Geary has long
known of the change that Auden later made and it has never made any sense to him.

i. We must love one another or die.

ii. We must love one another and die.

And it may be interesting to compare with still other possible versions. Auden rightly uses the strongest of the modals, 'must', and includes himself 'we'. That's different from the garden variety 'free choice' example, but I am reminded that in Geary's original post, the paraphrase indeed was:

iii. Love one another or die.

which could be interpreted as an imperative, and then we may have

iv. You may love one another or die.

And it may be interesting to compare with still other possible versions. Auden rightly uses the strongest of the modals, 'must', and includes himself 'we'. That's different from the garden variety 'free choice' example, but I am reminded that in Geary's original post, the paraphrase indeed was:

iii. Love one another or die.

which could be interpreted as an imperative, and then we may have

iv. You may love one another or die.

Geary: "I don't think Auden ever seriously thought of changing it to read:
"and/or". That's atrocious. Even Auden's choice of "and" seems to me to
diminish the soul of the poem. The "and die" version reads that there are two
things all human beings must do: one is "love one another" and the second is
"die." Both are inescapable for all human beings. But we that is *not* true.
Obviously, we do not have to "love another" -- in fact, we humans seldom do.
But yes, we all must die. The earth is going to go poof eventually, but that is
not germane to the poem. The poem says, Take care of one another, asshole
people, *or* war will kill us all long before the big poof. Not loving one
another will bring death to us all. You have a choice. That's the poem"

You're welcome.

It's interesting Geary should use 'choice', since
'free choice' is a phrase that philosophers (like Hans Kamp) have been used to
interpret utterances like Auden's. And for the record it should be pointed out
that he did allow the "or" version reprinted in a Penguin collection, AFTER the
"and" version reprint in the Williams collection.

Auden seems to be concerned with a conjunctive interpretation of a disjunctive construction. The relevant conjunctive interpretation is sometimes referred to as a "free choice effect" as attested when a disjunctive sentence is embedded within a modal operator. I will provide evidence that the relevant generalization extends (with some caveats) to all constructions in which a disjunctive sentence appears under the scope of an existential quantifier, as well as to seemingly unrelated

constructions in which conjunction appears under the scope of negation and a universal

quantifier.

Auden seems to be concerned with a conjunctive interpretation of a disjunctive construction. The relevant conjunctive interpretation is sometimes referred to as a "free choice effect" as attested when a disjunctive sentence is embedded within a modal operator. I will provide evidence that the relevant generalization extends (with some caveats) to all constructions in which a disjunctive sentence appears under the scope of an existential quantifier, as well as to seemingly unrelated

constructions in which conjunction appears under the scope of negation and a universal

quantifier.

It has been suggested that free choice effects
should be derived by the system that accounts for scalar implicature. However,
deriving a free-choice implicature (such as Auden's) is not a simple matter
within standard approaches to implicature computation. More specifically, free
choice seems to contradicts the Griceian attempts to deal with another
observation about disjunction due to Grice himself ("My wife is in the garden or
in the kitchen"). In response to this predicament, some have argued for a
system that derives the scalar implicature within the system, though in a
somewhat different manner, via a covert exhaustivity operator with meaning
somewhat akin to that of "only". The behaviour of a free-choice effect, as well
as Grice's observations about disjunction, follow from a fairly natural approach
to the meaning of the exhaustivity operator.

It is often claimed that the Griceian account of impicature follows from basic
truisms

about the nature of communication.

However, as is well known, one assumption is crucial, and far from trivial, namely the assumption that Grice’s Maxim of Quantity or Strength (his earlier Oxford lectures on implicature had none of this but an appeal to desiderata and principles: of candour, clarity, self-love and benevolence! -- the reference to QUANTITAS-QUALITAS-MODUS-RELATIO is a joke on Kant) should be stated with reference to a formally defined set of alternatives.

There is clearly no escape from formally defined alternatives.

However, if the perspective argued for here is correct, access to these alternatives should be limited to grammar. A quantity maxim which is not contaminated by syntactic stipulations (together with appropriately placed syntactic stipulations, i.e., within grammar) derives better empirical results.

Consider a simple disjunctive sentence:

When we hear such a sentence we draw a variety of inferences.

1. Auden talked to Chester or Christopher.

First, we conclude that (if the utterer is correct) Auden talked to Chester or to Christopher, a

conclusion, in and of itself, consistent with the possibility that Auden talked to both (basic

inference).

(The first to consider implicatures of 'disjunctions' was Grice, when criticising Strawson's Introduction to Logical Theory, in "Causal Theory of Perception": "My wife is in the garden or in the kitchen").

However, we typically also conclude, via implicature, and again assuming that the utterer’s utterance is correct, that this latter possibility was not attested.

Finally, we infer that the utterer’s beliefs (and this was Grice's focus in "Causal theory") don’t determine which person (i.e. Chester or Christopher) Auden talked to (ignorance inference).

The Inferences we draw from (1):

a. basic inference:

2a. Auden talked to Chester or Christopher (or both), i.e. Auden talked to Chester and/or Christopher.

b. implicature

2b Auden did NOT talk to both Chester and Christopher

c. ignorance inferences (Grice, "Causal Theory of Perception").

2c. The utterer doesn’t know that Auden talked to Chester

2c'. The utterer doesn’t know that Auden talked to Christopher.

The nature of the inferences in (2)a and (2)c seems rather straightforward.

The basic Inference, (2)a, is derived quite directly from the basic meaning of the sentence.

The Ignorance Inferences, (2)c, are, as Grice notes, not as direct, but, nevertheless, receive a fairly natural explanation.

They are derived straightforwardly from a general reasoning process about the belief states of utterers, along lines outlined by Grice's lectures on implicature (perhaps starting with "Causal Theory" -- although he is credited in a footnote to Strawon's logical textbook).

The source of the inference in (2)b, implicature, is less obvious.

The standard, neo-Gricean, approach captures this inference by enriching the set of assumptions that enter into the derivation of Ignorance Inferences, while various competing proposals attribute the

inference to a particular enrichment of the basic meaning.

Before one sees what is at stake, we should start with a formulation of what might be uncontroversial, namely the account of (2)c.

The basic idea is that communicative principles require utterers to contribute as much as possible to the conversational enterprise.

This idea is further elaborated when it is assumed that the goal of certain speech acts is to convey information, and that if all information is to be relevant, more is better.

Following Gazdar and Sauerland, some sometimes use the verb "know" to describe Ignorance Inferences.

This choice is problematic because of factivity inferences associated with "know", which are clearly inaccurate.

However, it’s not clear that there is a better choice.

"Believe" is problematic because of neg-raising ("I don't believe that p").

If some find factivity particularly disturbing (or neg-raising sufficiently innocuous), they favour "belief" (or 'doxastic', as I prefer) talk.

But these lexical choices can, however, be far from systematic.

The reader should bear all of this in mind and ignore factivity inferences associated with "know", as well as the neg-raising property of "believe".

Although some think that there is agreement that (2)c ought to be derived from principles of communication, there have been conflicting proposals concerning the precise derivation.

As we will see below, the complications could be argued to follow from the neo-Gricean perspective on implicature.

So, assume that two sentences are true and both contribute information that is completely relevant to the topic of conversation.

If one contains more information than the other (i.e., is logically stronger), use of the more informative one would constitute a greater contribution:

The conversational maxim regarding what Grice echoing (or making fun of Kant) calls quantity (in its basic version) may be formulated as:

If S1 and S2 are both relevant to the topic of conversation

and S1 is more informative than S2,

if the utterer believes that both are true,

the utterer should utter S1 rather than S2.

Typically, when (1) is uttered, the information conveyed by each of the disjuncts is relevant.

Furthermore, each disjunct is more stronger and informative than the entire disjunction (since "p" entails "p or q", but not vice versa.)

The fact that the utterer uttered the entire disjunction rather than just a disjunct, therefore, calls for an explanation.

If we, the people who interpret the utterance, assume that s obeys the Maxim of Quantity, we conclude, for each disjunct, p, that it is false to claim that the utterer believes that p is true, or if we keep to our convention of using the verb "know" instead of "believe", we can state this as a

conclusion that s does not know that p is true.

If we assume that the utterer believes that his utterance of the disjunction is correct, we derive the Ignorance Inferences.

But one logical property of the situation is worth focusing on.

When we conclude that the utterer does not believe that p is true, that is, in principle, consistent with two different states of affairs.

The utterer might believe that p is false, or, alternatively, he might have no conclusive opinion.

The reason we infer the latter is that the former would be inconsistent with our other inferences.

Under normal circumstances, we infer that the utterer believes that his utterance of "p or q" is true (Maxim of QUALITY, against a concoction by Grice echoing or making fun of Kant).

If we were to assume that utterer believes that p is false, we would have to conclude that he believes that q is true.

But that would conflict with our inference about q (based on the Maxim of Quantity).

Hence we must conclude, for each disjunct, that the utterer has no opinion as to whether or not it is true.

It's different with modal contexts (such as Auden, using 'must') or imperative contexts, where we have to generalise from truth-conditions to satisfaction-conditions (i.e. conditions where the states of affairs mentioned are factually satisfied).

Consider now whether we could extend this line reasoning to account for the implicature in (2)b.

Since we’ve already concluded that the utterer does not know that p is true and that the utterer does not know that q is true, it follows that the utterer does not know that the conjunction p and q is true.

This, again, is consistent with two different states of affairs.

The utterer might believe that p and q is false, or, alternatively, he might have no conclusive opinion.

If this time we could exclude the latter possibility, we would derive the implicature.

The problem is that basically the same line of reasoning we’ve employed above leads us exactly to the opposite conclusion, namely to the exclusion of the possibility that the utterer believes that p and q is false.

The idea is fairly simple.

The information that p and q is false, if true, would be relevant to the topic of conversation, hence the fact that utterer did not provide us with this information calls for an explanation.

The relevant notion of informativity for a pragmatic account should probably be that of contextual strength, i.e. logical strength given contextual presuppositions). This distintion has consequences on the theory of implicature with a potential argument against the Griceian account, I grant.

Once again, the natural explanation is that the utterer did not have the information, i.e., that s did not know that p and q is false.

In other words, instead of an implicature, we derive, once again, an Ignorance Inference: we

conclude that s does not know that p and q is true, and (exactly by the same type of

reasoning) that s does not know that p and q is false, i.e., we conclude that s does not

know whether or not p and q is true.

As far as we know, a version of this problem was first noticed in Kroch and stated in its most general form in class notes of Kai von Fintel and Irene Heim.

To appreciate the problem in its full generality, consider a general schema for deriving impicature in

response to s’s utterance of p (of, say, "I have 3 children").

We start by considering a more informative relevant utterance, p' (say, "I have 4 children"), and reason that if p' were true, and if s knew that p' were true, the Maxim of Quantity would have forced s to utter p' instead of p.

We then might reason that it is plausible to assume that s knows whether or not p' is true (say, that it is reasonable to assume that s knows how many children she has), and hence that s knows that p' is false.

The problem, however, is that there is always an equally relevant more informative utterance than p, namely p and not p' (in our case, I have exactly 3 children), call it p''. By

the same reasoning process, if p'' were true, and if s knew that p'' were true, the

communicative principles would have forced s to utter p'' instead of p.6 Furthermore, if s

knows the truth value of p, and of p', then s knows the truth value of p''. So, the same

reasoning process leads to the conclusion that s knows that p'' is false. The assumption

that the speaker knows whether or not p' is true, thus, leads to a contradiction and must,

therefore, be dropped.

This problem was dubbed the symmetry problem in class notes of Kai von Fintel and

Irene Heim. Whenever p is uttered, and fails to settle the truth value of a relevant

proposition, q, there will be two symmetrical ways of settling it, leading necessarily to an

Ignorance Inference. Stated somewhat differently, p is equivalent to the following

disjunction (p ∧ q) ∨ (p ∧ ¬q), and therefore should lead to Ignorance Inferences parallel

to those stated in (2)c.

1

The Griceians respond to this problem by a revision of the Maxim of Quantity.

Specifically, they suggest that the maxim doesn’t require speakers to utter the most

informative proposition that is relevant to the topic of conversation, but is more limited in

scope. The Maxim merely requires speakers to choose the most informative relevant

proposition from a formally defined set of alternatives. It does not require speakers to

consider all relevant propositions.

The assumptions made here about “relevance” are the following:

1. If p and q are both relevant, so is “p and q”.

2. if p is relevant, so is “not p”. (To say that p is relevant is to say that the question Is p true or false?

relevant.)

6 It is sometimes suggested that Grice’s Maxim of Manner (M) could be used to explain s’s avoidance of p.

Such a suggestion requires an ordering of linguistic expressions by which p' would be more optimal than p''

from M’s perspective. For arguments against obvious orderings (various measures of complexity), see

Matsumoto (1995), as well as Fox and Hackl (2005).

5

The common way to work this out, pioneered by Larry Horn (1972), starts out with

the postulation of certain sets of lexical items, Scalar Items, and sets of alternatives to

which the scalar items belong, which we will call Horn-Sets:7

Examples of Horn-Sets

a. {or, and}

b. {some, all}

c. {one, two, three,…}

d. {can, must}

These sets of lexical alternatives determine the set of (Horn) alternatives for a sentence

by a simple algorithm. The set of alternatives for S, Alt(S), is defined as the set of

sentences that one can derive from S by successive replacement of Scalar Items with

members of their Horn-Set.8

Alt(S) = {S': S' is derivable from S by successive replacement of scalar items with members of their Horn-Set}

The Maxim of Quantity can now be stated as follows:

Maxim of Quantity (Griceian version):

If S1 and S2 are both relevant to the

topic of conversation, S1 is more informative than S2, and S1∈Alt(S2), then, if the

speaker believes that both are true, the speaker should prefer S1 to S2.

Consider the sentence in (1). The postulated scalar item in this sentence is

disjunction, for which conjunction is lexically specified as the only alternative, (4)a. (1),

thus, has just one alternative (other than (1) itself):

Alt(1) = {(1), Sue talked to John and Fred }

When s utters (1), his addressee, h (for hearer), typically concludes (on the

assumption that s obeys the revised Maxim of Quantity) that s does not know that the

conjunctive sentence in Alt(1) is true, since this alternative sentence is more informative

than s’s utterance, and is typically relevant. If h assumes, further, that s has an opinion as

to whether or not the conjunctive sentence is true, h would conclude that s believes that it

is false. The neo-Griceans, thus, attribute a general tendency to addressees, namely the

tendency to assume that speakers are opinionated. I state Sauerland’s formulation of this

assumption in (8).

Opinionated Speaker (OS): When a speaker, s, utters a sentence, S, the addressee, h,

assumes, for every sentence S'∈Alt(S), that s’s beliefs determine the truth value of S',

unless this assumption about S' leads to the conclusion that s’s beliefs are

contradictory.

Usually called Horn-Scales for bad reasons, as discussed in Sauerland 2004.

This definition, which comes from Sauerland (2004), is implicit in much earlier work, and is of course

very similar to the definition of alternative sets in Rooth (1985).

Under the basic version of the Maxim of Quantity in (3), B-MQ, there was no way to maintain the assumption that the speaker is opinionated about any relevant sentence S' (not entailed by S). To repeat, B-MQ derived the symmetric results (a) that the speaker does not know that S and S' is true, and (b) that the speaker does not know that S and not S' is true. This, together with the assumption that the speaker knows that S is true (Quality), derived the conclusion that the speaker is not opinionated about S'.

By contrast, under the Neo-Gricean version of the Maxim of Quantity in (6), NGMQ, the assumption that the speaker is opinionated about various sentences (not entailed by S) is innocuous. NG-MQ does not always derive the inference that the speaker does not know that S and S' is true. It derives such an inference only when S and S' (or some equivalent sentence) is a member of Alt(S). Under such circumstances, the speaker could be opinionated about S' as long as Alt(S) does not have S and not S' as a member (nor some equivalent sentence). If S and not S' is not a member of Alt(S), NG-MQ does not derive the inference that the speaker does not know that S and not S' is true, and the

assumption that the speaker believes that S and S' is false could be made consistently.

To summarize, assume that a speaker s utters the sentence in (1), Sue talked to John or Bill. The addressee, h, assumes that s obeys the Maxim of Quality as well as the revised Maxim of Quantity, NG-MQ. Based on this assumption, h reasons in the following way:

Given NG-MQ, there is no X∈Alt(1), such that X is logically stronger than (1), and s thinks that X is true.

Alt(1) contains the conjunctive sentence Sue talked to John and Bill, which is logically stronger than s’s utterance. Hence, given 1, it’s not the case that s thinks that this conjunctive sentence is true.

3. Given OS, the default assumption is that s has an opinion as to whether Sue talked to John and Bill is true or false. Given 2 (the conclusion that it’s not the case that s thinks that the sentences is true), we can conclude that s thinks that

it is false.

So, by modifying the set of assumptions that derive Ignorance Inferences (replacing B-MQ with NG-MQ) one can account for the SI in (2)b. I would like at this point to discuss a possible alternative that keeps B-MQ in tact but instead enriches the set of syntactic representations available for (1). But it is worth pointing out first that, as things stand right now, our account of the Ignorance Inferences in (2)c is in jeopardy.

Specifically, it is incompatible with NG-MQ and our assumption in (4)a about the Horn-Set for disjunction. The account was crucially dependent on the assumption that the Maxim of Quantity would prefer the utterance of a disjunct to the utterance of a disjunction, an assumption incompatible with the way Alt(1) is defined on the basis of (4)a. One might respond to this problem with an independent (pragmatic) account for (2)c (or by enriching the Horn-Set for disjunction (Sauerland 2004). The latter will be discussed in greater detail later.

The alternative syntactic approach that I would like to defend is guided by the intuition that a principle of language use (such as the Maxim of Quantity) should not be sensitive to the formal (and somewhat arbitrary) definition of Alt(S).

If this intuition is correct, BMQ is to be preferred to NG-MQ.

However B-MQ derives Ignorance Inferences that contradict attested SIs. Therefore, if B-MQ is correct, something else is needed to derive SIs. More specifically, SIs must be derived from the basic meaning of the relevant sentences; otherwise the symmetry situation would yield unwanted Ignorance Inferences. The syntax of natural language has a covert operator which is optionally appended to sentences, and that this operator is responsible for SIs.

The guiding observation is that there is a systematic way to state the SI of a sentence, using the focus sensitive operator only. Consider the sentence in (9), which has the SI that

John didn’t buy 4 houses.

John bought three houses.

This SI could be stated explicitly using the focus sensitive particle only in association

with the numeral expression three.

John only bought THREE houses.

This observation extends to all SIs; SIs can always be stated explicitly with the focus

sensitive particle only, as long as the relevant scalar items bear pitch accent:

John did some of the homework (+> John only did SOME of the homework).

For all of the alternatives to ‘some’, d,

if the proposition that John did d of the homework is true,

then it is entailed by the proposition that John did some of the homework.

John talked to Mary or Sue (+> John only talked to Mary OR Sue).

For all of the alternatives to ‘or’, con, if the proposition that John talked to Mary con Sue is true

then it is entailed by the proposition that John talked to Mary or Sue.

This consideration would be weakened significantly if one could make sense of Alt(S) from the

perspective of a general theory of language use. For efforts along these lines, See Spector (2005).

Sentence that generates SIs usually contain scalar items,11 and in such cases it is always possible to state the SIs explicitly, by appending the operator only to the sentence and placing focal accent on the relevant scalar item:

The only implicature generalization (OIG): A sentence, S, as a default, licenses the implicature that (the speaker believes) onlyS', where S' is a modification of S with focus on scalar items.

From the Gricean perspective discussed in 1.2., it is pretty clear why (9)-(12) should obey the OIG.

As is commonly assumed, and as indicated by the paraphrases, the role of only is to eliminate alternatives. Furthermore, when focus is placed on scalar items, the relevant alternatives are precisely the Horn-alternatives that NG-MQ refers to.

However, the OIG suggests yet another possibility, namely that B-MQ is the right conversational maxim and SIs are derived within the grammar, namely by of a covert exhaustivity operator with a meaning somewhat a kin to that of only. Assume, for the moment, the semantics for only suggested by the paraphrases in (9)-(12). Specifically, assume that only combines with a sentence (the prejacent), p, and a set of alternatives, A, (determined by focus). The result of this combination is a sentence which presupposes that p is true and furthermore asserts that every true member of A is already entailed by p (i.e., that all non-weaker alternatives, all “real alternatives”, are false):

[[only]] (A)(pst) = λw: p(w) =1. ∀q∈NW(p,A): q(w)
=012

NW(p,A) = {q∈A: p does not entail q}

The exhaustivity operator, exh,
should mean the same, with one small modification. While with only the
requirement that the prejacent be true is a presupposition, with exh

this requirement should be part of the assertive component:

[[Exh]]
(A)(pst)(w) ⇔ p(w) & ∀q∈NW(p,A): ¬q(w)

Assume that natural
language has exh as a covert operator. Assume, further, that this operator
can append to a sentence, S, thereby yielding a stronger sentence S+
= Exh(Alt(S))(S).13 It is easy to see that such a representation would derive
both the “basic meanings” and the SIs of the sentences in (9)-(12).

