q) shows. This part of his theory demands serious revision. In particular, the syntactic
aspect of Gazdar’s notion that allows you to take subsentences out of their modal contexts
has to be removed. The problem concerning the incompetence that prevents Gazdar from
predicting the free choice inference for 4 _4 , on the other hand, seems to be less severe.
To avoid it one could change the generation process so as to not include the negative pcis
or weaken the notion of competence, for instance.
39To see this, the axiomatization of Zimmermann’s definition of competence in section 3.2.7 may be of
use.
40There is, of course, no difference in the results if we would assume as epistemic logic KD45.
21
An objection against the proposal of Gazdar of a totally different character is that it
lacks a precise motivation as a formalization of the theory of Grice. And again, it is above
all the syntactic aspect of his description of pcis that does not seem to fit. Gazdar sketches
a derivation of the implicatures from the maxims Q and T that is in its structure as old
as the theory of Grice itself: if the (rational and cooperative) speaker had known the truthvalue
of a subsentence of her utterance, then she should, in order to obey Q1 and T , have
used a sentence that semantically conveys this information. Hence, the speaker cannot know
whether the sub-sentence holds or not. Given the formulation of the maxims it is absolutely
unclear, why this reasoning should apply to sub-sentences of the utterance, and to those
only.
To develop our approach and overcome the deficits discussed above we will again start
from Grice and motivate a new description of the conversational implicatures due to T and
Q1. In the end we will see that the inferences we describe can be best understood as, again,
Gazdar’s clausal implicatures. In particular, all the appealing inferences he predicts are
maintained.
3.1.2 Technical Preliminaries
But before we can really start introducing the new approach, the basics of the formalization
of natural language we will use have to be laid out. This will be done here: We will make
our model-language precise, define its semantics and introduce some technical machinery
we will need later on.
Language Our language L is generated from a finite set of propositional atoms P =
{>,?, p, q, r, ...}, the logical connectives ¬,^,_, and ! and finitely many unary modal
operators {41,42...,4n}. We will use ri to shorten ¬4i¬ for all i 2 {1, ...n}. 3 is
the special modal operator 4i for some i 2 {1, ..., n} that refers to the belief state of the
speaker and 2 ¬3¬ expresses in turn that the speaker believes . 4 and r are used
as meta-variables over the set of modalities of L.
| · | : L −! N denotes the modal depth of formulas, defined recursively: for p 2 P and
, 2 L, |p| = |>| = |?| = 0, |¬ | = | _ | = | ^ | = max{| |, | |}, |4 | = |r | =
| | + 1.
We will frequently make use of sublanguages L0 L. For instance, the languages
L0
(n) = { 2 L0| | | n, n 2 N}, the set of L0-sentences with modal depth smaller or equal
to n, will play a central role. We call L(0), the language that contains the modal-free part
of L, the basic language. Another important construct we will use is 2L0 = {2 | 2 L0},
the language that shifts L0 into the belief state of the speaker.
We introduce the following abbreviations for certain L formulas:
(K) r( ! ) ! (r ! r ), (4) 2 ! 22 ,
(Dual) r $ ¬4¬ , (5) ¬2 ! 2¬2 ,
(D) 2 ! 3 .
22
Semantics A frame for L is an (n+1)-tuple of a set of worlds W and for every modality
4 of L a binary relation R4 over W. A model for L is a tuple consisting of a frame for
L and an interpretation function V for the non-logical vocabulary of L; a function from
p 2 P to characteristic functions over W. Let F = hW,R41 , ...,R4ni be a frame for L and
M = hF, V i a model. For w 2 W, R4i [w] denotes the set {v 2 W|hw, vi 2 R4i}. We call
the tuple s = hM,wi, where w 2 W, a state. Truth of a sentence of L with respect to a
state is defined along standard lines. We will give here only the definition of truth for a
modal formula 4 : M,w |= 4 iffdef there is a v 2 W such that v 2 R4[w] and M, v |= .
A sentence is valid on a class of states S (|=S ) iffdef for all s 2 S: s |= . A set of
formulas entails a formula relative to a class of states S ( |=S ) iffdef for all s 2 S:
s |= implies s |= . If = { }, we write |=S . F,w |= holds iffdef for all valuations
V : hF, V i,w |= . F |= iffdef for all w 2 W: F,w |= .
Without going into much details here, models for L can also be treated as models for the
first-order language L1 corresponding to L (see Blackburn et al. (2001), definition 2.44).
For 2 L1 we write Mk − iffdef is true in the model M = hF, V i. Fk − iffdef for all
valuations V : hF, V ik − .
For A1, ...An 2 L, StateKA1...An is the set of states s = hM,wi with M = hF, V i such
that for all i 2 {1, ..., n}: F,w |= Ai.41 We shorten |=StateKA1...An
with |=KA1...An. In this
thesis we will work with (subsets of) StateKD45. The sentences (D), (4), and (5) locally
correspond to well known first order formulas, what allows us to get a good grasp of the
structure of the states in StateKD45.
Fact 1 42 Let F = hW,R41 , ...,R4ni be a frame for L and w 2 W. Then the following
correspondences hold:
F,w |= (D) iff Fk − 9x : R3(w, x)
F,w |= (4) iff Fk − 8x, y : R3(w, x) ^ R3(x, y) ! R3(w, y)
F,w |= (5) iff Fk − 8x, y : R3(w, x) ^ R3(w, y) ! R3(x, y)
We know, hence, that the underlying frames of the states in StateKD45 are with respect
to R3 locally (in w) transitive, euclidical and non-blind43. Conceptually, that means that
we adopt a particular perspective on how the belief state of a speaker looks like: the speaker
has positive and negative introspective power and the absurd belief state is excluded. This
restriction has been necessary because of the limited resources of the study at hand.
The reader may be surprised by the choice to ask only for the local validity of the
formulas (D), (4), and (5). In consequence, we allow models where, for instance, the speaker
thinks it possible that some other agent takes him to be in a non-transitive, non-euclidian,
or absurd belief-state. To realize the idea that KD45 is a general restriction of what belief
states can possibly look like, one should impose the axioms globally on the underlying
frames. Then one should, of course, also single out the class of modal operators that
describe the belief state of other agents and impose parallel conditions on them. One
41Hence, Ai is valid in w on the frame underlying M.
42Proof: Because (D), (4), and (5) are Sahlqvist-formulas we know immediately that they have firstorder
correspondents (see Blackburn et al. (2001), theorem 3.54). Applying the Sahlqvist - van Benthem
algorithm gives us the claimed equivalencies.
43A state shM,wi is non-blind in w with respect to R3 of M iffdef R3[w] 6= ;.
23
reason why this is not done here is that in this paper we will never come in a situation
where we will talk about such deeply embedded belief. Hence, the restriction would never
show up. Furthermore, later on we will consider restrictions on states that are only plausible
when imposed locally.44
Another concept that will - in different disguises - play an important role in the theory
we are going to develop is n-bisimulation. Let hM,wi, hM0,w0i 2 S be two states in S. We
say that hM,wi, hM0,w0i are n-bisimilar (hM,wi =n hM0,w0i) if there exists a sequence of
binary relations Zn · · · Z0 with the following properties (for i + 1 n):
(i) wZnw0
(ii) If vZ0v0 then 8p 2 P : V (p)(v) = V 0(p)(v0)
(v and v0 agree on all proposition letters)
(iii) If vZi+1v0 and u 2 R4[v]
then there exists u0 with u0 2 R0
4 [v] and uZiu0 (forth)
(iv) If vZi+1v0 and u0 2 R0
4[v0]
then there exists u with u 2 R4[v] and uZiu0 (back).
We will frequently make use of the following well-known theorem:45
Theorem 1 Let L be a modal language with a finite set of atomic propositions: |P| = m 2
N. For two states s, s0 2 S where S is a set of states for L the following are equivalent.
(i) s =n s0
(ii) s and s0 agree on all ' 2 L with modal depth at most n.
Finally, we define for A1, ...,An 2 L the KA1....An-closure (CLKA1....An(M)) of a model
M = hW,R41 , ...,R4n, V i as a model M0 with M0 = hW,R0
41
, ...,R0
4n
, V i 2 StateKA1...An,
for all 4i, R4i R0
4i
, and there is no M00 = hW,R00
41
, ...,R00
4n
, V i 2 StateKA1...An such
that for some 4i R4i R00
4i
R0
4i
.
3.1.3 The Data in Overview
Our aim is to describe the diverse free choice inferences we met in section 2. Because we have
to account for inferences, the core of the proposal should be to define a notion of entailment,
lets call it | , such that the relevant inferences become valid with respect to it. Given
our formal model-language L as described above this means that the following entailment
statements have to be valid. Let p, q 2 L(0) be two logically independent propositions
({A|B} has to be read as ‘A is the premise or B is the premise’):
(D1) p _ q | 3p ^ 3q
44The choice to work with local restrictions has some consequences for the logic of |=S: the normal modal
logic with additional axioms (D), (4), and (5) is not sound with respect to StateKD45. However, it is not
difficult to find a sound and strongly complete proof system for this class of states: the only thing one has to
do is to restrict in the normal modal logic with additional axioms (D), (4), and (5) the rule of generalization
to formulas that are also derivable in the minimal normal modal logic K. Because syntactic considerations
play no role in the rest of the paper we abstain from discussing this topic in more detail.
45For a discussion see Blackburn et al. [2001], p. 74.
24
(D2) {3(p _ q)|3p _ 3q} | 3p ^ 3q
(D3) {4(p _ q)|4p _4q} | 4p ^4q
with competence of the speaker.
(D4) {4(p _ q)|4p _4q} | 3(4p ^ ¬4q) ^ 3(4q ^ ¬4p)
without competence of the speaker, 4 6= 3
(D5) {2(p _ q)|2p _ 2q?} | 3p ^ 3q
(D6) {r(p _ q)|rp _ rq?} | 4p ^4q
with competence of the speaker.
(D7) {r(p _ q)|rp _ rq} | 3(rp ^ ¬4q) ^ 3(rq ^ ¬4p)
without competence of the speaker, r 6= 2
(D8) 3(p ^ q) | 3(p ^ ¬q) ^ 3(¬p ^ q) ^ 3(¬p ^ ¬q)
(D9) 4(p ^ q) | 4(p ^ ¬q) ^4(¬p ^ q) ^4(¬p ^ ¬q)
with competence of the speaker.
The restriction to logically independent propositions reflects the observation that we
want the inferences only to be generated in case they are consistent with the context. For
instance, we don’t want (D1) to be valid if the antecedent is the logical form of (17), here
repeated as (22).
(22) Peter is in love or I’m a monkey’s uncle.
(D2), (D3), (D5) and (D6) are the core observations: the free choice inferences for
epistemic and deontic modalities. Zimmermann has added (D1) to the free choice inferences:
the intuition that simple occurrences of ‘or’ (no additional sentence operator is involved)
allow the inference that both disjuncts are considered to be possible by the speaker. (D4)
and (D7) describe the alternative free choice free interpretation for the deontic examples.
We observed that these readings are particularly forced if the speaker states explicitly her
incompetence and that they are not possible for epistemic modalities. (D8) and (D9),
finally, refer to one of the points of criticism Merin brought forward against the minimal
contraction approach: it cannot handle conjunction properly. As we will show, our approach
will predict exactly the reading van Rooy [2000] described as the normal interpretation of
such sentences, namely (D8) and (D9).
No claim will be made in advance with respect to the choice of scope on which each
inference is based. As we have noticed above, free choice as well as the incompetence
inference can come with the surface narrow scope form - but there may be a deletion
process working on deep structure, hence, this does not allow us to conclude that both
inferences have to be derivable from a narrow scope logical form. For surface wide scope
we noticed that free choice readings are more difficult to get and are reported to be absent
for all-quantifying modalities. However, because this was not tested properly, we will not
exclude this analysis for free choice. We will just see what our approach predicts.
25
3.2 Developing the Framework
3.2.1 A Gricean Notion of Entailment
Our starting point is Grice’s theory of conversational implicatures, especially the first submaxim
of Quantity (Q1): Make you contribution as informative as required (for the current
purpose of exchange).46 Because of the contrast with the second submaxim and the
applications Grice discusses, this maxim is generally understood as an imperative to maximize
the quantity of relevant information that is transmitted.47 But the speaker is not
allowed to provide any bit of relevant information she can think of. The maxim of Quality
(T ) restricts her contributions to facts she believes to hold and for whose truth she has
sufficient evidence. Both maxims can be brought together in the following interpretation
rule for a hearer:48
The contribution of a rational and cooperative speaker encodes
all of relevant information the speaker has; she knows only .
We now introduce a pragmatic notion of entailment, | , building on this rule of interpretation.
| is defined to hold in case is true on all models where represents all
the speaker knows. In other words, | describes the inferences an interpreter can draw from
an utterance if she assumes the speaker to behave rational and cooperative; in particular,
to obey T and Q1.
Fortunately, the question how to formalize that a certain formula expresses all an agent
knows has gotten some attention from logicians. Our account for the free choice inferences
highly relies on this work. More specifically we are going to use an approach introduced by
Halpern & Moses [1984] and generalized by van der Hoek et al. [1999, 2000].
The fundamental idea behind their formalization is strikingly simple. The states where
a sentence is all a speaker knows are taken to be those states where she is in the minimal
belief-state, given that she believes . Hence, we impose an order on all the states s
where the speaker believes her utterance , i.e. s |= 2 , that compares how much the
speaker believes in these states. And then we define as the class of pragmatic inferences
those sentences that hold on the minimal elements of this order. It is obvious that such a
formulation of a notion of entailment constitutes nothing more or less than an instance of
interpretation in preference structures. We summarize our considerations in the following
46We will not discuss why a rational cooperative speaker should obey the maxims of conversation. We
simply try to capture what an interpreter can infer if she assumes that the speaker does so.
47Although the restriction to relevant information has often been neglected in the literature.
