Free choice and non-alethic discourse: toppling a myth

In online study, Aloni writes:

"The law of propositional logic that states the deducibility of either A or B
from A is not valid for imperatives (Ross’s paradox, cf. [9])."

--- but people should be citing Hare, who, citing Grice, did state it IS VALID (regardless of 'queer' effects). Grice was a master of explaining the apparently odd as not really odd after all.

Aloni goes on:

The command
(or request, advice, etc.) in (1a) does not imply (1a) (unless it is taken in
its alternative-presenting sense), otherwise when told the former, I would be
justified in burning the letter rather then posting it.

(1)

a.

Post this letter!

6)

b.

Post this letter or burn it!

"Intuitively the most natural interpretation of the second imperative is as one
presenting a choice between two actions."

"Following [2] (and [6]) I call these
choice-offering imperatives."

"Another example of a choice-offering imperative is
(2) with an occurence of Free Choice ‘any’ which, interestingly, is licensed in
this context.

(2) Take any card!

Like (1a), this imperative should be interpreted as carrying with it a permission
that explicates the fact that a choice is being offered.
Possibility statements behave similarly (see [8]). Sentence (3b) has a reading
under which it cannot be deduced from (3a), and ‘any’ is licensed in (4).

(3) a. You may post this letter. 6) b. You may post this letter or burn it.

(4) You may take any card.

In [1] I presented an analysis of modal expressions which explains the phenomena
in (3) and (4). That analysis maintains a standard treatment of ‘or’
as logical disjunction (contra [11]) and a Kadmon & Landman style analysis
of ‘any’ as existential quantifier (contra [3] and [4]) assuming, however, an independently
motivated ‘Hamblin analysis’ for _ and 9 as introducing sets of
alternative propositions.

Modal expressions are treated as operators over sets
of propositional alternatives. In this way, since their interpretation can depend
on the alternatives introduced by ‘or’ (_) or ‘any’ (9) in their scope, we can
account for the free choice effect which arises in sentences like (3b) or (4). In
this article I would like to extend this analysis to imperatives. The resulting
theory will allow a unified account of the phenomena in (1)-(4). We will start
by presenting our ‘alternative’ analysis for indefinites and disjunction.


2 Indefinites and disjunction

Indefinites (e.g. ‘any’) and disjunction (e.g. ‘or’) have a common character reflected
by their formal counterparts 9 and _. Existential sentences and logical
disjunctions assert that at least one element of a larger set of propositions is
true, but not which one. Both constructions can be thought of as introducing a
set of alternative propositions and, indirectly, raising the question about which
of these alternatives is true. In what follows I propose a formal account of the
sets of propositional alternatives introduced by indefinites and ‘or’ (cf. [1]).
I recursively define a function [•]M,g whereM is a pair consisting of a set of
individuals D and a set of worlds W, and g is an assignment function. Function
[•]M,g maps formulae to sets of pairs hw, si consisting of a world w 2 W and
a sequence of values s, where the length of s is equivalent to the number n( )
of surface existential quantifiers in , – for atoms and negations, n( ) = 0; for
= 9x , n( ) = 1 + n( ), and for = 1 ^ 2, n( ) = n( 1) + n( 2). (By
[[ ]]M,w,g I refer, as standard, to the denotation of in M, w and g.)
Definition 1

1. [P(t1, ..., tn)]M,g = {hhi,wi | h[[t1]]M,w,g, ..., [[tn]]M,w,gi 2 [[P]]M,w,g};

2. [t1 = t2]M,g = {hhi,wi | [[t1]]M,w,g = [[t2]]M,w,g};

3. [¬ ]M,g = {hhi,wi | ¬9s : hs,wi 2 [ ]M,g};

4. [9x ]M,g = {hds,wi | hs,wi 2 [ ]M,g[x/d]};

5. [ ^ ]M,g = {hs1s2,wi | hs2,wi 2 [ ]M,g & hs1,wi 2 [ ]M,g}.

Disjunction _, implication ! and universal quantification 8 are defined as standard
in terms of ¬, ^ and 9. Truth and entailment are defined as follows.

Definition 2 [Truth and entailment]

(i) M,w |=g iff 9s : hs,wi 2 [ ]M,g;

(ii) |= iff 8M, 8w, 8g : M,w |=g ) M,w |=g .

