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Jon Barwise and John Perry have shown that replacing possible worlds
with situations as tools for semantic analysis yields promising treatments
of perception constructions (Barwise, 1981; Barwise and Perry, 1981;
Barwise and Perry, 1983). Barwise and Perry, however, have generally
restricted themselves to examining only quantifier-free (or, occasionally,
singly-qualified) sentences embedded in these constructions. They offer a
theory of only certain kinds of singular noun phrases. In this paper we
shall attempt to extend their treatment of naked infinitive perception
reports to handle embedded sentences containing an arbitrary number of
singular or plural quantifier phrases. After considering several options,
we shall recommend an approach to understanding quantifier phrases
within situation semantics that meets Barwise and Perry's criteria.
A theory capable of handling embedded sentences must specify
mechanisms for semantic evaluation very carefully. In a simple NI
perception sentence, according to Barwise and Perry, at least two situations must be invoked for this purpose: a situation in which the sentence
as a whole is to be evaluated, and another - a visual scene - in which at
least part of the embedded sentence takes a value. This raises a problem
familiar from work on iterated modalities in general. In a formula of
modal logic containing embedded modalities, one must specify very
carefully which world pertains to the evaluation of each atomic subformula. Similarly, in situation semantics, one must specify, in general,
which situation pertains to the evaluation of each expression in a sentence.
Our proposal consists of two parts. First, we construct a simplified
version of situation semantics incorporating a consistency principle
(suggested by Barwise (1981)). Second, we allow noun phrases to take
values in a situation other than that in which the sentence containing
them takes its value. We thus extend Barwise and Perry's notion of value
loading: they allow singular noun phrases to take values in contextually
determined "resource situations" in order to capture the extensionality
and transparency of perception reports. We extend their technique to all
noun phrases.
Our proposal is similar to one made by Richard Larson (Larson, 1983).
Larson allows different NPs within the same sentence to take values in
Linguistics and Philosophy 10 (1987) 567-596.
© 1987 by D. Reidel Publishing Company
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different situations to account for data concerning tense and adverbial
modification. If we are correct, even basic p h e n o m e n a concerning
determiners require a similar move. On Larson's proposal, n + 1 situations pertain to the evaluation of a sentence containing n NPs. We are
arguing here for a weaker thesis: quantifier phenomena in naked
infinitive perception constructions force one to allow NPs to take values
in contextually determined situations other than those in which the
sentences containing them take a value. T h a t is, we argue only that at
least three situations pertain to the evaluation of any naked infinitive
perception sentence: the situation in which the sentence as a whole takes
a value, the " s c e n e " or perceived situation, and a contextually determined resource situation. We do not of course mean to deny that more
complex phenomena, such as those Larson considers, may mandate the
use of further situations. 2
1. S I T U A T I O N
SEMANTICS
Situation semantics construes naked infinitive perception reports such as
(1)
Sheila sees Jack smile.
as having the analysis.
(2)
3 s(Sheila sees s & s c [[Jack smiles])
w h e r e ' s ' ranges over. situations and [[Jack smiles]] is the set of situations
"supporting" the truth of Jack smiles. Sheila sees Jack smile is true, then,
just in case Sheila sees a scene in which Jack smiles. Seeing, on this
analysis,'is a relation between an individual and a visual scene - that is, a
visually perceived situation, which is a complex of entities standing in
determinate relations to one another. T h e theory thus suggests, quite
intuitively, that people see portions of the world.
We represent sets of situations by means of situation types, defined as
follows: where U is a set of individuals (the domain) and Rn a set of
n-ary relations, the set S of situation types is the set of all partial
functions from Un(Rn x U n) into the set {0, 1}, where 1 represents truth.
A world type will be a maximally consistent union of situation types. A
situation type supports an atomic sentence of the form R"tl . . . tn just in
case it assigns the value 1 to the n + 1-tuple (R ", t l , . . . , tn). It supports a
negated atomic sentence just in case it assigns 0 to the appropriate
n + 1-tuple. As always, support for complex sentences must be defined
recursively. Barwise and Perry mean for "support" to capture the intuitive notion of a situation or limited set of circumstances making a
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sentence true. To avoid needless complication, however, we shall not
distinguish between situations and situation types, just as intensional
logicians usually do not distinguish world types, i.e., mathematical
representations of possible worlds, from worlds themselves. 3
To discuss adequately situation semantics' treatment of NI perception
sentences, we need to say how these constructions function in larger
contexts. That requires a definition of [[a sees 49]], the set of situations
supporting the truth of a sees 49. But this definition is also needed to
characterize a notion of validity appropriate to situation smantics. Barwise and Perry replace logical with "strong" implication (Barwise, 1981).
A set E of sentences strongly implies a sentence 49 if and only if every
situation supporting every member of E supports 49.
In standard "possible worlds" semantics, an inference is valid just in
case its conclusion is true in every world in which its premises are true.
But we need a measure of inference at least as restrictive as strong
implication in situation semantics for two reasons. First, strong validity
preserves " m e a n i n g " in situation semantics. The theory, given certain
pragmatic parameters, assigns sets of situations to sentences as their
semantic values. The conclusion of a strongly valid inference or
argument has as a semantic value a superset of the situations that support
the premises; strong validity therefore seems to capture the vague
intuition that the meaning of the conclusion is, in some sense, contained
in the collective meanings of the premises. But logical validity, given the
same conception of meaning, would not. Second, using logical implication to judge inferences involving NI perception constructions leads
to incorrect predictions - for instance, the inference from a sees 4) to a
sees (49 a n d qJ) or (4) a n d -70).
We must therefore define, in general, the set of situations supporting
an NI perception sentence. For Barwise and Perry, a sentence of the
form a sees 49 is supported by a situation s just in case a sees, in s, a
scene s' (contained in s) that supports 49. More formally,
(3)
s ~ [[a sees 49]]iff 3s' ~ s(s(([[see]], a, s')) = 1 & s ' e ~49]]).
Recall that situations are partial functions taking n + l - t u p l e s as
arguments. The first conjunct of (3) says that, in s, a sees a scene s'. In
(3), [[see]] is a binary relation between an individual and a situation. This
accords with the semantics' interpretation of see as a relation between an
individual and a visual scene. The second conjunct specifies that the
scene supports the truth of 49. (We cannot write the second conjunct in
the straightforward functional form of the first conjunct simply because 49
may not be atomic.) (3) requires, then, that the scene be a part of the
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situation s; without this stipulation, an important inference involving
naked infinitive perception constructions (namely, veridicality) would
fail.
Barwise holds that NI perception constructions validate a variety of
inferences. Many of these fail for epistemic perception reports. (1)
Veridicality. To say that NI perception constructions are veridical is to
say that inferences of the form a sees 49--~ 49 are valid. If Sheila sees Jack
smile, Jack smiles; how else could she see him smile? (2) Principle of
Substitution. Where t = t', Barwise counts valid inferences of the form a
sees 49(t)---~ a sees 49(t'). Suppose Jack is Harry's uncle. Then, if Sheila
sees Jack smile, Sheila sees Harry's uncle smile. (3) Exportation. Barwise
contends that singular existential quantifiers export; thus a sees some x
such that 49(x)---~ some x is such that a sees 49(x). Thus, if Sheila sees
somebody smile, there is somebody who she sees smile. (4) Negation. NI
perception constructions obey a principle of negation distribution: a sees
~ 4 9 - - ~ ( a sees 49). If Jack sees Harry not eat, then Jack does not see
Harry eat. (5) Disjunction Exploitation. If a sees (49 or ~O),then a sees 49
or a sees qJ. So, if Sheila sees Bill or Jack play tennis, she either sees Bill
play tennis or sees Jack play tennis. (6) Conjunction Exploitation. If a
sees (49 and ~0), then a sees 49 and a sees qJ. If Jack sees Sheila eat a hot
dog and drink a Coke, then Jack sees Sheila eat a hot dog and Jack sees
Sheila drink a Coke.
2.