This
would allow us to keep to the non-stipulative quantity maxim (B-MQ). The
cost lies, of course, in the stipulation of exh. There is a clear trade-off
here, one that suggests that no decision will be justifiable on a-priori
grounds. The goal of this paper is to

This statement is not always true,
nor is it predicted to be. Scalar Items generate alternatives,
but

alternatives could be specified in other ways as well: by pitch-accent or by an explicit question. See the discussion below.

λχ:ψ(χ).φ is a function defined only for objects
of which ψ is true.

This
assumption is most natural if Alt(S) is the focus value of S, which could follow
if scalar items are assumed to be inherently focused, see Krifka (1995) for a
possible implementation.

provide an empirical argument in favor of a theory in which B-MQ is at the heart of pragmatic reasoning and exh is responsible for SIs.

If such a theory is correct, we might think of exh as a
syntactic device designed (“by a super-engineer”) to facilitate communication
in a pragmatic universe governed by B-MQ.

Every conversational situation, C,
can be characterized by a set of sentences that are relevant at C, QC (for
question). An utterance of S at C will be associated with a set of Ignorance
Inferences determined by the set of sentences QS,C⊆QC, whose truth value
is not determined by S. B-MQ will derive the set of Ignorance Inferences that
correspond to QS,C, i.e., I-INF(S,C) = {¬Ksϕ: ϕ∈ QS,C}. Furthermore, if ϕ is
relevant to the topic of conversation, it seems that same would be true of ¬ϕ
(see note 5). Hence I-INF(S,C) = {¬Ksϕ: ϕ∈ QS,C} = {¬Ksϕ and ¬Ks¬ϕ : ϕ∈
QS,C}

Sometimes the set of Ignorance Inferences will be implausible and this
would motivate a new parse of the linguistic stimuli, one that involves an
exhaustive operator on top of S, i.e., S+. If no further stipulations are
added, the procedure should be able to apply recursively leading to S++,
S+++, etc.

This possibility will empirically distinguish our
syntactic perspective from the Neo-Gricean alternative.

There's core empirical phenomena to motivate exh,
namely the conjunctive interpretation of disjunction under existential modal
constructions (free choice, FC).

Furthermore, I will present an argument due to Kratzer and Shimoyama (2002) (K&S) and Alonso-Ovalle (2005), that FC should be derived by the system that derives SIs. In section 3, I will present evidence that FC arises in additional circumstances: when disjunction is embedded under certain other existential quantifiers and when conjunction is embedded under negation and universal quantifiers (under the sequence ¬∀).

There's a relevant observation about disjunction due to Chirchia, and in section 5 I will present the Neo-Gricean

response to
Chierchia’s observation and its failure to predict
FC phenomena. Finally, in section 6-10, a resolution is proposed based on
recursive exhaustification which extends to the phenomena discussed.

Consider the sentence in (16) when uttered by someone who’s understood to be an authority on the relevant rules and regulations, for example, a parent who is accustomed to specifying limits pertaining to the consumption of sweets.

You may love one another or die.

In such a context, the utterance would be an immediate inference.

This
inference, sometimes referred to as an inference of "free choice permission",
is not an expected

entailment of obvious candidates for the logical form of the utterance

You may love one another and you may die.

For example, it is not entailed by a plausible logical form

MAY [[we love one another] or [we die]]

What the utterance states is that the relevant rules do not prohibit the
disjunctive sentence (the

complement of allowed), or, in the terms of possible-world semantics, that there is a world consistent with the rules in which one of the disjuncts is true.

The utterance is thus equivalent to this below, which is
clearly weaker:

We may love one another or we may die.

The problem is to understand how a disjunctive logical form can be strengthened to yield the conjunctive inference, and this may have motivated the change in Auden.

In other words, we need to understand how a sentence that should
receive the modal logic formalization in (20)a – which is equivalent to (20)b
– justifies the FC inference in (20)c.

◊(p ∨ q)

◊p ∨ ◊q

◊p ∧ ◊q

Downward entailing operators, evidence
that Free Choice is an Implicature.

Free choice effects have been that arise when certain indefinite expressions are embedded

under existential modals, and presented a fairly strong argument that the effect should be

derived by the system that yields SIs.

The argument, which has been elaborated
and extended to disjunctive constructions is based
on the

observation that there are no traces of free-choice in certain downward entailing contexts.

If free-choice were to follow from
the basic meaning, we would expect the utterance to have a fairly weak
meaning.

It should be able to assert that no one is both allowed to love one another and allowed to die.

We would thus predict this to
be true in a situation in which everyone is allowed to one of the two things but no one has free-choice, i.e., in situations in which no one is
allowed to

decide what to do.

Such an interpretation, if
available, is extremely dispreferred.

To derive the natural interpretation, we must factor out whatever is responsible for free-choice.

The
subject is reconstructed into both disjuncts, and to is omitted.

This is done
for expository purposes and, of course, does not affect meaning.

Although Auden focuses on disjunction, the basic proposal can derive free-choice for
the relevant indefinites as long as we assume that the relevant indefinites have the following alternatives:
ALT(irgendein NP)= { irgendein NP': NP'⊆ NP}∪{all NP': NP'⊆ NP}.

Some think that the interpretation is available in contexts that, more generally, allow for “intrusive”

implicatures of the relevant sort.

None of my students did
SOME of the homework, They all did ALL of it.

No one is allowed to eat the cake OR the ice-cream. Everyone will be told what to eat.

No one is allowed to eat the cake or the ice-cream.

negation of free choice:

¬∃x[◊P(x) ∧ ◊Q(x)]

b

negation of standard meaning:

¬∃x◊(P(x) ∨ Q(x))

The natural interpretation is expected if free choice were to be derived as an implicature.

Although it is not yet clear how to derive free choice as an implicature, it is clear that if a derivation were available for the basic case, it would, nevertheless, not be available (at least not necessarily) for the other utterance.

This is seen most clearly under the Griceian approach to implicature.

Under this approach, an implicature is derived as a pragmatic strengthening of the basic meaning of a sentence.

The meaning is weaker than the basic meaning in and, therefore, can NOT be derived along Griceian lines.

Under the syntactic alternative, the
preference for (21)'a would be stated as a preference for stronger
interpretation (See Chierchia 2004, 2005). More specifically, assume,
contrary to what you might think at this point, that an exhaustive operator
can somehow derive the basic FC effect. We might then suggest that exh can
only be introduced if the overall result is a stronger proposition. This
could be motivated by the observation that as propositions get stronger fewer
Ignorance Inferences are derived by B-MQ. We might, thus, suggest that the
introduction of exh is subject to an economy

condition related to its functional motivation, namely to the elimination of Ignorance Inferences. (I.e., a sentence with exh must lead to fewer Ignorance Inferences than its counterpart without exh, see footnote 37.) Alternatively, we might suggest, in line with the neo-Griceans, that exh must be introduced in matrix position.

Be that as it may, it is reasonable to assume that the preference
for (21)'b would be predicted if FC could be derived as an SI, but not
otherwise. Quite independently of particular proposals, it is well known that
SIs tend to disappear in downward entailing environments (Gazdar 1979,
Chierchia 2004). The fact that FC appears to share this property with SIs
seems to be a good incentive to search for a theory that would derive the
effect as an SI.

The projection properties of FC seem to suggest that the effect should be derived as an SI. But how could we derive such a result? K&S made a very interesting proposal which is the basis for the proposal that I will make in section 6-10. But this will require quite a bit of ground work.

However, Kamp points out that it is easier for FC to “intrude” into the antecedent of a
conditional: If

you are allowed to eat the cake or the ice-cream, you are pretty lucky. If K&S are correct, the FC interpretation of the antecedent would require an analysis involving an embedded implicature but without the “meta-linguistic” feel that is sometimes associated with such intrusion. Unfortunately, I have nothing interesting to add.

The latter possibility seems less plausible given the accumulation of
evidence for “intrusive” or

“embedded” implicatures (at least in non-downward entailing environments), see Chierchia 2004. The discussion is of course, problematic, and more so from the Gricean perspective.

At this point, it is worth understanding what one might say in order to

derive SIs based on the neo-Gricean maxim of quantity (NG-MQ).

Quite generally, suppose that ϕ is the basic meaning of a sentence, S, and that our

goal is to derive a stronger meaning, ϕ', based on NG-MQ. The result could be achieved

if we proposed a set of alternatives of the following sort: ALT(S) := {S, S & not Sϕ' },

where the meaning of Sϕ' is ϕ'. If s was to utter S, the addressee, h, would conclude, based

on NG-MQ, that s does not believe that not Sϕ' is true. Furthermore, based on the

assumption that s is an opinionated speaker, h, would conclude that s believes that Sϕ' is

true.

More specifically, suppose that the alternatives of the sentence in (16), repeated

below, are the sentences in (22). This is slightly different from the general scheme for

deriving implicatures characterized above, but the basic idea is the same.18

(16) You’re allowed to eat the cake or the ice-cream.

(22) Alternatives needed to derive FC for (16) based on NG-MQ:

a. You are allowed to eat the cake or the ice-cream.

b. You are allowed to eat the cake but you are not allowed to the ice-cream

c. You are allowed to eat the ice cream but you are not allowed to eat the cake.

Based on NG-MQ, we would now derive the SI that (22)b and (22)c are both false,

which, together with (22)a, yields the FC inference. To see this, assume (22)a is true.

Now assume that one of the conjuncts in (17) is false, say that you cannot eat the icecream.

From this it follows that (22)b is true, contrary to the SI.

But of course this is not intended as a serious proposal. It follows from a general

algorithm that allows us to derive, on a case by case basis, any SI that we would like to,

and, hence, does not explain the particular SIs that are actualized (See Szaebo 2003). The

obvious way to turn this into a serious proposal is to show that the alternatives in (22) are

needed on independent grounds.

K&S, and in particular Alonso-Ovalle, propose a more natural set of alternatives,

namely the one in (23).

(23) Alternatives proposed by K&S/Alonso-Ovalle:

a. You are allowed to eat the cake or the ice-cream.

b. You are allowed to eat the cake.

c. You are allowed to eat the ice cream.

The set of alternatives in (23), in contrast to the one in (22), is consistent with a general

constraint on alternatives proposed in Matsumoto (1995).19 Furthermore, as we will see in

section 4, there is independent evidence for the type of Horn-Sets that would derive (a

18 It would instantiate the scheme if b and c were replaced by a single alternative, namely the disjunction of

the two.

19 Matsumoto argues that lexical items can be members of the same Horn-Set only if they denote functions

of the same monotonicity. ∨ is upward monotone with respect to both arguments, but ∧¬ is downward

monotone with respect to its right-hand argument. Skipping ahead to Sauerland’s Horn-set, L and R are

upward (as well as downward) monotone with respect to their immaterial arguments.

13

super-set of) the alternatives in (23). The problem, however, is that NG-MQ can not

derive the FC effect on the basis of (23). In fact, as we will see in greater detail in section

4.2., it derives Ignorance Inferences that directly conflict with FC.

K&S suggest, however, that FC should be derived from (23) based on a novel

principle, which they call anti-exhaustivity. When h interprets s’s utterance of (23)a, s

needs to understand why it is that s preferred this sentence to any of the alternatives. The

standard Neo-Gricean reasoning, which relates to the basic meaning of the alternatives,

would lead to the conclusion that s does not know/believe that any of the alternatives is

true. K&S, however, suggest that h might reason based on the strong meaning (basic

meaning + implicatures) of the alternatives. Specifically, K&S suggest that h would

attribute the choice of s to the belief that the strong meanings of (23)b and (23)c are both

false. Furthermore they assume that the strong meaning of (23)b and c is the basic

meaning of (22)b and c respectively.

As pointed out by Aloni and van Rooy (2005), this line of reasoning raises a question

pertaining to simple disjunctive sentences, such as (1). We would like to understand why

such sentences don’t receive a conjunctive interpretation via an anti-exhaustivity

inference of the sort outlined above. If each disjunct is an alternative to a disjunctive

sentence, why doesn’t the speaker infer that the exhaustive implicature of each disjunct is

false?

K&S provide an answer this question by postulating a covert modal operator for any

disjunctive sentence. I will not go over this proposal and the way it might address Aloni

and van Rooy’s objection. I would like, instead, to raise another challenge to K&S’s

basic idea. I think it is important to try to understand how the anti-exhaustive inference

fits within a general pragmatic system that derives Ignorance Inferences (as well as SIs).

Specifically, I think it is important to understand why NG-MQ does not lead to the

Ignorance Inferences in (24) (see section 4.2. for details).

(24) Predicted inferences of (16), based on (23) and NG-MQ:

a. s doesn’t know whether or not you can eat the cake.

b. s doesn’t know whether or not you can eat the ice cream.

This is a challenge that this paper attempts to meet. The idea, in a nut-shell, is to

eliminate NG-MQ in favor of the non-stipulative alternative B-MQ. However,

understanding how this is to work requires the introduction of a proposal made in

Sauerland (2004), which would be extracted from its neo-Gricean setting in order to meet

our goals. But before I get there, I would like to introduce an additional challenge.

Specifically, I would like to present a few other surprising inferences that are intuitively

similar to FC, and should, most likely, be derived by the same system.

3. Other Free Choice Inferences

In this section we will see effects that are very similar in nature to FC, but arise in

somewhat different syntactic contexts. These effects will argue for a fairly general

explanation of the basic phenomenon, one that is not limited to modal environments or to

disjunction.

FC Under negation and universal modals

Consider the sentence in (25) when uttered by someone who is understood to be an

authority on the relevant rules and regulations, for example, a parent who is accustomed

to assigning after-dinner chores.

(25) You are not required to both clear the table and do the dishes.

In such a context, (26) would normally be inferred by the addressee.

(26) You are not required to clear the table and you are not required to do the dishes.

This inference seems very similar to the FC inference drawn in (17) based on (16). To see

the similarity, notice that the basic meaning of (25) is predicted to be equivalent to the

disjunctive statement that you are allowed to either avoid clearing the table or evade

doing the dishes, (27)a, and that (26) is equivalent to the conjunction of two possibility

statements. (You are allowed to avoid clearing the table and you are allowed to avoid

doing the dishes, (27)b)

(27)a. Standard Meaning of (25)

¬(p ∧ q) ≡ ◊¬ (p ∧ q) ≡ ◊(¬p ∨ ¬q) ≡ ◊(¬p) ∨ ◊(¬q)

b. Free Choice Inference

◊(¬p) ∧ ◊(¬q)

Just as in (16), the basic meaning does not explain the inference, and the gap is formally

identical. Once again, we have to understand how a sentence that is equivalent to a

disjunctive construction can be strengthened to something equivalent to the

corresponding conjunction.

3.2. More generally under existential quantifiers

In the basic FC permission sentence in (16), disjunction appears in the scope of the

existential modal allowed. Furthermore, as is well-known, FC extends to all constructions

in which or is in the scope of an existential modal:

(28) a. The book might be on the desk or in the drawer.

(= The book might be on the desk and it might be in the drawer)

b. He is a very talented man. He can climb Mt. Everest or ski the Matterhorn.

(=He can climb Mt. Everest and he can ski the Matterhorn.)

What has not been discussed in any systematic way is that this type of conjunctive

interpretation extends also to some non-modal constructions:20

(29) a. There is beer in the fridge or the ice-bucket.

20While working on this paper, I have learned about two new papers about FC that make this same

observation: KIindinst (2005) and Eckardt (this volume).

15(= There is beer in the fridge and there is beer in the ice-bucket.)

b. Most people walk to the park, but some people take the highway or the scenic

route. (Irene Heim, pc attributed to Regine Eckardt, pc)

(= Some people take the highway and some people take the scenic route.)

c. This course is very difficult. In the past, some students waited 3 semesters to

complete it or never finished it at all. (Irene Heim, pc)

(= Some students waited 3 semesters to complete the course and some

students never finished it at all.)

It is thus tempting to suggest that conjunctive interpretations for disjunction are available

whenever disjunction is in the scope of an existential quantifier (with the domain of

quantification, worlds or individuals, immaterial). However, there are limitations:

(30) a. There is a bottle of beer in the fridge or the ice-bucket.

(≠ There is a bottle of
beer in the fridge and there is a bottle of beer in the icebucket.)

c. Someone took the highway or the scenic route.

(≠Someone took the highway and someone took the scenic route.)

d. This course is very difficult. In the past, some student waited 3 semesters to

complete it or never finished it at all.

(≠ some student waited 3 semesters to complete the course and some student

never finished it at all.)

As pointed out in Klindinst (2005), the relevant factor seems to be number marking on

the indefinite. We might, therefore, suggest the following generalization:

(31) Existential FC: A sentence of the form ∃x [P(x) ∨ Q(x)] can lead to the FC

inference, ∃xP(x) ∧ ∃x Q(x), as long as the existential quantifier, ∃x, is not

marked by singular morphology.

3.3. More generally, under negation and universal quantifiers

In (25) we saw an FC effect arising when conjunction is under the scope of negation and

a universal modal (under the sequence, ¬). As illustrated in (32), and stated in (33), the

effect arises also when is replaced by an ordinary universal quantifier:

(32) We didn’t give every student of ours both a stipend and a tuition waiver.

1. basic meaning: ¬∀x[P(x) ∧ Q(x)] ≡

∃x¬ [P(x) ∧ Q(x)] ≡

∃x [¬P(x) ∨ ¬Q(x)] ≡

∃x ¬P(x) ∨ ∃x¬Q(x)

2. Free Choice: ∃x ¬P(x) ∧ ∃x¬Q(x)

(33) Conjunctive FC: A sentence of the form ¬∀x[P(x) ∧ Q(x)] can lead to the FC

inference ∃x ¬P(x) ∧ ∃x¬Q(x).

16

In section 7-10 we will provide an account of our two generalizations ((31) and (33))

within a general theory of SIs that we will introduce in section 6, based on a discussion,

in section 4-5, of Sauerland’s approach to SIs. But before we move on, it is important to

rule out an alternative explanation of (25) and (32), in terms of wide scope conjunction.

To understand the concern, focus on (25). One might think that this sentence has a logical

form in which conjunction takes wide scope over the sequence ¬. If such a logical form

were available, the inference in (26) would follow straightforwardly from the basic

meaning, and would thus be unrelated to the FC effects that are distributed according to

(31).

But wide scope conjunction is not a probable explanation. One argument against such

an explanation is based on the sentences in (34). If conjunction could take scope over the

sequence ¬ in (25) (and over ¬∀ in (32)), we would expect it to be able to outscope

negation in (34), an expectation that is not born out.21

(34) a. I didn’t talk to both John and Bill.

b. We didn’t give both a stipend and a tuition waiver to every student.

What I think we learn from (34) is that conjunction can appear to outscope negation only

when a universal quantifier intervenes.22 This is expected if conjunction never outscopes

negation, and the generalization in (33) is real.

Another argument against wide scope conjunction comes from an additional inference

we draw from sentences such as (25) and (32). In both cases we draw the inference that

the alternative sentence with disjunction instead of conjunction is false. That is, we would

tend to draw (35)a as an inference from (25), and (35)b from (32). These inferences are

not expected if conjunction receives wide scope, but, as we will see later on, are expected

if the phenomenon is derived along with other FC effects.

(35) a. You are required to clear the table or do the dishes.

b. We gave every student of ours a stipend or a tuition waiver.

4. Chierchia’s Puzzle

The account of FC that I will develop will be based on a modification of a proposal made

in Sauerland (2004) to deal with a puzzle discovered in Chierchia (2004).23 To

understand the puzzle in greater detail, consider first (36) and its implicature that (36)' is

false.

(36) John did some of homework.

21 If the distributor both is omitted the resulting interpretation is equivalent to wide scope conjunction. The

correct account relies most likely on a “homogeneity” presupposition (Fodor 1976, Gajewski 2005).

22 In order to account for the difference between (32) and (34)b, we would also have to say that inversion of

the surface scope of conjunction and universal quantification is impossible in (34)b, a consequence of

Scope Economy, a principle I’ve argued for in Fox (2000).

23 See also Lee (1995), and Simons (2002).

(36)' John did all of the homework.

As outlined in 1.2., this implicature can be derived by NG-MQ under the assumption that

some and all are members of the same Horn-Set, (4)b,24 from which it follows that (36)'

is an alternative to (36). NG-MQ, together with the assumption of an opinionated

speaker, lead to the conclusion that the speaker believes that (36)' is false.