In this semantics, a formula is associated with a set of world-sequence
pairs, rather than, as usual, with a set of worlds. This addition is essential to
derive the proper set ALT( )M,g of alternative propositions induced by formula
, which is defined as follows.

Definition 3 ALT( )M,g = {{w | hs,wi 2 [ ]M,g} | s 2 Dn( )}.

For example, the set [P(x)]M,g = {hhi,wi | [[x]]M,w,g 2 [[P]]M,w,g} determines
the singleton set of propositions {that x is P}. More interestingly, the set
[9xP(x)]M,g = {hhdi,wi | d 2 [[P]]M,w,g} determines the set of alternatives
{that d1 is P, that d2 is P, . . . }, containing as many elements as there are
possible values for the quantified variable x.
On this account, the propositional alternatives introduced by a sentence
are defined in terms of the set of possible values for an existentially quantified
variable. To properly account also for the alternatives introduced by disjunctions,
I propose to add to our language, variables p, q ranging over propositions,
so that, for example, we can write 9p(_p^ p =^A) for A, where the operators _
and ^ receive the standard interpretation, so that, for example, [[_p]]M,g,w = 1 iff
w 2 g(p), and [[^A]]M,g,w = [[A]]M,g. In interaction with 9 or _, this addition, otherwise
harmless, extends the expressive power of our language in a non-trivial
way. Although the (a) and (b) sentences below are truth conditionally equivalent,
the sets of alternatives they bring about, depicted on the right column,
are not the same. While the (b) representations introduce singleton sets, the
(a) representations induce genuine sets of alternatives.

(5)

a.

9p(_p ^ (p =^A _ p =^B)) a0. A

B

b. 9p(_p ^ p =^(A _ B)) b0. A _ B
(6) a. 9p(_p ^ 9x(p =^A(x))) a0.

A(d1)

A(d2)
. . .
b. 9p(_p ^ p =^9xA(x)) b0. 9xA(x)

That these alternatives are needed is seen when we consider question semantics.
If we take questions ? to denote the sets of alternatives induced by ,
the pair in (5) allows us a proper representation for the ambiguity of questions
like ‘Do you want coffee or tea?’ between an alternative reading (expected answers:
coffee/tea), and a polar reading (expected answers: yes/no) (see [10]).
The sets of alternatives induced by (6a) and (6b) can serve as denotations for
constituent questions (e.g. ‘who smokes’) and polar existential questions (e.g.
‘whether anybody smokes’) respectively.

3 Imperatives

"While assertion have truth conditions, imperatives have compliance conditions."

"Someone cannot be said to understand the meaning of an imperative unless
he recognizes what has to be true for the command (or request, advice, etc.)
issued by utterance of it to be complied with."

"The framework presented in the
previous section supplies us with a straightforward method to characterize the
compliance conditions of imperative ! , namely by identifying them with the
set of alternatives induced by ."

"For example, the compliance conditions of the
imperative

Post this letter!

will be the singleton set containing the proposition

‘that the addressee posts the letter’.

1

Crucially choice-offering imperatives will
involve genuine sets of alternatives.

For example, the compliance conditions of

Post this letter or burn it!

on its choice-offering reading, will contain the two
propositions:

‘that the addressee posts the letter’ and ‘that the addressee burns
the letter’. Each of these propositions represents a possible way to comply with
the command (or request, advice, etc.) expressed by the imperative.

"Strictly speaking imperatives lack truth conditions."

"This would suggest
to identify their meaning with their compliance conditions."

"There is a sense,
1. We are bypassing the fact that imperatives deal with future actions, so the relevant proposition
here should be ‘that the addressee will post the letter’. See Rosja Mastop’s contribution
to this volume ‘Imperatives and Tense’.

----

however, in which the utterance of an imperative expresses some fact about the
desire state of the speaker."

"In order to account for this intuition, in this article,
I shall assume that imperatives ! denote propositions that specify desirable
situations."

---- cfr. Grice, "Probability, DESIRABILITY and mood operators".

"This means that they are interpreted with respect to a modal base Aw expressing the desires of (one of) the participants to the conversation at world w."