NOUN
PHRASES
AND
DETERMINERS
In this paper we shall focus on determiners in NI complements of
perception verbs. We hold that determiners denote relations between sets
(van Benthem, 1984). This allows us to treat a vast array of determiners
that lie beyond the resources of first-order logic: most, all but finitely
many, fewer than half, and at most a few, to name a few. It also permits a
categorization of NPs that proves very useful in drawing consequences
from theories of NI perception constructions. This is consistent with, but
does not commit us to, a Montague-style approach such as Barwise and
Cooper's (Barwise and Cooper, 1981).
We shall assume, for the sake of simplicity, that all NPs but proper
names are quantifiers and have the form [Determiner Noun]. That is, we
shall not consider NPs containing quantified relative clauses or partitive
constructions. We shall say that a determiner @ is monotonic increasing
(decreasing) if and only if [[~(A, B) and B ~ C(C _ B) implies [[@~(A, C).
We shall call a determiner ~ persistent just in case, if A _ B and
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~ ( A , C), then U~(B, C). Similarly, @ is antipersistent just in case, if
A ___B and [[@]](B, C), then ~ ( A , C). Otherwise, ~ is nonpersistent.
We shall distinguish three special classes of determiners: (1) existentials, (2) universals, and (3) negatives. These are by no means exhaustive.
A determiner is existential just in case it is monotonic increasing and
persistent, universal just in case it is monotonic increasing and antipersistent, and negative just in case it is monotonic decreasing and antipersistent. According to these definitions, some, (often) an, several, at least
n, a few, infinitely many and, on at least one reading, many and much
are existential; every, each, (often) any, all and all but finitely many are
universal; no, at most n, finitely many and, on at least one reading, few
and little are negative. We shall assume that quantifiers can be ordered
(semantically) in terms of scope within any sentence and appropriately
prenexed, to form an initial string of quantifiers ~ 1 . . - ~ n with determiners ~ . . . @,.4 We will often speak of a quantifier as existential,
universal or negative according to its determiner's properties.
We shall call a sentence ~b = ~ 1 . . . ~n4' an E-sentence if and only if
@1... ~ , are all existential. Similarly, ~b = R 1 . . . ~,~0 is an UNEsentence if and only if ~ , . . . , ~n are such that @1,..-, Ni-1 are universal, @i is negative, and ~i+1 . . . . . ~ , are existential, for 1 ~< i ~< n. ~b is
an F-sentence just in case it is quantifier-free; a U-sentence, just in case
all its determiners are universal; and a V-sentence, just in case all its
determiners are monotonic increasing except for perhaps an even number of determiners that are monotonic decreasing. Note that U-, F- and
E-sentences count as V-sentences.
3.
EMPIRICAL
EVIDENCE
We shall now outline criteria for successful accounts of NI perception
constructions. Barwise and Perry present a variety of inference patterns
that these constructions must obey. Here we shall focus on three: the
exportability of quantifiers, veridicality and negation.
Perry and Barwise limit their attention,to the exportation of existential
quantifiers (Barwise, 1981) and descriptions (Barwise and Perry, 1983);
here we shall consider exportability in a more general way. Under what
circumstances can one legitimately export noun phrases from the
embedded sentence of a NI perception sentence? That is, when is the
following inference valid?
(4)
a sees NP ~b
NP(a sees 4~)
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Consider this argument, which tests the exportation of an "existential"
NP.
(5)
Jack sees somebody run.
Somebody is such that Jack sees him run.
T h o u g h this may not be idiomatic, it seems valid. A more natural
exportation instance uses a relative clause:
(6)
Jack sees somebody run.
T h e r e is somebody who Jack sees run.
This strategy works only in some cases; in others we will need passivization to achieve a more natural effect.
(7)
Jack sees every frog turn into a prince.
E v e r y frog is seen by Jack to turn into a prince.
We contend that, in general, all NPs export. Consider the following
arguments:
(8)
(9)
(10)
(11)
(12)
(13)
Jack sees every child fall.
E v e r y child is seen by Jack to fall. (Universal)
Jack sees Sheila thank some donors.
T h e r e are some donors who Jack sees Sheila thank. (Existential)
Hans hears hardly any Huns humble Hamburg.
Hardly any Huns are heard by Hans to humble Hamburg.
(Negative)
Jack sees just one cat run.
T h e r e is just one cat that Jack sees run.
Jack sees a man walk into the hotel.
T h e r e is a man who Jack sees walk into the hotel.
Jack sees Sheila wink.
Sheila is seen by Jack to wink.
Most speakers find all the above valid. Some, however, find (7) and (8)
problematic. T h e y seem to feel that the premises of (7) and (8) talk about
fewer frogs and children, respectively, than their conclusions. This hesitation indicates that the conclusions of these arguments can be read as
stronger than the premises. We shall offer an explanation later. But our
primary goal is to account for most speakers' intuitions that all exportation instances are valid.
Perry and Barwise also contend that NI perception contexts are
veridical. T h e y assert that most inferences of the following form are
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valid:
(14)
a sees
Nevertheless, some instances of (14) plainly fail:
(15)
Sheila sees no student cheat on the exam.
No student cheats on the exam.
Any account of NI perception verbs must therefore characterize the
sentences for which (14) is strongly valid. These arguments contain
universal quantifiers:
(16)
(17)
(18)
Sheila hears every cat meow.
Every cat meows.
Sheila sees Fred feed each cat.
Fred feeds each cat.
Sheila sees each student make a mistake.
Each student makes a mistake.
We find these confusing. We are tempted to call them valid, but, for
reasons we shall explain later, feel some reluctance to judge in either
direction.
Inferences involving existential quantifiers, on the other hand, seem
valid:
(19)
(20)
Jack sees somebody kiss Sheila.
Somebody kisses Sheila.
Lulu listens to at least eleven lascivious linguists laugh.
At least eleven lascivious linguists laugh.
Finally, inferences involving quantifiers that are neither existential nor
universal are plainly bad:
(21)
(22)
(23)
Jack sees nobody kiss Sheila.
Nobody kisses Sheila.
Jack sees Sheila talk to few people.
Sheila talks to few people.
Jack sees just a few people talk to Sheila.
Just a few people talk to Sheila.
Two of these contain negative determiners; the last contains a determiner that is nonpersistent and not monotonic.
Finally, Barwise and Perry contend that NI perception verbs obey a
principle of negation distribution:
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a sees --n~b
~ ( a sees ~b)
Consider these arguments, all of which seem intuitively valid to us.
(25)
(26)
(27)
(28)
Sheila sees A1 do nothing.
Sheila doesn't see AI do anything.
Sheila sees not every runner finish.
Sheila doesn't see every runner finish.
Sheila hears AI fire fewer or more than two shots.
Sheila doesn't hear A1 fire exactly two shots.
Nancy sees numerous nimble nuns nibble nectarines.
Nancy doesn't see few nimble nuns nibble nectarines.
The principle of negation distribution thus seems valid regardless of the
quantificational structure of the embedded sentence.
4. W E A K
SITUATION
SEMANTICS
We have introduced a schema for the interpretation of NI perception
reports in situation semantics as well as our approach to determiners. We
shall now combine them, and turn to our major task: that of specifying
where each expression in an NI perception construction is to be evaluated.
In standard model-theoretic accounts that employ possible worlds, all
noun and verb phrases not involving modals, attitude or perception
verbs, or other intensional constructions take values in the world in
which the sentence itself takes its value. Thus Several Democratic candidates oppose tax indexing is true in a world w just in case several
candidates in w who are Democrats in w oppose tax indexing in w.
E m b e d d e d modalities, of course, complicate the picture. Several worlds
may pertain to the evaluation of the sentence; in giving a semantics, one
must indicate which pertains to each expression in the sentence.
Recall, however, that a sees ck gains support in a situation s just in case
a sees a scene in s that supports 4~. E v e n very simple NI perception
sentences thus raise a similar problem. Given a situation in which a
sentence as a whole takes a value, it remains an open question where its
constituents are to be evaluated. In this section, we shall explore several
possibilities for the evaluation of NPs in NI perception reports.
T o capture the range of possibilities, our rules will be schematic at this
stage. If 4~'s subject NP is a proper name, then our analysis is very
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simple: a scene s' supports 4> just in case
(29)
[[~I,(IPNH)
This schema may seem puzzling to those who have some acquaintance
with situation semantics, since the situation s' seems to have disappeared.
