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In Journal of Semantics 11 (1994): 365-406.
Contrast and Implication in Natural Language
Yoad Winter
Utrecht University
Mori Rimon
Hebrew University
Abstract In this paper we introduce a theoretical framework and a logical application for analyzing the se- mantics and pragmatics of contrastive conjunctions in natural language. It is shown how expressions like although, nevertheless, yet and but are semantically definable as connectives using an operator for implication in natural language and how similar pragmatic principles affect the behaviour of both con- trastive conjunctions and indicative conditionals. Following previous proposals, conditions on contrast in a conjunction are analyzed as presuppositions of the conjunction. Further linguistic evidence leads to a distinction between restrictive and non-restrictive connectives of contrast, and consequently between direct and indirect contrast, which are given a precise definition. A general interface for a theory of contrast using possible world semantics for implication is then presented. As a test case, we show how this interface is applicable to the semantics for conditionals that was introduced by Veltman in his article "Data Semantics and the Pragmatics of Indicative Conditionals" (1986). This application yields an extension of Veltman’s Data Logic, called Contrastive Data Logic. Once appropriate modifications are added to Veltman’s pragmatic considerations, we show that contrastive data logic provides an adequate tool for the analysis of substantial linguistic data concerning contrast and implication in natural language.
1 Introduction
Natural languages support their speakers with many ways of expressing contrast between elements in a sentence or in a discourse. Particles like ( al ) though, but, yet, nevertheless are some of the common expressions in English used for this purpose. Still, a formal semantic theory may find the notion of ’contrast’ quite obscure. The difficulty was wonderfully expressed by Frege in "On Sense and Reference":
"Subsidiary clauses beginning with ’ although ’ also express complete thoughts. This conjunction actually has no sense and does not change the sense of the clause but only illuminates it in peculiar fashion (similarly in the case of ’ but ’,’ yet ’). We could indeed replace the concessive clause without harm to the truth of the whole by another of the same truth value; but the light in which the clause is placed by the conjunction might then easily appear unsuitable, as if a song with a sad subject were to be sung in a lively fashion." (Frege, G., "On Sense and Reference", Geach and Black(eds.)(1970), pp. 73–74)
Here we would like to maintain that what was called by Frege "the light in which the clause is placed by the conjunction" can be explained using less mysterious terms from contemporary semantic and pragmatic theories. Sharing linguistic intuitions with some previous works, especially Anscombre & Ducrot(1977) and Lang(1984), we try to establish a logical framework that captures these intuitions. Essentially, we claim that the restrictions on contrastive conjunctions and the information they convey can be formulated as presuppositions of the conjunction. These presuppositions are stated using a schema that assumes a definition for the relation of implication between sentences/utterances in the language. Therefore, what we
first propose is a reduction of the problem of contrast in natural language into the much more investigated problem of implication, a manifold puzzle that was discussed in the large literature on indicative conditionals, counterfactuals and pragmatic implicatures. Then, our next step is to restrict this reduction– to find and to spell out general meaning (/use) postulates for the connectives of contrast and the implication operator they involve. The postulates are stated using the possibility and necessity operators of modal logic. The presuppositions for contrast together with these restrictions constitute what we call an " interface " to a formal theory of implication in possible world semantics. An interesting version of such a theory was presented by Veltman in his article "Data Semantics and the Pragmatics of Indicative Conditionals" (Veltman(1986)). We use the proposed interface to incorporate the connectives of contrast into Veltman’s Data Logic and prove the extended semantics for Contrastive Data Logic (CDL) to satisfy the restrictions of the interface, hence also the basic linguistic motivations. Many theories for implication in natural language are sensitive to the well-known "paradoxes" of implication: the counter-intuitive inferences that establish the truth of a conditional from the falsehood of the antecedent or truth of the consequent. We show how similar "paradoxes" of contrast are entailed by these problems. To account for such counter-intuitive results on implication in Data Logic Veltman uses a principle of " pragmatic unsoundness ". We add a principle of " pragmatic insufficiency ", that together with more specific pragmatic assumptions deals properly with the "paradoxes" of contrast, as well as other interactions in natural language between conditionals and modalities or contrastive elements. Some examples for this application of CDL are analyzed. The paper is organized as follows: Section 2 brings the essential linguistic data and presents a general intuitive definition for the presuppositions for contrast in a conjunction. In section 3 we formalize these intuitive ideas and establish the restrictions in an interface to a theory of implication, based on some more involved linguistic evidence. The "paradoxes" of contrast and their connection to the "paradoxes" of implication are presented and analyzed. In section 4, CDL, the application of the semantic interface to Data Logic is presented, and the required features in the interface are proved. Section 5 includes the proposed generalization to Veltman’s pragmatic framework. It is shown how the pragmatic principles are applied to handle the "paradoxes" of contrast and further linguistic evidence on the use of contrastive conjunctions in natural language.