48This paraphrase does not exactly capture what Grice says in his maxim Q2. According to his formulation
it may be the case that two possible utterances of the speaker are both maximally informative and relevant
but nevertheless convey different information. Our formulation implicitly assumes that there exists exactly
one maximum. We can make this claim because we will order sentences only with respect to the semantic
notion of entailment |=. Then it is the case that if two sentences are logical independent their conjunction
is more informative than both of them and therefore we can assume the existence of a unique maximum.
However, |= may not be an appropriate description of the order (how does it deal with the modification of
Grice: ‘as informative as required’, for instance). Nevertheless, if adaptations are necessary they can take
the work presented here as a starting point.
26
definition of a notion of pragmatic entailment.
Definition 1 (The Inference Relation | )
Let be a partial order on some class of states S. We define for sentences , 2 L:
| S iffdef 8s 2 S : [s |= 2 ^ 8s0 2 S : s0 |= 2 ) s s0] ) s |= .
This relation of entailment is the hart of the approach developed here.49
3.2.2 Information Orders. Introduction
The crucial question for an approach along these lines is whether we can find a convincing
definition of the order , hence, of what it should mean that in one state the speaker knows
more than in another. Until now we have said nothing on this point. This will be our next
topic.
Van der Hoek et al. [1999] introduce an appealing general characterization of such
orders. The underlying idea is very simple: we use our language to define the order. A
speaker is said to believe less in a state hM,wi than in a state hM0,w0i with respect to a
certain sublanguage L0 L, if the set of sentences of L0 she believes in the first state is a
subset of the set of sentences of L0 she believes in the second. To make this more precise
we define the notion of an information order.
Definition 2 (Information Orders)
A sublanguage L0 L defines with respect to a class of states S the relation iffdef
8s1, s2 2 S : s1 s2 , 8' 2 L0 : s1 |= ' ) s2 |= '. An order that is defined by a
sublanguage L0 L is called an information order.
The restriction to orders that have defining languages has some consequences for the
notions of entailment that can be defined in terms of them. For instance, these orders are
partial but not necessarily total orders. Hence, it may be the case that for a sentence
there exists more that one minimal state. Given the application we have in mind the orders
we are interested in compare states with respect to the beliefs of the speaker. Hence, we
will use information orders defined by languages 2L for L L. If for a sentence there
exist with respect to the order defined by such a belief-language more than one minimal
state this means that for the interpreter a speaker of can be in (with respect to 2L )
different minimal belief states - that she believes in the minimal belief states for different
things. But in such a situation the speaker has to have L -beliefs she did not communicate.
Thus, it is obvious for the interpreter that she did not obey the maxims Q1 and T (as
they are interpreted by this account). Such sentences should then be pragmatically not
well-formed. The notion of entailment that can be defined on basis of an information
order using definition 1 reflects this consideration: if there are different minimal states,
49Some reader may miss semantics in this definition: the truth conditions of the antecedent play no role.
This is true and the notion of pragmatic entailment defined above is not conservative with respect to the
semantic entailments of an utterance - hence, does not truly strengthen the semantic meaning of an utterance
as one might expect. However, this can be easily achieved by defining | s by 8s 2 S : s |= ( ^ 2 ) &
... . Also with respect to this notion, the results we are going to discuss will hold. (This is not trivial: by
extending the selection conditions, the set of minimal models may change.)
27
then | ?, because there will be no state that fulfills the antecedence in definition 1.
Halpern & Moses called such sentences, sentences that allow for different with respect to
2L incomparable minimal belief state, dishonest.
Definition 3 (Honesty)
A sentence 2 L is called honest with respect to a language 2L L and a set of states
S, iffdef {s 2 S|s |= 2 ^ 8s0 2 S : s0 |= 2 ) s s0} 6= ;, where is the order defined
by 2L .
There is an interesting and useful connection between being honest and another property
of formulas. Let’s say that a formula 2 L has the disjunction property with respect to a
language 2L L and a class of states S iffdef is satisfiable in S and for every finite set of
sentences 1, 2, ..., n 2 2L : if |=S 1 _ ... _ n then for some i 2 {1, ..., n}: |=S i.50
Now, assume that is honest with respect to 2L and S. This means that there exists a
state s 2 S such that s |= 2 and 8s0 2 S : s0 |= 2 ) s s0. Assume at the same time
that 2 does not have the disjunction property with respect to 2L and S. Then, there
is some sequence of sentences 1, ..., n 2 2L , such that 2 |=S 1 _ ... _ n but for no
i 2 {1, ..., n}: 2 |=S i. Of course, if 2 |=S 1 _ ... _ n then also s |= 1 _ ... _ n.
Because of the way the truth function of _ is defined, from this it follows that there is
some k 2 {1, ..., n} such that s |= k. However, by the choice of the 1, ..., n it cannot
be the case that 2 |= k. Hence, there has to be an s0 2 S such that s0 |= 2 and
s 6|= . But this impossible: we have chosen s such that 8s0 2 S : s s0 and, hence,
8 2 2L : s |= ) s0 |= ! This proves the following fact:
Fact 2 If is honest with respect to 2L (L L) and a class of states S, then 2 has
the disjunction property with respect to 2L and S.
This result will prove to be quite useful for establishing that certain formulas are not honest.
3.2.3 How to establish Properties of Minimal States
In the last two sections we introduced the central concepts of the new approach. We defined
a pragmatic notion of entailment - the notion by means of which we want to account for
the free choice inferences - by strengthening the inferences one can get from the premise
that the speaker believes what she said with taking the speaker to be in the minimal belief
state possible given that she believes what she said. And we characterized a class of orders
that can be used to compare the beliefs of the speaker. What else do we have to do before
we can see whether the approach gives us the free choice inferences?
Of course, we still have to choose a particular information order. A general characterization
is not enough. But before we are going to discuss this question, we will address
another problem: given a class of states and an information order, how can we establish
that certain inferences are valid with respect to the notion of entailment defined by this
order when applied to this class of states? How can we establish properties of minimal
states? This question is the topic of the present section.
50The definition can be easily extended to arbitrary languages.
28
We will use a state-construction technique to show that certain sentences hold on the minimal
states for a sentence . It will turn out that the same technique can also be used to
prove that a sentence is honest. Let S be a set of states and an information order on
S with defining language 2L . Assume that we could define a mapping
W
from the set of
subsets of S to S such that for arbitrary T S (i)
W
(T) 2 S and (ii) 8s 2 T :
W
(T) s,
hence
W
would map subsets T of S on a state that is -smaller that every s 2 T. Then, we
could in terms of this mapping characterize a class of honest formulas for the order-defining
language 2L .
Fact 3 If 2 is
W
-persistent, i.e. 8T S : (8s 2 T : s |=S 2 ) )
W
(T) |= 2 ), then is
honest with respect to S and 2L .
To see this take T = {s 2 S|s |= 2 }. W According to the assumptions we have made about
, we have
W
(T) 2 S and 8s 2 T :
W
(T) s. Furthermore, because 2 is
W
W -persistent:
(T) |= 2 . But this means that with
W
(T) we have found some state s 2 S such that
s |= 2 and 8s0 2 S : s0 |= 2 ) s s0. Hence, is honest with respect to S and 2L .
More relevant for our applications is the following fact.
Fact 4 If 8s 2 S : s |= 2 ^ ¬ ) [9s0 2 S : s0 |= 2 & s0 s], then | S .
Assume that |6 S . By definition, it follows that 9s 2 S : s |= 2 & [8s0 2 S : s0 |=
2 ) s s0] & s |= ¬ . But by the antecedent of fact 4, we know that 9s00 2 S : s00 |=
2 ^ s00 s - hence, contradiction.
We will see later on that an operation
W
that has the properties we have stated above
will also allow us to easily show the validity of the antecedent of fact 4. In sum, the main
work for establishing the free choice inferences will lay in finding such a mapping
W
. But it
will turn out that this is not a particularly difficult enterprise either. An adapted version
of the disjoint union of modal models will do the job.
3.2.4 The Basic Information Order
Now, it’s time to let all the techniques run on an example. We will, therefore, use this
section to study a simple but nevertheless for the free choice inferences very useful instance
of an information order: the basic information order.
The basic information order is the order defined by the language 2L(0).51
Definition 4 (The Basic Information Order 0)
For s, s0 2 StateKD45
52 we define
s 0 s0 iffdef 8' 2 2L(0) : s |=S ' ) s0 |=S '.
51This is one of the descriptions Halpern & Moses [1984] use for the definition of their notion of minimal
knowledge.
52As stated earlier, when discussing applications we will restrict our considerations to StateKD45.
29
0 compares how much the speaker knows on sentences that contain no modal operators.
Of course, this is a strong limitation of order-relevant information - and we will have to
extend the defining language to get all the intended inferences. But for the moment we will
stick to this simplification.
It is easy to see that the following simple fact holds.
Fact 5 For hM,wi, hM0w0i 2 StateKD45: hM,wi 0 hM0,w0i ,
8v0 2 R0
3[w0]9v 2 R3[w]8p 2 P : V (p)(v) = V 0(p)(v0).
Thus, we could have equivalently used the stated model-condition to define the order. To
give a rough reformulation of the statement, it says that a speaker knows in a state hM,wi
less than in a state hM0,w0i if for every epistemic possibility she sees in hM0,w0i there will
be an epistemic possibility of the speaker in hM,wi that gives exactly the same interpretation
to propositional atoms.
With the use of the basic information order we already can account for the inferences
(D1), (D2), (D5) and (D8) from section 3.1.3.
By inserting 0 in Definition 1 we get the first concrete instance of our pragmatic
entailment relation: | 0
KD45 holds iffdef on the 0-minimal set of StateKD45 where
the speaker believes , is valid.
How can we now, for instance for (D1), establish that the inference is valid for | 0
S?
Given the considerations of the last section, we want to use fact 4 for this purpose. But to
be able to do so we first have to show that its antecedent holds: we have to show that a state
s1 where the speaker believes the antecedent of (D1) but where the consequence 3p^3q is
not true is for 2(p_q) not minimal with respect to 0. Assume, without loss of generality,
that the consequence fails because s1 |= ¬3p. An idea how to find for s1 a strictly smaller s
could be to define it by adding to the epistemic possibilities the speaker distinguishes in s1
one where p holds. In the resulting s, ¬3p would be false and, hence, because ¬3p 2¬p
and 2¬p is in our order defining language 2L(0) it follows that ¬(s1 0 s). On the other
hand, because we only added an epistemic possibility where p holds, 2(p _q) is still true in
s, and in general the speaker should only have less L(0)-beliefs in s than in s1. In sum, s
would be a state such that s |= 2(p _ q) and s 0 s1, and the antecedent of fact 4 for (D1)
would be established.
In the following the argumentation sketched above is made more precise. For this
purpose we define the s used above as the result of a merge of s1 with a KD45-state where
2p holds. This operation will be a function with the properties described in the last section.
For the basic information order the merge comes down to a simple variation of the disjoint
union of modal models.
Definition 5 (The Merge Function
W
1)
Let T be a subset of StateKD45 indexed with the set I. Assume that the set of worlds of all
states in T are disjoint and w is a world not occurring in any domain.
W
1(T) is the state
hCLKD45(M0),wi, where M0 is defined as follows.
W0 =
S
i2I Wi [ {w},
R0
3 =
S
i2I Ri,3 [ {hw, vi|v 2 Ri,3[wi], i 2 I},
30
R0
4 =
S
i2I Ri,4 for 4 6= 3,
V 0(p) =
S
i2I Vi(p).
The picture below shows how the operator
W
1 merges two states s1 = hM1,w1i and s2 =
hM2,w2i to a state s = hM,wi =def
W
1({s1, s2}). As we can see, there are two ways in
which arrows are added to the epistemic accessibility relation of s: arrows from the new
point w to epistemically accessible points of s1 and s2 come by the definition of M0. Arrows
connecting the epistemic possibilities of different states with each other are added by the
closure operation that is applied to the model M0 in the definition.
w
w1
w2
by
construction
by
closure
R
"
"
"
"
"
"
"
b
b
b
b
b
b
b
```````
•
@@
•
@@
•
Figure 1:
The presentation of the states is highly
simplified. Only some essential R3-
connections are depicted. Thick lines represent
R3-arrows of the merged states s1
and s2, thin lines stand for R3- arrows inserted
by the merge operator. The arrowdirection
is always from left to right (connections
represented by curved lines hold
in both directions).
By definition,
W
1(T) 2 StateKD45. Its set of R3-accessible worlds is the union of
the epistemic alternatives of the speaker in every s 2 T with their respective valuation for
atomic propositions. But that means for hM,wi =def
W
1(T) that for every hMi,wii 2 T and
for every vi 2 Ri,3[wi] there is an epistemic alternative v 2 R3[w] that assigns in hM,wi
exactly the same valuation to atomic propositions as vi in hMi,wii: 8vi 2 Ri,3[wi]9v 2
R3[w]8p 2 P : V (p)(v) = Vi(p)(vi). But because this is exactly what defines the order 0
(see definition 2), we obtain: 8s 2 T :
W
1(T) 0 s.
Fact 6 For all T StateKD45 the following holds:
(i)
W
1(T) 2 StateKD45,
(ii) 8s 2 T :
W
1(T) 0 s.
Hence,
W
1 has all the properties to apply for the role of
W
in fact 3. Thus, we can use
the function to characterize a class of honest formulas for StateKD45 and 2L(0). And a
short consideration reveals that the antecedents of the inferences in (D1), (D2), (D5) using
the first antecedent, and (D8) are persistent under
W
1. Hence, they are all honest with
respect to 2L(0) and StateKD45. Take, for instance, the sentence =def 2(p _ q) and
assume hM1,w1i, hM2,w2i 2 StateKD45 make this sentence true. Hence, 8v1 2 R3[w1] :
M, v1 |= p _ q and the same for hM2,w2i.
W
1 maps those states on a new state hM,wi such
that R3[w] is simply the union of R1,3[w1] and R2,3[w2] - whose members all make p _ q
true. Of course, it will also hold that 8v 2 R3[w] : M, v |= p _ q. Hence, is persistent
under
W
1.