Consider next what happens when (36) is embedded as one of two disjuncts:

(37) John did the reading or some of homework.

This type of embedding was presented by Chierchia (2004) as a challenge to the neo-

Gricean derivation of implicatures. As Chierchia points out, (37)' should be an

alternative to (37), and it would therefore seem that (with the assumption that the speaker

is opinionated) we should derive the implicature that (the speaker believes that) (37)' is

false.

(37)' John did the reading or all of homework.

This implicature, however, is clearly too strong. If a disjunctive sentence is false, then

each of the disjuncts is false. When (37) is uttered, we do derive the inference that the

second disjunct of (37)' is false. However, we clearly do not derive a similar inference for

the first disjunct (which is also the first disjunct of (36)).

Chierchia’s challenge for the Neo-Griceans is to avoid the implicature that the first

disjunct of (36) is false while at the same time to derive the implicature that the stronger

alternative to the second disjunct is false:

(38) Let U be an utterance of p or q where q has a stronger alternative, q'.

a. Problem 1: to avoid the implicature of ¬p

b. Problem 2: to derive the implicature of ¬q'

Chierchia provides an account for the relevant generalization based on a recursive

definition of strengthened meanings. I will not discuss his account, since I can’t figure

out how to extend it to FC. Instead, I will discuss the neo-Gricean alternative, which also

fails to account for FC, but, which can, nevertheless, be modified in order to provide a

syntactic (non-Gricean) alternative that successfully extends to FC.25

5. Sauerland’s Proposal26

As pointed out at the end of section 1.2., the Horn-Set for disjunction in (4)a ({or, and})

cannot account for the Ignorance Inferences that are attested when a simple disjunctive

24 and that the set does not include the “symmetric alternative” to all, some but not all.

25 Chierchia himself developed an account of FC which is quite similar to the account proposed here and is

to some extent independent of his recursive procedure for implicature computation. Specifically, his

account, like mine, is based on operators that apply to a prejacent and a set of alternatives. However, the

crucial operator for him is an “anti-exhaustivity” operator, distinct from what might be responsible for

implicature computation.

26 Benjamin Spector made the same proposal, in a somewhat different (more generalized) format. A related

proposal can be found in Lee (1995).

18

sentence such as (1) is uttered. Sauerland suggests a remedy for this problem which also

resolves Chierchia’s puzzle.

To derive the appropriate Ignorance Inferences for (1), Sauerland suggests that the

alternatives for a disjunctive statement include each of the disjuncts in addition to the

corresponding conjunction:

p

(39) Alt(p∨q)= p∨q p∧q

q

These alternatives, which are plotted to represent logical strength,27 derive (based on NGMQ)

the following inferences with respect to a speaker, s, who utters p or q, inferences

which Sauerland calls Primary (or weak) Implicatures, PIs:

(40) PIs for p or q (based on NG-MQ)

a. s does not believe that p is true.

b. s does not believe that q is true.

c. s does not believe that p and q is true. Already follows from both a and b.

Given that s is assumed to believe that her utterance of p or q is true (Quality), we

derive the Ignorance Inferences discussed in section 1, that is, for each disjunct, we

derive the inference that the speaker does not know whether or not it is true. To derive

SIs, the principle of an Opinionated Speaker is employed, (8):

(8) Opinionated Speaker (OS): When a speaker, s, utters a sentence, S, the addressee, h,

assumes, for every sentence S'∈Alt(S), that s’s beliefs determine the truth value of S',

unless this assumption would lead to the conclusion that s’s beliefs are contradictory.

This principle asks us to scan the set of alternatives that are stronger than S, and to

identify those for which the assumption that the speaker is opinionated is consistent with

our prior inferences based on Quality and NG-MQ. For each such alternative, the speaker

is assumed to be opinionated, and given the relevant PI, a stronger inference is derived,

namely that the speaker believes that the relevant alternative is false, an inference which

Sauerland calls a Secondary Implicature (an SI, conveniently).

As mentioned above, (40)a,b together with Quality, lead to ignorance with respect to

p and to q. Hence, p and q is the only alternative for which the assumption that the

speaker is opinionated is consistent with prior inferences. Therefore, only one SI is

derived based on OS, namely the inference that the speaker believes that p and q is false:

(41) SI for p or q (based on OS)

s believes that p and q is false.

Sauerland, thus, derives the following definition for the two relevant sets of implicatures:

(42) When a speaker s utters a sentence A, the following implicatures are derived:

27 If x is to the left of y with a connecting line, then x is weaker than y.

19

a. PIs = {¬Bs(A'): A'∈ ALT(A) and A' is stronger than A}

b SIs = {Bs(¬A'): A'∈ ALT(A), A' is stronger than A, and

Bs

(A)∧∩PI ∧ Bs(¬A') is not contradictory}

Based on these definitions, a PI is derived for every alternative stronger than the assertion

and an SI for a subset of the stronger alternatives for which an Ignorance Inference hasn’t

already been derived (based on NG-MQ and Quality):28

(43) Implicatures for p∨q:

p

Alt(p∨q)= p∨q p∧q

q

a. PIs: ¬Bs(p), ¬Bs(q) The rest, ¬Bs(p∧q), follows

b SI: Bs¬(p∧q)

Sauerland shows that this rather principled approach solves Chierchia’s puzzle once

the lexical alternatives that derive the sentential alternatives in (39) are specified. The

basic intuition is fairly straightforward. An utterance of p or q derives Ignorance

Inferences that are inconsistent with the assumption that the speaker is opinionated about

p, thereby solving problem (38)a. Problem (38)b is solved as well, but seeing this

requires precision about the relevant lexical alternatives and the way they determine

sentential alternatives for complex disjunctions, such as (37).

The starting point is the observation that in order to derive (39) the alternatives for

disjunction must contain two lexical entries that are never attested:29

(44) Horn-Set(or) = {or, L, R, and}, where pLq = p and pRq = q.

These four alternatives, when combined with the alternatives for some ({some, all}),

yield 8 alternatives to (37), based on (5) above:30

(45) Alt(r or sh) = a. r ∨ sh

28 From now on, I will circle those alternatives for which an SI can be derived consistently with Quality and

NG-MQ.

29 Spector (2003, 2005) suggests a different perspective. Specifically, he suggests that alternative sets are

defined as the closure under ∧ and ∨ of the set of positive answers to a given question. The Sauerland

alternatives for John talked to Mary or Bill would, thus, be derived (along with other useful alternatives) if

the relevant question was who did John talk to?

What FC teaches us, if my proposal is correct, is that there is no closure under ∧. Some of what I say

could work if Sauerland’s alternatives were replaced by basic answers to a Hamblin-question closed under ∨ .

Conjunction in unembedded cases will be part of the basic Hamblin
denotation, derived from

quantification over pluralities). One would still have to make sense of second layers of exhaustivity (see

section 11.2 note 46.

30 r := John did the reading; sh := John did some of the homework; ah := John did all of the homework.

b. r L sh ≡ r

c. r R sh ≡ sh

d. r ∧ sh

e. r ∨ ah

f. r L ah ≡ r

g. r R ah ≡ ah

h. r ∧ ah

To see what PIs and SIs are derived, it is useful to plot the alternatives in a way that

indicates relative strength. But it is already easy to see how the two problems in (38) are

solved. To repeat, problem (38)a is solved based on the observation that the speaker

cannot believe that r is false if a PI ensures that she does not believe that sh is true and

Quality ensures that she believes that r or sh is true. Problem (38)b is solved based on the

observation that ah is a member of the alternative set (alternative g), and that an SI can be

derived for this alternative (consistent with prior inferences):

(46) Implicatures for r∨ sh:

ALT(r∨ sh) =

r

r∨ah r∧ah

r∨sh ah

r∧sh

sh

PI = ¬Bs(r∨ah), ¬Bs(sh), (the rest follow)

SI = Bs(¬ah), Bs(¬(r∧sh)) (the rest, Bs¬(r∧ah), follows)

The Horn-Set in (44) plays two independent roles for Sauerland. It provides NG-MQ with

the alternatives needed to derive the Ignorance Inferences for p∨q. These inferences

explain (within the Neo-Gricean paradigm) the lack of certain SIs when scalar items are

embedded within one of the disjuncts (problem (38)a). Furthermore, given (5), we can

generate alternatives for complex disjunctive sentences (e.g., q' for the sentence in (38))

that derive otherwise surprising SIs (problem (38)b).

However, the system makes a further prediction. Specifically, it predicts that in

certain contexts the two basic alternatives p and q will generate SIs rather than Ignorance

Inferences. The relevant case involves embedding of disjunction under an upward

monotone operator O such that O(p∨q) does not entail the disjunctive sentence

O(p)∨O(q). For such an operator, the following is not contradictory.

(47) O(p∨q) ∧ ¬O(p) ∧ ¬O(q) ∧ ¬O(p∧q)

Hence, if s utters O(p∨q), an SI would be generated for each of the stronger alternatives

(O(p), O(q), and O(p∧q)).

Evidence that this prediction is correct comes from (48) and (49), which naturally

yield the implicatures in (a) and (b).31

(48) You’re required to talk to Mary or Sue.

Implicatures:

a. You’re not required to talk to Mary.

b. You’re not required to talk to Sue.

(49)
Every friend of mine has a boy friend or a girl friend.

Implicatures:

a. It’s not true that every friend of mine has a boy friend.

b. It’s not true that every friend of mine has a girl friend.

These facts follow straightforwardly from the Sauerland scale:

∀xP(x)

(50) Alt(∀x(P(x)∨Q(x))= ∀x(P(x)∨Q(x)) ∀x(P(x)∧Q(x))

∀xQ(x)

PIs = ¬Bs(∀xP(x)), ¬Bs(∀xQ(x)) (the rest, ¬Bs∀x(P(x)∧Q(x)), follows)

SIs = Bs(¬∀xP(x)), Bs(¬∀xQ(x)) (the rest, Bs ¬∀x(P(x)∧Q(x)), follows)

5.2. But…what about FC?

Sauerland’s system makes yet another prediction about disjunction embedding, a

prediction which is in direct conflict with FC. If disjunction is embedded under an

upward monotone operator O such that O(p∨q) entails the disjunctive sentence

O(p)∨O(q), the system predicts Ignorance Inferences with respect to O(p) and O(q). The

reasoning is exactly identical to the basic case of unembedded disjunction: there is no

way to assume that the speaker is opinionated about one of the alternatives O(p) and O(q)

without contradicting the Primary Implicature that the speaker does not know that the

other disjunct is true (given Quality).

This does not seem to be the correct prediction for existential modals and plural

existential DPs (generalization (31)). These operators, under their basic meaning, are both

In Fox (2003) Fox points out this prediction, but was not sure about the empirical facts. He was convinced by conversations with Benjamin Spector and the discussion in Sauerland (2005).

commutative with respect to disjunction (◊(p∨q) ≡(◊p∨◊q); ∃x(P(x)∨Q(x)) ≡ ∃xP(x)∨

∃xQ(x)). Hence Ignorance Inferences are predicted.

(51) You may eat the cake or the ice-cream.

◊p

Alt(51)= ◊(p ∨ q) ◊(p∧q)

◊q

PIs = ¬Bs(◊p), ¬Bs(◊q), ¬Bs◊(p∧q)

SIs = Bs¬ ◊(p∧q)

We’ve seen good arguments that FC should be derived as an implicature. However, under

Sauerland’s system we derive Primary Implicatures (in bold) that contradict FC.

The situation is quite interesting. The alternatives that K&S and Alonso-Ovalle

appeal to in order to derive FC derive contradictory Ignorance Inferences in Sauerland’s

system. If K&S are right, Sauerland’s system needs to change. However, K&S’s insight,

if correct, needs to be embedded in a general system for implicature computation, one

that can account for Chierchia’s puzzle, as well as for the emergence of Ignorance

Inferences.

There are two problems with Sauerland’s system. On the one hand, it derives Ignorance

Inferences that directly contradict the attested FC effect. On the other hand, it does not

provide the basis for anti-exhaustivity, which, if K&S are correct, is at the heart of FC. I

will argue that the first problem teaches us that, contrary to the neo-Gricean assumption,

Primary Implicatures do not serve the foundation for the computation of SIs. Instead, SIs

are derived in the syntactic/semantic component via an exhaustive operator, as suggested

in section 1.3. Once a semantic representation is chosen, Ignorance Inferences are

computed by the pragmatic system, based on the non-stipulative maxim of quantity (BMQ).

Without an exhaustive operator, incorrect Ignorance Inferences are computed in FC

environments. However, once we modify the meaning of exh, based on Sauerland’s

insights, the inferences can be avoided by a sequence of two exhaustive operators, which

yield, in effect, anti-exhaustivity, thereby solving the second problem. Furthermore, it

turns out that FC is predicted in all the environments discussed in section 3.

Let’s start by reviewing our lexical entry for only and exh from section 1.3. These entries

(14) and (15), repeated below) derive strong meanings that are in most cases equivalent

to the basic meaning conjoined with the SIs derived by the neo-Gricean system.

(52)a. [[only]] (A)(pst) = λw: p(w) =1.
∀q∈NW(p,A): q(w) =0

NW(p,A) = {q∈A: p does not entail q}

b. [[Exh]] (A)(pst)(w) ⇔ p(w) & ∀q∈NW(p,A): ¬q(w)

23

However, predictions are sometimes different when the alternatives are not totally

ordered by entailment. In particular, for the sets of alternatives that Sauerland has

postulated, the lexical entries in (52) can derive contradictory results. To see this,

consider the following dialogue:

(53) A: John talked to Mary or Sue.

B: Do you think he might have spoken to both of them?

A: No, he only spoke to Mary OR Sue.

Under (52)a, A’s final sentence should presuppose that the prejacent, John spoke to Mary

or Sue, is true and that this is not the case for any of the (non-weaker) alternatives. Thus,

if Sauerland is right about the lexical alternatives for disjunction, the two alternatives in

(54) would both have to be false for the utterance to be true, which would, of course,

contradict the presupposition.

(54) a. John talked to Mary.

b. John talked to Sue.

This is a wrong result, which means that if Sauerland is right about the alternatives

for disjunction, the lexical entries in (52) probably need to be revised.32 A revision of this

sort is also needed based on much older observations due to Groenendijk and Stokhof

(1984):

(55) a. Who did John talk to?

Only Mary or SUE

b. Who did John talk to?

Only Some GIRL

Groenendijk and Stokhof (1984)

Let’s focus on (55)a. If (52)a is correct, the answer to the question should assert that

every alternative not entailed by the prejacent, John talked to Mary or Sue, is false. This

time the set of alternatives consists (most likely) of every proposition of the form John

talked to x based on the denotation of the question, and the fact that the whole DP Mary

or Sue is focused. In other words, the answer to the question in (55)a should entail the

propositions that John didn’t talk to Mary and that he didn’t talk to Sue, which should

contradict the presupposed prejacent.

Groenendijk and Stokhof (1984), who noticed the problem, suggested a modification

to the standard lexical entry for only, which was accommodated in Spector (2005) to the

syntax we are assuming (based on van Rooy and Schultz (2003)):

(56)a. [[only]] (A)(pst) = λw: p(w) =1. Minimal(w)(A)(p)

b. [[Exh]] (A)(pst)(w) ⇔ p(w) & Minimal(w)(A)(p)

32 Gennaro Chierchia (p.c.) points out that in the dialogue in (53) or might be receiving contrastive focus

with conjunction, with the other alternatives (L and R) inactive. This possibility will not be helpful in

explaining the avoidance of a contradiction in Groenendijk and Stokhof’s examples in (55).

Minimal(w)(A)(p) ⇔ ¬∃w'p(w')=1(Aw' ⊂ Aw)

Aω = {p∈A: p(ω)=1}

As pointed out by Spector (again, based on van Rooy and Schultz), this lexical entry can

solve Chierchia’s problem. However, it yields results that contradict FC (see note 41).

For this reason, I would like to suggest an alternative, one that is linked in a very direct

way to Sauerland’s proposal.

What we learn from Groenendijk and Stokhof is that there is something in the

meaning of only “designed” to avoid contradictions: only takes a set of alternatives A and

a prejacent p, and attempts to exclude as many propositions from A in a way that would

be consistent with the requirement that the prejacent be true. I would like to suggest that

the basic algorithm is Sauerland’s, i.e., that propositions from A are excluded as long as

their exclusion does not lead (given p) to the inclusion of some other proposition in A:

(57)a. [[only]] (A)(pst) = λw: p(w)
=1.

∀q∈ NW(p,A) [q is innocently excludable given A q(w) =0]

b. [[Exh]] (A)(pst)(w) ⇔ p(w) & ∀q∈NW(p,A)

[q is innocently excludable given A ¬q(w)]

q is innocently excludable given A if ¬∃q'∈ NW(p,A) [p∧¬q ⇒ q']

To see how this is supposed to work, consider an utterance of the disjunction p or q.

Consider first what happens without an exhaustive operator, under the basic syntactic

representation. Under such a representation, the sentence would assert that the disjunction

is true and would be consistent with the truth of the conjunction (inclusive or). By B-MQ

this would yield a variety of Ignorance Inferences, which might be implausible in a

particular context, and if so, would motivate the introduction of an exhaustive operator,

Exh(Alt(p or q))( p or q).

Under this alternative parse the sentence would assert that the prejacent p or q is true

and that every innocently excludable alternative is false. Assuming the Sauerland

alternatives, we derive the simple ExOR meaning. None of the disjuncts is innocently

excludable, since the exclusion of one will lead to the inclusion of the other, given the

prejacent. Once again, we will circle the innocently excludable alternatives.

p

(58) Alt(p∨q) = p∨q p∧q

q

Excluding p will necessarily include q while excluding q will necessarily include p.

p∧q is thus the only proposition in NW(p∨q, Alt(p∨q)) that can be innocently excluded

given the set of alternatives in (58). Thus, it is the only proposition that is excluded and

the derived meaning is the familiar ExOR.

Before moving to FC, I would like to show how the lexical entries in (57)b replicates

Sauerland results. But even before that, I would like to point out that Sauerland’s

algorithm is not totally contradiction free, and that his assumptions should therefore be

25

modified slightly. This modification would motivate a corresponding modification in

(57).

Consider the question answer pair in (59) from Groenendijk and Stokhof. Assume

that the alternatives for A is the Hamblin denotation of Q, Alt((59)A), in (60).

(59) Q: Who did Fred talk to?

A: Some GIRL

(60) Alt((59)A) = {that Fred talked to x: x is a person or a set of people}

Assume
also that there are more than two girls in the domain of quantification. If all
these

assumptions are correct, it would be possible (by Sauerland’s algorithm) to introduce an

SI of the form Bs¬ϕ, for every ϕ in Alt((59)A). Each SI of this sort is consistent with the

set of PIs and Quality. However, once all the SIs are collected, the result is contradictory.

The problem extends to the lexical entry for exh in (57) (and for only, if we look back at

(55)b). Every member of Alt((59)A) is innocently excludable. Hence, if we were to

append exh to (59)A, the result would be contradictory.

One way to deal with this problem is to assume that the set of alternatives is always

closed under disjunction (see Spector 2005, as well as footnote 29). An alternative, which

is available when exh is assumed, is to eliminate additional elements from the set of

innocently excludable propositions for a prejacent, p, given a set of alternatives A,

I-E(p,A):

(61)a. [[only]] (A)(pst) = λw: p(w) =1.

∀q∈ I-E(p,A) q(w) =0

b. [[Exh]] (A)(pst)(w) ⇔ p(w) & ∀q∈ I-E(p,A) ¬q(w)

I-E(p,A) = ∩{A'⊆A: A' is a maximal set in A, s.t., A'¬ ∪ {p} is consistent}

A¬= {¬ p: p∈A}

To
see if a proposition q is innocently excludable, we must look at every maximal
set of

propositions in A such that its exclusion is consistent with the prejacent. Every such set

could be excluded consistently as long as nothing else in A is excluded. Hence the only

propositions that could be excluded non-arbitrarily are those that are in every one of these

sets (the innocently excludable alternatives). Every proposition which is not in every such

set would be an arbitrary exclusion, since the choice to exclude it, will force us to include

a proposition from one of the other maximal exclusions (if the result is to be consistent),

and the choice between alternative exclusion appears arbitrary.33

To see what results is derived by this lexical entry, it is probably best to go through

the various cases we’ve discussed. Let’s first see how we would exhaustify p∨q given

The proposed mechanism for

exhaustification is reminiscent of what is needed for counterfactuals in the premise semantics developed by

Veltman (1977) and Kratzer (1981). In particular, the set of propositions that can be added as premises to a

counter factual antecedent p is ∩{A⊆C: A is a maximal set in C, s.t., A ∪ {p} is consistent} where C

is the set of all true propositions.

the Sauerland alternatives. The first step would be to identify the maximal consistent

exclusions given the prejacent p∨q. If p is excluded, q must be true and vice versa.