2 Linguistic evidence and some primary motivations
Let the notation CON^2 stand for the class of the CON nectives of CON trast, which consists at least of the following members: but, nevertheless, yet, although, though, even though. If we try to follow Frege’s line of thought in the above quotation, then the truth conditions of a contrastive conjunction p con q 1 are identical to those of the parallel conjunction p and q. One part of this identity is certainly true– by no doubt sentences like (1a) logically entail their and counterpart (1b):
(1) a. I love Venice but I would not like to be there again. b. I love Venice and I would not like to be there again.
Then we come to the question of whether boolean conjunction is the only truth conditional content of the CON^2 members. For instance- what is the problem in (1c), by contrast to the acceptable (1d):
(1) c.? I love Venice but I would like to be there again. d. I love Venice and I would like to be there again.
It seems plausible to claim that examples like (1c) cannot be considered a case where some truth condition of the sentence is false, at least not in the traditional definition for truth conditions. This claim is the
Reasonably, a rough formulation of the contrastive presupposition of a sentence in the form p yet q or q although p can be stated simply as:
(A) p implies not (q)
In (5a), p (= a= "it was cloudy") does not imply not (q) (= not (b) = "it was not raining"), so (A) does not hold and therefore the sentence is infelicitous. In (5b) p does imply not (q) (= b = "it was raining") and the sentence is felicitous. The same is with respect to (5c) and (5d). At this stage we are still not trying to explain what exactly do we mean in the relation "imply" between two sentences. The reader can think of this relation for the moment as something akin to "default" implication (as in an indicative conditional, e.g. "Normally, if it’s cloudy then it rains."). The recognized advantage of this vague notion is its intuitive appeal to speakers, and thus it may help us when trying to outline the informal presupposition schemata for members of CON^2. Later, when we use these schemata within a formal framework, the elusive nature of the notion "implication" will become our main concern. Clearly, (A) is too strong to account for certain other examples of contrastive conjunctions. For example:
(6) a.
p
[ We were hungry ] but
q
[ the restaurants were closed. ] b.?
p
[ We were hungry ] yet
q
[ the restaurants were closed. ] c.?
q
[ The restaurants were closed ] although
p
[ we were hungry. ]
In a "standard" context for (6a-c) p does not imply not (q), and as predicted by (A), (6b) and (6c) are indeed infelicitous. 4 The but conjunction in (6a), however, is still felicitous. Another example for this difference between but and other connectives of contrast is:
(7) a. It is raining but I took an umbrella. (Therefore, I won’t get wet). b.? It is raining; nevertheless , I took an umbrella. (Therefore, I won’t get wet). c.? I took an umbrella even though it is raining. (Therefore, I won’t get wet).
Once again, (A) is too strong for a but sentence like (7a) while it does predict the infelicity of (7b) and (7c). Also, the way it accounts for their infelicity is very intuitive; for instance- an expected reaction from a hearer of a sentence like (7b) or (7c) is: "What, for heaven’s sake, is so special in taking an umbrella when it rains outside ?". In the same way condition (A) also explains why when one of the conjuncts is negated in (7b) or (7c) the sentence becomes highly acceptable:
(7) d. It is not raining; nevertheless , I took an umbrella. e. It is raining; nevertheless , I did not take an umbrella.
Sentences as in (6) and (7) show that the case of but is different from those of yet, although, nevertheless, even though and other connectives of contrast. While the later require, as (A) claims, a direct relation between the conjuncts, in the former only some kind of an indirect relation between the conjuncts may be sufficient for coherence. In our proposed formalization for this "indirect contrast" we follow (independently) ideas of Lang in his discussion of German aber (see Lang(1984:pp.169-175)) and Anscombre & Ducrot in their account for the French mais (see Anscombre and Ducrot(1977), Ducrot et. al.(1980:pp.93-130)). 5 The basic idea is this: for a contrastive conjunction p con q to be felicitous in a given context there should be some statement which p implies and q denies. Let us use the notation r to represent the negation of such a statement. The statement p then implies not (r) and q implies r (q denies not (r)). The statement r is an overall implication of the whole contrastive sentence. Consider for example (6a): a possible r is "we didn’t eat" , as p (= "we wanted to eat") can imply "we ate" (= not (r)) and q (= "the restaurants were closed") implies "we didn’t eat" (r).