We will now again show that (D1) holds with respect to StateKD45 and | 0 but now
formally more precisely, using
W
1. By assumption, p and q are logical independent propositions.
Assume furthermore that there is a state s1 2 StateKD45, such that s1 |= 2(p _ q)
31
but s1 6|= 3p ^ 3q. Without loss of generality, we assume s1 |= 2¬p. Notice again that
2¬p 2 2L(0), whereby 2L(0) is the defining language of the information order 0. Because
p 6 ?, we can find a state s2 2 StateKD45 such that s2 |= 2p. It follows that s2 |= 2(p_q).
Let s =def
W
1({s1, s2}). Because 2(p _ q) is
W
1-persistent, s |= 2(p _ q). By construction,
we additionally have s |= 3p and hence s 6|= 2¬p. Thus, we found a formula in the defining
language that holds at s1 but not at s. Together with fact (6.ii), this allows us to conclude
s 0 s1. To summarize, we have found for every KD45-state verifying 2(p _ q) that does
not make the free choice inference true a strictly smaller KD45 state. Hence, we verified
the antecedent of fact 4 and can conclude that (D1) is valid with respect to StateKD45 and
| 0.
Notice, that (D1) is not necessarily valid for logical dependent p, q 2 L(0).53 Gazdar’s
inconsistency check on pcis now is a natural consequence of the non-monotonicity of the
notion of entailment we defined.
Let’s take as second example the inference (D5) with the first antecedent 2(p _ q). In
this case a state s1 that verifies the antecedent 22(p_q) ( KD45 2(p_q)) but falsifies the
conclusion 3p ^ 3q of (D5) will again (without loss of generality) imply the order-relevant
sentence 2¬p. To show that such a state s1 cannot be minimal with respect to the order 0,
this time, we have to merge s1 with a witness for 3p such that the stronger claim 22(p_q)
still holds in the merge. But the s2 we used above does already fulfill this requirement:
s2 |= 22(p _ q) and hence s =def
W
1({s1, s2}) |= 22(p _ q) because the sentence is
W
1
persistent. And again we can conclude from s 6|= 2¬p and fact (6.ii) that s 0 s1. Hence,
s1 is for 22(p _ q) not minimal with respect to 0 and (D5) follows by fact 4.
The validity of the inferences (D2) and (D8) can be proven by the same means. However,
the argument does not extend to (D5) using the second antecedent 2p _ 2q. The reason
is simply that 2p _ 2q is not
W
1-persistent. But the inference (D5) does hold! Actually
we obtain 2p _ 2q | 0
KD45 ?, because the formula 2p _ 2q is not honest with respect to
2L(0) and StateKD45, i.e. there is no state s 2 StateKD45 such that s |= 2(2p _2q) and
8s0 2StateKD45 : s0 |= 2(2p _ 2q) ) s 0 s0. How to see that 2p _ 2q is dishonest? If a
speaker believes 2p _ 2q she may, for instance, believe that p but not believe that q. Let
s1 be a state where this is the case. On the other hand, she may also believe that q but not
believe that p. Assume that this holds in s2. Because 2p,2¬p,2q,2¬q are in the orderdefining
language 2L(0) we have s1 6 0 s2 and s2 6 0 s1. Hence, if 2p_2q were honest, then
there would exists an s 2StateKD45 such that s |= 2(2p _ 2q) and s is 0-smaller than
both, s1 and s2. However, if s is smaller than s1, then s |=KD45 ¬2p because the speaker
can believe only less in s that she believes in s1. For the same reason s |=KD45 ¬2q. But
then s 6|=KD45 2(2p _ 2q) what contradicts our first assumption about s. Hence, 2p _ 2q
has to be dishonest with respect to 2L(0) and StateKD45. Or, in other words, a speaker
who utters 2p _ 2q cannot at the same time obey the maxims T and Q1: she cannot at
the same time believe in her utterance and say all she believes. This sentence is, given our
formalization of Grice’s maxims, pragmatically not well-formed.54
53If, for instance, p = ?, then we would not be able to find such an s2 as we used in the proof.
54The same point can be made much easier but less intuitively using the disjunction property. In
StateKD45 we have obviously: 2(2p _ 2q) |=KD45 2p _ 2q. Both, 2p and 2q, are in the order-defining
language 2L(0). In order to have the disjunction property, 2(2p _ 2q) should entail one of these sentences.
32
But this is not such a bad result! Sentences like (23a) and (23b) are indeed reported to
be odd. In particular, they do not allow a free choice reading.
(23) a. Mr. X must be in Amsterdam or Mr. X must be in Frankfurt.
b. I believe that A or I believe that B.
The approach at hand can explain these intuitions by referring to the notion of pragmatic
well-formedness that it allows to define.
This argument nicely illustrates the power of the notion of honesty. Normally, we are
not able to recognize whether a speaker is really obeying the maxims of Grice: whether
she does tell us the truth and gives us all the information she has. The interpreter has to
assume that the speaker is trustable. Dishonest sentences, however, directly compromise
the speaker. They can never be uttered in accordance with the Gricean maxims. They are
therefore also a very powerful testing device for the account presented here.
The information order 0 does not allow us to derive the other inferences in section 3.1.3 we
want to account for. In particular, we cannot account for free choice permission. The reason
is easily found: the basic information language does not contain any sentence that refers to
belief the speaker has about deontic modalities (or any other modality). Thus, if asked to
compare two states, the order is not able to see differences in belief about modalities. So
how can we expect any minimization effect on such kind of information? The strategy to
overcome the limitation we face here should be to extend the order defining language. This
will be done in the following sections.
However, a small modification of what we have said above is necessary. Perhaps the
reader has recognized that in using the basic information order we do have a certain minimization
effect on what the speaker beliefs on at least one modality: her own belief state.
But this is due to the particular logic of belief used here: KD45. This logic demands a close
relation between what the speaker believes and her beliefs on her beliefs: the speaker is assumed
to be competent on her beliefs. Therefore, minimizing the L(0)-beliefs of the speaker
will also minimize her beliefs on these beliefs. To make this a little bit more precise, consider
the language L defined by the following BNF: ' ::= p(p 2 L(0))|' ^' |' _' |2' .
This language allows to express beliefs of various depths but prohibits the occurrence of 2
under negation. Now, it can be shown that 8' 2 2L 9'0 2 2L(0) : ' KD45 '0. Hence, we
could have defined 0 equivalently using 2L . However, while | o also minimizes beliefs
the speaker has on what she does believe, it is easy to see that her beliefs on what she does
not believe do not count for the order.55
3.2.5 Information Orders with Modal Information
Of course, extending the defining language will in turn widen the class of properties of the
states that the information order is sensible to - this is actually our intention. But, to
Of course, this is not true. Hence, the formula cannot have the disjunction property and, therefore, using
fact 2, 2p _ 2q cannot be honest with respect to 2L(0) and StateKD45.
55Because of the introspective power of the agent minimizing beliefs on beliefs will actually result in
maximizing beliefs on what the speaker does not believe.
33
prove validity on minimal states with respect to such more complex orders we need to know
how the minimal states look like. We need a model-theoretic definition of the order. But
how to establish such results? Van der Hoek et al. [1999] provide an interesting technique
that simplifies this problem for certain order defining languages. They use the notion of
Ehrenfeucht - Fra¨ıss´e - Orders to this purpose:
Definition 6 (Ehrenfeucht - Fra¨ıss´e - Orders)
Let n, n 2 N, be an enumerable set of partial orders. We define the Ehrenfeucht-Fra¨ıss´e-order
as follows.
hM,wi hM0,w0i ,def 8n 2 N8v0 2 R0
3[w0]9v 2 R3[w] : hM, vi n hM0, v0i.
For this notion they prove the following result.56
Theorem 2 (Collecting)
Let L and {L
(n)|n 2 N} be a system of sublanguages such that for all n 2 N: L
(n) =
L \ L(n). If for all n 2 N the language L
(n) defines the order n and L is closed under
_, then 2L defines , i.e.
s s0 , 8 2 2L : s |= ) s0 |= .
Given this result, if we know the model conditions of the orders defined by the finite
sublanguages {L
(n)|n 2 N}, then we also know that the order defined by the language 2L
is the Ehrenfeucht- Fra¨ıss´e order of these ‘finite’ orders. And to establish model-conditions
for a finite language with finite modal depth is much less demanding.
3.2.6 The Objective Information Order
In this section we will study an appealing candidate for an information order that respects
the knowledge an agent has on other modalities. However, at the end we will see that the
notion of entailment defined by this order will not be able to give us in StateKD45 all the
inferences we are looking for.
To find a modality-sensible defining language we use the simple idea to extend the basic
language L(0) with all sentences of L that say something on modalities other than the
belief state of the speaker.
Definition 7 (The Objective Information Order o)
Let Lo L be the language defined by the BNF o ::= p(p 2 L(0))|¬ o| o ^ o|r ,r 6=
2( 2 L). We call Lo, following Halpern57, the objective information language and define
the objective information order, o, for s, s0 2 StateKD45 as follows:
s o s0 , 8' 2 2Lo : s |= ' ) s0 |= '.
56For a proof see van der Hoek et al. [1999]. The result relies on the finiteness property of the set of
atomic propositions.
57According to van der Hoek et al. [2000] the objective information order has been introduced by Halpern
in a paper where he extends the approach in Halpern & Moses [1984] to the multi-agent case.
34
Building on the result of the last section, van der Hoek et al. [1999] show that the
following equivalent model-theoretic definition of o can be given.
Fact 7 We define on KD45 for n 2 N the relations =−3
n as follows:
hM,wi =−3
0 hM0,w0i ,def 8p 2 P : V (p)(w) = V 0(p)(w0)
hM,wi =−3
n+1 hM0,w0i ,def
8<
:
hM,wi =−3
n hM0,w0i &
84 6= 38v 2 R4[w]9v0 2 R0
4 [w0] : hM, vi =n hM0, v0i (forth) &
84 6= 38v0 2 R0
4 [w]9v 2 R4[w] : hM, vi =n hM0, v0i (back)
Then o is the Ehrenfeucht- Fra¨ıss´e order defined by the orders =−3
n for n 2 N:
hM,wi o hM0w0i , 8n 2 N8v0 2 R0
3[w0]9v 2 R3[w] : hM, vi =−3
n hM0, v0i
The proof works by induction on the maximal modal depth n of the sublanguages
Lo(
n) = Lo \ L(n). For every n 2 N one has to show that Lo(
n) define the respective relation
=−3
n .58 Then one applies theorem 3 from section 3.2.5. Remember from section 3.1.2 that
=n stands for n-bisimulation between two states. The relation =−3
n is a weakening of =n:
R3/ R0
3 -branches starting at the root of two states that are compared are not considered.
In other words, the relation =−3
n ignores, the beliefs of the speaker on her own beliefs.
The extended order o defines, based on definition 1, a new notion of pragmatic entailment:
| o
KD45. The question we now have to address is: can this entailment relation account for
the inferences (D1) - (D9)? To investigate this problem we would like to use the same
techniques as in section 3.2.4 when we studied this question for the basic information order.
In particular, we would like to make use of fact 4. However, to establish the antecedent of
this fact, we cannot rely on the merge function
W
1 any longer. Remember that we need
our operation
W
to fulfill two conditions: (i) 8T StateKD45 :
W
(T) 2 StateKD45, and
(ii) 8s 2 T :
W
(T) o s. But for
W
1 we only showed the claim (ii) for the basic information
order 0. In this case the only thing we needed was that for every s0 = hM0,w0i 2 T and
every v0 2 R0
3 [w0] there exists for hM,wi =def
W
(T) a v 2 R3[w] that agrees with v0 in
the valuation of propositional atoms. For the objective information order, as fact 7 tells us,
we need much more. v and v0 have to agree on a great part of their generated submodels:
hM0, v0i =−3
n hM, vi for every n 2 N!
W
1 cannot warrant this. For suppose that in the
generated submodel of v0 in M0 there is a kind of circle leading back to a world v00 2 R0
3[w],
hence, there exists a sequence of accessibility relations hR0
1,4i
, ...,R0
n,4l
i of M0, 41 6= 3
and a sequence of worlds hv0
1, ..., v0
n+1i, such that 8i(1 i n) : v0
i+1 2 R4i [v0
i ] and v0
1 = v0
and v0
n+1 = v00. If you now merge hM0,w0i with other states using
W
1, then in the resulting
state hM,wi R3[v00] will be much bigger than R0
3 [v00]. Hence, in M v00 generates a different
submodel than in M0. But therefore, a relevant part of the submodel generated by v0 in the
new model M will change too and, even though =−3 ignores 3-branches starting from v0,
hM, vi 6 =−3 hM0, v0i. To make sure that we do not get in any trouble of this kind, we need
a new merge operation that does not pool the set of epistemical possible worlds directly.
Instead, we let it work on copies of them. We define:
58Essentially for the proof is the well-known result of modal logic that to decide whether a sentence of a
modal language with modal depth n 2 N is true it is sufficient to look n-deep into the model.
35
Definition 8 (The Merge Function
W
2)
Let T be a subset of StateKD45 indexed with I. Assume that the domains W of all states
s 2 T are disjunct and w is a world not occurring in any domain. Let ˜· be a copy-function, an
injection from
S
i2I Wi to a distinct set of worlds D.
W
2(T) is the state hCLKD45(M0),wi,
where M0 is defined as follows.
W0 =
S
i2I Wi [ { ˜ vi|vi 2 Ri,3[wi], i 2 I} [ {w},
R0
3 =
S
i2I Ri,3 [ {hw, ˜vi|v 2 Ri,3[wi], i 2 I},
R0
4 =
S
i2I Ri,4 [ {h ˜ vi, uii|vi 2 Ri,3[wi] ^ ui 2 Ri,4[vi] ^ i 2 I} for 4 6= 3,
V 0(p) =
S
i2I Vi(p) [ { ˜ vi|vi 2 Vi(p), i 2 I}.
The picture below demonstrates the working of the operator
W
2 for the merge of two states
s1 = hM1,w1i and s2 = hM2,w2i to a state s = hM,wi.
w
w1
w2
v1
˜v1
41
41
42
copy ?
copy 6
"
"
"
"
""
b
b
b
b
bb
``````
•
@
@
@
•
•
•
@
•
-
Figure 2:
Now, modalities different from 3 do occur in
the picture. They are represented by arrows annotated
with the modality to which they refer.
