From an online study by M. Aloni, "Implicatures of imperatives"
The article defines the relevance or utility of an imperative in terms of
how far it can help in increasing the probability of the occurrence of a
desirable future world.
In terms of this notion, Aloni accounts for
(i) the potential of imperatives to license
free-choice "any" in their scope;
and
(ii) the "free-choice" effects of disjunctive imperatives ("Mail those letters and post them!") and any-imperatives.
1 Choice-offering imperatives
1.1 "Or" in imperatives
It is a well known fact that or in imperatives can give rise to something that some people would call a "free-choice" effect. See(Ross, 1941, "Imperatives and logic" in Theoria, analysed along Griceian lines by Hare, 1967 -- incredibly how people think they can FAIL to quote Hare; A° quist, 1965; Hamblin, 1987)
and more recently (Aloni, 2003).
(1)
!(A _ B) ) 3A ^ 3B
As an illustration of (1), consider the following
example:
(2) SMITH:
Take her to Knightsbridge or Bond Street!
JONES STARTS TO LEAVE.
SMITH: (?)
Don’t you dare take her to Bond Street!
"Intuitively the most natural interpretation of Smith’s first imperative is as one presenting a "choice" between two different actions.
Smith’s subsequent imperative can be regarded as negating this choice, and, therefore, strikes us as out of place here.
The free choice inference in (1), however, is not always warranted as illustrated by the following example from Rescher and Robinson
(1964):
(3) TEACHER:
John, stop that foolishness or leave the room!
JOHN STARTS TO LEAVE.
TEACHER: Don’t you dare leave this room!
Examples like (3) suggest that we should treat "free-choice" effects as conversational implicatures, rather than semantic entailments.
In the classical literature (notably (A° quist, 1965 -- not citing Grice)), examples like
(3) has been presented as evidence in favour of an ambiguity between "choice"-offering and alternative-presenting disjunctive imperatives.
CANCELLATION
On a pragmatic approach, the failure of the "free-choice" implicatum in example (3)
can be explained as an implicature cancelation without multiplying the senses of imperative sentences.
A further indication that free choice effects of disjunctive imperatives are conversational implicatures is the fact that they disappear in
negative environments (e.g. Gazdar 1979).
"Don't mail those letters or don't burn them!"
(4) Don’t post this letter or burn it!
If "free-choice" implicata had the status of logical
entailment, then (4) could be used in a
situation in which one wants the letter to be
posted or burnt, but doesn’t want to leave the
choice to the hearer.
This is clearly not so.
------- the case of "any"
1.2 Any in imperatives
Another example of a ‘choice-offering’ imperative
is (5) with an occurrence of free
choice any which is licensed in this context.
(5) Take any card!
Like disjunctive imperatives, any-imperatives
should be interpreted as carrying with them
the inference that a choice is being offered.
(6) !(any x ) ) 8x3
As in the case of disjunctive imperatives,
the free choice effect in (6) disappears under
negation. One needs a special stress to retain
it, as in (8).1
(7) Don’t take any card!
(8) Don’t take just ANY card!
Contrary to disjunctive imperatives, however,
in a positive environment, the inference
in (6) is hard to cancel. Contrast (9) with (10).
(9) MARIA: Take any card!
YOU START TO TAKE A CARD.
MARIA: # Don’t you dare take the ace!
(10) MARIA: Take a card!
YOU START TO TAKE A CARD.
MARIA: (?) Don’t you dare take the ace!
1
The use of "any" illustrated in (8) have been called antiindiscriminative
in (Horn, 2000 -- but cfr. his more recent, "A brief history of negation") and anti-depreciative in
(Haspelmath, 1997).
On the present account, sentences like
(8) must be taken to involve a metalinguistic use of negation.
Imagine a context in which it is well known
that aces cannot be taken. In such a context,
Maria’s second imperative in (10) would be
natural.
In (9), however, it would be still out
of place. By using any, in (9), rather than a,
Maria conveys that no exceptions apply to her
prescription: even aces must be permissible
options.
This reduced tolerance of exceptions typical
of uses of any has been discussed in (Kadmon
and Landman, 1993).
On their account,
any has the effect of WIDENING the domain of
quantification compared to a standard use of
an indefinite noun phrase. Furthermore, domain
widening should be for a reason. Any
is licensed only in those cases where widening
the domain is functional, i.e., leads to a
STRENGTHENING of the statement made.