Similarly, in sentence (7a) a possible r can be " I will not get wet ". The conjunct p (rain) implies the possibility of getting wet ( not (r)). The conjunct q (umbrella) implies r, the negation of this possibility. The intention of a speaker in uttering any CON^2 sentence is always to make an argument in favour of a certain r. In the case of yet , although and most other members of CON^2 , r should be expressed explicitly as the conjunct q. In the case of but , r can also be introduced implicitly, using the context of utterance, as in (6a) and (7a). A general condition, presupposed for all the CON^2 sentences is therefore:
There exists a statement r s.t. in the context of utterance:
( C1 ) p implies not (r) and q implies r.
For a large sub-class of CON^2 (including nevertheless, although , etc.) ( C1 ) should be restricted as follows:
( C1 )[restricted] r = q
Condition ( C1 )[restricted] is exactly (A), since under any sense we can use "imply", it is certainly a reflexive relation, and then the requirement "q implies r" in ( C1 ) is trivially satisfied when r= q. In general, ( C1 ) by no way requires that r is unique. A but utterance may be vague with respect to the r that is intended to be implied. For example: both following inferences in (8) are felicitous:
(8) a. We were hungry but the restaurants were closed. Therefore,
r
[ we didn’t eat. ] b. We were hungry but the restaurants were closed. Therefore,
r
[ we didn’t eat in a restaurant. ]
The " therefore " context is a crucial test to distinguish between a possible r and an impossible one. For example:
(9) a. John is tall but he is clumsy. Therefore,
r
[ he is a lousy basketball player. ]
b. John is tall but he is clumsy.? Therefore, not^ (r^ )^ [ he is a good basketball player. ]
The " therefore " contexts in (9) are interesting also from another respect. We see here that as we claimed, r, which is implied by q, is an acceptable conclusion from the whole but conjunction. On the other hand, not (r) is implied by p, and both p and q are true, so what is the principle that allows r as a conclusion from p but q and does not allow r’s negation? We must conclude, as Anscombre and Ducrot observe too, that in some sense the implication "q implies r" is "stronger" than the implication "p implies not (r)" and an application of a principle like Modus Ponens is allowed only for the "stronger" implication. Another way to look at this fact is to say that q denies not (r) or that application of Modus Ponens to the second implication cancels conclusions obtained by Modus Ponens applied to the first implication. Whatever notion we choose, "strength" (of implications) or "cancellation" (of conclusions) should be explained formally. For the meantime, let us only state this as another informal condition on CON^2 sentences:
( C2 ) q’s implication of r is "stronger" than / "cancels" p’s implication of not (r).
Note that ( C2 ) should be trivially satisfied when ( C1 )[restricted] holds, since it should be guaranteed that the logical entailment between q and itself is a "stronger" implication than p’s implication of not (q). Evidently, p cannot logically entail not (q), to avoid a contradiction caused by the truth condition p and q. ( C2 ) can easily account for the strong asymmetries in the romantic implications between (10a) and (10b)^6 :
(10) a. There are many girls around but YOU are special. b. You ARE special but there are many girls around.
and for similar asymmetries in the more prosaic (11a-d):
(20) John did not waste his money but bought three books on the history of Lapland.
Sentence (20), opposed to (18), implies that the books that John bought were not a waste of money (rectification of the predicate in the first conjunct). There are also some evident syntactic differences between these two uses of but. To name just two of them: the gapping effects in conjunctions with but for rectification, and the requirement that the first conjunct includes an overt negation in such constructions. In some languages such as Spanish, Ger- man and Hebrew these two uses of but involve distinct lexical entries: pero / sino , aber / sondern , aval / ela , respectively. This distinction is discussed extensively in previous works including: Tobler(1896), Me- lander(1916), Abraham(1977), Anscombre and Ducrot(1977), Dascal and Katriel(1977), Horn(1985) and Horn(1989:pp.402-413), which contains a detailed bibliographic survey. Yet another use for but is in exceptive constructions. For example:
(21) Everyone but John came to the party.
