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Saturday, May 30, 2015

Auden and Grice


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.

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

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

(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.
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?
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).
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

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)]

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
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.
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

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

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).
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
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
(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
(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).
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:
(39) Alt(p∨q)= p∨q p∧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.
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
(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:
Alt(p∨q)= p∨q p∧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
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∨ah r∧ah
r∨sh ah
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.
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.

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:
(50) Alt(∀x(P(x)∨Q(x))= ∀x(P(x)∨Q(x)) ∀x(P(x)∧Q(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.
Alt(51)= ◊(p ∨ q) ◊(p∧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

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)
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

(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.
(58) Alt(p∨q) = p∨q p∧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
modified slightly. This modification would motivate a corresponding modification in
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,
(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.
(62) Alt(p∨q) = p∨q p∧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∨ah r∧ah
r∨sh ah
ah, r ∧sh, r∧ah are the proposition in Alt(ss∨b) that can be innocently excluded given the set of
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
(65) You’re required to talk to Mary or Sue.
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).
(67) Alt(∀x(P(x)∨Q(x))= ∀x(P(x)∨Q(x)) ∀x(P(x)∧Q(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
(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,

(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).


(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

(76) Alt(74)
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

◊(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).)
(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

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

(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

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

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 ∨ q)
C= ¬􀀀(p ∧ q)) ¬◊(p ∧ q) ¬◊(p ∨ 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:
¬􀀀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. 


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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