However, one must r e m e m b e r that it is a schema; letting y = s' would
yield something akin to Barwise and Perry's original proposal.
We shall assume that proper names take values globally; in accordance
with theories of direct reference, we shall say that a proper name
designates the same individual no matter where we evaluate it. For the
moment, however, we shall leave open the question of where verb
phrases should take values. When nontrivial, we shall place a subscript
on the evaluation function - as in '~tPly' above - to indicate that we
should evaluate qJ in the situation y. In (29), the subscript is a variable.,
We shall consider several ways of specifying its value.
In the general case, 4> will consist of a subject noun phrase and a verb
phrase. In accordance with an interpretation of determiners as relations
between sets, we shall analyze a scene's supporting 4> as
(30)
I~I(INI=, I~I,)
This says that a scene supports the complement of an NI perception
report just in case the determiner holds between the set of objects in x
satisfying the noun and the set of objects in y satisfying the verb phrase.
We will use the relational notation above as equivalent to a more
explicitly set-theoretic rendering, ([[Nlx, l~0Uy)e I91. Again to get a real
semantic rule for this schema, we must fill in values for 'x' and 'y'. How
we do this will play a large role in shaping our theory.
All " w e a k " versions of situation semantics allow semantic evaluation
with respect to proper portions of scenes. T h e y incorporate the basic
analysis of (3) and (30). In (3), s is a situation and s' a scene, that is, a
visible portion of s. T h e formula says that s supports the truth of a sees d?
if and only if a sees a visible portion of s that supports 4). Nothing there
demands that the scene include everything that a sees at that particular
moment.
Weak versions differ in their interpretations of 4~- Our schema, (30),
requires that we evaluate noun and verb phrases of the embedded
sentence in situations. We thus have several options.
S-l: Barwise and Perry may want to evaluate all of the embedded
sentence in the scene s'. A naive reading of (3) suggests that the NI
complement, 4~, should take a value in the visual scene. In S-l, then, 'x'
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and 'y' take the same value, the scene s'. S-1 does not distinguish the
interpretations of the noun and verb phrases, or of parts thereof and the
rest.
S-2: We might evaluate the embedded sentence's subject noun phrase
in s but its verb phrase in s'. In short, S-2 is identical to S-1 except that
'x' receives the value s rather than s'.
S-3: We might decide to evaluate both noun and verb phrase in the
scene s', as S-1 requires, with the exception that we evaluate all the
embedded sentence's quantifiers in s. S-3, then, assigns 'x' the value s
and 'y' the value s/s', to indicate that quantifiers are evaluated in s, the
remainder, in s'. S-2 differs from S-1 by evaluating the subject NP of the
embedded sentence in s; S-3 extends this strategem to all NPs. We turn
now to test three alternatives by examining their consequences concerning exportability, veridicality, and negation distribution.
5. E X P O R T A B I L I T Y
OF QUANTIFIERS
Given an appropriate semantic rule prenexing all quantifiers in order of
scope, (4)'s premise emerges in situation semantics as
(31)
3s' _c s(s((l[see], s', a)) = 1 & ~ll([[N]x, ~b]y)).
Under what circumstances does (31) entail:
(32)
l[~]([[N]]x,({b: 3s' ~_ s(s((l[see], s', a)) = 1 & [[qb]y(b))})?
In S-3, the set l[~bllyin (31) is a subset of the set in (32), so we obtain the
following result.
T H E O R E M 1: Exportation is valid in S-3 if ~ is monotonic increasing.
In S-2, theorem 1 holds only if the quantifier exports from the subject
noun phrase. Otherwise it will be evaluated relative to s' in (31) but to s
in (32). The set in (32) will still be larger, but the noun N will b e
evaluated in s.
T H E O R E M 2: Exportation is valid in S-2 if (1) if .~ exports from the
subject NP, ~ is monotonic increasing; (2) otherwise, ~ is monotonic
increasing and persistent.
In S-l, N takes a value in s'. This is impossible as we have written (32).
To determine when exportation holds in S-l, therefore, we must evaluate
N somewhere else (say, in s). This strategy will imply the exportability of
quantifiers that are both monotonic increasing and persistent.
In natural language, when do embedded quantifiers export? Recall that
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existentials, negatives and most other quantifiers do so readily; only
universals arouse some uneasiness. The weak versions of situation
semantics fail to explain these responses, since it can easily be shown that
they generally count inferences involving the exportation of quantifiers
with negative and nonpersistent determiners invalid. S-3 predicts exportation of existentials and universals while S-2 accepts only existentials.
S-1 fares no better than S-2. Plaintly, none of the weak systems examined
explains weak systems examined explains speakers' intuitions concerning
exportation.
6. V E R I D I C A L I T Y
Recall that inferences involving veridicality have the form
(33)
a sees ~b
4,.
Perry and Barwise construe (33) as
(34)
:Is' ~ s(s(([lsee~, s', a)) = 1 & s ' c [[q~)
s c U'~
For which ~b is this valid? Perry and Barwise restrict their discussion of
veridicality to "simple" sentences in (Barwise, 1981) and to "realistic"
sentences in (Barwise and Perry, 1981). Barwise and Perry call ~b monotonic increasing (or simple or realistic) if and only if, if s' ~_ s and s' 6 l[~b~,
then s E [[~b~ (Barwise and Perry, 1981). If 4~ in (34) is atomic or a truth
functional combination of atomic sentences, then veridicality holds.
Barwise and Perry, however, do not discuss our more general question.
A general examination of veridicality in NI perception reports requires
an analysis of the relation between the monotonicity of sentences and
that of quantifiers. Sentences that are strongly valid or "strongly contradictory" of course are trivially monotonic increasing and monotonic
decreasing, regardless of what determiners occur within them. Logically
valid sentences and quantifier-free sentences are all also trivially monotonic increasing. However, logically valid sentences can fail to be monotonic decreasing, since, though they are never false at any situation, they
may not be true at every situation. The real question, then, is how
quantified contingent sentences behave.
Because of the weakness of the semantics and the generality of the
theory about quantification, we cannot show any general results about
the class of all contingent sentences. We have separated out a certain
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subclass, which we shall call paradigmatically contingent (PC) sentences,
for which the following theorem holds.
T H E O R E M 3: Let tb be PC. T h e n ~b is monotonic increasing just in
case it is strongly equivalent to an F- or E-sentence. th is monotonic
decreasing just in case it is strongly equivalent to a UNE-sentence.
This theorem follows from an assumption, which we believe to be a
general feature of natural language: if ~O(a) is neither monotonic increasing nor monotonic decreasing, then there is no natural language determiner @ such that @(N, ~O) is monotonic increasing or monotonic
decreasing. E v e n without this assumption or the restriction to PC sentences, however, we can prove that if tb is an F or E-sentence, then it is
monotonic increasing and if ~b is a UNE-sentence, then tb is monotonic
decreasing.
In order to characterize PC sentences, we need two definitions.
D E F I N I T I O N 4: N positively covaries independently
VxVs(l[N]]s ___x ___U) -- > 3s'([[N]]s, = x & ~M]]~ = I[M]I~,))
of
M
iff
D E F I N I T I O N 5" N negatively covaries independently of M iff VxVs
(x _ ~N~, -- >3s'([[N]l,, = x & JIM]Is= ~M]~,)).
What independent covariance comes to is this: given a situation in which
N and M are interpreted, we may find other situations that add to or
subtract from the extension of N, without altering the extension of M. A
sentence q~ = @(N, ~b) is PC iff it is contingent and N and 6 positively
and negatively covary independently of each other. An example of a PC
sentence is All men are fat; an example of a contingent sentence that is
not PC is few animals are men. In what follows we shall restrict ourselves
to PC sentences.
T h e weakness of the connection between sentential monotonicity and
determiner type makes weak situation semantics unable to make very
general predictions concerning veridicality and negation. Recall that in
S-1 'x' and 'y' take the visual scene as values within perception contexts.
That is, all quantifiers take values in the visual scene. Outside the
perception context, ~b as a whole is evaluated in s. Therefore, since the
scene is always a subset of s, S-1 counts veridicality valid for just
monotonic increasing ~b, so we obtain
T H E O R E M 6 : S - 1 validates veridicality if and only if tb is strongly
equivalent to an F- or E-sentence.