For a semantic analysis of exceptive constructions see Hoeksema(1991) and von Fintel(1991) and their detailed bibliographic remarks. For a syntactic account see Reinhart(1991). These facts may suggest that the morpheme but in English is at least three-way lexically ambiguous. However, there are some strong relations between these three senses of but , which suggest that the last word on this issue has not yet been given. 7
Some writers (e.g. R. Lakoff(1971), Blakemore(1989)) have maintained that there are even two distinct senses for the use of but we labeled as contrastive^8. The distinction made is between using but for "semantic opposition" and but as marking "denial of expectation". The two alleged senses can be exemplified by the following sentences, slight variations of examples from Lang(1984):
(22) John is quick but Bill is slow.
(23) John is quick but he is no good at football.
While (22) was described as a "semantic opposition" use of but , which does not require any kind of world knowledge or contextual factors, (23) was considered as involving some further knowledge for modeling the "denial of expectation" in the second conjunct. We share with Lang the opinion that this distinction is theoretically problematic and not fully motivated by empirical data. For example: (22) might become infelicitous in cases where we are looking for a couple of persons, one quick and the other slow, and the sentence is given as an argument for John and Bill as an appropriate couple. It becomes hard to explain such facts with a theory that considers but ambiguous between "contrast" and "denial", since it has to be assumed then that the "contrastive" meaning of the connective disappears somehow in a situation like the above. Instead, we tend to prefer, with Lang, a theory that assigns but the same interpretation in all situations. Taking such a position, we have to show that sentences like (22) satisfy ( C1 ) and ( C2 ) in "normal" contexts. For example, if someone indicates that all the players in a team are quick, (22) can be used to deny this indication, provided that John and Bill are in the team. ( C1 ) and ( C2 ) are then satisfied with r= "not all the players in the team are quick". Thus, there are good reasons to stick to the null hypothesis (pace R. Lakoff and Blakemore) that no ambiguity whatsoever is to be attributed to the contrastive but. 9
An interesting discussion of the French connective mais , with which we share many linguistic intuitions (but not methodological assumptions or descriptive tools) appears in Anscombre and Ducrot(1977). A&D discuss mais within their framework of argumentative scales (ech`elles argumentatives). As defined in
Ducrot(1980:pp.15-18), two sentences p 1 and p 2 are placed by the speaker in the same argumentative
scale if there is a third sentence r for which he/she considers p 1 and p 2 as arguments. p 1 is said to be
argumentatively superior to p 2 with respect to r if accepting r as a conclusion from p 2 implies that r is
accepted as a conclusion from p 1 , but not the opposite. For A&D, an utterance of a sentence p mais q is
possible if p and q are on "opposite" argumentative scales, where p is an argument in favour of r and q is an argument in favour of r’s negation. Clearly, our analysis of but is on the same lines of A&D. Our main objection to A&D’s proposal is a conceptual one: it seems that argumentation in natural language is a name for a linguistic phenomenon and A&D’s framework is more like a detailed and careful description of this phenomenon. To say that a sentence like "she is tall" is an argument in favour of the sentence "she is a good basketball player" is simply to restate the intuitions based on our world knowledge. It is not an explanatory model of the facts. In A&D’s works it is assumed that argumentative scales exist but nothing in their discussion predicts which sentences are arguments in favour of which sentences in which contexts. Consequently, A&D’s framework is rather informal. What we try to do in the following sections is to use the linguistic intuitions (on which we agree with A&D in most cases) in order to provide a formal account of the informal notions "implication" and "strength of implication" in ( C1 ) and ( C2 ). We will try to show that there are strong relations between the kind of implication that exists in indicative conditionals and the implication operator that is needed to model contrast. In general, it is assumed that A&D’s description of argumentative scales should be predicted by comprehensive semantic and pragmatic theories of contrast and implication in natural language.
Before going on, we may summarize the main ideas of this section. The class of connectives of contrast can be divided into two sub-classes- restrictive and non-restrictive. Some examples follow (non-exhaustive):
- Restrictive:
(a) yet, nevertheless – conjunctions denoted p con q. (b) although , ( even ) though – conjunctions denoted q con p.
- Non-Restrictive: but – conjunctions denoted p con q.