Domain widening and strengthening (defined
in terms of entailment) explain the following
distribution facts:
(11) a. John did not take any card.
¬9x
b. # John took any card.
9x
Enlarging the domain of an existential in the
scope of negation does create a stronger statement
(example (11a)). In an episodic sentence,
it doesn’t (example (11b)).
It is easy to see, however, that this sort of
explanation does not extend directly to nondeclarative
cases.
Let us assume Groenendijk
and Stokhof’s (1984) notion of entailment
for interrogatives, and the
standard notion of
entailment for imperatives defined in terms
of inclusion of their compliance conditions.2
Then, widening the domain of an existential
in an interrogative or an imperative does not
create a stronger sentence, still any is licensed
in (12) and (13).
2
Imperative I entails I0 iff each way of complying with I
is a way of complying with I0. See e.g. (Hamblin, 1987).
(12) Did John take any card? ?9x
(13) Take any card! !9x
To explain (12), (van Rooij, 2003) proposed to
interpret "strength" in terms of relevance rather
than entailment, and provided a perspicuous
characterization of the relevance of a question
in terms of the decision theoretic notion of expected
utility.
In this article I would like to extend van
Rooij’s (2003) proposal to imperatives. In
order to do this, I will define a notion of
the relevance or utility of an imperative in a
context as a function of the probability of its
compliance and its desirability. According to
this notion, in example (13), domain widening
can lead to an interpretation with a higher expected
utility because it can increase the probability
of a positive response from the hearer.
In this sense, I would like to suggest, imperatives
meet Kadmon and Landman’s requirement
that domain widening should be functional.
Intuitively, by enlarging the domain
of an existential quantifier in an imperative
the speaker indicates that she will be pleased
by more ways of complying with her wishes.
This increases her chances that the hearer will
comply.
Note that domain widening increases
utility only in a situation in which no element
in the enlarged domain is ruled out as an option.
This allows us to derive from (13) the
permission to take any card as an implicature.
Since any can be used only in situations
where domain widening increases utility, this
explains why this implicature is hard to cancel.
Since existential sentences can be seen
as generalized disjunctive sentences, the free
choice implicatures of disjuntive imperatives
follow by the same reasoning.
In this case,
however, these implicatures can be canceled,
like in Rescher & Robinson’s example where
the implicated material was in conflict with
shared assumptions in the common ground.
The expected utility value of an imperative I is defined in terms of how far
I can help in increasing the probability of the
occurrence of a desirable future world.
Expected
utility values will be calculated with
respect to a state representing the speaker’s
beliefs and desires about the future.
2.1 States
A state is a pair (p, u) consisting of a probability
function p on the set W of possible
worlds and a utility function u.
The probability function p maps worlds to
numbers in the interval [0, 1], with the constraint
that
P
w2W p(w) = 1. Probability distributions
can be extended to subsets C of
W as follows: p(C) =
P
w2C p(w). In this
context, a world represents a way in which
things might turn out to be in the near future.
The probability function p represents the
belief of the agent with respect to the probability
of the occurrence of a world w. The
value p(w) may depend on a number of factors,
like physical possibility (relative to the
laws of nature), temporal possibility (possible
in the time), and, most important, active possibility
(relative to the willingness of the other
people to co-operate). If p (w) 6= 0 we will
say that w is possible in .
The utility function u is a mapping from W
to the set {0, 1} and expresses the desirability
of a world w. Desirable worlds obtain value
1, undesirable worlds, value 0.
As an illustration of these notions, consider
the following examples of a state (for simplicity
we are considering only four worlds,
where each world is indexed with the atomic
propositions holding in it. For example, in wq,
only q holds, and in w, no atomic proposition
holds):
(14) a.
p u
wq 1/2 1
wr 1/2 1
wqr 0 0
w 0 0
b.
p u
wq 0 1
wr 3/4 0
wqr 0 0
w 1/4 0
c.
p u
wq 1/6 1
wr 1/6 1
wqr 0 0
w 2/3 0
In order to understand this notion it might be
useful to ask ourself in which of these states
one would rather be. Intuitively, (14a) is the
best choice. Each world which is still possible
there, is also desirable.
State (14b) is
the worst choice, none of the possible worlds
is a desirable one. Finally, in (14c), which is
probably the most realistic option, some of the
possible worlds are desirable, some are not.