Hence, one maximal exclusion is {p, p∧q}, and the other is {q, p∧q}. The intersection is

p∧q, hence, Exh(Alt(p∨q))(p∨q) = (p∨q) ∧ ¬(p∧q) = p∇q.

p

(62) Alt(p∨q) = p∨q p∧q

q

We circle (with dotted-lines) the maximal exclusions consistent with the prejacent, and we circle

the intersection, the set of innocently excludable alternatives, with a completed line.

Exh(Alt(p∨q))(p∨q) = (p∨q) ∧ ¬(p∧q) = p∇q.

Consider now the exhaustification of (59)A under the assumption that the set of

alternatives is the Hamblin-denotation of the question, (60). Assume that there are three

girls in the domain of quantification, Mary, Sue, and Jane, and that there are no nongirls.

34 Every maximal exclusion will include every member of the Hamblin-set but one

of the following: (m) Fred talked to Mary, (s) Fred talked to Sue, and (j) Fred talked to

Jane. So the intersection of all-maximal exclusions, the set of innocently-excludable

alternatives, is the set of propositions of the form Fred talked to X, where X is a plurality

of girls:

(63) Alt((59)A) = {Fred talked to x: x a person or a set of people} =

m m&s

s m&j m&s&j

j s&j

Exh(Alt((59)A))((59)A) = Fred talked to some girl ∧ ¬(m&s) ∧ ¬(m&j) ∧ ¬(s&j)

= Fred talked to exactly one girl.

Consider again Chierchia’s sentence r∨sh and its Sauerland alternatives. To see which

alternative can be innocently excluded we have to identify the maximal (consistent)

34 Without non-girls, the answer is somewhat strange. That’s probably because questions presuppose that at

least one answer (in the Hamblin sense) is true, and, thus, without non-girls, the answer just repeats the

presupposition. Adding non-girls is thus crucial, but, it is trivial to see that it will not affect the result, in

any interesting way; propositions related to non-girls will be excluded and things will be more difficult to

draw, but other than that, it’s all the same.

exclusions. The set of innocently excludable alternative is the intersection. The reader can

consult the diagram in (64) to see that Sauerland’s results are replicated.

(64) Alt(r∨ sh) =

r

r∨ah r∧ah

r∨sh ah

r∧sh

sh

ah, r ∧sh, r∧ah are the proposition in Alt(ss∨b) that can be innocently excluded given the set of

alternatives:

Exh(Alt(r∨sh))(r∨sh) = (r∨sh)∧¬ ah ∧¬(r∧sh)

Consider next embedding under universal quantifiers. As discussed in section 5.1., such

embedding allows for the consistent exclusion of all the Sauerland-alternatives (other

than the prejacent). Hence, there is only one maximal exclusion, which is excluded by

exh:35

(65) You’re required to talk to Mary or Sue.

Implicatures:

a. You’re not required to talk to Mary.

b. You’re not required to talk to Sue.

(66) Every friend of mine has a boy friend or a girl friend.

a. It’s not true that every friend of mine has a boy friend.

b. It’s not true that every friend of mine has a girl friend.

35In conversation with Gennaro Chierchia, we’ve noticed that things are a little more complicated. As

things stand right now, Alt(∀x(P(x)∨Q(x)) contains additional members: ∃x(P(x)∨Q(x)), ∃x(P(x)), and

∃x(Q(x)). The latter two make it impossible to innocently exclude ∀xP(x) and ∀xQ(x). There are various

simple ways to correct for this problem. The obvious thing that comes to mind is to define Alt(S) so that it

includes only stronger sentences than S. However, this would be a problematic move given data that is not

discussed in this paper. Here’s another possibility: Alt(S) is the smallest set, s.t. (a) S∈ Alt(S), and (b) If

S’∈ Alt(S) and S” can be derived from S’ by replacement of a single scalar item with an alternative, and S’

does not entail S”, S” ∈ Alt(S).

28

∀xP(x)

(67) Alt(∀x(P(x)∨Q(x))= ∀x(P(x)∨Q(x)) ∀x(P(x)∧Q(x))

∀xQ(x)

Exh (Alt(∀x(P(x)∨Q(x)))(∀x(P(x)∨Q(x)))= ∀x(P(x)∨Q(x)) ∧¬∀xP(x) ∧¬∀xQ(x)

So the exhaustive operator as defined in (61)b, based on what’s needed for only, (61)a,

derives the same results as Sauerland’s system (with the exception of cases such as (59)

for which Sauerland’s system can derive contradictory implicatures). This is not

surprising. The set of innocently excludable proposition is (modulo (59)) precisely the set

of propositions for which Sauerland’s system yields SIs – for which SIs can be

introduced innocently.

However, there is an important architectural difference between the two systems, one

that relates to the division of labor between syntax/semantics and pragmatics. Under

Sauerland’s neo-Gricean system, NG-MQ (and the PIs that it generates) is the underlying

source of SIs. Under the syntactic alternative that we are considering, SIs have a syntactic

source, and can serve to avoid Ignorance Inferences, which are computed post

syntactically based on B-MQ. This architectural difference has empirical ramifications

for FC. We’ve already seen that Sauerland’s system predicts Ignorance Inferences that

contradict FC. We will see that under our syntactic alternative, the problem can be

avoided by recursive exhaustification.

Suppose that exh is a covert operator which can append to any sentence. It is reasonable

to assume that, in parsing (or producing) a sentence, exh will be used whenever the result

fairs better than its counter-part without exh. One way in which a sentence with exh

would be better than its exh-less counterpart is if the latter generates implausible

Ignorance Inferences based on B-MQ. We thus predict the following recursive parsing

strategy:

(68) Recursive Parsing-Strategy: If a sentence S has an undesirable Ignorance

Inference, parse it as Exh(Alt(S))(S).37

The core idea was developed during conversations with Ezra Keshet.

This should be modified to allow introduction of exh in a non-matrix position.

(i) Recursive Parsing-Strategy: If a sentence S has an undesirable Ignorance Inference, try to append exh

to some constituent X in S. I.e., modify the parse [S…X…] as follows: [S…Exh(Alt(X))(X)…].

We could also incorporate an economy condition of the sort alluded to in section 2.1.:

(ii) Condition on exh-insertion: exh can be appended to a constituent X, only if the resulting sentence

generates fewer Ignorance Inferences (based on B-MQ)

Consider the disjunctive sentence in (69)

(69) I ate the cake or the ice-cream.

If this sentence is parsed without an exhaustive operator, B-MQ will generate the

Ignorance Inference that the speaker doesn’t know what she ate (only that it included the

cake or the ice-cream or both). This inference might seem implausible, and the hearer

might therefore prefer the following parse, where C is the set of Sauerland-alternatives to

the disjunctive sentence.

(70) Exh(C)(I ate the cake or the ice-cream)

As we’ve seen already, the meaning of (70) is the ExOR meaning of (69). This meaning

will now generate (given B-MQ) the Ignorance Inference that the speaker doesn’t know

what she ate (only that, whatever it was, it included the cake or the ice-cream but not

both). This, again, might seem implausible, and the hearer might employ the parsing

strategy, again:

(71) Exh(C')[Exh(C)(I ate the cake or the ice-cream)]

where C'= Alt[Exh(C)(I ate the cake or the ice-cream)] = {Exh(C)(p): p∈C}

However, (71) ends up equivalent to (70). And further application of the parsing strategy

is not helpful either. It will, thus, follow that there is no way to avoid the (sometimes

undesirable) Ignorance Inference. To see this, we need to compute the set of alternatives,

C':

(72) C'= {1. Exh(C) (p ∨ q), 2. Exh(C)(p), 3. Exh(C)(q), 4. Exh(C)(p∧q)}

1. Exh(C) (p ∨ q) = (p ∨ q) ∧¬ (p∧q)

= (p ∧¬ q) ∨ ( q ∧¬ p)

2. Exh(C)(p)= p ∧¬ q

3. Exh(C)(q)= q ∧¬ p

4. Exh(C)(p∧q) = p∧q (can be ignored since already excluded by the prejacent)

Exh(C) (p ∨ q) = Exh(C)(p) ∨ Exh(C)(q)

Two simple observations are worth making. The first alternative, the prejacent of (71), is

equivalent to the disjunction of the second and third alternative, and the fourth alternative

is already excluded by the prejacent, and hence can be ignored. The relevant alternatives

are thus the following:

By the algorithm in (5).

2.Exh(C)(p)

(73) C' =Alt(Exh(p∨q)) = Exh(C)(p) ∨ Exh(C)(q)

3. Exh(C)(q)

Exh(C')[Exh(C) (p ∨ q)] = Exh(C) (p ∨ q) = (p ∨ q) ∧¬ (p∧q)

If 2 is excluded, 3 must be true, and vice versa. Hence, meaning does not change with a

second level of exhaustification, nor will it change when exh is appended yet another

time.39 There is thus no way to avoid what might be an undesirable Ignorance Inference.

Consider now (74).

(74) You may eat the cake or the ice-cream.

Without an exhaustive operator, this sentence will generate the Ignorance Inference that

the speaker doesn’t know what one is allowed to eat (only that the allowed things include

the cake or the ice-cream or both). This might seem implausible, and the hearer might opt

for another parse:

(75) Exh(C)(You may eat the cake or the ice-cream)

Given the Sauerland alternatives for disjunction, the set of alternatives, C, would be the

following:

(76) Alt(74)

◊p

C= ◊(p ∨ q) ◊(p∧q)

◊q Notice ◊(p ∨ q) ⇔ ◊p ∨ ◊q but (crucially)

◊(p∧q) <≠> ◊p ∧ ◊q

39The theorem in 1 is completely trivial, and the one in 2 (due to Benjamin Spector, p.c.) is less so:

1. Let C be a set of alternatives, Si, such that for each i exhaustification is trivial (i.e., Exh(C)(Si)⇔ Si),

then for each i, 2nd exhaustification is trivial (i.e., Exh(C')(Exh(C)(Si)) ⇔ Exh(C)(Si), where C' =

{Exh(C)(S): S∈C})

2. due to Spector: Let C be a set of finite alternatives, Si , then there is an n∈N, s.t. ∀m>n,

Exhn(C)(Si) = Exhm(C)(Si)

Exhn(C)(Si) := Exh(C')Exhn-1(C)(Si),

where C' = {Exhn-1(C)(S): S∈C}, and Exh1(C)(S) = Exh(C)(S).

40 The set of alternatives is actually larger, including a variant of each alternative in C with a universal

modal replacing the existential modal. This does not affect our results as the reader can verify. See the

appendix, as well as (84) and (85) where a parallel computation is carried out with the full set of

alternatives.

◊(p∧q) is the only proposition in Alt(◊(p∨q)) that can be innocently excluded given the

set of alternatives (excluding ◊p will necessarily include ◊q while excluding ◊q will

necessarily include ◊p). Hence, the meaning of (75) in our modal logic formalization is

◊(p ∨ q) ∧¬ ◊(p∧q).

Crucially (75) is consistent with the free choice possibility, ◊p∧◊q, though it, of

course, does not assert free choice.41 This new meaning will now generate the Ignorance

Inference that the speaker doesn’t know what one is allowed to eat (only that the allowed

things include the cake or the ice-cream but not both). This might seem implausible, and

the hearer might employ the parsing strategy again:

(77) Exh(C')[Exh(C)(You may eat the cake or the ice-cream)]

where C'={Exh(C)(p): p∈C}

This time, second exhaustification has consequences. To see this, we need to compute the

meanings of the various alternatives:

(78) C'= {1. Exh(C) (◊(p ∨ q)), 2. Exh(C)(◊p), 3. Exh(C)( ◊q), 4. Exh(C) (◊(p∧q))}

1. Exh(C) (◊ (p ∨ q)) = ◊(p ∨ q) ∧¬ ◊ (p∧q), crucially

≠ (◊p ∧¬ ◊q) ∨ (◊q ∧¬ ◊p)

2. Exh(C)(◊p)= ◊p ∧¬ ◊q

3. Exh(C)(◊q)= ◊q ∧¬◊ p

4. Exh(C) ◊(p∧q) = ◊(p∧q) (can be ignored since already excluded by the prejacent)

◊p ∧¬ ◊q

C'= ◊(p ∨ q)) ∧¬ ◊ (p∧q)

◊q ∧¬◊ p

There are now two propositions in C' that can be innocently
(and non-trivially) excluded.

(Excluding Exh(C)(◊p) will not necessarily include Exh(C)(◊q), and excluding

Exh(C)(◊q) will not necessarily include Exh(C)(◊p).)

Hence,

(79) Exc(C’)(Exh(C) (◊(p∨q))) = ◊(p ∨ q)) ∧¬ ◊ (p∧q) and

¬(◊p ∧¬ ◊q) and

¬(◊q ∧¬◊ p)

41 This exemplifies the difference between the lexical entry we are considering and the Groenendijk and

Stokhof-type alternative in (56). Under (56), (75) would express a stronger proposition ◊(p ∨ q) ∧¬

(◊p∧◊q), which will be inconsistent with FC.

= ◊(p) ∧ ◊ (q) and

¬ ◊ (p∧q)

We thus derive the FC effect for cases in which disjunction is embedded under existential

modals.42

The key to the distinction between disjunction embedded under an existential modal and

unembedded disjunction is that in the latter case the strongest alternative ◊(p∧q) is

stronger than the conjunction of the two other alternatives ◊p and ◊q. Hence, the first

layer of exhaustification is consistent with the later exclusion of Exh(C)◊p and

Exh(C)◊q.43 This answers Aloni and van Rooy’s (2005) objection to Kratzer and

Shimoyama (section 2.3.), and extends to account for embedding under existential

quantifiers:

(80) a. There is beer in the fridge or the ice-bucket.

b. People sometimes take the highway or the scenic route (Irene Heim, pc

attributing Regine Eckardt, pc)

c. This course is very difficult. Last year, some students waited 3 semesters to

complete it or never finished it at all. (Irene Heim, pc)

Here, too, first exhaustification will be fairly weak ∃x(Px∨Qx) ∧¬ ∃x (Px∧Qx) consistent

with later exclusion of Exh(C)∃xPx and Exh(C)∃xQx, the cumulative effect of which

entails ∃xPx∧∃xQx.

At the moment the system makes wrong predictions for embedding under singular

indefinites:

(81) a. There is a bottle of beer in the fridge or the ice-bucket.

b. This course is very difficult. Some student waited 3 semesters to complete it

or never finished it at all.

Right now, an FC effect is expected for this case as well. However, the expectation

changes once an independently needed difference between plural and singular indefinites

42 As pointed out in Simons (2005), ◊(p ∨ q) sometimes yields FC without the inference that ◊ (p∧q) is

false. A solution to this problem will be discussed in section 11.1.

43 The following is easily verifiable:

Let C= {w, s, n, e} be a diamond set of alternatives going stronger from w to e (w is weaker than s, n, and

e; s and n are logically independent and weaker than e), where w entails (s∨n). With such alternatives, 2nd

exhaustification of w is vacuous (Exh2(C)(w) ⇔ Exh(C)(w)) iff e ⇔ s&n. Furthermore, when 2nd

exhaustification of w is not vacuous, Exh2(C)(w) ⇔ s∧n∧¬e.

is factored in. Consider the sentences in (82). These sentences have the indicated

implicature that the alternative assertion involving quantification over plural individuals

is false:

(82) a. There is a bottle of beer in the fridge.

Implicature: there aren’t two bottles of beer in the fridge.

b. Some student talked to Mary

Implicature: It’s not true that two students talked to Mary.

This implicature leads to the conclusion that a singular indefinite is a scalar item, with a

plural (or dual) indefinite as an alternative:

(83) Horn-Set(Some NP-sing) = {Some NP-sing (henceforth ∃1),

two NPs (henceforth ∃2)}

With this Horn-Set, exh would, of course, derive the correct implicature for (82). But,

interestingly, we also explain the lack of FC in (81). To see this consider (81)a, and the

set of alternatives derived by (5), (84). The alternative-set includes alternatives of the sort

we’ve considered in (76) (upper diamond of (84)). But it also includes alternatives

generated by replacing ∃1 with ∃2 (lower diamond of (84)).

(84) Alt((82)a)

∃1xP(x)

∃1x (P(x)∧Q(x))

∃1xQ(x)

∃2xP(x)

C= ∃1x(P(x) ∨ Q(x)) ∃2x(P(x) ∨ Q(x)) ∃2x (P(x)∧Q(x))

∃2xQ(x)

The set of innocently-excludable alternatives contains ∃1x (P(x)∧Q(x)) as well as

∃2x(P(x)∧Q(x)). Hence the exhaustification of (82)a is the following:44

(85) Exh(C)( ∃1x(P(x) ∨ Q(x))) = ∃1x(P(x) ∨ Q(x)) &

¬∃1x(P(x) ∧ Q(x)) &

¬∃2x(P(x) ∨ Q(x))

This explanation,
however, depends on the assumption that the alternatives for ∃1 cannot be
inactive

when exh associates with ∨. An assumption of this sort was argued for on independent grounds in

Chierchia (2005).

⇒ ¬(∃1xP(x) ∧ ∃1x Q(x))

This representation already contradicts the FC effect, a situation
which, of course, cannot

change by further exhaustification. We thus derive the generalization in (31)

It is of course important to return to our computation of basic FC and make sure that

nothing changes when universal quantifiers are introduced as alternatives to existential

quantifiers (4)b,d (see footnote 40). I leave this as a task for the interested reader, though

an equivalent computation will be carried out in (87) and (88), below, and the appendix

will contain a theorem that will make all of our results transparent with fewer

computations.

Multiple exhaustification also accounts for the generalization in (33), i.e., it generates FC

effects for the sequences ¬ ∧ and ¬ ∀∧ (introduced in 3.1 and 3.3.) I will illustrate this

for ¬ ∧, and allow the reader to verify that nothing changes when is replaced with ∀.

Consider (25), repeated below as (86), with its FC inference, which (to repeat) is not

predicted by the basic meaning.

(86)You are not required to both clear the table and do the dishes.

1. basic meaning: ¬(p ∧ q) ≡ ◊¬ (p ∧ q) ≡ ◊(¬p ∨ ¬q) ≡ ◊(¬p) ∨ ◊(¬q)

2. Free Choice: ◊(¬p) ∧ ◊(¬q)

Once again, FC will follow after two layers of exhaustivity are computed. Let’s start with

the first layer, which we compute based on the alternatives generated by Sauerland’s

Horn-Set {∧, L, R, ∧} and the traditional Horn-Set {◊, }, (5)d.

(87) Alt((86))

¬p

¬( p ∨ q)

¬q

¬◊p

C= ¬(p ∧ q)) ¬◊(p ∧ q) ¬◊(p ∨ q)

¬◊q

Exh(C)(¬(p ∧ q)) = ¬(p ∧ q)) &

( p ∨ q) &

◊(p ∧ q)

If we decide to add another layer of exhaustification, we get the following result:

(88)

¬p ∧ q ∧ ◊(p ∧ q)

C' = ¬(p ∧ q)) ∧ ( p ∨ q) ∧ ◊(p ∧ q)

¬q ∧ p ∧ ◊(p ∧ q)

Exh(C')[Exh(C)(¬(p ∧ q))] =

¬(p ∧ q)) & ( p ∨ q) & ◊(p ∧ q) & ¬(¬p ∧ q) & ¬(¬q ∧ p)

This yields the FC effect, based on the following equivalences:

¬(p ∧ q)) ≡ ◊¬p ∨ ◊¬q

¬(¬p ∧ q) ≡ ¬(◊¬p ∧ ¬◊¬q)

¬(¬q ∧ p) ≡ ¬(◊¬q ∧ ¬◊¬p)

We have seen how our two generalizations about the distribution of
FC

((31) and (33)) can be derived based on recursive exhaustification under a Sauerlandinspired

meaning for exh. But before concluding, I would like to discuss two apparent

predictions of the account which are somewhat problematic.

11.1. ¬◊(p ∧q)

The lack of a conjunctive interpretation for p∨q was derived in section 7 on the basis of

the observation that the first layer of exhaustification excludes p∧q, an exclusion which

cannot be overridden at the second level of exhaustification. The situation changes in FC

environments, by the introduction of appropriate operators. When p∨q is embedded under

an existential quantifier, e.g. ◊ (p∨q), the first level of exhaustification excludes ◊ (p∧q),

a relatively weak exclusion, i.e. consistent with ◊ p∧◊ q. Hence it is possible (at the

second level of exhaustification) to innocently exclude the exhaustive interpretation of ◊p

and of ◊q.