A presupposition for CON^2 conjunctions guarantees a relation of contrast between p and q:
There exists a statement r s.t. in the context of utterance:
( C1 ) p implies not (r) and q implies r. ( C2 ) q’s implication of r is "stronger" than / "cancels" p’s implication of not (r).
For the restrictive sub-class of CON^2 , direct contrast must exist between the conjuncts. Direct contrast is established when conditions ( C1 ) and ( C2 ) hold with an r that is logically equivalent to q. Under this restriction ( C1 ) and ( C2 ) should boil down to:
( C1 )[restricted] p implies not (q)
A relation of contrast which is not direct is called indirect contrast. The non-restrictive members of CON^2 allow also indirect contrast between the conjuncts. In English, the most typical representative for the non-restrictive subclass of CON^2 is the connective but^10 , when it is used in a sentential contrastive conjunction^11. These are the key notions which we are going to need in order to provide a formal theory of contrast in natural language.
- Possible world semantics: A weak implication is an implication in another possible world: x weakly
implies y in a world w iff x!y in some world w’ accessible to w.
In what follows we investigate only the second option. There is of course a lot to say also on the first one. We do not attempt to do it in this paper.
We are talking then possible worlds. Let us use the notations 3 and 2 as the possibility and necessity
operators, ^; _; : as usual, and ) as logical entailment (in terms of truth conditions). From ( C1 ) and ( C2 )
we can draw the following definition:
( D 1 ) The contrast relation: A proposition r establishes contrast between two (ordered) propositions p
and q iff 3 (p! :r ) ^ (q! r ) is true. This relation is denoted by Θr (p; q).
Definition ( D 1 ) introduces the notion of contrast in terms of the implication connective and the modal
operator 3. We will not pay much attention to the "kind" of world where the weak implication is realized.
For instance, in (5b) this world can be called "standard", in (6), "expected". We maintain that such typology of possible worlds is linguistically meaningful only with a theory of how certain expressions ( to want , to expect , etc.) affect the accessibility relations of the "actual" world. We do not wish to impose this problem
on the other major problems with respect to CON^2. Therefore we only use the possibility operator 3 with no
specification of the type of possible worlds quantified over. Definition ( D 1 ) can now be applied to derive from ( C1 ) and ( C2 ) the definitions of the truth conditions and the presuppositions of contrastive conjunctions 12.
( D 2 ) p but q , p ^ q
pres (p but q) = exists r s.t. Θr (p; q)
( D 3 ) p nvs q , p ^ q
pres (p nvs q) = Θq (p; q)
How are these definitions interpreted in a certain semantics depends on the exact definitions for the
notations used ( pres , !, 3 ). Obviously, in proposing an "interface" one should take into account the
restrictions on the interpretation of these notions that stem from linguistic evidence. Such restrictions should guarantee that any proposed semantics that uses this interface is adequate to linguistic data and not just a sterile formal machinery. We therefore turn now to show some motivations or meaning postulates (" M ’s") that the above definitions for the truth conditions and presuppositions of but and nvs conjunctions should satisfy within any specific proposed semantic structure.
3.2 Semantic restrictions on the interpretation of (D 1 ) -(D 3 )
The first restriction is quite abstract: in order to guarantee that the intuition in ( C2 ) is satisfied we must make
sure that whenever the truth condition p^q and the presupposition Θr (p; q) of a contrastive conjunction are
satisfied, :r is only a weak implication of p, and not (accidentally) a strong one:
( M 1 ) Θr (p; q) ^ p ^ q ) :(p! :r ) (if r establishes contrast between true p and q then p does
not imply :r)
This should follow trivially from definitions if! satisfies Modus Ponens.
A second more concrete M is based on the linguistic fact that any restrictive member of CON^2 (represented by nvs ) can always be replaced by a non-restrictive but , while the opposite is not necessarily true (witness (6) and (7)):
( M 2 ) pres (p nvs q) ) pres (p but q) (the presupposition for direct contrast between p and q entails the
presupposition for indirect contrast)
pres (p but q) 6 ) pres (p nvs q) (the converse is not necessarily true)
The first entailment is of course immediate by definition, the second is not. Restriction ( M 1 ) together with ( M 2 ) guarantee that the intuition spelled out in ( C2 ) is satisfied by both ( D 2 ) and ( D 3 ): ( M 1 ) entails that definition ( D 2 ) satisfies ( C2 ) and ( M 2 ) claims that ( D 3 ) is stronger than ( D 2 ). The next M is on the interactions between the connectives of contrast and the possibility and necessity operators. Consider for example the sentences: (24) a. It is possible for us to swim; nevertheless , we don’t. b. It is possible for us to swim; nevertheless , it isn’t necessary. (25) a. The coin wasn’t in the drawer even though it could have been there. b. The coin didn’t have to be in the drawer even though it could have been there.