Otherwise the same conventions as for the last
picture are in force. The graphic illustrates how
the use of copies for the merged states breaks
the circles that caused the problem we discussed
above. If we had merged s1 and s2 directly
using
W
1, then hM1, v1i 6 =−3
3 hM, v1i, because
wWe would have obtained that in hM,wi =def
1({s1, s2}) from v1 one can reach by 41, 42
and 3 all elements of R2,3[w2]. However, this is
not possible in M1.
For ˜ v1 this problem does not occur. Notice that even though hM, v1i =−3
n hM, ˜ v1i for n 2 N
- hence, they make exactly the same sentences in Lo true, ˜ v1 differs from v1 by not seeing
in M under any modality 4 6= 3 any other point of R3[ ˜ v1].
Fact 8 For all T StateKD45 the following holds:
(i)
W
2(T) 2 StateKD45,
(ii) 8s 2 T :
W
2(T) o s.
The proof is quite simple given what we have said above. Take an arbitrary T
StateKD45, hMi,wii 2 T and let hM,wi =def
W
2(T). We have to show that 8n 2 N8vi 2
Ri,3[wi]9v 2 R3[w] : hM, vi =−3
n hMi, vii. We know that by construction 8vi 2 Ri,3[wi] :
˜ vi 2 R3[w]. But this copy ˜ vi of vi is a v 2 R3[w] such that hM, vi −3
n hMi, vii for all
n 2 N by the way the construction is set up.
Together with fact 3 this result allows us to establish a general class of 2Lo-honest formulas:
If 2s is
W
2-persistent, then is 2Lo-honest. Actually, all antecedents of the
36
inferences in the data-section, beside the part of (D5) we already disqualified in the last
section, are honest with respect to 2Lo .
Parallel to the argumentation in section 3.2.4, we can establish the validity of the free
choice inferences using fact 4. However, not all inferences from section 3.1.3 will come out
as valid. While we obtain (D1), (D2), (D4), (D5), (D7), and (D8), the inferences (D3),
(D6), and (D9) do not hold.
Let’s start by discussing one inference we already had as valid given the basic information
order: (D2). Let p, q 2 L(0) be two logically independent propositions. Let s1 2 StateKD45
be a model of the formula that verifies 23(p _ q) but fails the conclusion of (D2), hence,
without loss of generality, s1 |= 2¬p, where 2¬p 2 2Lo, our order defining language.
Because p and q are logically independent, p is a contingent fact and, hence, there will be a
state s2 2 StatesKD45 such that s2 |= 3p. It follows that s2 |= 3(p_q) and, by the validity
of axiom (5) in StateKD45, s2 |= 23(p _ q). Let s =def
W
2({s1, s2}). Because 23(p _ q)
is
W
2-persistent, we obtain that s |= 23(p _ q). Additionally, s |= 3p by construction, and
hence, s o s1. By fact 4, 3(p _ q) | o
KD45 3p ^ 3q.
However, we will not be able to conclude by a parallel argument that in the set of ominimal
states for 24(p _ q), 4p holds and hence to account for (D3). The reasoning fails
the moment we want to conclude that on a state s2 where 4p is valid, 24p is valid too. In
the derivation above we could do the step from 3p to 23p because of axiom (5): because
we assumed the speaker to have negative introspective power. We have nothing similar for
4.
Instead of (D3) we do get (D4), the incompetence inference. Let p, q 2 L(0) be two logical
independent propositions and let s1 be a model of the formula that verifies 24(p _ q)
but fails the conclusion of (D4). Without loss of generality, s1 |= 2(¬4p _ 4q). The
formula 2(¬4p _ 4q) is an element of the order defining language 2Lo. p ^ ¬q is consistent
and hence, there will be a state s2 2StatesKD45 such that s2 |= 2r(p ^ ¬q). It
follows that s2 |= 24(p _ q). Let s =
W
2({s1, s2}). Then, by persistence of the antecedent
we have s |= 24(p _ q). Additionally, s |= 3(4p ^ ¬4q) by construction and hence,
s 6|= 2(¬4p_4q). This allows us to conclude that s o s1. Hence, by fact 4, (D4) is valid.
To summarize, on StateKD45 the incompetence inference (D4) is valid with respect to
| o while the free choice permission (D3) is not. Actually this is not such a bad result.
As observed in the literature, the incompetence reading does exist. Additionally, the free
choice reading seem to be bound to the assumption that the speaker is competent. We did
not assume any connection between the deontic facts and the knowledge of the speaker here
- hence, we should not obtain free choice permission. On the other hand, for the epistemic
modality we build competence into the system - by assuming the axioms (4) and (5) to be
valid. As Zimmermann did, we can explain the observation that epistemic free choice seems
always valid by claiming that interpreters always take a speaker to have full introspective
power.
The aim of the next section has to be to generalize the concept of competence behind
introspection to other modalities - here we will rely on the work of Zimmermann [2000] -
and then investigate whether we can make the inferences (D3) and (D6) valid this way.
37
3.2.7 Competence and Multi-Modal Belief
As we have observed in the last section we will be only able to derive the intended inferences
in the given setting if we assume additional restrictions on the class of models on which
utterances are interpreted. There has to be some kind of connection between what is valid
on a certain modality and the speaker’s beliefs with respect to this modality: the speaker
has to be assumed to be in some sense competent on the relevant modality.
In how we make this idea precise we rely on Zimmermann’s [2000] formalization of
competence. As stated in section 2.2, Zimmermann, building on a proposal of Groenendijk
& Stokhof [1984], defines competence by the following first-order model condition.59
Definition 9 (Competence)
A speaker is strongly competent in a state hM,wi 2 StateKD45 (where M = hF, V i and
F = hW,R4i , ...,R4ni) with respect to a modality 4 iffdef
8v 2 W : v 2 R3[w] ) (R4[v] = R4[w]).
It is easy to prove that this condition is characterized in modal propositional logic by
the two axioms (C14) and (C24) described below, i.e. a speaker is strongly competent
in some state s = hM,wi if the underlying frame locally (hence, in w) satisfies (C14) and
(C24).
(C14) r ! 2r
(C14) will be called the axiom of positive competence. Intuitively it expresses that if a
sentence holds in the R4-admissible worlds then the speaker knows this. If one substitutes
2 for r one obtains the positive introspection axiom (4). Using the same strategy as in the
proof of fact 1 one can show that the following holds for any frame F = hM,R41 , ...,R4ni
for the language L and any point w in this frame.
F,w |= C14 iff Fk − 8v 2 W : v 2 R3[w] ) (R4[v] R4[w])
We turn to the second axiom.
(C24) ¬r ! 2¬r
(C24) will be called the axiom of negative competence. Intuitively it expresses that the
speaker knows also for every sentence that is not valid on the R4 accessibility relation that
this is the case. It comes as no surprise that this is a multi-modality generalization of
negative introspection or euclidicity:
F,w |= C24 iff 8v 2 W : v 2 R3[w] ) (R4[v] R4[w])
59The (intensional) predicate w x.P (w)(x) in his definition is instantiated here with the characteristic
function of the R4-accessibility relation: w v.wR4v.
38
As these considerations show, the definition of competence used by Zimmermann [2000]
is a generalization of the concept of full introspective power to arbitrary modalities. Now
we have to investigate whether imposing this generalized model condition on the class of
states that is considered (hence, to work on StateKD45C14C24 instead of on StateKD45)
allows us to account for the free choice permission as much as assuming introspective power
of speaker allowed us to account for the epistemic free choice inferences. Thus, the question
is whether (D3) and (D4) are valid conditional on assuming the speaker to be competent
in the sense defined above.
Unfortunately, if we use the notion of entailment defined by the objective information
order o on the set StateKD45C14C24 we obtain a system that is much too strong: if the
speaker is assumed to be competent on 4, then every consistent sentence is dishonest
with respect to StateKD45C14C24 and o. Or, in other words, given the way | o interprets
the maxims T and Q1 a speaker competent on 4 as formalized in C14 and C24 cannot
utter any consistent sentence and be obeying these maxims. The reason for this is that
| o demands a strongly competent speaker to provide so much information that there is
no finite sentence to encode it all. In some more detail: for every ' 2 L both, 2r' and
2¬r' are in the order defining language 2Lo. Recall that this means that the order o
prefers states where these sentences are false. But if a speaker is strongly competent with
respect to 4 then for every ' 2 L: 2r' and 2¬r', cannot be false at the same time
(given the formalization we have chosen a speaker strongly competent on 4 knows for every
' 2 L which of r' and ¬r' is true; hence, either 2¬r' or 2r' holds). That means
that there will be at least two incompatible classes of minimal states: one where 2r' is
true, and one where 2¬r' holds. But this would unmask the speaker to be dishonest.
Hence, if a competent speaker does not want to be trapped by not obeying the maxims T
and Q1 (as formalized in | o) then she already has to exclude one of the possibilities with
her utterance of : 2 |=KD45C14C24 2r' or 2 |=KD45C14C24 2¬r'. But the same
argument applies for every ' 2 L! Thus, for every sentence ' her utterance has to convey
whether she believes r' or ¬r'. However, there can be no finite and consistent sentence
that is that strong: that decides every sentence valid on the R4 accessible worlds. Hence,
the general dishonesty result.60
Given these results we have to accept that the combination of strong (positive and
negative) competence with respect to some modality 4 with a defining language that takes
every fact on 4 to be relevant for the order does not result in an appropriate notion of
pragmatic entailment. Two ways to improve the situation can be considered: (1) we can
restrict the order defining language 2Lo and, hence, weaken the pragmatic entailment, or
(2) we can weaken our notion of competence. As we will see, both choices can be turned
into a successful approach to the free choice inferences.
60Exactly the same point can also be made by using the disjunction property of honest formulas.
Parallel to the situation in StateKD45, StateKD45C14C24 has theorems (T1) 2r $ r (Proof:
())2r )D 3r ,Def ¬2¬r )C24 r ,(() by C14) and (T2) 2¬r $ ¬r (Proof: ())
2¬r $Def ¬3r !D ¬2r !C14 ¬r , (() by C24) for the modality 4. Hence, we have
|=KD45C14C24 2r _2¬r (Proof: |=KD45C14C24 _¬ . Instantiate : |=KD45C14C24 2r'_¬2r'.
Using T1 and T2 : |=KD45C14C24 2r' _ 2¬r'), without the disjuncts being theorems, too. Because formulas
that are honest with respect to StateKD45C14C24 and o have to fulfill the disjunction property
it follows that if is o-honest in StateKD45C14C24 then for all 2 L: |=KD45C14C24 2r or
|=KD45C14C24 2¬r , which is impossible to fulfill with a consistent and finite sentence.
39
3.3 Applying the Framework
3.3.1 Solution 1: Weakening the Order
In this section we will introduce a first combination of a logic and a notion of entailment
that gives us the free choice inferences. It will be derived from the framework of the
previous section by weakening the information order o on which the notion of entailment
| o was based. However, the notion of competence as used by Zimmermann [2000] will be
kept. For the moment, we will assume that the language L has only one modality r 6= 2.
Furthermore, we assume that the speaker is strongly competent with respect to 4. Hence,
we are working in StateKD45C14C24. We will come back to the multi-modal case at the
end of this section.
One way to look at the problem we ended up with in the last section is that we chose a
too strict formalization of the conversational maxims T and Q1. As they are described by
| o a speaker who wants to obey the maxim has to give nearly every information she has.
The only thing we excluded from Lo were the beliefs the speaker has about she does not
believes. But all other sentences counted as relevant for the information order. Perhaps we
can obtain a more natural notion of pragmatic entailment when we allow the speaker to
withhold more information. The problem, then, becomes to find the right restriction that
fits our intuitions about when utterances give sufficient information. A promising approach
is to generalize the restrictions we have already working for 2-sentences in Lo to every
modality the speaker is competent on. Hence, we systematically remove all sentences from
Lo where r is the modality with widest scope and occurs under negation. We call the new
information order that is defined by this language the positive information order: 61
Definition 10 (The Positive Information Order +)
Let L+ L be language defined by the BNF '+ ::= p(p 2 L(0))|'+ ^ '+|'+ _ '+|r',r 6=
2(' 2 L). We call L+, the positive information language and define the positive information
order, +, for s, s0 2 StateKD45C14C24 as follows:
s + s0 , 8' 2 2L+ : s |= ' ) s0 |= '.
This order asks the speaker to give all information she has on what is necessary with respect
to R4, but she may know more about what is possible in R4 than she says. Also for the
positive information order an equivalent model-theoretic definition can be given.
Fact 9 We define the relations .3
n for n 2 N on StateKD45C14C24 as follows:
hM,wi .−3
0 hM0,w0i ,def 8p 2 P : V (p)(w) = V 0(p)(w0)
hM,wi .−3
n+1 hM0,w0i ,def
hM,wi .−3
n hM0,w0i &
8v0 2 R0
4 [w]9v 2 R4[w] : hM, vi =n hM0, v0i (back)
Then + is the Ehrenfeucht- Fra¨ıss´e order defined by the orders .−3
n for n 2 N:
hM,wi + hM0w0i , 8n 2 N8v0 2 R0
3[w0]9v 2 R3[w] : hM, vi .−3
n hM0, v0i
61Notice, that this notion differs from what van der Hoek et al. [2000] call the positive information order.
40
It does not come not as a surprise that the order .−3
n differs from the relation =−3
n of
fact 7 only in the absence of the forth condition - which is responsible for perseverance of
sentences of the form 2¬r , 2 L. Because the order + has not been discussed by van
der Hoek et al. the proof of fact 9 is given here. We show by induction on n 2 N that the
languages L+
(n) = L(n) \ L+ define the relation .−3
n . Then, the claim follows immediately
by theorem 3.
For n = 0, L+
(0) = L(0), s .−3
0 s0 , s =0 s0, and the claim follows from theorem 1.