This proposal makes a clear prediction, or at least so it seems. Specifically, it predicts

that FC will always be accompanied by the anti-conjunctive inference ¬◊ (p∧q).

However, it has been claimed that this prediction is false.

Specifically, a sentence can be presented that
can produce a free-choice effect while lacking an anti-conjunctive inference.

(89) You may love one another or die.

Possible Reading:

You may love one another and you may die, compatible with permission to do both.

Interestingly, people seem to have a different feelings when asked about the free-choice effect that arises for the sequence ¬∀∧. Consider, again:

You are not required to both love on another and die.

It seems quite hard to get rid of the inference that you are required to either love one another

or die.

More specifically, although judgments of this sort or
notoriously difficult, there doesn’t seem to be an interpretation which
involves FC (i.e. entails that each disjunct is such that you are allowed to
avoid it), which does not, at the same time, entail that at least one of the disjuncts is requited.

Each of the disjuncts could be exhaustified separately.

Assume,
as we have assumed above, that the alternatives for Exh could be determined
(at least in the absence of scalar items) based on focus.

It might now receive the following parse, where C' and C'' are determined based on scalar items (as outlined above), and C1, C2 are determined based on the focus value of the relevant prejacent .

Exh(C'')(Exh(C')(◊(Exh(C1)(we must love one another) or Exh(C2)(we must die)))).

Assume, further, that sing and dance are focused so
that C1 = C2.

To simplify the exposition (but with no loss of generality)
let’s assume that C1,2 has only two members {p = we must love one another, q = we must die},
with the following result:

p!:=Exh(C1)(p) = p ∧¬q

q!:=Exh(C2)(q) = q ∧¬ p

we derive this reading:

Exh(C'')(Exh(C')(◊(
p!∨q!))) = ◊( p!∨q!) ∧¬◊(p!∧ q!) ∧ ◊(p!) ∧ ◊(q!) =◊( p!∨q!) ∧ ◊(p!) ∧
◊(q!)

The analysis of free-choic crucially depends on the assumption that in the relevant sentences

disjunction receives narrow scope relative to the relevant existential quantifier.

This assumption is
corroborated by a contrast.

Note that both is crucial.

We may love one another die (free-choice effect)

Either we may love one another or die (no free-choice effect)

It has been pointed
out that either marks the scope of disjunction in constructions.

The fact that free-choice is present only for "We may love one another or die" corroborates the
scopal

assumptions made so far.

However, free-choice seems to be available in:

You may love one another or you may die.

We leave this as an unresolved problem, noting that the behaviour with indefinites is different.

Cfr

Some students waited 3 semester to complete this course or never finished it at all. (free-choice)

Some students either waited 3 semester to complete this course or never finished it at all. (free-choice)

Either some students waited 3 semester to complete this course or some students never finished it at all. (no free-choice)

A free-choice effect depends on the nature of the alternatives (e.g., E must be stronger than the conjunction of N and S. The correlation with scope is predicted on the basis of the algorithm that determines alternatives.

At every level of exhaustfication , alternatives are determined on the basis of the structure of the prejacent.

If the sentence turns
out to be indicative, i.e., if it turns out that free-choice is possible even when disjunction has scope over the relevant existential
quantifies, it would be possible to capture the facts with a relatively simple
modification of the system we've proposed.

We've assumed that
Alt(S) is determined either contextually or by an algorithm which is crucial for the recursive step.

However, we
could define a special rule for recursive exhaustification which would allow
the rule to apply even when alternatives are contextually determined.

Suppose
that a sentence S is uttered in a context in which Q is the salient set
of alternatives.

If S were to be exhaustified (relative to Q), the syntactic
representation would be Exh(Q)(S).

We could now define the second layer of exhaustification as follows: Exh2(S):= Exh(C)[Exh(Q)(S)], where C=
{Exh(Q)(ϕ): ϕ∈Q}. Now (92) could receive an FC interpretation if Q could be the
set of sentences of the form

You can eat x.

where x denotes a singular or
plural individuals, perhaps with closure under disjunction.

It's been argued that a 'free choice' effect arises in two seemingly unrelated contexts.

First, when disjunction is embedded under a non-singular existential quantifier.

Second, when a conjunction is embedded
under a universal quantifier which is, itself, commanded

by negation.

The free-choice effect follows from a method for exhaustification inspired by a solution to the disjunctional puzzle, a method in which the notion
of an innocently excludable alternative plays a central role.

However, the
proposal can work only if the basic idea is removed from
its Griceian setting.

The reason for this is rather plain.

Under the Griceian assumptions, implicature is derived as strengthening of inferences that
follow from NG-MQ, a maxim which would derive Ignorance Inferences based on
the symmetric alternatives generated by disjunction.

Hence the Griceian
assumption derives the ignorance implicature that conflict with the empirically
attested free-choice effect.

A necessary conclusion, given the alternatives
for disjunction, is that NG-MQ cannot be correct, and that there can be no
primary implicature which is computed on the basis of the relevant
alternatives.

The conclusion, itself, is an immediate consequence of a system
in which pragmatic reasoning is based on all relevant alternative assertions
(BMQ), i.e. a pragmatic system which can only derive an ignorance implicature.

If B-MQ is correct, implicature must be derived within grammar.

The grammatical mechanism needed for FC seems to be an exhaustive operator, which can apply recursively to a single linguistic expression, based on a lexical entry.

If this is correct, it might be useful to ask questions about possible external/functional motivations for "exh."

We hinted at the possibility that exh is needed to solve a communication problem
that will arise very often in a pragmatic universe governed by
BMQ.

We can prove a rather simple theorem which should allow the reader to understand the results described in this view with fewer computations.

We define an FC interpretation which we call AnEx (for Anti-exhaustivity), and prove that this

interpretation, if consistent, is the result of the 2nd layer of exhaustification. Let C be a set of propositions with p∈C, I = I-E(p,C) ≠∅, I'= (C I {p}) ≠ ∅, AnEx = ∩{¬ExhC(q):q∈I'}∩ ExhC(p)

Claim: If AnEx ≠∅ (is consistent), Exh2, C(p) =AnEx.. Proof: ExhC(p) entails ¬q, for all q∈I (by definition of Exh). Hence, ExhC(p) entails ¬ ExhC(q), for all q∈I (¬q entails ¬ ExhC(q)). Hence, AnEx entails ¬ExhC(q), for all q∈I (AnEx has ExhC(p) as a conjunct). Hence, AnEx =∩{¬ExhC(q):q∈I'}∩{¬ExhC(q):q∈I}∩ExhC(p)=∩{¬ExhC(q): q∈ C {p}} ∩ ExhC(p). Hence, If AnEx is consistent, I-E(ExhC(p), C') = C' {ExhC(p)} (where C': {ExhC(q):q∈C}). Hence, Exh2 C(p) = ∩{¬ExhC(q):q∈ C' {ExhC(p)}} ∩ ExhC(p) = AnEx (by definition of exh)

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about the nature of communication.

However, as is well known, one assumption is crucial, and far from trivial, namely the assumption that Grice’s Maxim of Quantity or Strength (his earlier Oxford lectures on implicature had none of this but an appeal to desiderata and principles: of candour, clarity, self-love and benevolence! -- the reference to QUANTITAS-QUALITAS-MODUS-RELATIO is a joke on Kant) should be stated with reference to a formally defined set of alternatives.

There is clearly no escape from formally defined alternatives.

However, if the perspective argued for here is correct, access to these alternatives should be limited to grammar. A quantity maxim which is not contaminated by syntactic stipulations (together with appropriately placed syntactic stipulations, i.e., within grammar) derives better empirical results.

Consider a simple disjunctive sentence:

When we hear such a sentence we draw a variety of inferences.

1. Auden talked to Chester or Christopher.

First, we conclude that (if the utterer is correct) Auden talked to Chester or to Christopher, a

conclusion, in and of itself, consistent with the possibility that Auden talked to both (basic

inference).

(The first to consider implicatures of 'disjunctions' was Grice, when criticising Strawson's Introduction to Logical Theory, in "Causal Theory of Perception": "My wife is in the garden or in the kitchen").

However, we typically also conclude, via implicature, and again assuming that the utterer’s utterance is correct, that this latter possibility was not attested.

Finally, we infer that the utterer’s beliefs (and this was Grice's focus in "Causal theory") don’t determine which person (i.e. Chester or Christopher) Auden talked to (ignorance inference).

The Inferences we draw from (1):

a. basic inference:

2a. Auden talked to Chester or Christopher (or both), i.e. Auden talked to Chester and/or Christopher.

b. implicature

2b Auden did NOT talk to both Chester and Christopher

c. ignorance inferences (Grice, "Causal Theory of Perception").

2c. The utterer doesn’t know that Auden talked to Chester

2c'. The utterer doesn’t know that Auden talked to Christopher.

The nature of the inferences in (2)a and (2)c seems rather straightforward.

The basic Inference, (2)a, is derived quite directly from the basic meaning of the sentence.

The Ignorance Inferences, (2)c, are, as Grice notes, not as direct, but, nevertheless, receive a fairly natural explanation.

They are derived straightforwardly from a general reasoning process about the belief states of utterers, along lines outlined by Grice's lectures on implicature (perhaps starting with "Causal Theory" -- although he is credited in a footnote to Strawon's logical textbook).

The source of the inference in (2)b, implicature, is less obvious.

The standard, neo-Gricean, approach captures this inference by enriching the set of assumptions that enter into the derivation of Ignorance Inferences, while various competing proposals attribute the

inference to a particular enrichment of the basic meaning.

Before one sees what is at stake, we should start with a formulation of what might be uncontroversial, namely the account of (2)c.

The basic idea is that communicative principles require utterers to contribute as much as possible to the conversational enterprise.

This idea is further elaborated when it is assumed that the goal of certain speech acts is to convey information, and that if all information is to be relevant, more is better.

Following Gazdar and Sauerland, some sometimes use the verb "know" to describe Ignorance Inferences.

This choice is problematic because of factivity inferences associated with "know", which are clearly inaccurate.

However, it’s not clear that there is a better choice.

"Believe" is problematic because of neg-raising ("I don't believe that p").

If some find factivity particularly disturbing (or neg-raising sufficiently innocuous), they favour "belief" (or 'doxastic', as I prefer) talk.

But these lexical choices can, however, be far from systematic.

The reader should bear all of this in mind and ignore factivity inferences associated with "know", as well as the neg-raising property of "believe".

Although some think that there is agreement that (2)c ought to be derived from principles of communication, there have been conflicting proposals concerning the precise derivation.

As we will see below, the complications could be argued to follow from the neo-Gricean perspective on implicature.

So, assume that two sentences are true and both contribute information that is completely relevant to the topic of conversation.

If one contains more information than the other (i.e., is logically stronger), use of the more informative one would constitute a greater contribution:

The conversational maxim regarding what Grice echoing (or making fun of Kant) calls quantity (in its basic version) may be formulated as:

If S1 and S2 are both relevant to the topic of conversation

and S1 is more informative than S2,

if the utterer believes that both are true,

the utterer should utter S1 rather than S2.

Typically, when (1) is uttered, the information conveyed by each of the disjuncts is relevant.

Furthermore, each disjunct is more stronger and informative than the entire disjunction (since "p" entails "p or q", but not vice versa.)

The fact that the utterer uttered the entire disjunction rather than just a disjunct, therefore, calls for an explanation.

If we, the people who interpret the utterance, assume that s obeys the Maxim of Quantity, we conclude, for each disjunct, p, that it is false to claim that the utterer believes that p is true, or if we keep to our convention of using the verb "know" instead of "believe", we can state this as a

conclusion that s does not know that p is true.

If we assume that the utterer believes that his utterance of the disjunction is correct, we derive the Ignorance Inferences.

But one logical property of the situation is worth focusing on.

When we conclude that the utterer does not believe that p is true, that is, in principle, consistent with two different states of affairs.

The utterer might believe that p is false, or, alternatively, he might have no conclusive opinion.

The reason we infer the latter is that the former would be inconsistent with our other inferences.

Under normal circumstances, we infer that the utterer believes that his utterance of "p or q" is true (Maxim of QUALITY, against a concoction by Grice echoing or making fun of Kant).

If we were to assume that utterer believes that p is false, we would have to conclude that he believes that q is true.

But that would conflict with our inference about q (based on the Maxim of Quantity).

Hence we must conclude, for each disjunct, that the utterer has no opinion as to whether or not it is true.

It's different with modal contexts (such as Auden, using 'must') or imperative contexts, where we have to generalise from truth-conditions to satisfaction-conditions (i.e. conditions where the states of affairs mentioned are factually satisfied).

Consider now whether we could extend this line reasoning to account for the implicature in (2)b.

Since we’ve already concluded that the utterer does not know that p is true and that the utterer does not know that q is true, it follows that the utterer does not know that the conjunction p and q is true.

This, again, is consistent with two different states of affairs.

The utterer might believe that p and q is false, or, alternatively, he might have no conclusive opinion.

If this time we could exclude the latter possibility, we would derive the implicature.

The problem is that basically the same line of reasoning we’ve employed above leads us exactly to the opposite conclusion, namely to the exclusion of the possibility that the utterer believes that p and q is false.

The idea is fairly simple.

The information that p and q is false, if true, would be relevant to the topic of conversation, hence the fact that utterer did not provide us with this information calls for an explanation.

The relevant notion of informativity for a pragmatic account should probably be that of contextual strength, i.e. logical strength given contextual presuppositions). This distintion has consequences on the theory of implicature with a potential argument against the Griceian account, I grant.

Once again, the natural explanation is that the utterer did not have the information, i.e., that s did not know that p and q is false.

In other words, instead of an implicature, we derive, once again, an Ignorance Inference: we

conclude that s does not know that p and q is true, and (exactly by the same type of

reasoning) that s does not know that p and q is false, i.e., we conclude that s does not

know whether or not p and q is true.

As far as we know, a version of this problem was first noticed in Kroch and stated in its most general form in class notes of Kai von Fintel and Irene Heim.

To appreciate the problem in its full generality, consider a general schema for deriving impicature in

response to s’s utterance of p (of, say, "I have 3 children").

We start by considering a more informative relevant utterance, p' (say, "I have 4 children"), and reason that if p' were true, and if s knew that p' were true, the Maxim of Quantity would have forced s to utter p' instead of p.

We then might reason that it is plausible to assume that s knows whether or not p' is true (say, that it is reasonable to assume that s knows how many children she has), and hence that s knows that p' is false.

The problem, however, is that there is always an equally relevant more informative utterance than p, namely p and not p' (in our case, I have exactly 3 children), call it p''. By

the same reasoning process, if p'' were true, and if s knew that p'' were true, the

communicative principles would have forced s to utter p'' instead of p.6 Furthermore, if s

knows the truth value of p, and of p', then s knows the truth value of p''. So, the same

reasoning process leads to the conclusion that s knows that p'' is false. The assumption

that the speaker knows whether or not p' is true, thus, leads to a contradiction and must,

therefore, be dropped.

This problem was dubbed the symmetry problem in class notes of Kai von Fintel and

Irene Heim. Whenever p is uttered, and fails to settle the truth value of a relevant

proposition, q, there will be two symmetrical ways of settling it, leading necessarily to an

Ignorance Inference. Stated somewhat differently, p is equivalent to the following

disjunction (p ∧ q) ∨ (p ∧ ¬q), and therefore should lead to Ignorance Inferences parallel

to those stated in (2)c.

1

The Griceians respond to this problem by a revision of the Maxim of Quantity.

Specifically, they suggest that the maxim doesn’t require speakers to utter the most

informative proposition that is relevant to the topic of conversation, but is more limited in

scope. The Maxim merely requires speakers to choose the most informative relevant

proposition from a formally defined set of alternatives. It does not require speakers to

consider all relevant propositions.

The assumptions made here about “relevance” are the following:

1. If p and q are both relevant, so is “p and q”.

2. if p is relevant, so is “not p”. (To say that p is relevant is to say that the question Is p true or false?

relevant.)

6 It is sometimes suggested that Grice’s Maxim of Manner (M) could be used to explain s’s avoidance of p.

Such a suggestion requires an ordering of linguistic expressions by which p' would be more optimal than p''

from M’s perspective. For arguments against obvious orderings (various measures of complexity), see

Matsumoto (1995), as well as Fox and Hackl (2005).

5

The common way to work this out, pioneered by Larry Horn (1972), starts out with

the postulation of certain sets of lexical items, Scalar Items, and sets of alternatives to

which the scalar items belong, which we will call Horn-Sets:7

Examples of Horn-Sets

a. {or, and}

b. {some, all}

c. {one, two, three,…}

d. {can, must}

These sets of lexical alternatives determine the set of (Horn) alternatives for a sentence

by a simple algorithm. The set of alternatives for S, Alt(S), is defined as the set of

sentences that one can derive from S by successive replacement of Scalar Items with

members of their Horn-Set.8

Alt(S) = {S': S' is derivable from S by successive replacement of scalar items with members of their Horn-Set}

The Maxim of Quantity can now be stated as follows:

Maxim of Quantity (Griceian version):

If S1 and S2 are both relevant to the

topic of conversation, S1 is more informative than S2, and S1∈Alt(S2), then, if the

speaker believes that both are true, the speaker should prefer S1 to S2.

Consider the sentence in (1). The postulated scalar item in this sentence is

disjunction, for which conjunction is lexically specified as the only alternative, (4)a. (1),

thus, has just one alternative (other than (1) itself):

Alt(1) = {(1), Sue talked to John and Fred }

When s utters (1), his addressee, h (for hearer), typically concludes (on the

assumption that s obeys the revised Maxim of Quantity) that s does not know that the

conjunctive sentence in Alt(1) is true, since this alternative sentence is more informative

than s’s utterance, and is typically relevant. If h assumes, further, that s has an opinion as

to whether or not the conjunctive sentence is true, h would conclude that s believes that it

is false. The neo-Griceans, thus, attribute a general tendency to addressees, namely the

tendency to assume that speakers are opinionated. I state Sauerland’s formulation of this

assumption in (8).

Opinionated Speaker (OS): When a speaker, s, utters a sentence, S, the addressee, h,

assumes, for every sentence S'∈Alt(S), that s’s beliefs determine the truth value of S',

unless this assumption about S' leads to the conclusion that s’s beliefs are

contradictory.

Usually called Horn-Scales for bad reasons, as discussed in Sauerland 2004.

This definition, which comes from Sauerland (2004), is implicit in much earlier work, and is of course

very similar to the definition of alternative sets in Rooth (1985).

Under the basic version of the Maxim of Quantity in (3), B-MQ, there was no way to maintain the assumption that the speaker is opinionated about any relevant sentence S' (not entailed by S). To repeat, B-MQ derived the symmetric results (a) that the speaker does not know that S and S' is true, and (b) that the speaker does not know that S and not S' is true. This, together with the assumption that the speaker knows that S is true (Quality), derived the conclusion that the speaker is not opinionated about S'.

By contrast, under the Neo-Gricean version of the Maxim of Quantity in (6), NGMQ, the assumption that the speaker is opinionated about various sentences (not entailed by S) is innocuous. NG-MQ does not always derive the inference that the speaker does not know that S and S' is true. It derives such an inference only when S and S' (or some equivalent sentence) is a member of Alt(S). Under such circumstances, the speaker could be opinionated about S' as long as Alt(S) does not have S and not S' as a member (nor some equivalent sentence). If S and not S' is not a member of Alt(S), NG-MQ does not derive the inference that the speaker does not know that S and not S' is true, and the

assumption that the speaker believes that S and S' is false could be made consistently.

To summarize, assume that a speaker s utters the sentence in (1), Sue talked to John or Bill. The addressee, h, assumes that s obeys the Maxim of Quality as well as the revised Maxim of Quantity, NG-MQ. Based on this assumption, h reasons in the following way:

Given NG-MQ, there is no X∈Alt(1), such that X is logically stronger than (1), and s thinks that X is true.

Alt(1) contains the conjunctive sentence Sue talked to John and Bill, which is logically stronger than s’s utterance. Hence, given 1, it’s not the case that s thinks that this conjunctive sentence is true.

3. Given OS, the default assumption is that s has an opinion as to whether Sue talked to John and Bill is true or false. Given 2 (the conclusion that it’s not the case that s thinks that the sentences is true), we can conclude that s thinks that

it is false.

So, by modifying the set of assumptions that derive Ignorance Inferences (replacing B-MQ with NG-MQ) one can account for the SI in (2)b. I would like at this point to discuss a possible alternative that keeps B-MQ in tact but instead enriches the set of syntactic representations available for (1). But it is worth pointing out first that, as things stand right now, our account of the Ignorance Inferences in (2)c is in jeopardy.