The notion of the value of a state defined in
the following paragraph is meant to capture
these intuitions.
2.2 The value of a state
We can think of a state = (p, u) as a degenerate
decision problem in which the set of
alternative actions has just one element. Following
the standard notion of expected utility
in Bayesian decision theory, I define the value
of a state as follows:
(15) V ( ) =
P
w2W(p (w) × u (w))
The value of a state expresses the probability
in of the occurrence of a desirable world.
A state with value 1 is one in which each possible
world is also desirable, e.g. (14a) above.
A state with value 0 is one in which none of
the possible worlds are desirable, e.g (14b).
More realistic states are those in which the
value lies between 0 and 1, like (14c) above
with value (1/6 + 1/6) = 1/3.
In order to increase the value of a state,
an agent may do different things. She might
change her desire or, better, she might act in
order to change her probability function, for
example, by using an imperative. Declaratives
do not have the power to change the probability
of a future world, imperatives do. The
goal of a declarative is to update an information
state. The goal of an imperative is to enlarge
the chance of the occurrence of a desirable
world.
In what follows I will characterize the expected
utility of an imperative in a state in
terms of how far it can help in increasing the
value of . More precisely, the expected utility
value of an imperative I will be defined in
terms of the utility value and the probability of
the proposition CI expressing the compliance
conditions of I.
2.3 Compliance conditions
"Declaratives have truth-conditions, interrogatives have "answerhood-conditions", imperatives have "compliance"-conditions."
"Someone
cannot be said to understand the meaning of
an imperative I unless she recognizes what
has to be TRUE for the command (or request,
advice, etc.) issued by an utterance of I to
be complied with."
"I shall identify the compliance
conditions C! of imperative ! with the
proposition expressed by .3
For example,
(16) I:
‘Kill Bill!’
CI : ‘That the hearer kills Bill’
(17) I: ‘Kill Bill or John!’
CI : ‘That the hearer kills Bill or John’
3
AGAINST THE RADICAL:
see (Mastop, 2005) or (Portner, 2004) who, among others, have argued that an imperative is better analyzed in terms of an "action" or a "property" rather than a full "proposition".
----- Jones perhaps should like the bit above.
2.4 Utility value of a proposition
Following (van Rooij, 2003), we define the
utility value UV (C, ) of a proposition C in a
state as the difference between the value of
after updating with C and before updating
with C, where updates are defined in terms of
Bayesian conditionalizations.
(18)
UV (C, ) = V ( /C) − V ( )
where /C = (pC, u) and pC is the old probability
function p conditionalized on C, that is,
for each world w:
(19) pC(w) = p(w & C)/p(C)
The utility value of a proposition C in a state
expresses how much an update with C can
enlarge the value of .4
As an illustration, let us calculate the utility
value of the following three propositions
in the state (14c) above.
(20) q _ r, q, ¬q
In order to do this we need to update (14c)
(rewritten as in (21)) with the propositions
in (20) and calculate the value of the resulting
states.
(21)
p u
wq 1/6 1
wr 1/6 1
wqr 0 0
w 2/3 0
(22) a. /(q _ r)
p u
wq 1/2 1
wr 1/2 1
wqr 0 0
w 0 0
4
This notion is different from the value of sample information
of statistical decision theory, e.g. (Raiffa and
Schlaifer, 1961).
b. /q
p u
wq 1 1
wr 0 1
wqr 0 0
w 0 0
c. /¬q
p u
wq 0 1
wr 1/5 1
wqr 0 0
w 4/5 0
States (22a) and (22b) have value 1. State
(22c) has value 1/5. Since V ( ) = 1/3, we
obtain for our three propositions the following
utility values:
(23) a. UV (q _ r, ) = 1 − 1/3 = 2/3
b. UV (q, ) = 1 − 1/3 = 2/3
c. UV (¬q, ) = 1/5 − 1/3 = −2/15
We can now define the expected utility
value of imperatives.
2.5 Expected utility of imperatives
The expected utility value of an imperative I
is defined as the product of the utility value
and the probability of its compliance conditions
CI .
(24) EUV (I, ) = UV (CI , ) × p (CI)
The expected utility of imperative I in depends
not only on the utility value of CI ,
UV (CI , ), formalizing how much closer to
your goal the imperative would lead you, if
accepted, but also on the probability of its acceptance,
p (CI).