Assume L+
(n) would define .−3
n . We show that L+
(n+1) defines .−3
n+1. Thus:
hM,wi −3
n+1 hM0,w0i , 8' 2 L+
(n+1) : hM,wi |= ' ) hM0,w0i |= ' (*).
(“)”) Notice first that for all 2 L+
(n+1) we can find a sentence
Vl
i=1
Wk
j=1 'i,j 2
L+
(n+1), where for all i, j either 'i,j 2 L(0) or 'i,j = r i,j for i,j 2 L(n), such that
Vl
i=1
Wk
j=1 'i,j .62 Now, assume that for some 2 L+
(n+1) : M0,w0 6|= . We have
to show that then also M,w 6|= , given that hM,wi .−3
(n+1) hM,w0i. M0,w0 6|= ,
9i(1 i l)8j(1 j k) : M0,w0 6|= 'i,j . Firstly, assume 'i,j 2 L(0). Then, because
hM,wi .−3
(n+1) hM,w0i entails hM,wi .−3
0 hM,w0i, M,w 6|= 'i,j . On the other hand, if
'i,j = r i,j , then 9v0 2 R0
4 [w0] : M0, v0 6|= . But hM,wi .−3
(n+1) hM0,w0i entails also,
that 8v0 2 R0
4 [w0]9v 2 R4[w] : hM, vi =n hM0, v0i. By theorem 1, we can conclude
that 9v 2 R4[w] : M, v 6|= i,j and, hence, M,w 6|= r i,j . In sum we have shown that
9i(1 i l)8j(1 j k) : M,w 6|= 'i,j and, thus, M,w 6|= .
(“(”) Assume, that the left-side of the condition (*) does not hold. Hence, for some
hM,wi, hM0,w0i 2 StateKD45C14C24 either (a) hM,wi 6.−3
n hM0,w0i, or (b) 9v0 2 R0
4[w0]
8v 2 R4[w] : hM, vi 6 =n hM0, v0i. We show that in both cases the right side of (*) does not
hold, either. In case (a) the induction assumption allows us to conclude that there exists
' 2 L+n L+ n+1 such that M,w |= ', while M0,w0 6|= '. Hence, the claim. In case (b) by
theorem 1 it follows that there exists a v0 2 R0
3[w0] such that for every v 2 R4[w] there
exists a ' 2 L(n) such that M, v |= ' but M0, v0 6|= '. Because L(n) is finite, we can find a
finite sequence h'1, ..., 'mi with '1, ..., 'm 2 L(n) such that 8v 2 R4[w] : M, v |=
Wm
i=1 'i
and M0, v0 6|=
Wm
i=1 'i. Hence, M,w |= r
Wm
i=1 'i, but M0,w0 6|= r
Wm
i=1 'i. However,
r
Wm
i=1 'i is an element of L+ n+1. Thus, we conclude that hM,wi 6.−3
n+1 hM0,w0i. q.e.d.
In terms of the information order + again a notion of entailment | + can be defined
using definition 1. Now, we want to study which of the inferences of section 3.1.3 are valid
with respect to this notion. A first thing that catches the eye is that on StateKD45C14C24
the formula rp _ rq is not honest with respect to 2L+. The sentence has incomparable
minimal states: one, where rp but not rq holds and the speaker is informed about this,
and one, where rq but not rp holds - again known by the speaker.63
62This is a straightforward consequence of how L+ is defined.
63Using the disjunction property of honest formulas one argues as follows: the problem is - as one might
expect - that by (T1) and (T2) (see footnote 56) 2(rp _rq) |=KD45C14C24 2rp _ 2rq (Proof: 2(rp _
rq) 2(¬rp ! rq) )K 2¬rp ! 2rq )T2 ¬rp ! 2rq rp _ 2rq )T1 2rp _ 2rq) but none
of the disjuncts is entailed. Hence, the formula fails the disjunction property of the language 2L+ and
StateKD45C14C24 and cannot, given fact 2, be honest.
41
Next, we want to establish the inferences (D3) and (D6) with respect to | +. If we
want to use the same kind of argument we employed earlier we need again a proper mergeoperation
W
3 on StateKD45C14C24. Thus, we need (i) 8T StateKD45C14C24 :
W
(T) 2
StateKD45C14C24, and (ii) 8s 2 T :
W
3(T) + s. Notice that if we would drop the
competence axioms and apply | + to StateKD45 we could again use
W
2 as merge operation.
Condition (ii) would still be fulfilled because + is a weakening of the order o. Hence,
s o s ) s + s0. However, if we take the axioms C14 and C24 to be valid, then we have
to face the fact that
W
2(T) is normally not an element of StateKD45C14C24. Therefore,
we have to define a new merge function
W
3 - but notice that
W
3 differs from
W
2 only in the
stronger closure applied to M0.
Definition 11 (The Merge Function
W
3)
Let T a subset of StateKD45C14C24 indexed with I. Assume that the domains Ws of all
states s 2 T are disjunct and w is a world not occurring in any domain.
W
3(T) is defined
as the state hCLKD45C14C24(M0),wi where M0 is defined as in definition 8.
Now property (i) is immediately warranted for
W
W 3, but we might have lost (ii): 8s 2 T :
3(T) + s. Fortunately, it turns out that we did not.
Fact 10 For all T StateKD45C14C24 the following holds:
(i)
W
3(T) 2 StateKD45C14C24,
(ii) 8s 2 T :
W
3(T) + s.
Intuitively, this can be seen by the observation that the only difference between
W
3 W and
2 is that for
W
3 extra arrows are added to R4[w] and R4[v] for v 2 R3[w] in
W
2(T).
But about these extra arrows + does not care - exactly because we dismissed the forth
condition when going from =−3
n to .−3
n . Hence, we get
W
3(T) + W
2(T) + s for all
s 2 T.
Based on
W
3 we can use fact 3 to establish: If 2 is persistent under taking
W
3 then
is +-honest with respect to StateKD45C14C24. The antecedents of (D3) and (D6)
are persistent under union
W
3 and we can continue to the question central to the whole
discussion: do they | +-entail their consequences? The answer is affirmative. We will only
discuss (D6), (D3) works along the same lines. Let s1 be a state for the honest formula
2r(p_q) that does not satisfy the consequence 4p^4q of (D6). Without loss of generality,
we assume that s1 |= r¬p. By C14, we can conclude that s1 |= 2r¬p, whereby 2r¬p 2
2L+. Because p 6 ? there will be a state s2 2 StateKD45C14C24 such that s2 |= 2rp
and, in consequence, s2 |= 2r(p _ q). Let s =def
W
3({s1, s2}). By the
W
3-persistence
of 2r(p _ q) we have s |= 2r(p _ q) and by construction we know s |= 34p. Hence,
s 6|= 2r¬p, which allows us to conclude, together with fact 10.ii, that s + s1. Now, we
can apply fact 4 and arrive at (D6).
As an aside: in this system (D4) and (D7) will not be valid, of course. According to
KD45C14C24, incompetence of the speaker is not possible and hence the conclusions of
(D4) and (D7) are not consistent with C14 and C24, let alone valid on minimal models.
In this section we restricted our considerations to a language that contained only one additional
modal operator: r. As we have seen, in this case if the hearer thinks the speaker
42
to be competent on 4 in the sense of C14 and C24, then | + predicts the free choice
inferences (D3) and (D6) to be valid. Hence, if we interpret 4 as ‘may’ we do obtain free
choice permission. Of course, it would be preferable if we could extend the account to a
richer language with more modalities - including, for instance, the belief states of other
speakers or different kinds of deontic options. But then, one would also like to consider
the possibility that the speaker is also competent with respect to these modalities. Such
an extension is easily possible. And it is also not difficult to imagine how a language 2L
should be defined that allows, for instance, free choice inferences for a set of modalities 1
on which the speaker is competent and no free choice for the modalities: L should look like
Lo, except that we exclude for r 2 1 sentences with negative occurrences of r not embedded
under another modality. The respective model-conditions are also straightforward, as
is the validity of the free choice inferences (D3) and (D6) for r 2 1 and the incompetence
inferences (D4) and (D7) for r 2 2.
It is even possible to have a notion of entailment that is independent of the choice of 1
and nevertheless provides the free choice inferences for modalities the speaker is competent
on. This is the entailment defined by a language that allows no modality with widest scope
to occur under negation: the BNF is ' ::= p(p 2 P)|' _ '|' ^ '|r ,r 6= 2( 2 L). Let
us call it | ++. One important difference to | + is that with respect to | ++ (D4) and
(D7) are not valid for any modality on which the speaker is not taken to be competent (as
described by the axioms C14,C24).64
3.3.2 Solution 2: Weakening the Logic
Recall the discussion at the end of section 3.2.7: applying | o on StateKD45C14C24 will
predict dishonesty for all consistent sentences of the language L. The problem was that,
on the one hand, a speaker who is competent on a modality r knows everything that can
be expressed on this modality. On the other hand, the information order o also demands
her to say everything she knows on r. Both obligations together can never be met by
uttering a consistent sentence. In the last section we discussed a way out of this unpleasant
situation by weakening the notion of entailment | o. Another possibility is, of course, not
to weaken what the speaker has to communicate, but to take him to be less competent.
Parallel to the last section, where we removed sentences from the language that concerned
knowledge of the speaker on what is not necessary with respect to the modality r, we now
consider the case of an interpreter that does not take the speaker to be competent on these
sentences: we weaken KD45C14C24 to KD45C14. In this case, the speaker knows of
excluded 4-options that they are excluded, but she may take it possible that more options
are excluded than actually are. For instance, applied to deontic options, she may take it as
possible that more obligations exist than actually do.
On StateKD45C14 | o will not disqualify every consistent sentence as dishonest. But
do we obtain the free choice inferences? Once more we run our machinery. Again, we have
64Given the problem that brought us to consider + one may also think about using an order that instead
of skipping the forth condition of o skips the back condition. This would result in minimizing what the
speaker takes to be possible and hence maximizing her belief. This approach does not lead to an appeling
concept of honesty. Because the speaker’s belief can be maximized in many different ways, honesty becomes
a very rare thing again.
43
to adapt our merge-operator65: we define
W
4 simply by weakening the closure conditions
in definition 11 to KD45C14. We can establish again the following fact:
Fact 11 For all T StateKD45C14 the following holds:
(i)
W
4(T) 2 StateKD45C14,
(ii) 8s 2 T :
W
4(T) o s.
In consequence, we have again our result about the Lo-honesty of
W
4-persistent formulas.
It turns out that with the exception of 2p _ 2q all antecedents of the inferences in section
3.1.3 are honest. Particularly, rp _ rq is honest in StateKD45C14 with respect to 2L+.
This may be a bit surprising on first glance, but take a look at the states sketched in figure
3. While there is no state in StateKD45C14C24 that is o-smaller than s1 and s2, s3 2
StateKD45C14 is o- smaller than both. Actually, s3 is the result of merging s1 and s2
using
W
4. s3, however, while in StateKD45C14 is not an element of StateKD45C14C24.
Thus, the sentence rp _rq becomes honest because in StateKD45C14 less restrictions are
imposed on the competence of the speaker.
@ @I
@@R
•
@ @I
@@R
•
@ @I
@@
-
@@R
-
•
s1 s2
s3
w1 w2
w3
p
p,q
p
p,q
p,q
q
p,q
q
p
q
p,q
p
p,q
p,q
q
Figure 3:
In this picture, arrows present
4- admissible worlds, lines 3-
admissible worlds. Again, only
R3-arrows from the root of the
state are drawn. Furthermore,
only information on the deontic
possibility of p and q is specified.
Not mentioning a proposition
letter at a world means that
the proposition is not true there.
Also in this system we can account for the free choice inferences (D3) and (D6) (with
respect to the latter now for both antecedents). To show that a state s1 in StateKD45C14
that entails, for instance, 24(p _ q) but not the conclusion 4p ^ 4q cannot be minimal
with respect to o and StateKD45C14 one can use the
W
4 merge with the same kind of s2
as used in the last section to prove the validity of (D6).
But actually, the inferences (D4) and (D7) are valid too! This could have been expected,
because in both cases the sentences you obtain by negating their conclusions are part of the
order defining language 2Lo. Hence, the interpreter tries to make them false when interpreting
the speaker using | o, and, hence the consequences true. On the other hand, the
consequences are also consistent with the logic and the assumption that the speaker believes
the respective antecedents. Therefore, in sum, the free choice inferences occur together with
the inference that the speaker is not competent on the modality. In particular, it is predicted
the interpreter will learn more about what is valid on R4 than she thinks the speaker knows.
65W
3 is not appropriate because it fails condition (ii) in fact 11.
W
2, on the other hand, fails condition (i).
44
We conclude this section with two final remarks. First, in the discussion above we assumed
that the speaker is taken to be competent on one modality r. The discussed results
all extend to the case that the logic is enriched with C140 for other modalities 40 of the
language.
Furthermore, it is interesting to look at the second possibility to weaken the logic: by
giving up C14 and keeping strong negative competence C24. There seems to be nothing
that speaks generally against such a concept of competence. However, it will not allow us
to account for the data we want to capture. The order + minimizes also what the speaker
knows about what is possible on 4-accessible worlds. Hence, in the minimal states as much
as possible statements of the form 3r will be valid. Following C24 the interpreter thinks
that all the speaker takes possibly to be valid on R4 is actually valid (because if not, the
speaker would know so). Hence, from 3r the interpreter infers r . This leads to results
that are again much too strong. For instance, 4p _ 4q will be dishonest, because the
sentences are consistent with rp ^ r¬q as well as with rq ^ r¬p.