Specifically, it is incompatible with NG-MQ and our assumption in (4)a about the Horn-Set for disjunction. The account was crucially dependent on the assumption that the Maxim of Quantity would prefer the utterance of a disjunct to the utterance of a disjunction, an assumption incompatible with the way Alt(1) is defined on the basis of (4)a. One might respond to this problem with an independent (pragmatic) account for (2)c (or by enriching the Horn-Set for disjunction (Sauerland 2004). The latter will be discussed in greater detail later.

The alternative syntactic approach that I would like to defend is guided by the intuition that a principle of language use (such as the Maxim of Quantity) should not be sensitive to the formal (and somewhat arbitrary) definition of Alt(S).

If this intuition is correct, BMQ is to be preferred to NG-MQ.

However B-MQ derives Ignorance Inferences that contradict attested SIs. Therefore, if B-MQ is correct, something else is needed to derive SIs. More specifically, SIs must be derived from the basic meaning of the relevant sentences; otherwise the symmetry situation would yield unwanted Ignorance Inferences. The syntax of natural language has a covert operator which is optionally appended to sentences, and that this operator is responsible for SIs.

The guiding observation is that there is a systematic way to state the SI of a sentence, using the focus sensitive operator only. Consider the sentence in (9), which has the SI that

John didn’t buy 4 houses.

John bought three houses.

This SI could be stated explicitly using the focus sensitive particle only in association

with the numeral expression three.

John only bought THREE houses.

This observation extends to all SIs; SIs can always be stated explicitly with the focus

sensitive particle only, as long as the relevant scalar items bear pitch accent:

John did some of the homework (+> John only did SOME of the homework).

For all of the alternatives to ‘some’, d,

if the proposition that John did d of the homework is true,

then it is entailed by the proposition that John did some of the homework.

John talked to Mary or Sue (+> John only talked to Mary OR Sue).

For all of the alternatives to ‘or’, con, if the proposition that John talked to Mary con Sue is true

then it is entailed by the proposition that John talked to Mary or Sue.

This consideration would be weakened significantly if one could make sense of Alt(S) from the

perspective of a general theory of language use. For efforts along these lines, See Spector (2005).

Sentence that generates SIs usually contain scalar items,11 and in such cases it is always possible to state the SIs explicitly, by appending the operator only to the sentence and placing focal accent on the relevant scalar item:

The only implicature generalization (OIG): A sentence, S, as a default, licenses the implicature that (the speaker believes) onlyS', where S' is a modification of S with focus on scalar items.

From the Gricean perspective discussed in 1.2., it is pretty clear why (9)-(12) should obey the OIG.

As is commonly assumed, and as indicated by the paraphrases, the role of only is to eliminate alternatives. Furthermore, when focus is placed on scalar items, the relevant alternatives are precisely the Horn-alternatives that NG-MQ refers to.

However, the OIG suggests yet another possibility, namely that B-MQ is the right conversational maxim and SIs are derived within the grammar, namely by of a covert exhaustivity operator with a meaning somewhat a kin to that of only. Assume, for the moment, the semantics for only suggested by the paraphrases in (9)-(12). Specifically, assume that only combines with a sentence (the prejacent), p, and a set of alternatives, A, (determined by focus). The result of this combination is a sentence which presupposes that p is true and furthermore asserts that every true member of A is already entailed by p (i.e., that all non-weaker alternatives, all “real alternatives”, are false):

[[only]] (A

this requirement should be part of the assertive component:

alternatives could be specified in other ways as well: by pitch-accent or by an explicit question. See the discussion below.

provide an empirical argument in favor of a theory in which B-MQ is at the heart of pragmatic reasoning and exh is responsible for SIs.

Furthermore, I will present an argument due to Kratzer and Shimoyama (2002) (K&S) and Alonso-Ovalle (2005), that FC should be derived by the system that derives SIs. In section 3, I will present evidence that FC arises in additional circumstances: when disjunction is embedded under certain other existential quantifiers and when conjunction is embedded under negation and universal quantifiers (under the sequence ¬∀).

There's a relevant observation about disjunction due to Chirchia, and in section 5 I will present the Neo-Gricean

Consider the sentence in (16) when uttered by someone who’s understood to be an authority on the relevant rules and regulations, for example, a parent who is accustomed to specifying limits pertaining to the consumption of sweets.

In such a context, the utterance would be an immediate inference.

entailment of obvious candidates for the logical form of the utterance

complement of allowed), or, in the terms of possible-world semantics, that there is a world consistent with the rules in which one of the disjuncts is true.

The problem is to understand how a disjunctive logical form can be strengthened to yield the conjunctive inference, and this may have motivated the change in Auden.

◊p ∨ ◊q

◊p ∧ ◊q

Free choice effects have been that arise when certain indefinite expressions are embedded

under existential modals, and presented a fairly strong argument that the effect should be

derived by the system that yields SIs.

observation that there are no traces of free-choice in certain downward entailing contexts.

decide what to do.

Some think that the interpretation is available in contexts that, more generally, allow for “intrusive”

No one is allowed to eat the cake OR the ice-cream. Everyone will be told what to eat.

No one is allowed to eat the cake or the ice-cream.

b

Although it is not yet clear how to derive free choice as an implicature, it is clear that if a derivation were available for the basic case, it would, nevertheless, not be available (at least not necessarily) for the other utterance.

This is seen most clearly under the Griceian approach to implicature.

Under this approach, an implicature is derived as a pragmatic strengthening of the basic meaning of a sentence.

The meaning is weaker than the basic meaning in and, therefore, can NOT be derived along Griceian lines.

condition related to its functional motivation, namely to the elimination of Ignorance Inferences. (I.e., a sentence with exh must lead to fewer Ignorance Inferences than its counterpart without exh, see footnote 37.) Alternatively, we might suggest, in line with the neo-Griceans, that exh must be introduced in matrix position.

The projection properties of FC seem to suggest that the effect should be derived as an SI. But how could we derive such a result? K&S made a very interesting proposal which is the basis for the proposal that I will make in section 6-10. But this will require quite a bit of ground work.

you are allowed to eat the cake or the ice-cream, you are pretty lucky. If K&S are correct, the FC interpretation of the antecedent would require an analysis involving an embedded implicature but without the “meta-linguistic” feel that is sometimes associated with such intrusion. Unfortunately, I have nothing interesting to add.

“embedded” implicatures (at least in non-downward entailing environments), see Chierchia 2004. The discussion is of course, problematic, and more so from the Gricean perspective.

At this point, it is worth understanding what one might say in order to

derive SIs based on the neo-Gricean maxim of quantity (NG-MQ).

Quite generally, suppose that ϕ is the basic meaning of a sentence, S, and that our

goal is to derive a stronger meaning, ϕ', based on NG-MQ. The result could be achieved

if we proposed a set of alternatives of the following sort: ALT(S) := {S, S & not Sϕ' },

where the meaning of Sϕ' is ϕ'. If s was to utter S, the addressee, h, would conclude, based

on NG-MQ, that s does not believe that not Sϕ' is true. Furthermore, based on the

assumption that s is an opinionated speaker, h, would conclude that s believes that Sϕ' is

true.

More specifically, suppose that the alternatives of the sentence in (16), repeated

below, are the sentences in (22). This is slightly different from the general scheme for

deriving implicatures characterized above, but the basic idea is the same.18

(16) You’re allowed to eat the cake or the ice-cream.

(22) Alternatives needed to derive FC for (16) based on NG-MQ:

a. You are allowed to eat the cake or the ice-cream.

b. You are allowed to eat the cake but you are not allowed to the ice-cream

c. You are allowed to eat the ice cream but you are not allowed to eat the cake.

Based on NG-MQ, we would now derive the SI that (22)b and (22)c are both false,

which, together with (22)a, yields the FC inference. To see this, assume (22)a is true.

Now assume that one of the conjuncts in (17) is false, say that you cannot eat the icecream.

From this it follows that (22)b is true, contrary to the SI.

But of course this is not intended as a serious proposal. It follows from a general

algorithm that allows us to derive, on a case by case basis, any SI that we would like to,

and, hence, does not explain the particular SIs that are actualized (See Szaebo 2003). The

obvious way to turn this into a serious proposal is to show that the alternatives in (22) are

needed on independent grounds.

K&S, and in particular Alonso-Ovalle, propose a more natural set of alternatives,

namely the one in (23).

(23) Alternatives proposed by K&S/Alonso-Ovalle:

a. You are allowed to eat the cake or the ice-cream.

b. You are allowed to eat the cake.

c. You are allowed to eat the ice cream.

The set of alternatives in (23), in contrast to the one in (22), is consistent with a general

constraint on alternatives proposed in Matsumoto (1995).19 Furthermore, as we will see in

section 4, there is independent evidence for the type of Horn-Sets that would derive (a

18 It would instantiate the scheme if b and c were replaced by a single alternative, namely the disjunction of

the two.

19 Matsumoto argues that lexical items can be members of the same Horn-Set only if they denote functions

of the same monotonicity. ∨ is upward monotone with respect to both arguments, but ∧¬ is downward

monotone with respect to its right-hand argument. Skipping ahead to Sauerland’s Horn-set, L and R are

upward (as well as downward) monotone with respect to their immaterial arguments.

13

super-set of) the alternatives in (23). The problem, however, is that NG-MQ can not

derive the FC effect on the basis of (23). In fact, as we will see in greater detail in section

4.2., it derives Ignorance Inferences that directly conflict with FC.

K&S suggest, however, that FC should be derived from (23) based on a novel

principle, which they call anti-exhaustivity. When h interprets s’s utterance of (23)a, s

needs to understand why it is that s preferred this sentence to any of the alternatives. The

standard Neo-Gricean reasoning, which relates to the basic meaning of the alternatives,

would lead to the conclusion that s does not know/believe that any of the alternatives is

true. K&S, however, suggest that h might reason based on the strong meaning (basic

meaning + implicatures) of the alternatives. Specifically, K&S suggest that h would

attribute the choice of s to the belief that the strong meanings of (23)b and (23)c are both

false. Furthermore they assume that the strong meaning of (23)b and c is the basic

meaning of (22)b and c respectively.

As pointed out by Aloni and van Rooy (2005), this line of reasoning raises a question

pertaining to simple disjunctive sentences, such as (1). We would like to understand why

such sentences don’t receive a conjunctive interpretation via an anti-exhaustivity

inference of the sort outlined above. If each disjunct is an alternative to a disjunctive

sentence, why doesn’t the speaker infer that the exhaustive implicature of each disjunct is

false?

K&S provide an answer this question by postulating a covert modal operator for any

disjunctive sentence. I will not go over this proposal and the way it might address Aloni

and van Rooy’s objection. I would like, instead, to raise another challenge to K&S’s

basic idea. I think it is important to try to understand how the anti-exhaustive inference

fits within a general pragmatic system that derives Ignorance Inferences (as well as SIs).

Specifically, I think it is important to understand why NG-MQ does not lead to the

Ignorance Inferences in (24) (see section 4.2. for details).

(24) Predicted inferences of (16), based on (23) and NG-MQ:

a. s doesn’t know whether or not you can eat the cake.

b. s doesn’t know whether or not you can eat the ice cream.

This is a challenge that this paper attempts to meet. The idea, in a nut-shell, is to

eliminate NG-MQ in favor of the non-stipulative alternative B-MQ. However,

understanding how this is to work requires the introduction of a proposal made in

Sauerland (2004), which would be extracted from its neo-Gricean setting in order to meet

our goals. But before I get there, I would like to introduce an additional challenge.

Specifically, I would like to present a few other surprising inferences that are intuitively

similar to FC, and should, most likely, be derived by the same system.

3. Other Free Choice Inferences

In this section we will see effects that are very similar in nature to FC, but arise in

somewhat different syntactic contexts. These effects will argue for a fairly general

explanation of the basic phenomenon, one that is not limited to modal environments or to

disjunction.

FC Under negation and universal modals

Consider the sentence in (25) when uttered by someone who is understood to be an

authority on the relevant rules and regulations, for example, a parent who is accustomed

to assigning after-dinner chores.

(25) You are not required to both clear the table and do the dishes.

In such a context, (26) would normally be inferred by the addressee.

(26) You are not required to clear the table and you are not required to do the dishes.

This inference seems very similar to the FC inference drawn in (17) based on (16). To see

the similarity, notice that the basic meaning of (25) is predicted to be equivalent to the

disjunctive statement that you are allowed to either avoid clearing the table or evade

doing the dishes, (27)a, and that (26) is equivalent to the conjunction of two possibility

statements. (You are allowed to avoid clearing the table and you are allowed to avoid

doing the dishes, (27)b)

(27)a. Standard Meaning of (25)

¬(p ∧ q) ≡ ◊¬ (p ∧ q) ≡ ◊(¬p ∨ ¬q) ≡ ◊(¬p) ∨ ◊(¬q)

b. Free Choice Inference

◊(¬p) ∧ ◊(¬q)

Just as in (16), the basic meaning does not explain the inference, and the gap is formally

identical. Once again, we have to understand how a sentence that is equivalent to a

disjunctive construction can be strengthened to something equivalent to the

corresponding conjunction.

3.2. More generally under existential quantifiers

In the basic FC permission sentence in (16), disjunction appears in the scope of the

existential modal allowed. Furthermore, as is well-known, FC extends to all constructions

in which or is in the scope of an existential modal:

(28) a. The book might be on the desk or in the drawer.

(= The book might be on the desk and it might be in the drawer)

b. He is a very talented man. He can climb Mt. Everest or ski the Matterhorn.

(=He can climb Mt. Everest and he can ski the Matterhorn.)

What has not been discussed in any systematic way is that this type of conjunctive

interpretation extends also to some non-modal constructions:20

(29) a. There is beer in the fridge or the ice-bucket.

20While working on this paper, I have learned about two new papers about FC that make this same

observation: KIindinst (2005) and Eckardt (this volume).

15(= There is beer in the fridge and there is beer in the ice-bucket.)

b. Most people walk to the park, but some people take the highway or the scenic

route. (Irene Heim, pc attributed to Regine Eckardt, pc)

(= Some people take the highway and some people take the scenic route.)

c. This course is very difficult. In the past, some students waited 3 semesters to

complete it or never finished it at all. (Irene Heim, pc)

(= Some students waited 3 semesters to complete the course and some

students never finished it at all.)

It is thus tempting to suggest that conjunctive interpretations for disjunction are available

whenever disjunction is in the scope of an existential quantifier (with the domain of

quantification, worlds or individuals, immaterial). However, there are limitations:

(30) a. There is a bottle of beer in the fridge or the ice-bucket.

c. Someone took the highway or the scenic route.

(≠Someone took the highway and someone took the scenic route.)

d. This course is very difficult. In the past, some student waited 3 semesters to

complete it or never finished it at all.

(≠ some student waited 3 semesters to complete the course and some student

never finished it at all.)

As pointed out in Klindinst (2005), the relevant factor seems to be number marking on

the indefinite. We might, therefore, suggest the following generalization:

(31) Existential FC: A sentence of the form ∃x [P(x) ∨ Q(x)] can lead to the FC

inference, ∃xP(x) ∧ ∃x Q(x), as long as the existential quantifier, ∃x, is not

marked by singular morphology.

3.3. More generally, under negation and universal quantifiers

In (25) we saw an FC effect arising when conjunction is under the scope of negation and

a universal modal (under the sequence, ¬). As illustrated in (32), and stated in (33), the

effect arises also when is replaced by an ordinary universal quantifier:

(32) We didn’t give every student of ours both a stipend and a tuition waiver.

1. basic meaning: ¬∀x[P(x) ∧ Q(x)] ≡

∃x¬ [P(x) ∧ Q(x)] ≡

∃x [¬P(x) ∨ ¬Q(x)] ≡

∃x ¬P(x) ∨ ∃x¬Q(x)

2. Free Choice: ∃x ¬P(x) ∧ ∃x¬Q(x)

(33) Conjunctive FC: A sentence of the form ¬∀x[P(x) ∧ Q(x)] can lead to the FC

inference ∃x ¬P(x) ∧ ∃x¬Q(x).

16

In section 7-10 we will provide an account of our two generalizations ((31) and (33))

within a general theory of SIs that we will introduce in section 6, based on a discussion,

in section 4-5, of Sauerland’s approach to SIs. But before we move on, it is important to

rule out an alternative explanation of (25) and (32), in terms of wide scope conjunction.

To understand the concern, focus on (25). One might think that this sentence has a logical

form in which conjunction takes wide scope over the sequence ¬. If such a logical form

were available, the inference in (26) would follow straightforwardly from the basic

meaning, and would thus be unrelated to the FC effects that are distributed according to

(31).

But wide scope conjunction is not a probable explanation. One argument against such

an explanation is based on the sentences in (34). If conjunction could take scope over the

sequence ¬ in (25) (and over ¬∀ in (32)), we would expect it to be able to outscope

negation in (34), an expectation that is not born out.21

(34) a. I didn’t talk to both John and Bill.

b. We didn’t give both a stipend and a tuition waiver to every student.

What I think we learn from (34) is that conjunction can appear to outscope negation only

when a universal quantifier intervenes.22 This is expected if conjunction never outscopes

negation, and the generalization in (33) is real.

Another argument against wide scope conjunction comes from an additional inference

we draw from sentences such as (25) and (32). In both cases we draw the inference that

the alternative sentence with disjunction instead of conjunction is false. That is, we would

tend to draw (35)a as an inference from (25), and (35)b from (32). These inferences are

not expected if conjunction receives wide scope, but, as we will see later on, are expected

if the phenomenon is derived along with other FC effects.

(35) a. You are required to clear the table or do the dishes.

b. We gave every student of ours a stipend or a tuition waiver.

4. Chierchia’s Puzzle

The account of FC that I will develop will be based on a modification of a proposal made

in Sauerland (2004) to deal with a puzzle discovered in Chierchia (2004).23 To

understand the puzzle in greater detail, consider first (36) and its implicature that (36)' is

false.

(36) John did some of homework.

21 If the distributor both is omitted the resulting interpretation is equivalent to wide scope conjunction. The

correct account relies most likely on a “homogeneity” presupposition (Fodor 1976, Gajewski 2005).

22 In order to account for the difference between (32) and (34)b, we would also have to say that inversion of

the surface scope of conjunction and universal quantification is impossible in (34)b, a consequence of

Scope Economy, a principle I’ve argued for in Fox (2000).

23 See also Lee (1995), and Simons (2002).

(36)' John did all of the homework.

As outlined in 1.2., this implicature can be derived by NG-MQ under the assumption that

some and all are members of the same Horn-Set, (4)b,24 from which it follows that (36)'

is an alternative to (36). NG-MQ, together with the assumption of an opinionated

speaker, lead to the conclusion that the speaker believes that (36)' is false.

Consider next what happens when (36) is embedded as one of two disjuncts:

(37) John did the reading or some of homework.

This type of embedding was presented by Chierchia (2004) as a challenge to the neo-

Gricean derivation of implicatures. As Chierchia points out, (37)' should be an

alternative to (37), and it would therefore seem that (with the assumption that the speaker

is opinionated) we should derive the implicature that (the speaker believes that) (37)' is

false.

(37)' John did the reading or all of homework.

This implicature, however, is clearly too strong. If a disjunctive sentence is false, then

each of the disjuncts is false. When (37) is uttered, we do derive the inference that the

second disjunct of (37)' is false. However, we clearly do not derive a similar inference for

the first disjunct (which is also the first disjunct of (36)).

Chierchia’s challenge for the Neo-Griceans is to avoid the implicature that the first

disjunct of (36) is false while at the same time to derive the implicature that the stronger

alternative to the second disjunct is false:

(38) Let U be an utterance of p or q where q has a stronger alternative, q'.

a. Problem 1: to avoid the implicature of ¬p

b. Problem 2: to derive the implicature of ¬q'

Chierchia provides an account for the relevant generalization based on a recursive

definition of strengthened meanings. I will not discuss his account, since I can’t figure

out how to extend it to FC. Instead, I will discuss the neo-Gricean alternative, which also

fails to account for FC, but, which can, nevertheless, be modified in order to provide a

syntactic (non-Gricean) alternative that successfully extends to FC.25

5. Sauerland’s Proposal26

As pointed out at the end of section 1.2., the Horn-Set for disjunction in (4)a ({or, and})

cannot account for the Ignorance Inferences that are attested when a simple disjunctive

24 and that the set does not include the “symmetric alternative” to all, some but not all.

25 Chierchia himself developed an account of FC which is quite similar to the account proposed here and is

to some extent independent of his recursive procedure for implicature computation. Specifically, his

account, like mine, is based on operators that apply to a prejacent and a set of alternatives. However, the

crucial operator for him is an “anti-exhaustivity” operator, distinct from what might be responsible for

implicature computation.