Definition

4

[Imperatives]


[! ]M,g

= {hhi,wi | 8 2 ALT( )M,g : 9w0 2 Aw :

w0 2 & 8w0 2 Aw : 9 2 ALT( )M,g : w0 2 }

"On this account, !’ is an operator over the set of propositional alternatives introduced
in its scope."

"Imperative ! is true in w iff (i) every alternative induced by
is compatible with the desire state Aw."

(ii) the union of all these alternatives
is entailed by Aw.

"Intuitively, clause (ii) expresses the fact that if I say

Post the letter or burn it!

then, in each of my desirable worlds, it should hold that
either the letter is posted or burnt.

Clause (i)

expresses the fact that, in this
case, my desires must be consistent with both options.


In this framework we can give a straightforward treatment of ‘embedded
uses’ of imperatives like in


Vincent wants you to post this letter.


We first define
a relation of entailment between desire states and imperatives, as follows.

State
entails ! , |=M,g! iff 9w : M,w |=g! and Aw = . We then assume that
a sentence like

Vincent wants !’ is true in w iff Vincent’s desire state in w
entails ! .


Let us see now how the choice-offering imperatives discussed in the introductory
part of the article are analyzed in this framework.

Applications

Example (7) is ambiguous between a choice-offering reading, represented
in (7a), and an alternative-presenting reading in (7b).

(7) Post this letter or burn it!

a. !9p(_p ^ (p =^A _ p =^B)) a0. A

B

b. !9p(_p ^ p =^(A _ B)) b0. A _ B

The choice-offering reading involves the set containing the two propositions:
‘that the addressee posts the letter’ and ‘that the addressee burns the letter’,
both expressing a possible way of complying with the imperative. The weaker
reading in (7b) instead induces the singleton set containing the proposition ‘that
the addressee posts the letter or burns it’. Since, by clause (i) of our definition,
all the alternatives induced by the embedded clause must be consistent with
the modal base, only on this second reading is the sentence compatible with a
subsequent imperative: ‘Do not burn the letter!’ Assuming a standard treatment
of 3 and 2, the following holds:

(8) a. !9p(_p ^ (p =^A _ p =^B)) |= 3A,3B,2(A _ B)

b. !9p(_p ^ p =^(A _ B)) 6|= 3A,3B

Example (9a) is analyzed as in (9b) which induces the set containing the
propositions ‘that the addressee takes the ace of hearts’, ‘that the addressee
takes the king of spades’, . . .

(9) a. Take any card!

b. !9p(_p ^ 9x(p =^A(x))) b0.
A(d1)
A(d2)
. . .
Compare (9) with the following two examples where no choice is being offered:
(10) a. Take every card!
b. !9p(_p ^ (p =^8xA(x))) b0. 8xA(x)
(11) a. Take a card!
b. !9p(_p ^ p =^9xA(x)) b0. 9xA(x)

In principle our semantics predicts (11b) as second possible reading for
sentence (9a). Intuitively, however, (9a) never obtains such a ‘pure’ existential
meaning. Imperative ‘Do not take the ace!’ would never be acceptable after
(9a). Our representation (9b) accounts for this fact, because it entails that any
card may be taken. Representation (11b), instead, lacks this entailment.
(12) a. !9p(_p ^ 9x(p =^A(x))) |= 8x3A,29xA
b. !9p(_p ^ p =^9xA(x)) 6|= 8x3A

In order to explain why reading (11b) is not available for sentence (9a), I will use
Kandom and Landman’s analysis of any (see [7]). According to their account,
any phrases are indefinites which induce maximal widening of the domain as
part of their lexical meaning. Crucially this widening should be for a reason,
namely, they propose, the strengthening of the statement made. If we define the
strength of an imperative in terms of entailment, |=, in the ‘pure’ existential
reading (11b), widening the domain would weaken the statement. This explains
why this reading is not available for the any-sentence (9a). But what about the
‘free choice’ reading in (9b)? Why is this available? Unfortunateky widening the
domain in this case does not make our statement stronger. None of the wide
or the narrow interpretation of sentence (9b) entail the other. We lack then an
explanation of why (9a) can be interpreted at all. In order to solve this problem
we have to say something more about in what sense an imperative can be said
to be stronger than another.