3.3.3 Solution 3: The Combination
Finally, it is interesting to note, that also the combination of | + with StateKD45C14,
hence, the combination of weakening the order and weakening the notion of competence
allows us to derive the free choice inferences. The results from the last sections can be
transfered nearly directly. One point that may need some discussion is the fact that also
with respect to + we can use the merge
W
4 without losing fact (11.ii) for the positive
information order. But this follows immediately from the fact that s o s0 ) s + s0. The
only other point that may be problematic by going from o to + is that the sentences
we used to show that states that do not verify the conclusion of our inferences are not
minimal are no longer in the order defining language. It appears that this is no problem for
the free choice inferences (D1), (D2), (D3), (D5) with the first antecedent, (D6) for both
antecedents and also for the inferences concerning the conjunction (D8) and (D9). The
relevant sentences 2¬p, 2r¬p and 2r¬(p ^ ¬q) are all in 2L+. Again, this combination
of a concept of competence and a notion of entailment differs from the one in section 3.2.6
in not disqualifying the second antecedent of (D6) as dishonest. From the solution studied
in the last section this approach differs by not making the incompetence inferences (D4)
and (D7) valid. Assume, for instance, that (D7) does not hold in some s1 2 StateKD45C14.
This means in particular that s1 6|= 3(rp^¬rq). Hence, s1 |= 2(¬rp_rq). This is not a
sentence of 2L+ and hence a state that simply differs from s1 in not entailing 2(¬rp_rq)
will not be strictly smaller than s1 with respect to +.
Just for the sake of completeness, we will discuss the inference (D9). Assume, there
exists s1 2 StateKD45C14 such that s1 |= 24(p ^ q) and, without loss of generality,
s1 6|= 4(p ^ ¬q). By C14 it follows that s1 |= 2r(¬p _ q). The latter sentence is in the
order defining language 2L+. Because p and q are logically independent from each other,
there are states such that p ^ q as well as p ^ ¬q is true. Take a s2 2 StateKD45C14 such
that s2 |= 2(rp^4q^4¬q). It follows that s2 |= 24(p^q). Hence, for s =def
W
4({s1, s2})
it hold that s |= 24(p ^ q). Furthermore, by construction, s 6|= 2r(¬p _ q) and together
with fact 11(ii) (8s 2 T :
W
4(T) + s) we can conclude that s + s1. In combination with
fact 4 this proves the claim.
45
4 Discussion of the Proposal
The topic of this last chapter is an evaluation of the account developed in the thesis. We
start with comparing the three combinations of a notion of pragmatic entailment and a class
of states on which the entailment is evaluated that turned out to predict the free choice
inferences. After that we will evaluate the approach in general.
4.1 Comparing the three proposed Approaches
At the end of the third section we discussed three combinations of a notion of entailment
(capturing conversational implicatures due to the maxims Q1 and T ) and a class of states
on which the notion of entailment is applied (modeling assumptions about the competence
of the speaker) that all predicted the free choice inferences. In the following we will broaden
our view and evaluate the plausibility of the accounts with respect to other questions. As
we will see they do not always perform equally good. In the end we will come up with a
ranking of the three approaches concerning their general adequacy and choose one of them
as the most convincing account for free choice inferences.
4.1.1 Strong Competence and Weak Order
The first successful system, discussed in section 3.3.1, took a very strong perspective on
competence: both, the positive (C14) as well as the negative (C24) competence axiom
were assumed to be valid in the context of interpretation. This is what we called strong
competence of the speaker with respect to 4. To avoid general dishonesty we had, in
turn, to weaken what counts as the relevant information the speaker has to convey and,
hence, to restrict the language that defined the pragmatic entailment. We discussed two
possible adaptations of the objective information language 2Lo. In variant 1 we removed for
modalities 4 the speaker is assumed to be competent on sentences from Lo where a negative
occurrence of r is not embedded under another modal operator; this led to the notion | +.
In a second variant all sentences that contain a negative occurrence of a modal operator
in wide scope position are removed from Lo - irrespective of competence; the resulting
notion of entailment was | ++. Both system are, despite the weaker information orders,
quite strong in their predictions. For instance, we obtain that an utterance of the sentence
rp _ rq66 made by a competent speaker is pragmatically not well-formed. In particular,
no free choice inferences can be derived. The reason is simple: the sentence is not honest
with respect to the order defining language 2L+/ 2L++ and StateKD45C14C24. Or, to
put it in other words, given the way | + and | ++ formalize the conversational maxims T
and Q1 a competent speaker uttering rp _ rq cannot be obeying these maxims.67
Another pleasant property of the combination of a weak information order with a strong
notion of competence is that it suggests a straightforward way how to account for the
possible cancellation of free choice inferences. As we observed in section 2 free choice
readings do not occur in case the speaker explicitly mentions her incompetence (cf. the
examples (8) and (9) here repeated as (24) and (25)).
66This is the second antecedent in the inference schema (D6), which we flagged with a question mark.
67Recall that all three accounts we discussed predict the sentence 2p _ 2q to be pragmatically not wellformed
for exactly the same reason.
46
(24) You may take an apple or a pear - but I don’t know which.
(25) You may go to Shoal Creek or go to Shingle Creek. But stay away from the dangerous
one.
What does this approach predict in such a situation? Of course, any statement of the
speaker by which she claims to be (partly) incompetent will be inconsistent with taking
her to be strongly competent. Hence, an interpreter that takes the speaker to be strongly
competent on the deontic options will not be able to directly incorporate the information
of the examples (24) and (25) in her belief state. She has, presumably, first to weaken her
belief about the competence of the speaker and to give up strong competence. In this thesis
we cannot discuss the topic of belief change, so we can only suggest that this adaption of
the belief state of the interpreter leads to a new state on which | + and | ++ will not make
the free choice inference valid.68,69
Already in section 3.3.1 we hinted at a difference between the two notions of weak
entailment, | + and | ++. The inference schemas (D4) and (D7), which we wanted to
obtain for modalities the speaker is not taken to be competent on, are not valid for | ++.
To give a concrete example, it is predicted that a possibly incompetent speaker of (25)
takes it as possible that the addressee may go to Shoal creek, but it is not predicted that
the speaker takes it as possible that the addressee may go to Shoal Creek and may not
go to Shingle Creek. This seems contra intuitions, according to which the sentence is
understood as implying that the addressee may go only to one of the creeks and the speaker
does not know which one. The reason why this inference is missing is that sentences like
2(4p^¬4q)) are not elements of the order defining language L++ and, hence, not subject
to minimization. Therefore, we cannot derive that the speaker takes 4p ^ ¬4q to be an
epistemic possibility and, in turn, establish the consequence of (D4). Even though this may
be no serious defect of | ++ we have an alternative notion, | +, that can account for (D4)
and (D7).
One might argue that also | + has its weak point: it depends on what the hearer believes
about the competence of the speaker. One may ask why an interpreter should adapt the
completeness expectations she has for the utterance of the speaker this way. The following
observation hints at an explanation. If the speaker is believed to be strongly competent
68Notice that whether we obtain this result indeed depends on the way the belief change of the interpreter
is modeled. If, for instance, in reaction to (24) or (25) the interpreter adopts a belief state where not C24
but still C14 is valid, then | + and | ++ would still derive the free choice inferences. However, this belief
adaption is arguably to specific. Why should the interpreter choose to give up only C24?
69Of course, this outline of an explanation of the cancellation observation builds on the assumption that
the interpreter first takes the speaker to be strongly competent. Only in these situations can we assume that
belief change is necessary. But if competence is assumed to be something the hearer learns from incoming
information it may be the case that she has only partial knowledge about the competence of the speaker but
still enough to derive the free choice inferences. Then the speaker’s announcement of incompetence may not
be strong enough to contradict the belief of the interpreter and the free choice inferences may still go trough.
An example for such a situation is again the case where the hearer thinks C14 to be valid but does not
know about C24. A sentence like (24) does not stand in conflict with such a belief state. And, as we know
from section 3.3.3, application of | +, | ++ on this belief state would still render the free choice inferences
valid. An interesting option how to exclude such a situation (and they seem to be contra intuition) is to
take also competence to be an assumption the hearer makes about the speaker. We will come back to this
perspective in section 4.2.
47
with respect to some modal operator 4, then | + together with the assumption of strong
competence are already enough to infer all the speaker’s beliefs on Lo − L+, the part left
out of the language. Therefore, in case strong competence of the speaker is commonly
known a speaker does not need to mention these beliefs and a hearer will not expect her
to do so. That the speaker will not give this information seems even more reasonable given
conversational maxims as the second submaxim of Quantity. It asks the speaker only to
communicate what is necessary. There is no time to go into more details on this point. It
has to be left for further research. But so far there seems to be no true objection against
| +.
4.1.2 Weak Competence and Weak Order.
We come now to another approach to the free choice inferences, which was discussed in
section 3.3.3. As we observed there, also the application of the weak notions of entailment
| +/ | ++ to a class of states where the speaker is only taken to be positive competent on
some modality 4 allows to account for the free choice inferences.
Because the notion of entailment used remains the same, what was said above on this
topic applies here as well. But there are some negative consequences of dropping C24 that
have to mentioned. First of all, we lose the suggested account for cancellation in case the
speaker mentions her incompetence. The reason is that the logic KD45C14 leaves the
speaker elbow-room to be in some sense incompetent on 4 and this space is not occupied
by the information order. This has the effect that in +/ ++-minimal states the speaker
does not have to have complete beliefs on what does and does not hold on R4. Therefore,
if the speaker claims to be incompetent on the modality 4 this does not necessarily conflict
with what the hearer has to think of the beliefs of the speaker in order to derive the free
choice inferences. Hence, no inconsistency does occur that forces the interpreter to give up
C14 and, in consequence, the free choice inferences may still go through.70
We will not dwell on possible solutions for this problem but instantly turn to another
suspicious property of this approach that is strongly related. Without C24 we are no longer
able to derive by | +, | ++ from 4(p_q) that 2(4p^4q), hence, that the speaker knows
the free choice inference to be valid. (To illustrate this point, figure 4 sketches two states
for 4(p _ q) that are minimal with respect to + but where C24 does not hold. In s2
2(4p ^ 4q) is not true.) This result does not seem to be correct. Intuitively, we take a
speaker of 4(p _ q) to know that the addressee has free choice.71
Finally, this variant of the approach loses, compared with the one discussed above, the
possibility to disqualify sentences like def rp _ rq as pragmatically not well-formed.
However, there is a straightforward way to repair this defect. In a minimal state for such a
sentence it will be the case that the utterance itself, , is not true.72 This is certainly not a
70Notice, that this is particularly true for the two examples of cancellation discussed here : (24) and (25).
71Notice that the loss of this particular information seems also to be responsible for the difficulties to
account for cancellation. If the speaker is assumed to be aware of the information she implies, then continuation
with “ ... but I don’t know which” would be inconsistent with uttering 4(p _ q) (something along the
same lines can be said about (25)). Hence, in this case one may again be able to account for the cancellation
effects using belief change.
72In the minimal belief state of this sentence the speaker distinguishes an epistemic possibility where
rp ^ ¬rq holds and one where rq ^ ¬rp is true. Hence, ¬2rp ^ ¬2rq. By C14 (which we assume to be
48
s1
H HY
@@R
p *
q
p,q
p
q w1
•
s2
H HY
A
A
A
-
@@R
*
HHj
-
q
p
p,q
p
q w2
p
q
•
Figure 4:
The conventions for this figure are the same as for figure
3: arrows present 4-admissible worlds, lines 3-
admissible worlds. Only 3-connections from the root of
the state are plotted. Furthermore, only information on
the deontic possibility of p and q is given. Absence of a
proposition letter at a world means that the proposition
does not hold there.
plausible model for the utterance. Already at the beginning of section 3.2.1 we discussed the
possibility to strengthen our notion of pragmatic entailment by incorporating the semantic
meaning of the utterance: to take the truth of the utterance to be part of the requirements
minimal states have to fulfill. It turns out that with respect to such an extended notion
of entailment the sentence rp _ rq would again become dishonest. This can be seen as
follows. The first thing to notice is that for two states s1, s2, such that s1, s2 |= rp _ rq
they either entail rp or rq. Assume now that s1 is minimal for rp _ rq with respect to
+. Without loss of generality, s1 |= rp. We obtain s1 + s2 ) s2 |= 2rp (this follows
straightforward from C14 and the fact that rp is in the order defining language). However,
there are states s3 making rp _ rq true that do not entail 2rp: take, for instance, the s3
sketched in figure 5. Hence, there are states s3 such that for a minimal state s1: s1 6 + s3.
In turn, there cannot be an unique minimal belief state. This implies that with respect to
the set of states where 4p _ 4q is true the sentence is dishonest with respect to L+ and
StateKD45C14.
s3
H HY
@@
-
*
HHj
q
p,q
q
q
p,q
w2
•
Figure 5:
4.1.3 Weak Competence and Strong Order
Finally, in section 3.3.2 we discussed the possibility to account for the free choice inferences
by combining positive competence with the notion of entailment | o based on the strong
objective information order. This approach faces similar problems as does the last one
and actually even in increasing strength. Because the order minimizes not only belief on
R4-necessities but now also on R4-possibilities and again positive competence does not lay
restrictions on the speaker’s belief about what is possible on R4-admissible worlds, a hearer
infers using | o that the speaker is as far as possible incompetent on R4-possibilities. For
instance, remember the states s1 and s2 sketched in figure 3. According to the objective
valid in this variant of the approach) it follows that ¬rp ^ ¬rq. Thus, ¬(rp _ rq).
49
information order, s2 o s1 and even s2 is not minimal for 4(p _ q) with respect to o.
Particularly, it turns out that using | o a speaker of 4(p _ q) while implying free choice
permission is inferred not to know that this inference holds. These predictions are clearly
not in accordance with intuitions.
Given this problem it is not surprising that also the difficulties to predict cancellation of
the free choice inferences in case incompetence is stated transfers from the system discussed
in the section above. And, as much as for this approach, the pleasant dishonesty properties
of the first system are not valid.73
Finally, this system predicts that a strongly competent speaker is always dishonestly
withholding information. Again, this goes against intuitions.
4.1.4 Conclusions
Let us summarize the discussion so far. While all three accounts we ended up with in
section 3 predict the free choice inferences, some differences occur in other respects. Only
the first system is able to disqualify the marked inferences in (D5) and (D6) because of
dishonesty. However, we have seen that a straightforward modification of the definition of
the notion of pragmatic entailment may extend the results also to the other two systems.