26 Benjamin Spector made the same proposal, in a somewhat different (more generalized) format. A related

proposal can be found in Lee (1995).

18

sentence such as (1) is uttered. Sauerland suggests a remedy for this problem which also

resolves Chierchia’s puzzle.

To derive the appropriate Ignorance Inferences for (1), Sauerland suggests that the

alternatives for a disjunctive statement include each of the disjuncts in addition to the

corresponding conjunction:

p

(39) Alt(p∨q)= p∨q p∧q

q

These alternatives, which are plotted to represent logical strength,27 derive (based on NGMQ)

the following inferences with respect to a speaker, s, who utters p or q, inferences

which Sauerland calls Primary (or weak) Implicatures, PIs:

(40) PIs for p or q (based on NG-MQ)

a. s does not believe that p is true.

b. s does not believe that q is true.

c. s does not believe that p and q is true. Already follows from both a and b.

Given that s is assumed to believe that her utterance of p or q is true (Quality), we

derive the Ignorance Inferences discussed in section 1, that is, for each disjunct, we

derive the inference that the speaker does not know whether or not it is true. To derive

SIs, the principle of an Opinionated Speaker is employed, (8):

(8) Opinionated Speaker (OS): When a speaker, s, utters a sentence, S, the addressee, h,

assumes, for every sentence S'∈Alt(S), that s’s beliefs determine the truth value of S',

unless this assumption would lead to the conclusion that s’s beliefs are contradictory.

This principle asks us to scan the set of alternatives that are stronger than S, and to

identify those for which the assumption that the speaker is opinionated is consistent with

our prior inferences based on Quality and NG-MQ. For each such alternative, the speaker

is assumed to be opinionated, and given the relevant PI, a stronger inference is derived,

namely that the speaker believes that the relevant alternative is false, an inference which

Sauerland calls a Secondary Implicature (an SI, conveniently).

As mentioned above, (40)a,b together with Quality, lead to ignorance with respect to

p and to q. Hence, p and q is the only alternative for which the assumption that the

speaker is opinionated is consistent with prior inferences. Therefore, only one SI is

derived based on OS, namely the inference that the speaker believes that p and q is false:

(41) SI for p or q (based on OS)

s believes that p and q is false.

Sauerland, thus, derives the following definition for the two relevant sets of implicatures:

(42) When a speaker s utters a sentence A, the following implicatures are derived:

27 If x is to the left of y with a connecting line, then x is weaker than y.

19

a. PIs = {¬Bs(A'): A'∈ ALT(A) and A' is stronger than A}

b SIs = {Bs(¬A'): A'∈ ALT(A), A' is stronger than A, and

Bs

(A)∧∩PI ∧ Bs(¬A') is not contradictory}

Based on these definitions, a PI is derived for every alternative stronger than the assertion

and an SI for a subset of the stronger alternatives for which an Ignorance Inference hasn’t

already been derived (based on NG-MQ and Quality):28

(43) Implicatures for p∨q:

p

Alt(p∨q)= p∨q p∧q

q

a. PIs: ¬Bs(p), ¬Bs(q) The rest, ¬Bs(p∧q), follows

b SI: Bs¬(p∧q)

Sauerland shows that this rather principled approach solves Chierchia’s puzzle once

the lexical alternatives that derive the sentential alternatives in (39) are specified. The

basic intuition is fairly straightforward. An utterance of p or q derives Ignorance

Inferences that are inconsistent with the assumption that the speaker is opinionated about

p, thereby solving problem (38)a. Problem (38)b is solved as well, but seeing this

requires precision about the relevant lexical alternatives and the way they determine

sentential alternatives for complex disjunctions, such as (37).

The starting point is the observation that in order to derive (39) the alternatives for

disjunction must contain two lexical entries that are never attested:29

(44) Horn-Set(or) = {or, L, R, and}, where pLq = p and pRq = q.

These four alternatives, when combined with the alternatives for some ({some, all}),

yield 8 alternatives to (37), based on (5) above:30

(45) Alt(r or sh) = a. r ∨ sh

28 From now on, I will circle those alternatives for which an SI can be derived consistently with Quality and

NG-MQ.

29 Spector (2003, 2005) suggests a different perspective. Specifically, he suggests that alternative sets are

defined as the closure under ∧ and ∨ of the set of positive answers to a given question. The Sauerland

alternatives for John talked to Mary or Bill would, thus, be derived (along with other useful alternatives) if

the relevant question was who did John talk to?

What FC teaches us, if my proposal is correct, is that there is no closure under ∧. Some of what I say

could work if Sauerland’s alternatives were replaced by basic answers to a Hamblin-question closed under ∨ .

quantification over pluralities). One would still have to make sense of second layers of exhaustivity (see

section 11.2 note 46.

30 r := John did the reading; sh := John did some of the homework; ah := John did all of the homework.

b. r L sh ≡ r

c. r R sh ≡ sh

d. r ∧ sh

e. r ∨ ah

f. r L ah ≡ r

g. r R ah ≡ ah

h. r ∧ ah

To see what PIs and SIs are derived, it is useful to plot the alternatives in a way that

indicates relative strength. But it is already easy to see how the two problems in (38) are

solved. To repeat, problem (38)a is solved based on the observation that the speaker

cannot believe that r is false if a PI ensures that she does not believe that sh is true and

Quality ensures that she believes that r or sh is true. Problem (38)b is solved based on the

observation that ah is a member of the alternative set (alternative g), and that an SI can be

derived for this alternative (consistent with prior inferences):

(46) Implicatures for r∨ sh:

ALT(r∨ sh) =

r

r∨ah r∧ah

r∨sh ah

r∧sh

sh

PI = ¬Bs(r∨ah), ¬Bs(sh), (the rest follow)

SI = Bs(¬ah), Bs(¬(r∧sh)) (the rest, Bs¬(r∧ah), follows)

The Horn-Set in (44) plays two independent roles for Sauerland. It provides NG-MQ with

the alternatives needed to derive the Ignorance Inferences for p∨q. These inferences

explain (within the Neo-Gricean paradigm) the lack of certain SIs when scalar items are

embedded within one of the disjuncts (problem (38)a). Furthermore, given (5), we can

generate alternatives for complex disjunctive sentences (e.g., q' for the sentence in (38))

that derive otherwise surprising SIs (problem (38)b).

However, the system makes a further prediction. Specifically, it predicts that in

certain contexts the two basic alternatives p and q will generate SIs rather than Ignorance

Inferences. The relevant case involves embedding of disjunction under an upward

monotone operator O such that O(p∨q) does not entail the disjunctive sentence

O(p)∨O(q). For such an operator, the following is not contradictory.

(47) O(p∨q) ∧ ¬O(p) ∧ ¬O(q) ∧ ¬O(p∧q)

Hence, if s utters O(p∨q), an SI would be generated for each of the stronger alternatives

(O(p), O(q), and O(p∧q)).

Evidence that this prediction is correct comes from (48) and (49), which naturally

yield the implicatures in (a) and (b).31

(48) You’re required to talk to Mary or Sue.

Implicatures:

a. You’re not required to talk to Mary.

b. You’re not required to talk to Sue.

Implicatures:

a. It’s not true that every friend of mine has a boy friend.

b. It’s not true that every friend of mine has a girl friend.

These facts follow straightforwardly from the Sauerland scale:

∀xP(x)

(50) Alt(∀x(P(x)∨Q(x))= ∀x(P(x)∨Q(x)) ∀x(P(x)∧Q(x))

∀xQ(x)

PIs = ¬Bs(∀xP(x)), ¬Bs(∀xQ(x)) (the rest, ¬Bs∀x(P(x)∧Q(x)), follows)

SIs = Bs(¬∀xP(x)), Bs(¬∀xQ(x)) (the rest, Bs ¬∀x(P(x)∧Q(x)), follows)

5.2. But…what about FC?

Sauerland’s system makes yet another prediction about disjunction embedding, a

prediction which is in direct conflict with FC. If disjunction is embedded under an

upward monotone operator O such that O(p∨q) entails the disjunctive sentence

O(p)∨O(q), the system predicts Ignorance Inferences with respect to O(p) and O(q). The

reasoning is exactly identical to the basic case of unembedded disjunction: there is no

way to assume that the speaker is opinionated about one of the alternatives O(p) and O(q)

without contradicting the Primary Implicature that the speaker does not know that the

other disjunct is true (given Quality).

This does not seem to be the correct prediction for existential modals and plural

existential DPs (generalization (31)). These operators, under their basic meaning, are both

In Fox (2003) Fox points out this prediction, but was not sure about the empirical facts. He was convinced by conversations with Benjamin Spector and the discussion in Sauerland (2005).

commutative with respect to disjunction (◊(p∨q) ≡(◊p∨◊q); ∃x(P(x)∨Q(x)) ≡ ∃xP(x)∨

∃xQ(x)). Hence Ignorance Inferences are predicted.

(51) You may eat the cake or the ice-cream.

◊p

Alt(51)= ◊(p ∨ q) ◊(p∧q)

◊q

PIs = ¬Bs(◊p), ¬Bs(◊q), ¬Bs◊(p∧q)

SIs = Bs¬ ◊(p∧q)

We’ve seen good arguments that FC should be derived as an implicature. However, under

Sauerland’s system we derive Primary Implicatures (in bold) that contradict FC.

The situation is quite interesting. The alternatives that K&S and Alonso-Ovalle

appeal to in order to derive FC derive contradictory Ignorance Inferences in Sauerland’s

system. If K&S are right, Sauerland’s system needs to change. However, K&S’s insight,

if correct, needs to be embedded in a general system for implicature computation, one

that can account for Chierchia’s puzzle, as well as for the emergence of Ignorance

Inferences.

There are two problems with Sauerland’s system. On the one hand, it derives Ignorance

Inferences that directly contradict the attested FC effect. On the other hand, it does not

provide the basis for anti-exhaustivity, which, if K&S are correct, is at the heart of FC. I

will argue that the first problem teaches us that, contrary to the neo-Gricean assumption,

Primary Implicatures do not serve the foundation for the computation of SIs. Instead, SIs

are derived in the syntactic/semantic component via an exhaustive operator, as suggested

in section 1.3. Once a semantic representation is chosen, Ignorance Inferences are

computed by the pragmatic system, based on the non-stipulative maxim of quantity (BMQ).

Without an exhaustive operator, incorrect Ignorance Inferences are computed in FC

environments. However, once we modify the meaning of exh, based on Sauerland’s

insights, the inferences can be avoided by a sequence of two exhaustive operators, which

yield, in effect, anti-exhaustivity, thereby solving the second problem. Furthermore, it

turns out that FC is predicted in all the environments discussed in section 3.

Let’s start by reviewing our lexical entry for only and exh from section 1.3. These entries

(14) and (15), repeated below) derive strong meanings that are in most cases equivalent

to the basic meaning conjoined with the SIs derived by the neo-Gricean system.

(52)a. [[only]] (A

NW(p,A) = {q∈A: p does not entail q}

b. [[Exh]] (A

23

However, predictions are sometimes different when the alternatives are not totally

ordered by entailment. In particular, for the sets of alternatives that Sauerland has

postulated, the lexical entries in (52) can derive contradictory results. To see this,

consider the following dialogue:

(53) A: John talked to Mary or Sue.

B: Do you think he might have spoken to both of them?

A: No, he only spoke to Mary OR Sue.

Under (52)a, A’s final sentence should presuppose that the prejacent, John spoke to Mary

or Sue, is true and that this is not the case for any of the (non-weaker) alternatives. Thus,

if Sauerland is right about the lexical alternatives for disjunction, the two alternatives in

(54) would both have to be false for the utterance to be true, which would, of course,

contradict the presupposition.

(54) a. John talked to Mary.

b. John talked to Sue.

This is a wrong result, which means that if Sauerland is right about the alternatives

for disjunction, the lexical entries in (52) probably need to be revised.32 A revision of this

sort is also needed based on much older observations due to Groenendijk and Stokhof

(1984):

(55) a. Who did John talk to?

Only Mary or SUE

b. Who did John talk to?

Only Some GIRL

Groenendijk and Stokhof (1984)

Let’s focus on (55)a. If (52)a is correct, the answer to the question should assert that

every alternative not entailed by the prejacent, John talked to Mary or Sue, is false. This

time the set of alternatives consists (most likely) of every proposition of the form John

talked to x based on the denotation of the question, and the fact that the whole DP Mary

or Sue is focused. In other words, the answer to the question in (55)a should entail the

propositions that John didn’t talk to Mary and that he didn’t talk to Sue, which should

contradict the presupposed prejacent.

Groenendijk and Stokhof (1984), who noticed the problem, suggested a modification

to the standard lexical entry for only, which was accommodated in Spector (2005) to the

syntax we are assuming (based on van Rooy and Schultz (2003)):

(56)a. [[only]] (A

b. [[Exh]] (A

32 Gennaro Chierchia (p.c.) points out that in the dialogue in (53) or might be receiving contrastive focus

with conjunction, with the other alternatives (L and R) inactive. This possibility will not be helpful in

explaining the avoidance of a contradiction in Groenendijk and Stokhof’s examples in (55).

Minimal(w)(A)(p) ⇔ ¬∃w'p(w')=1(Aw' ⊂ Aw)

Aω = {p∈A: p(ω)=1}

As pointed out by Spector (again, based on van Rooy and Schultz), this lexical entry can

solve Chierchia’s problem. However, it yields results that contradict FC (see note 41).

For this reason, I would like to suggest an alternative, one that is linked in a very direct

way to Sauerland’s proposal.

What we learn from Groenendijk and Stokhof is that there is something in the

meaning of only “designed” to avoid contradictions: only takes a set of alternatives A and

a prejacent p, and attempts to exclude as many propositions from A in a way that would

be consistent with the requirement that the prejacent be true. I would like to suggest that

the basic algorithm is Sauerland’s, i.e., that propositions from A are excluded as long as

their exclusion does not lead (given p) to the inclusion of some other proposition in A:

(57)a. [[only]] (A

∀q∈ NW(p,A) [q is innocently excludable given A q(w) =0]

b. [[Exh]] (A

[q is innocently excludable given A ¬q(w)]

q is innocently excludable given A if ¬∃q'∈ NW(p,A) [p∧¬q ⇒ q']

To see how this is supposed to work, consider an utterance of the disjunction p or q.

Consider first what happens without an exhaustive operator, under the basic syntactic

representation. Under such a representation, the sentence would assert that the disjunction

is true and would be consistent with the truth of the conjunction (inclusive or). By B-MQ

this would yield a variety of Ignorance Inferences, which might be implausible in a

particular context, and if so, would motivate the introduction of an exhaustive operator,

Exh(Alt(p or q))( p or q).

Under this alternative parse the sentence would assert that the prejacent p or q is true

and that every innocently excludable alternative is false. Assuming the Sauerland

alternatives, we derive the simple ExOR meaning. None of the disjuncts is innocently

excludable, since the exclusion of one will lead to the inclusion of the other, given the

prejacent. Once again, we will circle the innocently excludable alternatives.

p

(58) Alt(p∨q) = p∨q p∧q

q

Excluding p will necessarily include q while excluding q will necessarily include p.

p∧q is thus the only proposition in NW(p∨q, Alt(p∨q)) that can be innocently excluded

given the set of alternatives in (58). Thus, it is the only proposition that is excluded and

the derived meaning is the familiar ExOR.

Before moving to FC, I would like to show how the lexical entries in (57)b replicates

Sauerland results. But even before that, I would like to point out that Sauerland’s

algorithm is not totally contradiction free, and that his assumptions should therefore be

25

modified slightly. This modification would motivate a corresponding modification in

(57).

Consider the question answer pair in (59) from Groenendijk and Stokhof. Assume

that the alternatives for A is the Hamblin denotation of Q, Alt((59)A), in (60).

(59) Q: Who did Fred talk to?

A: Some GIRL

(60) Alt((59)A) = {that Fred talked to x: x is a person or a set of people}

assumptions are correct, it would be possible (by Sauerland’s algorithm) to introduce an

SI of the form Bs¬ϕ, for every ϕ in Alt((59)A). Each SI of this sort is consistent with the

set of PIs and Quality. However, once all the SIs are collected, the result is contradictory.

The problem extends to the lexical entry for exh in (57) (and for only, if we look back at

(55)b). Every member of Alt((59)A) is innocently excludable. Hence, if we were to

append exh to (59)A, the result would be contradictory.

One way to deal with this problem is to assume that the set of alternatives is always

closed under disjunction (see Spector 2005, as well as footnote 29). An alternative, which

is available when exh is assumed, is to eliminate additional elements from the set of

innocently excludable propositions for a prejacent, p, given a set of alternatives A,

I-E(p,A):

(61)a. [[only]] (A

∀q∈ I-E(p,A) q(w) =0

b. [[Exh]] (A

I-E(p,A) = ∩{A'⊆A: A' is a maximal set in A, s.t., A'¬ ∪ {p} is consistent}

A¬= {¬ p: p∈A}

propositions in A such that its exclusion is consistent with the prejacent. Every such set

could be excluded consistently as long as nothing else in A is excluded. Hence the only

propositions that could be excluded non-arbitrarily are those that are in every one of these

sets (the innocently excludable alternatives). Every proposition which is not in every such

set would be an arbitrary exclusion, since the choice to exclude it, will force us to include

a proposition from one of the other maximal exclusions (if the result is to be consistent),

and the choice between alternative exclusion appears arbitrary.33

To see what results is derived by this lexical entry, it is probably best to go through

the various cases we’ve discussed. Let’s first see how we would exhaustify p∨q given

exhaustification is reminiscent of what is needed for counterfactuals in the premise semantics developed by

Veltman (1977) and Kratzer (1981). In particular, the set of propositions that can be added as premises to a

counter factual antecedent p is ∩{A⊆C: A is a maximal set in C, s.t., A ∪ {p} is consistent} where C

is the set of all true propositions.

the Sauerland alternatives. The first step would be to identify the maximal consistent

exclusions given the prejacent p∨q. If p is excluded, q must be true and vice versa.

Hence, one maximal exclusion is {p, p∧q}, and the other is {q, p∧q}. The intersection is

p∧q, hence, Exh(Alt(p∨q))(p∨q) = (p∨q) ∧ ¬(p∧q) = p∇q.

p

(62) Alt(p∨q) = p∨q p∧q

q

We circle (with dotted-lines) the maximal exclusions consistent with the prejacent, and we circle

the intersection, the set of innocently excludable alternatives, with a completed line.

Exh(Alt(p∨q))(p∨q) = (p∨q) ∧ ¬(p∧q) = p∇q.

Consider now the exhaustification of (59)A under the assumption that the set of

alternatives is the Hamblin-denotation of the question, (60). Assume that there are three

girls in the domain of quantification, Mary, Sue, and Jane, and that there are no nongirls.

34 Every maximal exclusion will include every member of the Hamblin-set but one

of the following: (m) Fred talked to Mary, (s) Fred talked to Sue, and (j) Fred talked to

Jane. So the intersection of all-maximal exclusions, the set of innocently-excludable

alternatives, is the set of propositions of the form Fred talked to X, where X is a plurality

of girls:

(63) Alt((59)A) = {Fred talked to x: x a person or a set of people} =

m m&s

s m&j m&s&j

j s&j

Exh(Alt((59)A))((59)A) = Fred talked to some girl ∧ ¬(m&s) ∧ ¬(m&j) ∧ ¬(s&j)

= Fred talked to exactly one girl.

Consider again Chierchia’s sentence r∨sh and its Sauerland alternatives. To see which

alternative can be innocently excluded we have to identify the maximal (consistent)

34 Without non-girls, the answer is somewhat strange. That’s probably because questions presuppose that at

least one answer (in the Hamblin sense) is true, and, thus, without non-girls, the answer just repeats the

presupposition. Adding non-girls is thus crucial, but, it is trivial to see that it will not affect the result, in

any interesting way; propositions related to non-girls will be excluded and things will be more difficult to

draw, but other than that, it’s all the same.

exclusions. The set of innocently excludable alternative is the intersection. The reader can

consult the diagram in (64) to see that Sauerland’s results are replicated.

(64) Alt(r∨ sh) =

r

r∨ah r∧ah

r∨sh ah

r∧sh

sh

ah, r ∧sh, r∧ah are the proposition in Alt(ss∨b) that can be innocently excluded given the set of

alternatives:

Exh(Alt(r∨sh))(r∨sh) = (r∨sh)∧¬ ah ∧¬(r∧sh)

Consider next embedding under universal quantifiers. As discussed in section 5.1., such

embedding allows for the consistent exclusion of all the Sauerland-alternatives (other

than the prejacent). Hence, there is only one maximal exclusion, which is excluded by

exh:35

(65) You’re required to talk to Mary or Sue.

Implicatures:

a. You’re not required to talk to Mary.

b. You’re not required to talk to Sue.