As an illustration consider again our state
, with value 1/3:
p u
wq 1/6 1
wr 1/6 1
wqr 0 0
w 2/3 0
Suppose one wants to increase V ( ) by using
an imperative. The notions defined above
can help us in making predictions on which
imperative one should choose. We have three
reasonable options:
(25) a. !q
Post this letter!
b. !r
Burn this letter!
c. !(q _ r)
Post this letter or burn it!
"To see which is the best choice let us calculate
their expected utility."
"In order to do so we need to determine the utility values and
the probabilities of the propositions expressing
their compliance conditions, namely q, r,
and q _ r."
"As we have already seen, these three propositions
obtain equivalent utility values since
updating with any of them leads to a state
of value 1."
(26)
a.
UV (q, ) = UV (r, ) = 2/3
b.
UV (q _ r, ) = 2/3
The probabilities, however, of the three
propositions crucially differ, giving for the
three imperatives the following expected utilities:
(27)
a.
EUV (!q, ) = 2/3 × 1/6 = 1/9
b.
EUV (!r, ) = 2/3 × 1/6 = 1/9
c.
EUV (!(q_r), ) = 2/3×1/3 = 2/9
"Among the options which have the potential
to maximally increase the value of , !(q _ r)
is the one with the highest probability of being
accepted."
"Therefore, !(q _r) is recommended
as the best choice in this case."
3 Applications
In this section we discuss two applications of
the previously defined notions.
The first application
concerns the potential of imperatives to
license free choice any.
The second concerns
the free choice effects of or and any imperatives.
3.1 "Any" in imperatives
The utility value of a disjunction UV (A _ B)
can never be higher than the utility values of
both its disjuncts.
(28) For no state :
UV (A_B, ) > UV (A, ), UV (B, )
"In declaratives, disjunctions cannot increase
relevance."
"The use of "or," in alethic utterances,
usually signals either lack of information
(it is unknown which of the disjuncts is
true) or lack of relevance (none of the disjuncts
would be strictly more relevant)."
"In non-alethic utterances (!-utterances in Grice, 1989), however, disjunctions can
be used to increase relevance.
The example
discussed in the previous section, has shown
that the expected utility of a disjunctive imperative
EUV
(!(A _ B))
can be higher than the
expected utility value of any of its disjuncts:
(29) There is a state :
EUV (!(A _ B), ) > EUV (!A, ) &
EUV (!(A _ B), ) > EUV (!B, )
Since existential sentences can be treated as
generalized disjunctions:
(30) 9x (a) _ (b) _ (c) _ ...
we can then conclude that domain widening
can increase the relevance of an existential
imperative (!9x ), but not of an existential
declarative (9x ).
This explains why any is
licensed in (31a), while it is out in (31b).
(31) a. Take any card!
b. # John took any card.
In (31a), domain widening can increase relevance
because it can increase the probability
that the hearer will comply. In (31b), it cannot.
The utility of a declarative is not a function
of its probability.
With imperatives, but not with declaratives,
a weaker option can be more relevant than a
stronger alternative.
3.2
"Free-choice"
implicatures
"On this account, free choice effects are derived
as implicatures arising from the following
Gricean reasoning."
"For ease of exposition
we only consider the case of disjunction."
(32)
The utterer utterered
!(A _ B)
rather than the shorter
!A
or
!B.
Why?
!A and !B must have had a lower expected utility.
A disjunctive imperative !(A_B) has a higher expected
utility than !A and !B only in
a situation in which both disjuncts
are allowed.
Then A and B must both be allowed.
To formalize (32), I first define the following
semantics for deontic 3, to be read as
‘It is allowed’, and 2, to be read as
‘it is obligatory’:
(i) |= 3 iff 9w : u(w) = 1 & w 2 [ ];
(ii) |= 2 iff 8w : u(w) = 1 ) w 2 [ ].
is allowed in iff there is at least one desirable
world in in which is true. is obligatory
in iff in each desirable world in , is
true.
Building on ideas from (Schulz, 2003), I then define the implicatures of an imperative I as the sentences NOT ENTAILED by I holding in
all /I where is an optimal states for I.
(33) I implicates , I | ,
I 6|= & 8 2 opt(I) : /I |=
An optimal state for I is one in which I is the choice with highest expected utility
among a set of alternatives.