Furthermore, the intuitions on the well-formedness of sentences like rp _ rq were not
without some doubts. Therefore, the force of this argument in favor of the first account
is not particularly strong. However, both other accounts give unintuitive results for the
hearers picture of the belief state of the speaker after minimization: while intuitive in case
free choice permission obtains the speaker is taken to know this, the system combining
| + with positive competence is not able to make this prediction, and | o even allows to
infer the opposite. These problems lead also to difficulties with the suggested account for
cancellation if incompetence is stated. We conclude that therefore the first approach should
be preferred. This still allows us to choose between | + and | ++. As we have seen, | ++
fails some of the inference schemes we want to be valid. Therefore, | + has to be selected
as the notion of pragmatic entailment that gives the best results. For the rest of the paper
we will always refer to this notion and evaluate it using the strong version of competence.
4.2 Evaluating the Approach
While in the last section we discussed the three successful ways in which the proposed pragmatic
account to the free choice inferences can be spelled out, now we turn to a general
evaluation of the plausibility of this pragmatic approach.
If one considers the list of pragmatic inferences we wanted to obtain (see section 3.1.3)
the approach defended in this thesis is very successful: we were able to come up with a
formally precise Gricean notion of entailment that, indeed, makes all the inferences on this
list valid. But there are also some problems, or at least, open questions connected to the
way free choice inferences are approached here. One point that needs further discussion is
73However, also the proposed solution discussed above extends to the present system. The argumentation
can be copied directly. Because the defining language of the objective information order is an extension of
the order defining language for the positive information order, the latter order is an extension of the former
one.
50
the status of the speaker’s competence. The way it is formalized here (as part of the belief of
an interpreter) has difficulties to account for some further observations. First, when we look
at the data it catches the eye that taking the speaker to be competent seems to be rather
the rule than the exception. Free choice is the normal interpretation disjunctions undergo;
that this reading is not intended has to be signaled explicitly by mentioning incompetence.
Because in the proposed account the free choice inferences depend on competence it is predicted
that the hearer normally believes the speaker to be competent. On the other hand,
one glance on the formalization makes it quite clear that competence is a strong restriction
on the belief of the interpreter. One might question whether in every situation where a free
choice interpretation occurs indeed the hearer has the factual evidence that the speaker is
competent on the matter of discourse. Against this background, it seems more appropriate
to understand also the speaker’s competence as a default assumption of the interpreter.
And remember that also the suggested explanation for cancellation of free choice inferences
works much better with this perspective on competence.
An adaption of the presented interpretation function that reflects such a competence
assumption can be easily given. But this change in the theory provokes some troublesome
questions. For instance, how to motivate such a competence default? None of the Gricean
maxims seems to contain a competence assumption. Is it nevertheless something that is
reasonable to assume in certain situations? Or is it a linguistic convention?
We will leave these questions unanswered and come to another, more serious problem of
the proposed account. As the reader may have recognized, the approach presented here - at
least in the way it is spelled out in section 3 - strongly over-generates. What we mean with
over-generate in this context is that besides free choice much more pragmatical inferences
are predicted and many of them are not in accordance with intuitions. The concrete problem
can be formulated particularly clearly using the order defining languages. The reader
will remember that in case 2L is used to define an information order , everything that
is taken to be in L is subject to belief-minimization and hence, if possible, predicted not to
be decided in the speaker’s belief state. It seems that we took too much of L to be relevant
for the order, hence, we chose an L that is too big. The language fragment L+ contains,
for instance, all atomic propositions P. We obtain, therefore, for arbitrary p, q, r 2 P:
4(p _ q) | + 3r ^ 3¬r ^ 34r ^ 34¬r. And under the additional assumption that the
speaker is competent on 4, 4(p _ q) | + 4r ^ 4¬r.74 Or, to use more natural examples,
given that the sentences Aunt Hetty is making apple pie, Mr. X is in Berlin and You take
a banana can be expressed in our formal language, we obtain, for instance, that (26a) | +-
entails (26b), (26c) | +-entails (26d), and (26e) | +-entails, assuming C1may and C2may
to be valid, (26f). These predictions are certainly wrong.
(26) a. Tomorrow we have better wether or we have to stop.
b. The speaker doesn’t know whether aunt Hetty is making apple pie.
c. Mr. X might be in Amsterdam or in Frankfurt.
d. Mr. X might be in Berlin.
74The same holds for | o and | ++.
51
e. You may take an apple or a pear.
f. You may take a banana.
What we, thus, need to make the approach to the free choice inference defended here
work is a general description of that part of L+ with respect to which the belief of the
speaker really should be minimized.
But even if we had such an description of the order defining language, we still might
be in serious trouble. The point is that because of the specific way in which we motivated
our approach not any description will do. The defined notion of entailment was set out as
formalizing the concept of conversational implicature; as description of what it means to interpret
a speaker as obeying certain Gricean maxims. Hence, whatever language restrictions
we impose, they have to have a Gricean motivation.
In the remainder of this section we will discuss some possible ways to approach this
problem and connected difficulties. However, no final solution will be given. Further investigations
have to show whether the challenge of adequate language restrictions can be
answered or whether the problem of over-generation reveals that, in the end, we chose the
wrong approach.
How could a Gricean description of the order defining languages look like? Let’s start
with a negative example: with a kind of description that can not be motivated this way.
Intuitively, there seems to be a close connection between the inferences we want to obtain
(and, hence, the sentences that should define the order) and what has been actually said by
the speaker. Given this observation, a first idea one might have is to determine the order
defining language using the logical form of the utterance. Gazdar’s calculation of potential
clausal implicatures comes close to such a way of proceeding. But, as already mentioned
when discussing his approach, if it is really the form of the uttered sentence that matters
how should a Gricean motivation for such a dependency look like?75 Is it not the case that
then by definition the generated inferences belong - as form-properties - to the sphere of
semantics? Therefore, we have to conclude that an approach along these lines would not
be in accordance with the general understanding of free choice inferences advanced here.
To find an appropriate description of the order defining language a promising strategy
might be to start directly from Grice and asks oneself what kind of order defining languages
can get a motivation from his theory of conversational implicatures. Or, to reformulate the
question, what part of the belief of the speaker would this theory predict to be exhausted
by her contribution? In the formulation of Q1 Grice explicitly marks that the speaker only
has to be complete with respect to relevant information. In the maxim of Relevance (and
also in the second submaxim of Quantity) he even asks the speaker not to give irrelevant information.
Hence, a restriction of the order defining language to sentences that are relevant
would fit into a Gricean setting.
There can be two different ways of how to incorporate relevance in our description of
the free choice inferences. First, what the speaker thinks to be relevant may be taken by the
interpreter to be a subjective variable that she can access only indirectly - by the utterance
75We would clearly need more information about the form than pure complexity. Hence, a motivation via
Grice’s maxim of Brevity would not be an option.
52
made. In this case the order defining language could be defined as those set of propositions
the utterance made is in some sense relevant to, the facts the speaker is giving information
about. Second, relevance may also be an inter-subjective contextual factor that can be
manipulated by the discourse participants, for instance, by asking questions. In this case
the order defining language is defined as those sentences that serve the commonly known
goal of discourse at the moment when the speaker makes her contribution.
For both perspectives on relevance there exists a wide range of proposals how to make
them precise. To give a simple example how to formalize the first perspective on relevance:
one could, for instance, require that the utterance of the speaker gives information about a
sentence in case the logical form of the utterance and this sentence are logically dependent
on each other. This would, of course be a very coarse-grained notion of relevance. For
the second kind of approach we can make use of the well-developed theories of relevance
determined in terms of contextual given questions or decision problems (e.g. Groenendijk &
Stokhof [1984], van Rooy [2003], van Rooy & Schulz [submitted]). We will use here a rather
simple concept of relevance proposed in this area to illustrate some aspects of relevancebased
approaches to the order defining language. Let us assume that the meaning of a
question is a partition Q of the logical space, hence, a set of sets of states (cf. Groenendijk
& Stokhof [1984]). We define a sentence ' to be about Q ( 2About(Q)) iffdef 8q 2 Q :
(|=q ') or (|=q ¬').76 Our notion of entailment | is then defined for answers to the
question Q by the language 2L where L contains those and only those sentences that are
about Q, (L =defAbout(Q)).
What would such an approach predict, for instance, for an utterance of (26e)? It seems
quite natural to assume in context of the utterance: ‘You may take an apple or a pear’ a
question of the form ‘Which fruits may I take?’. The set of sentences that are about this
question are then L = { you make take an apple (4apple), you may not take an apple
(¬4apple), you make take a banana (4banana), ...}.
A first thing to notice is that this language - and this would also have been the case if
we had used most other notions of relevance - is closed under negation. As we have already
seen when discussing the combination of the objective information language and strong
competence this results in an unintuitively strong notion of dishonesty. To formulate the
problem in some generality:
Fact 12 If (i) the speaker is assumed to be strongly competence with respect to two sentences
1 and 2 (hence, i ! 2 i and ¬ i ! 2¬ i are valid for i = 1, 2), and 1 and 2 and
(ii) their respective negations are elements of L , then the disjunction 1 _ 2 is dishonest
with respect to 2L .
There are different ways to see this.77 Intuitively, the problem is simply that the sentence
2( 1 _ 2) has three different and incomparable minimal models in this setting: one where
1 ^ ¬ 2 holds, one where 2 ^ ¬ 1 is true, and, finally, one where 1 ^ 2 is true. To
give a concrete example and also to illustrate how serious this problem is for our matters,
take 1 = 4p and 2 = 4q. If we assume the speaker to be strongly competent on 4/r
76This notion of aboutness has been independently introduced by Lewis [1998] and Groenendijk [1999].
77You can, for instance, use the disjunction property for 2L . Using strong competence one can derive
2( 1 _ 2) ` 2 1 _2 2 while the antecedent does not entail one of the disjuncts. Hence, it cannot be honest
with respect to 2L .
53
and 4p,4q,¬4p,¬4q 2 L (notice that 4p,4q,¬4p,¬4q 2 About(Q)) then 4p _ 4q
( 4(p _ q)) would turn out to be dishonest.
As with the more general problem of combining strong competence with the objective
information language we have two perspectives now: (i) we can weaken the competence assumption,
or (ii) we can weaken the order defining language. In the first part of section 4 we
have argued that an account that keeps strong competence but describes the order defining
language to be (a subpart of) 2L+ should be preferred. Of course, we can now postulate,
for instance, that the order defining language is the set 2Neg, where Neg= About(Q)\L+
and, hence, for the example we have discussed: Neg = {¬4apple,¬4banana, ...}. Indeed,
in terms of 2Neg we can account for the free choice inferences. But again: what is the
(Gricean) motivation for taking this intersection? Actually, one may argue that Pos =
About(Q)− Neg = {4apple,4banana, ...} is a much more natural candidate in this respect.
For instance, the sentences in Pos are prototypical optimal answers to the question
‘What fruits may I take?’, what leads Hamblin to propose this set to be the meaning of
the question. Of course, minimizing belief with respect to Pos does not give the desired
inferences.78 But a simple adaption brings us back on the right track. We only have to
change how a language defines an information order. First, it is easy to prove that the
following fact holds.
Fact 13 . Let L− be the language defined by the BNF '− ::= p(p 2 L(0))|'− ^ '−|'− _
'−|4',4 6= 3(' 2 L). We define further
s vo s0 iffdef8' 2 Lo : s0 |= 3' ) s |= 3',
s v− s0 iffdef8' 2 L− : s0 |= 3' ) s |= 3'.
Then, it holds 8s 2 StateKD45 : s o s0 , s vo s0 and s + s0 , s v− s0.
In consequence, the notions of entailment that can be defined in terms of the interchangeable
orders (using definition 1) are equivalent too. Hence, minimizing the beliefs of
the speaker with respect to 2Lo/ 2L+ provides exactly the same results as maximizing her
epistemic possibilities with respect to the languages 2Lo/ 2L−. In the same way it can be
shown that vpos defined by Pos is equivalent to neg defined by Neg. And therefore, also
from Pos the free choice inferences can be obtained if we take the perspective of maximizing
the epistemic possibilities of the speaker.79
So far, things are going quite well: we have seen that some problems relevance based
accounts face on the first view can easily be solved. A next question one might want to ask
is which of the two ways to incorporate relevance in the proposed account for the free choice
inferences should actually be adopted: taking relevance to be a contextually given factor
78The order defined by Pos using definition 2 (section 3.2.2) produces inferences that are inconsistent with
a free choice interpretation: having 4apple in the order defining language will lead to an order that prefers
belief states where the speaker does not believe that 4apple is true. In case the speaker is C2may-competent
we even obtain 4(apple _ pear) | pos r¬apple ^ r¬pear.
79As a final observation on this topic, the Gricean argument sketched in section 4.1.1 to motivate the
language restriction from Lo to L+ is independent of the choice of minimizing belief with respect to Neg or
maximizing epistemic possibilities with respect to Pos and, hence, still an option to explain these necessary
language restrictions.
54
or to be determined by the utterance of the speaker. It is important to notice that there
are subtile differences between both perspectives. They make, for instance, different predictions
in case the speaker gives information on contextually irrelevant sentences. In this
case, the latter notion would give rise to implicatures that are not predicted by the former
one. A different situation occurs when the speaker gives no information on sentences that
are contextually relevant. Then, using the first concept of relevance as basis would give rise
to implicatures that do not occur with a language founded on what the speaker is talking
about. We can use these situations where both approaches make different predictions to
test them against each other.
The dialogue fragment (27) is set up as example for a speaker giving contextually irrelevant
information.
(27) A: Was it raining this morning?
B: Yes, and aunt Hetty is making apple pie or muffins.
Even though the response of B is quite strange, especially in those contexts that interests
us here: where aunt Hetty’s activities are absolutely irrelevant for answering A’s question,
intuitively, B is taken not to know what aunt Hetty is preparing for tea. Hence, it seems
as if a formulation of the order defining languages in terms of what the speaker is talking
about is more appropriately describing our intuitions. One may counter with the suggestion
that the speaker’s utterance of the disjunction may change what counts as relevant in the
discourse. Hence, that what the speaker is talking about can be used to repair situations
where the interpreter’s picture of what is relevant and the contribution of the speaker are
not in accordance. One may furthermore suggest that the speaker’s strategy in such a
situation is to add the disjuncts to the set of relevant sentences and, hence, the order
defining language. ‘But why?’ one might ask. In particular, why adding the disjuncts and
not the whole disjunction?