(66) Every friend of mine has a boy friend or a girl friend.

a. It’s not true that every friend of mine has a boy friend.

b. It’s not true that every friend of mine has a girl friend.

35In conversation with Gennaro Chierchia, we’ve noticed that things are a little more complicated. As

things stand right now, Alt(∀x(P(x)∨Q(x)) contains additional members: ∃x(P(x)∨Q(x)), ∃x(P(x)), and

∃x(Q(x)). The latter two make it impossible to innocently exclude ∀xP(x) and ∀xQ(x). There are various

simple ways to correct for this problem. The obvious thing that comes to mind is to define Alt(S) so that it

includes only stronger sentences than S. However, this would be a problematic move given data that is not

discussed in this paper. Here’s another possibility: Alt(S) is the smallest set, s.t. (a) S∈ Alt(S), and (b) If

S’∈ Alt(S) and S” can be derived from S’ by replacement of a single scalar item with an alternative, and S’

does not entail S”, S” ∈ Alt(S).

28

∀xP(x)

(67) Alt(∀x(P(x)∨Q(x))= ∀x(P(x)∨Q(x)) ∀x(P(x)∧Q(x))

∀xQ(x)

Exh (Alt(∀x(P(x)∨Q(x)))(∀x(P(x)∨Q(x)))= ∀x(P(x)∨Q(x)) ∧¬∀xP(x) ∧¬∀xQ(x)

So the exhaustive operator as defined in (61)b, based on what’s needed for only, (61)a,

derives the same results as Sauerland’s system (with the exception of cases such as (59)

for which Sauerland’s system can derive contradictory implicatures). This is not

surprising. The set of innocently excludable proposition is (modulo (59)) precisely the set

of propositions for which Sauerland’s system yields SIs – for which SIs can be

introduced innocently.

However, there is an important architectural difference between the two systems, one

that relates to the division of labor between syntax/semantics and pragmatics. Under

Sauerland’s neo-Gricean system, NG-MQ (and the PIs that it generates) is the underlying

source of SIs. Under the syntactic alternative that we are considering, SIs have a syntactic

source, and can serve to avoid Ignorance Inferences, which are computed post

syntactically based on B-MQ. This architectural difference has empirical ramifications

for FC. We’ve already seen that Sauerland’s system predicts Ignorance Inferences that

contradict FC. We will see that under our syntactic alternative, the problem can be

avoided by recursive exhaustification.

Suppose that exh is a covert operator which can append to any sentence. It is reasonable

to assume that, in parsing (or producing) a sentence, exh will be used whenever the result

fairs better than its counter-part without exh. One way in which a sentence with exh

would be better than its exh-less counterpart is if the latter generates implausible

Ignorance Inferences based on B-MQ. We thus predict the following recursive parsing

strategy:

(68) Recursive Parsing-Strategy: If a sentence S has an undesirable Ignorance

Inference, parse it as Exh(Alt(S))(S).37

The core idea was developed during conversations with Ezra Keshet.

This should be modified to allow introduction of exh in a non-matrix position.

(i) Recursive Parsing-Strategy: If a sentence S has an undesirable Ignorance Inference, try to append exh

to some constituent X in S. I.e., modify the parse [S…X…] as follows: [S…Exh(Alt(X))(X)…].

We could also incorporate an economy condition of the sort alluded to in section 2.1.:

(ii) Condition on exh-insertion: exh can be appended to a constituent X, only if the resulting sentence

generates fewer Ignorance Inferences (based on B-MQ)

Consider the disjunctive sentence in (69)

(69) I ate the cake or the ice-cream.

If this sentence is parsed without an exhaustive operator, B-MQ will generate the

Ignorance Inference that the speaker doesn’t know what she ate (only that it included the

cake or the ice-cream or both). This inference might seem implausible, and the hearer

might therefore prefer the following parse, where C is the set of Sauerland-alternatives to

the disjunctive sentence.

(70) Exh(C)(I ate the cake or the ice-cream)

As we’ve seen already, the meaning of (70) is the ExOR meaning of (69). This meaning

will now generate (given B-MQ) the Ignorance Inference that the speaker doesn’t know

what she ate (only that, whatever it was, it included the cake or the ice-cream but not

both). This, again, might seem implausible, and the hearer might employ the parsing

strategy, again:

(71) Exh(C')[Exh(C)(I ate the cake or the ice-cream)]

where C'= Alt[Exh(C)(I ate the cake or the ice-cream)] = {Exh(C)(p): p∈C}

However, (71) ends up equivalent to (70). And further application of the parsing strategy

is not helpful either. It will, thus, follow that there is no way to avoid the (sometimes

undesirable) Ignorance Inference. To see this, we need to compute the set of alternatives,

C':

(72) C'= {1. Exh(C) (p ∨ q), 2. Exh(C)(p), 3. Exh(C)(q), 4. Exh(C)(p∧q)}

1. Exh(C) (p ∨ q) = (p ∨ q) ∧¬ (p∧q)

= (p ∧¬ q) ∨ ( q ∧¬ p)

2. Exh(C)(p)= p ∧¬ q

3. Exh(C)(q)= q ∧¬ p

4. Exh(C)(p∧q) = p∧q (can be ignored since already excluded by the prejacent)

Exh(C) (p ∨ q) = Exh(C)(p) ∨ Exh(C)(q)

Two simple observations are worth making. The first alternative, the prejacent of (71), is

equivalent to the disjunction of the second and third alternative, and the fourth alternative

is already excluded by the prejacent, and hence can be ignored. The relevant alternatives

are thus the following:

By the algorithm in (5).

2.Exh(C)(p)

(73) C' =Alt(Exh(p∨q)) = Exh(C)(p) ∨ Exh(C)(q)

3. Exh(C)(q)

Exh(C')[Exh(C) (p ∨ q)] = Exh(C) (p ∨ q) = (p ∨ q) ∧¬ (p∧q)

If 2 is excluded, 3 must be true, and vice versa. Hence, meaning does not change with a

second level of exhaustification, nor will it change when exh is appended yet another

time.39 There is thus no way to avoid what might be an undesirable Ignorance Inference.

Consider now (74).

(74) You may eat the cake or the ice-cream.

Without an exhaustive operator, this sentence will generate the Ignorance Inference that

the speaker doesn’t know what one is allowed to eat (only that the allowed things include

the cake or the ice-cream or both). This might seem implausible, and the hearer might opt

for another parse:

(75) Exh(C)(You may eat the cake or the ice-cream)

Given the Sauerland alternatives for disjunction, the set of alternatives, C, would be the

following:

(76) Alt(74)

◊p

C= ◊(p ∨ q) ◊(p∧q)

◊q Notice ◊(p ∨ q) ⇔ ◊p ∨ ◊q but (crucially)

◊(p∧q) <≠> ◊p ∧ ◊q

39The theorem in 1 is completely trivial, and the one in 2 (due to Benjamin Spector, p.c.) is less so:

1. Let C be a set of alternatives, Si, such that for each i exhaustification is trivial (i.e., Exh(C)(Si)⇔ Si),

then for each i, 2nd exhaustification is trivial (i.e., Exh(C')(Exh(C)(Si)) ⇔ Exh(C)(Si), where C' =

{Exh(C)(S): S∈C})

2. due to Spector: Let C be a set of finite alternatives, Si , then there is an n∈N, s.t. ∀m>n,

Exhn(C)(Si) = Exhm(C)(Si)

Exhn(C)(Si) := Exh(C')Exhn-1(C)(Si),

where C' = {Exhn-1(C)(S): S∈C}, and Exh1(C)(S) = Exh(C)(S).

40 The set of alternatives is actually larger, including a variant of each alternative in C with a universal

modal replacing the existential modal. This does not affect our results as the reader can verify. See the

appendix, as well as (84) and (85) where a parallel computation is carried out with the full set of

alternatives.

◊(p∧q) is the only proposition in Alt(◊(p∨q)) that can be innocently excluded given the

set of alternatives (excluding ◊p will necessarily include ◊q while excluding ◊q will

necessarily include ◊p). Hence, the meaning of (75) in our modal logic formalization is

◊(p ∨ q) ∧¬ ◊(p∧q).

Crucially (75) is consistent with the free choice possibility, ◊p∧◊q, though it, of

course, does not assert free choice.41 This new meaning will now generate the Ignorance

Inference that the speaker doesn’t know what one is allowed to eat (only that the allowed

things include the cake or the ice-cream but not both). This might seem implausible, and

the hearer might employ the parsing strategy again:

(77) Exh(C')[Exh(C)(You may eat the cake or the ice-cream)]

where C'={Exh(C)(p): p∈C}

This time, second exhaustification has consequences. To see this, we need to compute the

meanings of the various alternatives:

(78) C'= {1. Exh(C) (◊(p ∨ q)), 2. Exh(C)(◊p), 3. Exh(C)( ◊q), 4. Exh(C) (◊(p∧q))}

1. Exh(C) (◊ (p ∨ q)) = ◊(p ∨ q) ∧¬ ◊ (p∧q), crucially

≠ (◊p ∧¬ ◊q) ∨ (◊q ∧¬ ◊p)

2. Exh(C)(◊p)= ◊p ∧¬ ◊q

3. Exh(C)(◊q)= ◊q ∧¬◊ p

4. Exh(C) ◊(p∧q) = ◊(p∧q) (can be ignored since already excluded by the prejacent)

◊p ∧¬ ◊q

C'= ◊(p ∨ q)) ∧¬ ◊ (p∧q)

◊q ∧¬◊ p

(Excluding Exh(C)(◊p) will not necessarily include Exh(C)(◊q), and excluding

Exh(C)(◊q) will not necessarily include Exh(C)(◊p).)

Hence,

(79) Exc(C’)(Exh(C) (◊(p∨q))) = ◊(p ∨ q)) ∧¬ ◊ (p∧q) and

¬(◊p ∧¬ ◊q) and

¬(◊q ∧¬◊ p)

41 This exemplifies the difference between the lexical entry we are considering and the Groenendijk and

Stokhof-type alternative in (56). Under (56), (75) would express a stronger proposition ◊(p ∨ q) ∧¬

(◊p∧◊q), which will be inconsistent with FC.

= ◊(p) ∧ ◊ (q) and

¬ ◊ (p∧q)

We thus derive the FC effect for cases in which disjunction is embedded under existential

modals.42

The key to the distinction between disjunction embedded under an existential modal and

unembedded disjunction is that in the latter case the strongest alternative ◊(p∧q) is

stronger than the conjunction of the two other alternatives ◊p and ◊q. Hence, the first

layer of exhaustification is consistent with the later exclusion of Exh(C)◊p and

Exh(C)◊q.43 This answers Aloni and van Rooy’s (2005) objection to Kratzer and

Shimoyama (section 2.3.), and extends to account for embedding under existential

quantifiers:

(80) a. There is beer in the fridge or the ice-bucket.

b. People sometimes take the highway or the scenic route (Irene Heim, pc

attributing Regine Eckardt, pc)

c. This course is very difficult. Last year, some students waited 3 semesters to

complete it or never finished it at all. (Irene Heim, pc)

Here, too, first exhaustification will be fairly weak ∃x(Px∨Qx) ∧¬ ∃x (Px∧Qx) consistent

with later exclusion of Exh(C)∃xPx and Exh(C)∃xQx, the cumulative effect of which

entails ∃xPx∧∃xQx.

At the moment the system makes wrong predictions for embedding under singular

indefinites:

(81) a. There is a bottle of beer in the fridge or the ice-bucket.

b. This course is very difficult. Some student waited 3 semesters to complete it

or never finished it at all.

Right now, an FC effect is expected for this case as well. However, the expectation

changes once an independently needed difference between plural and singular indefinites

42 As pointed out in Simons (2005), ◊(p ∨ q) sometimes yields FC without the inference that ◊ (p∧q) is

false. A solution to this problem will be discussed in section 11.1.

43 The following is easily verifiable:

Let C= {w, s, n, e} be a diamond set of alternatives going stronger from w to e (w is weaker than s, n, and

e; s and n are logically independent and weaker than e), where w entails (s∨n). With such alternatives, 2nd

exhaustification of w is vacuous (Exh2(C)(w) ⇔ Exh(C)(w)) iff e ⇔ s&n. Furthermore, when 2nd

exhaustification of w is not vacuous, Exh2(C)(w) ⇔ s∧n∧¬e.

is factored in. Consider the sentences in (82). These sentences have the indicated

implicature that the alternative assertion involving quantification over plural individuals

is false:

(82) a. There is a bottle of beer in the fridge.

Implicature: there aren’t two bottles of beer in the fridge.

Implicature: It’s not true that two students talked to Mary.

This implicature leads to the conclusion that a singular indefinite is a scalar item, with a

plural (or dual) indefinite as an alternative:

(83) Horn-Set(Some NP-sing) = {Some NP-sing (henceforth ∃1),

two NPs (henceforth ∃2)}

With this Horn-Set, exh would, of course, derive the correct implicature for (82). But,

interestingly, we also explain the lack of FC in (81). To see this consider (81)a, and the

set of alternatives derived by (5), (84). The alternative-set includes alternatives of the sort

we’ve considered in (76) (upper diamond of (84)). But it also includes alternatives

generated by replacing ∃1 with ∃2 (lower diamond of (84)).

(84) Alt((82)a)

∃1xP(x)

∃1x (P(x)∧Q(x))

∃1xQ(x)

∃2xP(x)

C= ∃1x(P(x) ∨ Q(x)) ∃2x(P(x) ∨ Q(x)) ∃2x (P(x)∧Q(x))

∃2xQ(x)

The set of innocently-excludable alternatives contains ∃1x (P(x)∧Q(x)) as well as

∃2x(P(x)∧Q(x)). Hence the exhaustification of (82)a is the following:44

(85) Exh(C)( ∃1x(P(x) ∨ Q(x))) = ∃1x(P(x) ∨ Q(x)) &

¬∃1x(P(x) ∧ Q(x)) &

¬∃2x(P(x) ∨ Q(x))

when exh associates with ∨. An assumption of this sort was argued for on independent grounds in

Chierchia (2005).

⇒ ¬(∃1xP(x) ∧ ∃1x Q(x))

change by further exhaustification. We thus derive the generalization in (31)

It is of course important to return to our computation of basic FC and make sure that

nothing changes when universal quantifiers are introduced as alternatives to existential

quantifiers (4)b,d (see footnote 40). I leave this as a task for the interested reader, though

an equivalent computation will be carried out in (87) and (88), below, and the appendix

will contain a theorem that will make all of our results transparent with fewer

computations.

Multiple exhaustification also accounts for the generalization in (33), i.e., it generates FC

effects for the sequences ¬ ∧ and ¬ ∀∧ (introduced in 3.1 and 3.3.) I will illustrate this

for ¬ ∧, and allow the reader to verify that nothing changes when is replaced with ∀.

Consider (25), repeated below as (86), with its FC inference, which (to repeat) is not

predicted by the basic meaning.

(86)You are not required to both clear the table and do the dishes.

1. basic meaning: ¬(p ∧ q) ≡ ◊¬ (p ∧ q) ≡ ◊(¬p ∨ ¬q) ≡ ◊(¬p) ∨ ◊(¬q)

2. Free Choice: ◊(¬p) ∧ ◊(¬q)

Once again, FC will follow after two layers of exhaustivity are computed. Let’s start with

the first layer, which we compute based on the alternatives generated by Sauerland’s

Horn-Set {∧, L, R, ∧} and the traditional Horn-Set {◊, }, (5)d.

(87) Alt((86))

¬p

¬( p ∨ q)

¬q

¬◊p

C= ¬(p ∧ q)) ¬◊(p ∧ q) ¬◊(p ∨ q)

¬◊q

Exh(C)(¬(p ∧ q)) = ¬(p ∧ q)) &

( p ∨ q) &

◊(p ∧ q)

If we decide to add another layer of exhaustification, we get the following result:

(88)

¬p ∧ q ∧ ◊(p ∧ q)

C' = ¬(p ∧ q)) ∧ ( p ∨ q) ∧ ◊(p ∧ q)

¬q ∧ p ∧ ◊(p ∧ q)

Exh(C')[Exh(C)(¬(p ∧ q))] =

¬(p ∧ q)) & ( p ∨ q) & ◊(p ∧ q) & ¬(¬p ∧ q) & ¬(¬q ∧ p)

This yields the FC effect, based on the following equivalences:

¬(p ∧ q)) ≡ ◊¬p ∨ ◊¬q

¬(¬p ∧ q) ≡ ¬(◊¬p ∧ ¬◊¬q)

¬(¬q ∧ p) ≡ ¬(◊¬q ∧ ¬◊¬p)

((31) and (33)) can be derived based on recursive exhaustification under a Sauerlandinspired

meaning for exh. But before concluding, I would like to discuss two apparent

predictions of the account which are somewhat problematic.

11.1. ¬◊(p ∧q)

The lack of a conjunctive interpretation for p∨q was derived in section 7 on the basis of

the observation that the first layer of exhaustification excludes p∧q, an exclusion which

cannot be overridden at the second level of exhaustification. The situation changes in FC

environments, by the introduction of appropriate operators. When p∨q is embedded under

an existential quantifier, e.g. ◊ (p∨q), the first level of exhaustification excludes ◊ (p∧q),

a relatively weak exclusion, i.e. consistent with ◊ p∧◊ q. Hence it is possible (at the

second level of exhaustification) to innocently exclude the exhaustive interpretation of ◊p

and of ◊q.

This proposal makes a clear prediction, or at least so it seems. Specifically, it predicts

that FC will always be accompanied by the anti-conjunctive inference ¬◊ (p∧q).

However, it has been claimed that this prediction is false.

You may love one another and you may die, compatible with permission to do both.

You are not required to both love on another and die.

It seems quite hard to get rid of the inference that you are required to either love one another

or die.

It might now receive the following parse, where C' and C'' are determined based on scalar items (as outlined above), and C1, C2 are determined based on the focus value of the relevant prejacent .

Exh(C'')(Exh(C')(◊(Exh(C1)(we must love one another) or Exh(C2)(we must die)))).

p!:=Exh(C1)(p) = p ∧¬q

q!:=Exh(C2)(q) = q ∧¬ p

The analysis of free-choic crucially depends on the assumption that in the relevant sentences

disjunction receives narrow scope relative to the relevant existential quantifier.

We may love one another die (free-choice effect)

Either we may love one another or die (no free-choice effect)

assumptions made so far.

However, free-choice seems to be available in:

You may love one another or you may die.

We leave this as an unresolved problem, noting that the behaviour with indefinites is different.

Some students waited 3 semester to complete this course or never finished it at all. (free-choice)

Some students either waited 3 semester to complete this course or never finished it at all. (free-choice)

Either some students waited 3 semester to complete this course or some students never finished it at all. (no free-choice)

A free-choice effect depends on the nature of the alternatives (e.g., E must be stronger than the conjunction of N and S. The correlation with scope is predicted on the basis of the algorithm that determines alternatives.

At every level of exhaustfication , alternatives are determined on the basis of the structure of the prejacent.

It's been argued that a 'free choice' effect arises in two seemingly unrelated contexts.

by negation.

If B-MQ is correct, implicature must be derived within grammar.

The grammatical mechanism needed for FC seems to be an exhaustive operator, which can apply recursively to a single linguistic expression, based on a lexical entry.

If this is correct, it might be useful to ask questions about possible external/functional motivations for "exh."

We can prove a rather simple theorem which should allow the reader to understand the results described in this view with fewer computations.

We define an FC interpretation which we call AnEx (for Anti-exhaustivity), and prove that this

interpretation, if consistent, is the result of the 2nd layer of exhaustification. Let C be a set of propositions with p∈C, I = I-E(p,C) ≠∅, I'= (C I {p}) ≠ ∅, AnEx = ∩{¬ExhC(q):q∈I'}∩ ExhC(p)

Claim: If AnEx ≠∅ (is consistent), Exh2, C(p) =AnEx.. Proof: ExhC(p) entails ¬q, for all q∈I (by definition of Exh). Hence, ExhC(p) entails ¬ ExhC(q), for all q∈I (¬q entails ¬ ExhC(q)). Hence, AnEx entails ¬ExhC(q), for all q∈I (AnEx has ExhC(p) as a conjunct). Hence, AnEx =∩{¬ExhC(q):q∈I'}∩{¬ExhC(q):q∈I}∩ExhC(p)=∩{¬ExhC(q): q∈ C {p}} ∩ ExhC(p). Hence, If AnEx is consistent, I-E(ExhC(p), C') = C' {ExhC(p)} (where C': {ExhC(q):q∈C}). Hence, Exh2 C(p) = ∩{¬ExhC(q):q∈ C' {ExhC(p)}} ∩ ExhC(p) = AnEx (by definition of exh)

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