S-3 differs considerably from S~I concerning veridicality. We have
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seen that S-3 allows exportation of monotonic increasing quantifiers. It
thus permits exportation of existentials and universals. For a quantified
sentence 4' = ~4~ where qJ possibly also contains quantifiers, (34) emerges
as
(35)
3s* _c s(s((l[seel], s*, a)) = 1 & [[%]]([N]s, [[0]]s/~*)
~gNNL ~4,~s)
In S-3, all quantifiers (noun phrases) are evaluated in s, and the
quantifer-free portion of the VP is evaluated in s*. Recall also that
s * c s. Consider a series of quantifiers with sets as arguments. The
quantifier-free portion of the embedded sentence will be monotonic
increasing. If [[9]] holds between the set of objects satisfying N in s and
the set of objects denoted by 0 in s*, it will continue to hold between the
former set and the latter's superset [[0Is. So any sentence with all
monotonic increasing determiners will validate (35) according to S-3.
In this case, we can show that two monotonic decreasing determiners
have the effect of a pair of monotonic increasing determiners. This
argument is slightly more complex in nature. Let ~1 and ~2 be any two
"negative" quantifiers and let 4' = ~lRztP. By definition, 91 and 92 are
monotonic decreasing and antipersistent. The inference we want to
investigate is:
(36)
3s* c s(s((~see], s*, a)) = 1 &
~l]](~Nl~s, {b: I92~([N2~, {c: [O],/s*(b, c)})}))
~9,~(~Nl~s, {b: [[~2~(~N2]]s, {c: [[0~s (b, c)})})
Assume that this inference fails. Then some scene in s supports the NI
perception complement, but s does not. There are two ways in which the
inference can fail in virtue of the fact that ~1 is negative: (i) the set of
objects that satisfy N1 in the conclusion is not a subset of the set of
objects that satisfy N1 in the premise; (ii) the set of objects that satisfy the
verb phrase of 4) in the conclusion is not a subset of the set of objects
that satisfy the verb phrase of 4' in the premise. Clearly, only (ii) is a real
possibility. Consequently, for some d in s, [~2l](~Nz~s, [[t)~, (d, c)) but
~2]](~Nz~s, ~O~/~,(d, c)). Since (i) s* _c s, (ii) 92 is monotonic decreasing, (iii) the situation in which all quantifiers are evaluated remains
unchanged and (iv) only the situation in which the quantifier-free portion
of tp is evaluated changes, however, this is impossible. Consequently, our
claim is proven and we have the following theorem:
T H E O R E M 7 : S - 3 validates veridicality if 4) is strongly equivalent to a
V-sentence.
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NICHOLAS ASHER AND DANIEL BONEVAC
S-2 has features of both S-1 and S-3. With regard to subject noun
phrases, it is just S-3; with regard to quantifiers in the verb phrase, it
produces the results of S-l, assuming that verb phrase quantifiers evaluated relative to s* in the premise are evaluated relative to s in the
conclusion. Special features of S-3, like the cancellation of pairs of
negative determiners, do not hold in S-2, because subject noun phrases
and noun phrases in the verb phrase will be evaluated in different
situations. Thus,
T H E O R E M 8 : S - 2 validates veridicality if and only if ~b is strongly
equivalent to either an F-sentence or an E-sentence with a monotonic
increasing but possibly nonpersistent determiner in the subject noun
phrase.
Recall that veridicality inferences seem fine for F- and E-sentences,
somewhat questionable for U-sentences, and terrible for UNE-sentences
and sentences involving most nonpersistent determiners. S-1 accounts for
the validity of veridicality arguments involving E-sentences and for the
invalidity of those involving UNE-sentences. But S-2 distinguishes the
subject NP from other NPs in a way that the evidence does not seem to
justify. Neither theory can explain why
(37)
(38)
Jack sees every fisherman catch some trout.
Every fisherman catches some trout.
Jack hears Sheila laugh at everyone.
Sheila laughs at everyone.
seem questionable, but nonetheless acceptable. S-1 condemns both
completely, while S-2 counts the former valid but the latter bad. Finally,
S-3 validates veridicality for just V-sentences. It explains why F- and
E-sentences are unproblematic and why UNE-sentences are bad, but not
why U-sentences arouse discomfort.
7. N E G A T I O N
Recall that inferences involving the principle of negation distribution are
of the form,
(39)
a sees 7~b
~(a sees ~b).
Situation semantics interprets (39) as
(40)
3s' ~ s(s((~see~, s', a)) = 1 & s ' c ~ b ~ )
7 3 s ' _ s(s(~see~, s', a)) = i & s'~ ~b])
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We can get a countermodel for (40) by picking two scenes, s' and s",
such t h a t a sees both s' and s", s'~[[~b~, s"6H~b~, and s', s " _ s.
Obviously it cannot happen both that s c ~b]] and that s ~ ~ b ~ . So
assuming the existence of a countermodel leads to a contradiction if we
can move from s' ~ [[-qth]] to s ~ [[-qth] and from s" ~ [[th] to s ~ [[~b]. These
inferences hold if both ~b and ~th are monotonic increasing sentences.
(40) is valid in S-1-S-3, therefore, for all atomic sentences and their
truth-functional combinations. The only quantified sentences (given the
restrictions we have made) that are monotonic increasing are E-sentences; negations of these will be strongly equivalent to UNE-sentences
and hence will not in general be monotonic increasing.
T H E O R E M 9 : S - 1 - S - 3 validate negation distribution if ~b is strongly
equivalent to an F-sentence.
But all negation instances seem intuitively valid. Again no weak situation
semantics accounts for our intuitive responses.
These systems have a problem with negation distribution because, in
general, there is no relation between the scene in the premise and the
scene in the conclusion. Perry and Barwise point out that the negation
principle relies on two applications of veridicality, for ~b and -~th. But
these systems permit veridicality inferences only for sentences with
existential or monotonic increasing determiners; their negations will not
have a similar form. So there are no quantified sentences for which these
systems sanction both veridicality inferences.
To see how we can construct countermodels, suppose th =
~1 . . . . , ~n~O, where ~b is quantifier-free. We can specify two scenes, s'
and s", both of which a sees and both of which are contained in the
situation s. So long as all the determiners Di (in ~i) are monotonic
increasing, constructing a countermodel appears to require finding
objects X 1 . . . . . . X n such that ( x l , . . . , xn) c ~O]]s, and ( x l , . . . , x~) ¢ [[@]]s,,.
Since situations are partial functions and neither s' nor s" has to be a
subset of the other, this is possible.
8. S T R O N G
SITUATION
SEMANTICS
Exportability inferences seem valid, most of the time, no matter what
quantifier is exported from where. Negation distribution similarly appears
valid for all sentences, whether quantified or not. In S-3, countermodels
to such arguments look peculiar, requiring that the perceiver see more
than one scene at the same time and in the same situation. Most speakers
of natural language seem to assume that this does not happen. We shall
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BONEVAC
call "strong" any version of situation semantics prohibiting semantic
evaluation in proper portions of scenes.
Barwise and Perry offer a principle for possible addition to situation
semantics stating that a person in a given situation sees at most one scene
at a t i m e / W e shall call 'S-4' the system resulting from the addition of
this strong consistency principle to situation semantics, specifically, to S-3.
This permits a vast simplification of our definition of ~a sees ~]]. In weak
systems we had to say that this gains support in s just in case there is a
scene s' that a sees in s and that supports ~. Now, hOwever, we may
speak of the scene a sees in s, rather than merely some scene a sees in s.
As a result, we can interpret perception verbs as functions from situations and individuals to situations.
This simplification allows us to say that situation s supports a sees d~
just in case
(41)
//see]](s, a) c ~]].
This provides a simpler and more natural formulation of situation
semantics, and it allows situation semantics to reflect more accurately
many speakers' intuitions.
Given (41), exportation amounts to
(42)
~]([[N]]s, [[~b]][see~(s,a))
so we get:
T H E O R E M 1 0 : S - 4 universally validates exportation.