In this framework, we have a number of alternative options for defining
the relative strength of imperatives. Entailment is one possibility. The following
two are other particularly interesting options.

1. !A | 1!B iff 8 2 ALT(A) : 9 2 ALT(B) : ;

2. !A | 2!B iff 8 2 ALT(B) : 9 2 ALT(A) : .

"Intuitively, imperative !A is as strong1 as !B, !A | 1!B iff each way of complying
with !A is also a way of complying with !B. Whereas !A | 2!B holds iff any
way of complying with !B is part of a strategy to comply with !A. If ! | 1!
and ! | 2! , then ! |=! ."

"If !A and !B denote singleton sets, | 1 and | 2 (and |=) define the same
notion. For example, imperative (13a) is stronger than (13b) according to both
notions. Indeed, every way of satisfying (13a) satisfies (13b), and to satisfy (13b)
is part of a strategy to satisfy (13a).

(13) a. Put all books in your bag! b. Put the Tractatus in your bag!

Once choice-offering imperatives enter the picture though, the two notions
give opposite results (by !(A _c B) I refer to the free choice reading of a
disjunctive imperative e.g. (7a)):

(14)

a.

Post this letter!

b. Post this letter or burn it!

c.

!A | 1!(A _c B) and !(A _c B) 6| 1!A

d.

!A 6| 2!(A _c B) and !(A _c B) | 2!A

e.

!A 6|=!(A _c B) and !(A _c B) 6|=!A


Sentence (14a) is strictly stronger1 than (14b), because posting the letter is a
way to satisfy (14b), but burning the letter is not a way to satisfy (14a). On
the contrary, sentence (14b) is strictly stronger2 than (14a), because posting
the letter is part of a strategy to satisfy (14b), but there is a way to satisfy the
latter, namely burning the letter, which is not part of a strategy to satisfy (14a).
Going back to our example (9), in the ‘pure’ existential readings in (11b),
widening the domain makes our statement weaker according to all notions |=,
| 1 and | 2. This explains why this reading is not available for the any-sentence
in (9). In the ‘free choice’ reading in (9b), widening makes the statement weaker
according to notion | 1, but stronger according to notion | 2. This, I suggest,
supplies enough reason for widening to occur.

References


[1] Maria Aloni.

Free choice in modal contexts.

In Arbeitspapiere des Fachbereichs
Sprachwissenschaft. University of Konstanz, 2003.

[2] Lennart Aquist.

Choice-offering and alternative-presenting disjunctive
commands. Analysis, 25:185–7, 1965.


[3] Veneeta Dayal.

Any as inherently modal. Linguistics and Philosophy,

21:433–476, 1998.

[4] Anastasia Giannakidou.

The meaning of free choice. Linguistics and Philosophy,
24:659–735, 2001.

Grice, H. P. 1961. The causal theory of perception. "My wife is in the kitchen or in the bathroom"

---- 1966. Logic and Conversation: The Oxford lectures

--- 1971. Probability, desirability and mood operators.

--- 2001. Aspects of reason. On Satisfactoriness of "!p v !q" as entailed by !p -- criticised online by Harman, "Mail those letters or post them!"
--

HARE, R. M.
---- Some alleged differences between imperatives and indicatives. Mind 1961. Citing Grice extensively.

---

[5] Charles L. Hamblin.

Questions in Montague English. Foundation of Language,
10:41–53, 1973.

[6] Charles L. Hamblin. Imperatives. Basil Blackwell, 1987.

[7] Nirit Kadmon and Fred Landman. Any. Linguistics and Philosophy,
16:353–422, 1993.

[8] Hans Kamp.
Free choice permission. Proceedings of the Aristotelian Society,
74:57–74, 1973.
---------------------- citing Grice, "be brief!" etc.


[9] Alf Ross. Imperatives and logic. Theoria, 7:53–71, 1941.
---- repr. "Philosophy of Science", 1944.


[10] Arnim von Stechow.

Focusing and backgrounding operators. In Werner
Abraham, editor, Discourse Particles, number 6, pages 37–84. John Benjamins,
Amsterdam, 1990.

[11] Ede Zimmermann.

"Free choice disjunction and epistemic possibility."

Natural
Language Semantics, 8:255–290, 2000