(34)
opt(I) = { | 8I0 2 alt(I) :
EUV (I) > EUV (I0)}
Now, it is easy to prove that a disjunctive
imperative !( 1 _ 2) has a higher expected
utility than any of its disjuncts ! i only in a
state in which each i is possible, p([ i]) 6= 0,
and allowed, 9w : u(w) = 1 & w 2 [ i].
If we assume as set of alternatives for a disjunctive
imperative !(A_B), the set {!A, !B},
and for an existential imperative !9Dx the
set {!(9Zx ) | Z D}, it then follows
that choice-offering imperatives implicate that
each alternative way of complying with them
is allowed."
(35)
a.
!(A _ B) | 3A ^ 3B
b.
!9x | 8x3
On this account, all disjunctive and indefinite
imperatives induce a free choice effect.
"Like all implicatures, this effect disappears under
the scope of negation."
As it is easy to see, reconstructing
the optimal state for
!¬(A _ B)
or
!¬9x
does not yield any free choice inference.
In the case of positive disjunctive or
a-imperatives, free choice effects can be canceled
depending on the circumstances of the
utterance (examples (1), (3) and (10)).
In the
case of AFFIRMATIVE "any"-imperatives, free choice
effects cannot be canceled.
This fact can be
explained if we assume that any is felicitous
only in contexts in which domain widening
is functional, i.e. it increases relevance.
In
a context in which not all elements in the enlarged
domain are permitted options, domain
widening would be unjustified and any would
be infelicitous.
I have defined the expected utility of an imperative
in terms of how far it can help in increasing
the probability of the occurrence of
a desirable world.
This notion has been then
applied to explain: (i) the potential of imperatives
to license any in their scope; and (ii)
the free choice effects of disjunctive and anyimperatives.
"Any" is licensed in imperatives, because enlarging
the domain of an existential quantifier
in an imperative can increase its expected
utility.
In this sense, imperatives meet Kadmon
and Landman’s requirement that domain
widening should be for a reason.
Free choice effects have been derived as
implicatures defined in terms of what must
hold in a state in order for the used imperative
to have maximal expected utility in that
state.
References
Maria Aloni.
On choice-offering imperatives.
In Paul Dekker and Robert van Rooij, editors, Proceedings
of the 14th Amsterdam Colloquium. ILLCUniversity
of Amsterdam.
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Choice-offering and
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Analysis,
25:182–184.
Grice, H. P. 1961. The causal theory of perception. The locus classicus for the study of "or" in declarative discourse and its non-truth-functional implicata of ignorance ("My wife is in the kitchen or in the garden").
--- 1966. Logic and conversation: the Oxford lectures. (Prior to the better known William James ones)
--- 1967. Logic and conversation. Repr. in WoW.
--- 1971. Probability, desirability, and mode operators.
--- 1977. Alethic and non-alethic satisfactoriness. Repr. in 2001. Aspects of reason. Reviewed by Harman, "Mail those letters or burn them!"
--- 2001. Aspects of reason. Oxford: Clarendon Press.
Jeroen Groenendijk and Martin Stokhof. 1984.
Studies
on the Semantics of Questions and the Pragmatics of
Answers.
Ph.D. thesis, University of Amsterdam.
Charles L. Hamblin. 1987.
Imperatives.
Basil Blackwell.
Martin Haspelmath. 1997. Indefinite Pronouns. Oxford
University Press, Oxford.
L. Horn. 2000.
Pick a theory (not just any theory):
Indiscriminatives and the free-choice indefinite.
In L. Horn and Y. Kato, editors,
Studies in
Negation and Polarity. Oxford U. Press.
---- A brief history of negation.
Nirit Kadmon and Fred Landman. 1993.
Any.
Linguistics
and Philosophy, 16:353–422.
Rosja Mastop. 2005.
What can you do? Imperative
mood in semantic theory.
Ph.D. thesis, University
of Amsterdam.
Paul Portner. 2004.
The semantics of imperatives
within a theory of clause types.
In Kazuha Watanabe
and Robert B. Young, editors, Proceedings of
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CLC Publications.
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MA.
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Can
one infer commands from commands.
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24:176–179.
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Imperatives and logic.
Theoria, 7:53–
71.
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case study on the paradox of free choice permission.
Master thesis. University of Amsterdam.
Robert van Rooij. 2003.
Negative polarity items in
questions: Strength as relevance. Journal of Semantics,
20:239–273.
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