Second, S-4 universally validates the negation principle. (33) becomes
(43)
//seek(s, a) E ~-n~b~
~see~(s, a) ¢ [[4~
Because we have characterized situations as partial functions, we get:
T H E O R E M 1 1 : S - 4 universally validates negation distribution.
A weaker though less elegant principle would do the same work as
(55). S-4 requires that a person in a given situation see only one scene at
a time. We might demand instead that if a person in a given situation sees
more than one scene at a time, he/she also sees the union of those scenes.
Then, to get the effect of S-4, we could add to our definition of [[a sees
4~]] a clause requiring that the embedded sentence must be evaluated at
the union. This weak consistency principle has the effect of its stronger
cousin without implying that a person in a situation sees at most one
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scene at a time. It allows the perceiver to see several scenes, but specifies
that one of them, namely, the union of all he/she sees, is solely relevant
to the semantics. It also has the advantage of being consistent with a
fairly intuitive principle, namely, that anyone who sees a scene sees all of
its subscenes. The addition of the weak consistency principle allows all
quantifiers to export and universally validates the negation principle. Call
this system, resulting from the addition of the weak consistency principle
to S-3, S-5. 6
T H E O R E M 1 2 : S - 5 universally validates exportation and negation distribution.
The addition of either consistency principle to S-3, however, does not
affect S-3's treatment of veridicality.
9.
FLEXIBLE
SITUATION
SEMANTICS
Although many English speakers assent to all instances of exportation
and the negation principle, some do not. An adequate semantic theory of
NI perception contexts should indicate why these differences occur and
why many arguments we have disucssed seem indeterminate or ambiguous.
An obvious suggestion stems from the previous section: perhaps only
some speakers approve of the consistency principle. If this is correct,
then exportability should seem most problematic when the determiner
involved is not monotonic increasing. In fact, however, such inferences
involving no, few, etc., seem fine; all, each, and other universal (and
thus monotonic increasing) determiners encounter resistance. In addition, disagreement arises about certain veridicality inferences - involving universal determiners especially - in ways that any proposal concerning consistency principles cannot explain. Finally, the suggestion implies
that determiners that are not monotone in any direction - exactly n, for
example - should make sentences containing them resist inferences based
upon the negation principle. But (27) above appears just as acceptable as
the related arguments involving monotonic quantifiers.
Why does the following argument appear problematic?
(44)
Sheila sees every frog jump.
Every frog jumps.
And why does the next argument seem far more acceptable?
(45)
Sheila sees every frog in the contest jump.
Every frog in the contest jumps.
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Clearly, the latter argument specifies a domain ("the contest") while the
former does not. When we hear Sheila sees every frog jump we assume
that Sheila sees every frog in some situation jump, but the situation is
largely indeterminate. It does somehow pertain to Sheila; the situation
must contain her and something perceptible to her. Every frog jumps, in
contrast, indicates nothing whatever about the intended domain: every
frog in the universe jumps? Every frog in Texas? Every frog in the
jumping contest? (45), however, declares that every frog in the contest
jumps. We assume that the situation relevant to evaluating the argument
comprises both the contest and Sheila's perception of it. If this analysis is
on the right track, then we should look to interpretations of quantifiers
(in particular of the common noun phrases constituting them) for an
explanation of differences in speakers' reactions. By now, this should not
be very surprising, since varying the evaluation of the embedded sentence in general has played a major role in determining the class of
inferences situation semantics sanctions.
We propose, therefore, an emendation to S-4 which allows the evaluation of quantifiers within the embedded sentence to vary. We propose to
evaluate quantifiers in a contextually specified resource situation s that
can vary from premise to conclusion. Perry and Barwise require that
singular noun phrases can be value loaded at some contextually specified
resource situation, other than the scene or situation of evaluation for the
whole sentence, in order to validate the principle of substitutivity within
NI perception contexts and to capture referential readings of singular
terms (Barwise and Perry, 1983). Our proposal extends value loading to
all noun phrases. Unlike Barwise and Perry, however, we do not believe
that in NI perception reports noun phrases occur "value free"; noun
phrases are always value loaded at some resource situation.
Our use of a resource situation reflects the fact that speakers can
exploit one situation in describing another. In discourse, speakers tend to
construct a resource situation shared by the parties to the discourse
(Lewis, 1979, Kamp, 1981). Like any other situation, the resource
situation may be partial. The information it provides about the objects it
mentions may be very incomplete in comparison with the information
available in the possible world as a whole. Since it is pragmatically
defined, furthermore, the resource situation may vary - and, in particular,
grow - as discourse proceeds.
The contextual domain of the conclusion of an argument, therefore,
may be larger than that of any premise. To account for the function of
resource situations, we need a theory of discourse such as that of
(Karttunen, 1976) or (Kamp, 1981). Here we will simply illustrate the
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585
thesis with some examples, and draw some implications for situation
semantics' handling of NI perception contexts.
According to Kamp and Karttunen, certain linguistic expressions
introduce points of reference - discourse referents or reference m a r k e r s into a discourse. The discourse referents already introduced and available to parties to a discourse constitute the contextual d o m a i n . Indefinite
descriptions such as a w o m a n typically introduce an object into the
contextual domain which can then serve as a subject for anaphora: I m e t
a w o m a n at the A I m e e t i n g last w e e k . She s e e m e d very astute. Most of the
time, discourse proceeds by augmenting the contextual domain and the
operative resource situation. Occasionally, of course, one party may
correct or disagree with another, so the process is not entirely ampliative.
Nevertheless, the resource situation tends to grow as discourse develops
among the participants a larger and larger body of shared information.
Argument tokens in natural language differ in some ways from those of
the logician. They clearly have temporal or spatial extension; the sentences constituting the argument appear in succession, in some particular
order. Consequently, the sentences that appear first use the current
resource situation, while later sentences may use a situation larger or
smaller than the initial one. We should admit the possibility, then, that
the resource situation will change as the argument proceeds.
This suggests that certain parts of the conclusion of an argument in
natural language may demand evaluation in a resource situation different
from that operative throughout some of the premises. If the conclusion
appears last in the succession, we should expect that its resource situation
will be at least as large as that for any premise. Even if it appears earlier,
a fair evaluation of the argument will demand consideration of such a
situation; the discourse referents introduced by subsequent premises
play a role in justifying the conclusion, and the argument may suffer an
injustice if they are not, as it were, admitted into evidence.
The operative resource situation may thus remain constant or grow
larger as an argument proceeds from premises to conclusion. In discourse, resource situations may also shrink. One speaker may correct or
disagree with another; these activities may remove from the contextual
domain an object previously admitted or some property attributed to one
of its members. But we contend that arguments in the logical sense do
not exhibit such shrinkage. If corrections or disagreements occur as a
speaker presents an argument, the argument itself is the finished product
of these modifications, not their history. Consequently, we shall consider
only arguments in which resource situations remain constant or grow.
We shall therefore characterize a new notion of implication, which is
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NICHOLAS ASHER AND DANIEL BONEVAC
slightly stronger (i.e. more restrictive) than strong implication. Recall
that one sentence strongly implies another if, and only if, every situation
supporting the former supports the latter. To account for the role of
resource situations, we shall say that one sentence monotonically implies
another just in case every situation supporting the former, interpreted
with the help fo a resource situation s, also supports the latter, interpreted with the help of a resource situation at least as large as s. More
generally, a set of sentences E monotonically implies a sentence 4) if and
only if every situation supporting each member of E, interpreted with the
help of some resource situation, also supports 4), interpreted with the
help of a resource situation at least as large as that of any premise.
Given the generally augmentative nature of ~iiscourse, we contend that
monotonic implication offers a novel and accurate way of evaluating
natural language inferences. Ideally, the inferences Barwise and Perry
take as characteristics of NI perception constructions should be monotonically as well as strongly valid.
We shall d e f i n e ' s c [[a sees 4)1]' as
(46)
~see]](s,a) ~ l[4)]]s
where we evaluate quantifiers in 4) at s, some nonempty resource
situation. Essentially, we propose to value load all NPs at a situation s
which may differ from the situation in which the sentence containing
them takes a value. Which situation s is may depend on many elements,
including previous discourse, speaker "connections," or the context of
utterance.