We leave this question unanswered and consider the second test condition. Examples
where the speaker provides (given the semantics assumed here) no information about contextually
relevant sentences can be easily found. Take a situation where (26e) is uttered in
the context of (28).
(28) A: Which fruits may I take?
B: You may take an apple or a pear.
In this situation standard notions of contextual determined relevance (in particular, the
one described above) would predict ‘You may take a banana’ to be a relevant sentence.
However, the speaker B does not provide any information about its truth. What are the
intuitions about this example? Do we infer that the speaker does not know whether it is
true that the addressee A may take a banana or is nothing concluded about the speaker’s
beliefs on the permissibility of taking bananas? The dominant opinion in the literature on
comparable examples (e.g. Groenendijk & Stokhof [1984], van Rooy & Schulz [submitted1]
and [submitted2]) is that while there are contexts in which the answer of B is taken to
represent all the speaker knows (and, hence, the approach using contextually determined
relevance would make the correct predictions), in most situations the interpreter of B’s
utterance will conclude that an apple and a pear is all the addressee may take (hence, taking
55
a banana is prohibited) and this is known to the speaker. Thus, it seems that neither of the
relevance approaches gives in general the right result: while using the utterance to determine
what counts as relevant is unable to account for those cases where indeed the speaker is
inferred to be incompetent about the banana, determining the order defining language with
contextual relevance gives in other contexts a language that is still not restricted enough.
In the latter case the predictions made seem to interfere with the so called exhaustive
interpretation of utterances. The following dialogue fragment gives a standard example for
this notion.
(29) Paul: Who knows the answer?
Paula: John and Mary.
In most contexts Paula’s answer in (29) is interpreted as conveying that John and Mary are
the only people who know the answer. Although the sentence ‘John and Mary know the
answer’ does not bear this information if occurring in isolation (hence, one could argue, it
is not conveyed by its semantic meaning in the strict sense of the word), used as answer it is
understood as giving all those and only those who know the answer. In the same sense (26c)
on page 65 is understood as giving all whereabouts of Mr. X the speaker thinks possible
and (26e) to give a full list of what the addressee may take from the fruit bowl.
When considered more closely, exhaustive interpretation seems to be simply a stronger
variant of the non-monotonic reasoning we modeled in chapter 3. Again, the absence of
information is made meaningful. But instead of inferring from the fact that the speaker
did not claim that some sentence p holds that she does not know whether it does, here
the stronger inference is derived that p does not hold - negation as failure. With respect
to which sentences this inference is made seems to be strongly dependent on what is relevant
in the context of utterance (cf. van Rooy & Schulz [submitted1]).80 Actually, taking
exhaustivity to minimize the set 2Pos described above while at the same time assuming
strong competence on the extension of the question predicate yields in many cases a correct
description. So, let’s adopt for the moment the thesis that exhaustivity can be described
in this way.
We have the following options now. We can first adopt the position that what the speaker
is talking about determines the order defining languages. Then we have to come up with
something else to account for the incompetence readings this approach cannot account for.
Independent of the fact that until now we have said nothing about how to make what the
speaker is talking about precise there are some questions that such a way of proceeding raises.
As may have become clear in section 3 the class of inferences described by our notion of
pragmatic entailment can be seen as generalization of Gazdar’s clausal implicatures. But
there are also strong connections between exhaustive interpretation and Gazdar’s notion
of scalar implicatures (see van Rooy & Schulz [submitted1]). Gazdar took both classes of
conversational implicatures to be due to the first submaxim of Quantity (and the maxim
80Perhaps the context dependence is even stronger here than with free choice inferences. Consider (30).
(30) A: Who is coming for tea?
B: Mary. And aunt Hetty is making [apple pie]F .
For this example an exhaustive interpretation of the second sentence of B seems hardly possible.
56
of Quality). We added in this section a third maxim to the derivation: the maxim of Relevance.
One question that has to be answered now is why we need two different notions of
relevance for both classes of implicatures.
A second option is to take for both: free choice inferences and exhaustification, the
same notion of relevance: contextual determined relevance. But this would mean, given the
preliminary description of exhaustive interpretation we have adopted, that in both cases
2Pos is the order-defining language. It is difficult to see how then the mispredictions we
discussed in connection with example (28) can be excluded. This can, for instance, not
be done simply by executing one inference-generating process after the other (as Gazdar
proposed for his description of clausal and scalar implicatures). As we have described both
processes they work in completely opposite ways on 2Pos: while to obtain the free choice
inferences we have to select the maximal elements of the resulting order, we described exhaustivity
as doing exactly the opposite: looking for its minimal states. It is obvious that,
therefore, no matter which order is adopted, there will be no contribution of the process
running as second.
Of course, this whole discussion depends strongly on the provisional description of the
exhaustive interpretation we have adopted. Therefore, we conclude that in order to find a
solution for the problem we started with: the problem of how to restrict the order defining
languages, first, we have to closely study the phenomenon of exhaustive interpretation and
its interaction with the free choice inferences. But this would go beyond the scope of this
thesis and has, therefore, to be left for another occasion.
57
5 Conclusions
This thesis proposed a formalization for parts of Grice’s theory of conversational implicatures.
The proposal builds on work of Halpern & Moses [1984] recently generalized by van
der Hoek et al. [1999, 2000]. The core of the approach consists of a notion of pragmatic
entailment that is intended to capture the conversational implicatures due to the first submaxim
of Quantity and parts of the maxim of Quality introduced by Grice. Technically, the
defined notion of entailment represents an instance of reasoning in preferential structures:
a sentence is said to pragmatically entail a sentence if in all states where the speaker
is in the unique minimal belief state given that she believes also is valid.
The proposed formalization was then tested by trying to account in terms of it for the
free choice inferences. In particular, we tried to describe free choice permission, a well-known
problem for many semantic and pragmatic theories. It was shown that given contextual
information about the competence of the speaker - and here we build strongly on work of
Zimmermann [2000] - the approach allows us to derive the free choice inferences. Hence, the
introduced formalization of the theory of Grice allows us to describe free choice inferences
as conversational implicatures certain utterances of competent speakers come with.
Actually, three accounts for the free choice inferences were obtained, differing in how
the central notions of competence and pragmatic entailment are spelled out. One of them
appears to be particularly promising. It takes as relation comparing the belief state of
the speaker the positive information order, +, which demands the speaker to give for a
modality 4 she is competent on only the information she has on what according to her
belief is necessary the case on the set of R4-admissible words. She does not have, however,
to convey also her beliefs on what is possible on this set of worlds. The resulting notion of
entailment was then combined with the strong version of the competence assumption. This
system allows us to account for all inferences that have been worked out in section 2 to be
characteristic for the free choice phenomenon.
As has been mentioned earlier the proposed notion of pragmatic entailment is closely related
to Gazdar’s [1979] description of clausal implicatures; not only with respect to the inferences
predicted but also in the way they are thought to be linked to the theory of Grice. A
question that this observation immediately provokes is which account should be preferred.
When discussing Gazdar’s approach we found some of its properties problematic. First, he
was not able to account for free choice inferences of modalities different from the one which
refers to the speaker’s belief state. Furthermore, Gazdar predicts some inferences that are
not reasonable. For instance, he predicts for the sentence 4(p _ q) the potential clausal
implicature 3p. Finally, we complained that the syntactic moment in his description of
potential clausal implicatures (he restricted them to subsentences of the sentence uttered)
has not been motivated with Grice’s theory and that it is also questionable whether such a
motivation can be given at all. While the first type of critics does not apply to the proposal
made here, the second one does. Also the approach developed in section 3 has to fight with
mispredictions. Actually, exactly the same inference we complained about when discussing
Gazdar is also predicted here. And furthermore, in some sense the account is also subject
to the third type of critic on Gazdar: when introducing the approach in section 3 we did
not give a Gricean motivation for the choice of the order defining languages. In section 4.2
58
some attempts for an improvement on this point were made. We discussed the possibility
to determine the order defining languages in terms of relevance. But to be able to see
whether such a way of proceeding is successful it appeared to be necessary first to discuss
the phenomenon of exhaustive interpretation. However, this could not be achieved in the
scope of this thesis and has to be left for another occasion.
To summarize, even though we improve on Gazdar’s approach with respect to some
points the proposal made here is still not fully satisfying. Further research has to reveal
whether it is possible to overcome the remaining shortcomings.
Another topic that needs attention in further work is to what extend the presented formalization
of the theory of Grice (if correct) can and should be extended to inferences due to
other conversational maxims or other phenomena analyzed as conversational implicatures.
Particularly interesting in this respect is an extension to the class of scalar implicatures.
As the reader might remember, Gazdar took these inferences to be consequences of the first
submaxim of Quantity (Q1) too. If he is right in this point one would expect that a correct
formalization of the implicatures due to this maxim should also account for the scalar
implicatures. We already noticed, when discussing the approach of Gazdar, that clausal
and scalar implicatures are clearly distinguished by a different grade of strength: while the
former only convey that certain statements are not believed by the speaker, scalar implicatures
claim that the speaker believes other sentences not to hold.81 How can inferences so
different in character be due to the same maxim?
Interestingly, some linguists working on conversational implicatures (e.g. Soames [1982])
proposed that the strength of Q1 implicatures depends on whether the speaker is taken to be
competent on the subject of conversation. This would suggest that the approach defended
here can also account for scalar implicatures: we already can describe clausal implicatures,
so the only thing we have to do is to apply our pragmatic notion of entailment in contexts
where the speaker is assumed to be strongly competent on the subject of discourse. The approach
to exhaustivity we adopted in section 4.2 realizes this program. We have seen there
that such a way of proceeding leads to certain problems with respect to the free choice
inferences. In van Rooy & Schulz [submitted2], it has been argued that this simple account
for exhaustivity will in general not work. The central problem is that typical scalar implicatures
are also allowed for utterances that are obviously not made by a strongly competent
speaker (if competence is understood as described in section 3.2.7). For more discussion on
the question of how to extend the present approach to scalar implicatures see van Rooy &
Schulz [submitted2].
We close the thesis by going back to the examples that led von Wright [1969] to speak
of a paradox of free choice permission and discuss the explanation given by the account
presented here for von Wright’s paradox.
The problem von Wright faced was that he saw no way to describe a logic of the expressions
involved in permission and obligation sentences that can account for the way we
81The reader may notice the parallels between the description of scalar implicatures given here and the
characterization of the inferences due to exhaustive interpretation, that can be found in section 4.2. One of
the central claims made in van Rooy & Schulz [submitted1] is that both types of inferences are to a wide
extend identical.
59
understand these sentences. In particular, for him there seemed to exist no possibility to
have free choice permission valid together with other necessary inferences without that the
obtained logical system gives rise to absurd predictions. Let us take as an example for such
mispredictions the following chain of reasoning.82
(31) (1) Mr. X may take a taxi.
(2) Mr. X takes a taxi implies that Mr. X takes a taxi or a boat.
(3) Hence, Mr. X may take a taxi or a boat.
(4) Hence, Mr. X may take a boat.
Intuitively, it should not be possible to conclude from (1) that (4) is valid. However,
in a deontic logic (like a logic was supposed to look like at the times of von Wright) that
allows free choice permission as a valid inference (31) can be turned into an argument: (2)
is valid as a tautology of propositional logic. From the premise (1) we conclude by (2) and
a rule of substitution truth conditional weaker sentences in modal contexts that (3) holds.
Then, the free choice inference allows us to conclude from (3) to (4).
Let us first recapitulate the position of Zimmermann [2000] regarding this problem.
First, in contrast to the early position of Kamp [1973], he does not detect the problem in
using a descriptive logic to reason over deontic sentences, but he objects that the wrong
translation of natural language sentences in a descriptive logic is used. More particularly,
Zimmermann claims that the interpretation rule for ‘or’ is wrong. If this is corrected in the
way he proposes (A or B is interpreted as 3A ^ 3B) (3) will no longer be derivable from
premises as (1) and (2).83 On the other hand, Zimmermann predicts that the step from
(3) to (4) is context dependent: it can be made only in case it is known that the speaker is
competent with respect to the deontic options ‘may’ refers to.
What has the account presented in this thesis to say about the paradox? First, together
with Zimmermann, the roots of the paradox are not located in the special features of
the semantics of non-assertive speech acts. But in contrast to him we did not blame the
assumptions von Wright made on the semantic meaning of the utterance involved. In
particular, we adopted an interpretation of ‘or’ as inclusive disjunction. With respect to
semantics, in turn, (2) and the inference to (3) are valid, but the step from (3) to (4) is not
permitted. In this thesis free choice permission is analyzed as a pragmatic inference that
is based on reasoning about the beliefs of a competent speaker that observes the maxims
of conversation in uttering (3). Only on the level of pragmatics an interpreter may reason
from observing an utterance of (3) to the truth of (4). Hence, there is no paradox of deontic
logic but confusion about semantic and pragmatic inferences. Notice finally, that (3) as
an utterance of a speaker who also uttered (1) or concluded (3) from (1) does not allow
us to derive (4). In both cases, given that the speaker of (3) believes (1) to be true, (3)
cannot be all she believes. Therefore the speaker was dishonest with her utterance and an
interpretation according to the maxims of conversation is not possible. Hence, given an
utterance of (31) (1)-(3) from the same speaker (4) is not a valid pragmatic inference.
82This example differs from the one we discussed in section 2. It is a weaker argument for the paradox
than the earlier one because it involves stronger assumptions on the logic of permission/obligation sentences.
Nevertheless, it is more popular in the literature.
83It may even be the case that (2) is not valid anymore. This depends on the modal concept of a belief
state that is used. In S5 and KD45, at least, p ! (3p ^ 3q) does not hold.
60
Acknowledgments
I would like to thank my supervisor Frank Veltman, Paul Dekker and Robert van Rooy
for comments and support during the preparation of the thesis.
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