We offer now general rules of interpretation for NI perception reports
incorporating the use of resource situations. A simple NI perception
report has the form
(47)
NPI[v~,I Vp[s,NPzVP2]],
where NP is a noun phrase, Vp an NI perception verb, VP a verb phrase
and S' an incomplete sentence. Again, we'distinguish two cases: (i) where
NP2 is a quantifier phrase of the form [ ~ 2 N 2 ] , and (ii) where NP2 is a
proper name. For case (i) we have:
(48)
~[VpNP2VP2Ils(a) iff I[~]([[N2]s, I[VP2]I~/W,.~,a))
which says, roughly, that an NI perception sentence holds of an individual in a situation just in case the relation denoted by the determiner
holds of the set denoted by N2 when evaluated at a contextually specified
situation s and the set of objects satisfying the verb phrase, the common
noun phrases in the quantifiers of which are evaluated at s and the
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quantifier-free remnants of which are evaluated at the scene the individual sees in the original situation. When the subject noun phrase of
the NI complement is a proper name we can simplify (48) to (49):
(49)
[VpNP2VP2~ (a) iff ~VP2~,/~vpn~,~)([NP2~).
We shall call the resulting version of situation semantics S-6.
First, consider exportability of quantifiers. Recall that the form of this
inference is
(50)
a sees ~ 0
a sees ~O.
In S-6 exportability inferences turn out to have the structure
(51)
U~HNL, M s/~soo~<s,o~)
~n(~NL,, ~q,Bs,.oo~(~,.3.
Our "flexible situation semantics" allows the resource situation to grow
from premises to conclusion. We find that exportation inferences provide
a paradigmatic case of this. Exportation (perhaps like topicalization)
imparts an emphasis which suggests that the exported quantifier demands
evaluation elsewhere. When a quantifier is embedded within an NI
perception context, speakers find it easy to read the quantifier restricted
to a resource situation that almost always contains the scene. When the
quantifier is exported, however, speakers find it difficult not to interpret
the quantifier in an expanded resource situation. This difficulty shows up
in speakers' hesitations concerning the exportation of quantifiers with
monotonic increasing but antipersistent determiners, i.e. uv ~'rsals. We
therefore conclude that instances of exporation that are monotonically
valid should meet with universal acceptance. On the other hand,
speakers may evaluate both premise and conclusion using the same
resource situation. Instances of exportation in which the resource situation is held cons+ant from premise to conclusion are strongly valid. The
theory predict~ ..~t these should not be quite as acceptable to all
speakers as m~,,otonically valid instances. Nevertheless, they should not
strike speakers as implausible.
Since the monotonic or strong validity instances of (51) depends only
on the relations of s to s', we quickly conclude
T H E O R E M 13:
(1)
(2)
S-6 strongly validates all instances of exportation.
S-6 monotonically validates exportation if @ is existential.
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BONEVAC
Intuitions concerning these sorts of cases suggest that s is almost always
at least as large in the conclusion as in the premise. This is only natural,
since the premise forms p'~rt of the conclusion's context, and intervening
premises can introduce new elements of the domain. We might expect,
then, that existentials export in almost every context, and that universals
export in many. This accords well with our intuitions. 7
Second, consider veridicality. S-6 renders (33) as
(52)
[[N]]([[N~s, [II//]]s/[[see](s, a))
II II(IINII,,
In S-6, two elements of (52) can change from premise to conclusion.
First, the set assigned to q~ grows from premise to conclusion, since
l[see]](s, a ) _ s. Recall that V-sentences, in effect, contain determiners
that are all monotonic increasing. If we assume that the resource situation is held constant, then instances of (52) will be valid if and only if 4' is
strongly equivalent to a V-sentence. If we also allow the resource
situation to grow from premise to conclusion, then only sentences containing determiners that are persistent and monotonic increasing will
validate (52). Thus,
T H E O R E M 14:
(1)
(2)
S-6 strongly validates veridicality if 4' is strongly equivalent to
a V-sentence;
S-6 monotonically validates veridicality if 4' is strongly
equivalent to an F- or E-sentence.
S-6 predicts, then, that veridicality inferences will succeed unquestionably for all embedded sentences that are either quantifier-free or Esentences, and will feel somewhat more dubious for other V-sentences.
Veridicality inferences involving other sorts of quantified sentences,
especially UNE-sentences, should seem blatantly bad. Once again, this
harmonizes with our intuitions.
Finally, consider the negation principle. S-6 assigns (39) the form
(53)
~II~]I(~N~, [[tOll,/~,~B(,,,o)
ll,,Oll,,/u,ooB(, o>)
(53) requires some discussion. Say that a determiner @ has a negation,
and a dual, ~ - n . We define their truth
conditions in the following way:
-n~, a complement, ~ ,
(1)
(2)
(3)
[[~([[N]], ~b~) iff ~ ( ~ N ~ , ~b~)
[[~-q~([[Nl],~qJ]])iff [[~]]([[N]],[[~qJ]])
[[~-7~]([[N]],~q~])iff ~ ] ( ~ N ] , ~ b ] )
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To give some classical examples, suppose ~ is s o m e . Then its negation is
no, its complement is s o m e n o t (i.e., not all), and its dual is all. If ~ is no,
then its negation is s o m e , its complement, all, and its dual, n o t all. s
T H E O R E M 15:
(1)
(2)
S-6 strongly validates all instances of negation distribution.
S-6 monotonically validates negation distribution if th is an For UNE-sentence.
Nothing like exportation or topicalization contradicts speakers' desires
to maintain a constant evaluation throughout premise and conclusion
here, so we can let s = s'. If we do allow s' to be larger, then (53) will hold
provided that 74, is monotonic increasing. We know, by the above
theorem, that -~b is monotonic increasing if and only if -q~b is an F- or
E-sentence; it will hold, then, if 4) itself is either an F- or a UNEsentence. We might expect that negation distribution instances would
meet with strong and universal acceptance for quantifier-free qS; acceptance almost as strong for UNE-sentences; and still strong acceptance for other quantified sentences. But these distinctions appear to
be extremely thin to the extent that they are recognized at all. This too
seems to square with linguistic intuition.
We have thus offered six options for treating quantified sentences in NI
perception constructions within the rubric of situation semantics. We find
S-6 the most accurate in reflecting our linguistic intuitions. Situation
semantics in this form is an extremely powerful tool for dealing with
extensional perception contexts and promises fruitful extension to other
features of natural language. But our analysis should make it clear that
the theory requires supplementation with a rigorous analysis of how
contexts function and develop in discourse.
10. APPENDIX
Our theorem concerning the relation of montonicity to determiners bears
some resemblance to two theorems of standard model theory, due to Los,
Robinson and Tarski. Monotonic increasing and decreasing sentences
are preserved under extensions of models and submodels, respectively.
And the existential and universal quantifiers of first-order logic are
existential and universal in our broader sense. In what follows we restrict
ourselves only to PC sentences, though we shall examine a possible
conjecture about PC sentences and the broader set of contingent sentences.
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R E M A R K 1: Situation semantics allows us many options concerning the
interpretation of quantified sentences. We will adopt a "classical" interpretation of quantified sentences. Recall that determiners denote relations between sets and that [IN]Is and [[~O]]sare sets of individuals or tuples
of individuals that satisfy the respective predicates in s. So s ~ [[~(N, ~0)]]
if [[@~(~N]]s,~b]~); s e [[-7~(N, ~)~ otherwise.
R E M A R K 2: It is natural to ask how broad the class of PC sentences is.
One might suppose that every quantified contingent sentence D(N, g~) is
strongly equivalent to D'(N', ~'), which is PC. Because of problems
involving the relative readings of determiners like many, most and few
and the nonintersective components (for instance, some adjectives) of
common noun phrases, considerable ingenuity is required to guarantee
the possibility of positive, independent covariance. In some instances, we
can construct PC strong equivalents for contingent sentences that are not
PC. Consider for instance the sentence all dancers are good dancers; it is
contingent, but yet the common noun phrase does not negatively covary
independently of the verb phrase. However, the sentence all dancers are
good at dancing is strongly equivalent to it and its subject common noun
phrase and verb phrase do covary independently of each other. On the
other hand it seems very difficult to find a strong equivalent of few
primates are men that has the required properties of covariance.
R E M A R K 3: Suppose g~(a) is a monotonic increasing sentence. Then, if
s c ~ ( a ) ] then Vs'___ s s' e ~ ( a ) ~ . This is also true whatever name we
substitute for ' a ' in ~(a). Consequently, if s ___s', [[g~]s___[[~]]s'. Suppose
s " _ s and s c ~(a)]l. s" is a subfunction of s, and so contains fewer
entries than s. Now suppose s ¢ ~ ( c ) ] . ~(c) is monotonic increasing if
~(a) is. Then s"¢ ~(c)~. This holds true for whatever name we substitute for 'c' in ~(c). So, ~ ] ~ , , c ~ ] ~ . Suppose ~(a) is a monotonic
decreasing sentence. By similar arguments, we can show that for s' ~_ s,
[[~b]~---[[~l]~,, and for s c_ s", ~b]s,,~_]~b]]~. Note that if ~b(a) is monotonic
decreasing, the elements of the tuples in ~b]]~ need not themselves figure
in the entries in s; rather ~b]] grows as it is evaluated in situations with
fewer and fewer entries.
D E F I N I T I O N 16: s # is a positive reduct of s with respect to N 1 , . . . , Nm
just in case [[N1]]~,'---[[NI~. . . . . . [[N,,~ = ~N,,]]s, and s # contains no other
entries other than those concerning the positive extensions of
N1 . . . . , Nm.
A positive reduct can always be formed by paring or thinning entries in
s. This allows us to conclude s # ~_ s.
To prove the theorem, we need several lemmas.
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L E M M A 17: F-sentences are monotonic increasing.
L E M M A 18: Let ~ be monotonic increasing. Then ~b = @(N, if) is
monotonic increasing iff ~ is existential.
Proof: Say that ff is monotonic increasing. To show the left-to-right
direction, assume that ~b is monotonic increasing. Suppose s'~ ~qb]. A
positive reduct of s', s ÷, verifies q~, since s' does. Since ~b is monotonic
increasing, if s + __ s, s c l[~b~. But s ~ I[~b~just in case I[~]](I[N~)~,[[~s). Let
[[0~+ __c_X. Since ~b is PC, $ positively covaries independently of N, and
there is an s* such that I[0]~* = X, but I[N1]s.=I[N~+. Further, we can
construct s* from s ÷ by simply adding entries to s +, since ~b is monotonic
increasing. Thus, s + _ s*; and so @ is monotonic increasing. Additionally, N is a simple common noun phrase and is by hypothesis quantifier
free, so N is monotonic increasing. Suppose again that s e [[4~1].A positive
reduct of s with respect to N and ~, s #, verifies ~b since s does. Now let
IIN~,~ __ X. Since ~b is PC, N positively covaries independently of 0So there is an s" such that IIN1]~,,=X and [[qJ]~ =l[O]~,,. By a similar
argument to the one given above, s # __ s", and so ~ is persistent as well.
So ~ is existential. To show the right-to-left direction, suppose that ~ is
existential, that s ' c s, and that s ' c l[~bll. Then [[D](IIN]~,, ~ff]ls,). O and N
are all monotonic increasing, so ~]([[N~s,, [~b]l~). Therefore s c ~b~, so ~b
is monotonic increasing.
D E F I N I T I O N 19: ;~ qua, tifier is limited itt it is monotonic decreasing
and persistent.
L E M M A 20: Let ~ be monotonic decreasing. Then ~b = ~(N, 0) is
monotonic increasing iff ~ is limited.
Proof: Assume that ~ is monotonic decreasing. To establish the leftto-right direction, let ~b be monotonic increasing. Assume that [ [ ~
(I[N]I~,, I[0]~,). A positive reduct of s', s ÷, verifies ~b since s' does. Then, if
s + ___s, [[@]I([[N]~,[[qJl]~). Let X _c [[q~l,- Since ~b is PC, there is an s* such
that [[~]~. = X and ~N]]~ = I[Nlls.. Further, we can construct s* from s ÷ by
adding entries, so that s+~_ s*. Since ff is monotonic decreasing, by
remark 3, ~ is thus monotonic decreasing. To show persistence, we use
exactly the same argument as in the previous lemma. Therefore !~ is
limited. The right-to-left direction is again similar to that in the previous
lemma.
L E M M A 21: Let q~ be monotonic increasing. Then ~b = ~(N, ~b) is
monotonic decreasing itt ~ is negative.
Proof: Assume that ~ is monotonic increasing. For the left-to-right
direction, assume that ~b is monotonic decreasing and let s! e limb]. Let
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X c_ [[~b~s,. Since th is PC, there is an s* such that [[~b~s.= X and [[N]]~.=
f[N]]~,. Further, we can construct s*, again by thinning entries from s',
since ~0 is monotonic increasing, so s*__c_s'. Since 4~ is monotonic
decreasing, s*~ [l~b~ and so @ is monotonic decreasing. Additionally,
since N is a simple common noun phrase, it is monotonic increasing. So
by a similar argument to the one given above, we can show @ must be
antipersistent. The opposite direction is easy.
L E M M A 22: Let ~0 be monotonic decreasing. Then ~b = ~ ( N , ~O) is
monotonic decreasing itI ~ is universal.
Proof: Suppose ~b is monotonic decreasing. For the left-to-right direction, assume that th is too and that s' ~ I[~b~. Let [[~0]]s,_c X. Since ~b is PC,
then there is an s* such that [[q,]]s. = X and I[Nlt~. = [[N~,,. We can construct s* by thinning entries from s', since ~Ois monotonic decreasing, so
that s * _ s'. Since 4~ is monotonic decreasing, s*clth~ and so @ is
monotonic increasing. That ~ is antipersistent follows for the same
reasons as in the previous lemma. ~ is therefore universal. The other
direction is easy,
L E M M A 23: If th is strongly equivalent to an F- or E-sentence, then ~b
is monotonic increasing.
L E M M A 24: If th is strongly equivalent to a UNE-sentence, then ~b is
monotonic decreasing.
Both these lemmas follow quickly from preceding ones.
We now prove some general equivalence lemmas concerning determiners:
L E M M A 25: ~ is universal iff -n~ is limited.
Proof: (Left to right) Assume @ is universal, but that -n~ is not
limited. Then 7 @ must be either nonpersistent or not monotonic
decreasing. Suppose that it is nonpersistent. Then there are sets X, Y and
B such that X _ Y, -n~(Y, B) but not ~ ( Y , B). By our classical theory
of determiners, ~(Y, B). But, since ~ is antipersistent, ~ ( X , B), which is
a contradiction. So, suppose that ~fl0 is not monotonic decreasing. Then
there are sets W, Z and A such that -n~(A, Z) and not -nfl0(A, W)
although W c Z. Again, by our classical theory, @(A, W). But, since ~ is
monotonic increasing, @(A, Z), contradiction. The other direction is
similar.
L E M M A 2 6 : @ 7 is limited iff
Proof: The truth conditions
determiner of the form @-I(N,
@([[N], [[~0]). Suppose now that
~ is existential.
of a sentence ~b containing a limited
qJ) are given as follows: ~ ( [ [ N ] , [[~O]])iff
@7 is limited and that ~-n([[Nll,, i[qJ~s) for
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some s. By the truth conditional equivalence given, ~(~N],,,[[-n~O],).
Suppose [[-ntp]s ___X. Assuming that 4) is PC, there is an s* such that
X = [ [ ~ s * and ~N~s =[[N~.. Suppose [[O]]~.___~qt~s. Then [[~(~N]]s.,
[[t)]~,), since ~ - l is monotonic decreasing. But then [[~](HN],*, [hqJ]~.),
and so @ is monotonic increasing. Suppose it is not the case that
~qj],, _c ~O]]s. Take a positive reduct of s*, s # with respect to N and -n0.
Given that s # is a positive reduct only with respect to N and ~ ,
~tp~=O___~O]]~. Then [[@-7~(~N~s*, ~0~,~), since ~-q is monotonic
decreasing. But then ~@]]([[NB~, [[~]1~), and so ~ is monotonic increasing. It is easy to show that ~ is persistent, if @-n is, given the truth
conditional equivalence for ~ - l . So @ is existential.
Going the other way, suppose @ is existential. Again it is trivial to
show that @7 is persistent if ~ is. To show that ~-1 is monotonic
decreasing, suppose that ~(~N~s, [[~O]]~) for some s. By the truth conditional equivalence given, ~-n([[N~s, [[qt]]~). Suppose X _ [[Oils. Assuming
that 4) is PC, there is an s* such that X = [[~b~. and IN~ = [[N]],.. Suppose
[[~0~s* -~ [[q~H~.Then ~]l([[N~s*, ~-n~0~.), since ~ is monotonic increasing.
But then ~]](~N]s*, [[0~,*), and so ~
is monotonic decreasing. Suppose it is not the case that [[~O]]~._~~0~- Take a positive reduct of s*, s ÷,
with respect to N, ~Oand -q~b. Construct a set of entries s # from s + such
that
[[N~ = ~N]]s+= [[N~,, H~O]]s~= ~t)]]~ = [[qJ]]~*,
and
[[-nq~s~=
[[~OUs + U~-nO]]~. We must show that s # is a partial function. The one
constraint that we must check is ~-~O~s~ 7/~q~]~ = 0. Since s is a situation,
~ ] ] ~ f-I ~]~ = 0, But [[~0]~ _ [[~]]~ and [[01]~+= 0. So the constraint is
satisfied and s # is a situation. But ~Ot]~,~--~-n~0]]~ and so ~(~[N~*,
[[-nq~s#), and then ~-~(~N]]~, [[~0~s~) and since ~b]]~ = ~0~., ~ is monotonic decreasing. Consequently, ~ is limited.
L E M M A 27: ~
is negative iff ~ is existential.
Proof: Easy, given our classical theory of determiners.
The following corollary now follows from the last three lemmas,
assuming that determiners are defined as relations on a well-formed
universe of sets:
T H E O R E M 28: The negation of a UNE-sentence is strongly equivalent
to an E-sentence.
To prove the main part of the theorem, we must do an induction on the
number of quantifiers in 4).
The proof relies on the following conjecture, which we take to be a
universal feature of natural language:
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P R O P O S I T I O N 29: Let ¢ be neither monotonic increasing nor monotonic decreasing. Then 4' = ~ ¢ is neither monotonic increasing nor
monotonic decreasing.
That is, there is no natural language quantifier that can transform a
nonmonotonic sentence into a monotonic one. We hypothesize that no
substitution of a quantifier for 'Jack' in
(54)
Jack owns exactly one cat.
yields a monotonic sentence. Though we have no formal proof of the
nonexistence of such a quantifier, we cannot see how a natural language
could contain one.
Proof: Let 4' = ~1 . . . ~.¢ = ~IXBasis: Let n = 0. ~ is an F-sentence, and is monotonic increasing. So
the result is trivial.
Induction: Let n > 0. Assume the theorem for all sentences with k < n
quantifiers. Case 1: Suppose 4' is monotonic increasing. Case la: Suppose X is monotonic increasing. Then X is strongly equivalent to an F- or
E-sentence by the inductive hypothesis. But then @1 is existential, by
Lemma 18, so 4' is strongly equivalent to an E-sentence. Case lb:
Suppose X is monotonic decreasing. Then ~1 is limited, by Lemma 20
and so monotonic decreasing but persistent. X is strongly equivalent to a
UNE-sentence, by hypothesis. But the substitution of any limited determiner for a negated universal determiner preserves strong equivalence
by L e m m a 25. So 4' is strongly equivalent to the negation of a UNEsentence, i.e., to an E-sentence by Theorem 28. Case lc: Suppose X is
nonmonotonic. This is absurd, by Proposition 29. Case 2: Let 4' be
monotonic decreasing. Case 2a: Suppose X is monotonic decreasing. By
L e m m a 22, 91 must be universal; by hypothesis, X is equivalent to a
UNE-sentence. So 4' is equivalent to a UNE-sentence. Case 2b: Suppose
X is monotonic increasing. By L e m m a 21, @1 is negative; by hypothesis X
is equivalent to an F- or E-sentence. So 4' is again equivalent to a
UNE-sentence. Case 2c: Suppose X is neither monotonic increasing or
decreasing. Proposition 29 implies that this is absurd. Following from the
above is our theorem:
T H E O R E M 30: Let 4, be PC. Then 4, is monotonic increasing just in
case it is strongly equivalent to an F- or E-sentence. 4' is monotonic
decreasing just in case it is equivalent to a UNE-sentence.
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NOTES
1 We are grateful to Hanno Beck, Irene Heim, Hans Kamp, Richard Larson, Stanley Peters
and Richmond Thomason for their help and comments on an earlier draft of this paper, and
to the Center for Cognitive Science of the University of Texas at Austin for its research
support.
2 For another approach adopting much the same strategy, but within Montague grammar,
see (Enc, 1981).
3 In (Barwise and Perry, 1983), situations are concrete; situation types sort them according
to various uniformities; abstract situations (or states of affairs) are located situation types;
and courses of events are sets of abstract situations. Since we shall not concern ourselves
with matters of tense, we shall ignore these last two categories.
4 Our assumption excludes "branching" or "partially ordered" qnantifiers; see (Hintikka,
1979).
5 In the language of situation semantics, we express this principle as follows:
(55)
VsVs', s" ~_ s((s[Tsee~, s', a) = 1 & s([[see~, s", a) = 1) ~ s' = s")
6 The special principle for S-5 is:
(56)
3s' ~ s(sl[see],s',a)= l & s' ~l[da~ & Vs" c_s((s(l[see~,s",a)= l)
~s"~s')).
7 Negative quantifiers, intuitively, export with little trouble. We can explain the validity of
arguments like (10) by noting that, although S-6 does not universally validate negative
quantifier exportations, countermodels have a peculiar form; ruling them out requires a
revision to S-6 that goes beyond the scope of this paper. See (Asher and Bonevac, 1985b).
8 According to our truth conditions, (53)'s premise is strongly equivalent to its conclusion.
Some speakers find this appropriate. Others obtain a reading in which (53) does not invert.
We conjecture that they read negative quantifiers as ambiguous. Situation semantics
provides a way of analyzing this ambiguity; investigating it here, however, would take us far
afield. For more on this question, see (Asher and Bonevac, 1985a; Asher and Bonevac,
forthcoming).
REFERENCES
Asher, N. and D. Bonevac: forthcoming, 'Negative Quantifiers'.
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Asher, N. and D. Bonevac: 1985, 'How Extensional is Extensional Perception?', Linguistics
and Philosophy 8, 203-228.
Barwise, J.: 1981, 'Scenes and Other Situations', Journal of Philosophy 78, 369-397.
Barwise, J. and R. Cooper: 1981, 'Generalized Quantifiers in Natural Language', Linguistics and Philosophy 4, 159-219.
Barwise, J. and J, Perry: 1981, 'Situations and Attitudes', Journal of Philosophy 78,
668-691.
Barwise, J. and J. Perry: 1983, Situations and Attitudes, MIT Press, Cambridge, Massachusetts.
Enc, M.: 1981, Tense Without Scope: An Analysis of Nouns as Indexicals, Ph.D. thesis,
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Hintikka, J.: 1979, 'Quantifiers versus Quantification Theory', in E. Saarinen (ed.), GameTheoretical Semantics, D. Reidel, Dordrecht, pp. 49-80.
Kamp, H.: 1981, 'A Theory of Truth and Semantic Representation', in Formal Methods in
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the Study of Language, J. Groenendijk, th. Janssen and M. Stokhof (ed.) Mathematisch
Centrum Tracts, Amsterdam, pp. 277-322.
Karttunen, L.: 1976, 'Discourse Referents', in J. D. McCawley (ed.), Syntax and Semantics,
Academic Press, New York, pp. 363-386.
Larson, R.: 1983, Restrictive Modification: Relative Clauses and Adverbs, Ph.D. thesis,
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Lewis, D.: 1979, 'Scorekeeping in a Language Game', Journal of Philosophical Logic 8,
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van Benthem, J.: 1984, 'Questions about Quantifiers', Journal of Symbolic Logic 49,
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Center for Cognitive Science
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University of Texas
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U.S.A.