The pattern of the a sentences above is ( 3 p) nvs :p and the pattern of the b’s is ( 3 p) nvs : 2 p (the
corresponding but patterns are consequences, respectively, of these two by ( M 2 )). It seems very plausible to conclude that these two patterns are always felicitous whenever their conjuncts are true:
( M 3 ) ( 3 p) ^ :p ) pres (( 3 p) nvs :p)
or alternatively:
( M 3 ’ ) ( 3 p) ^ : 2 p ) pres (( 3 p) nvs : 2 p)
The reason we provide here two alternatives for ( M 3 ) is that although in most possible world semantics ( M 3 )
is more general than ( M 3 ’ ) since usually 2 p ) p and then :p ) : 2 p, in Veltman’s data semantics
that we use in section 4, the opposite (p ) 2 p) holds for descriptive (= non-modal) p’s. We want the more
general among the ( M 3 )’s to hold, whatever possible world semantics is used.
We turn now to some restrictions on the implication connective (!). The first restriction is trivial: In
section 2 we required that ( C1 ) holds whenever ( C1 )[restricted] (p weakly implies :q) is satisfied. This
means we claim that Θq (p; q) is entailed by 3 (p! :q) (the converse is by definition). Therefore we
require that the proposition (q! q) in Θq (p; q) is vacuously satisfied, so! must be a reflexive operator.
This is a reasonable assumption to make in any other respect we can think of.
A more interesting restriction on! is motivated by the following example:
(26) (F, John’s father, to D, a doctor, after D operated John’s leg) F: John walks now very slowly, doctor.
D: Yes indeed, p^ [ your son walks slowly ] but q^ [ he walks !]
The plausible interpretation of the reply of the doctor is with an r as "the operation was successful"^13. We
are faced then with the following puzzle: How comes p ) q ("John walks slowly" logically entails "John
walks"), q!r ("John walks" implies that "the operation was a success"), and still, we cannot conclude that
p!r (certainly, "John walks slowly" does not imply here that "the operation was successful").
Two conclusions are possible: One possibility is that logical entailment is not a special case of !, i.e.
there can be sentences p and q s.t. p ) q and still in some possible world (p! q) is false. An alternative
conclusion can be that! is not transitive. Actually, both these possibilities might seem plausible (but see
sub-section 5.4): if we think of! as some kind of a "cognitive reasoning" relation, then it should be neither
transitive (why should we be capable of obtaining all the transitive closure of implications that we already know ?) nor containing the relation of logical entailment (why should we be capable of obtaining all the logical consequences of a statement ?). We may conclude that the analysis of contrast provides us with the following restriction on implication:
(31) John sneezes. a.? Therefore, if John does not sneeze then Eric Satie is a German philosopher. b.? Therefore, if John does not sneeze then Eric Satie is a French composer.
(32) Eric Satie is a French composer. a.? Therefore, if John sneezes then Eric Satie is a French composer. b.? Therefore, if John does not sneeze then Eric Satie is a French composer.
In general: the problem is to account for what is wrong (if anything) in the following inferences:
(I 1 )
:x
x! y
(I 2 )
y
x! y
These are problems for material implication^15 , but they are shared by many other common accounts of implication in natural language. Our point is that when this kind of puzzle is created by the definition for
the meaning of! it causes similar problems for the analysis of the felicity of contrastive conjunctions. For
instance: suppose that someone knows that there is a possibility that a statement p is false ( 3 :p). Will
(s)he then be willing to accept any sentence of the form p nvs q as felicitous, as the trivial satisfaction of
( D 3 ) ( 3 (p! :q)) predicts? Suppose you know there exists a possibility that John does not sneeze and
you tell it to somebody. Will you be ready to accept the following response as felicitous?
(33)? Generally, there’s a possibility that John doesn’t sneeze. Actually, he sneezes; nevertheless, Hans Eisler was a German composer.
The inferences in question, which in most possible world semantics are derivable from (I 1 ) and (I 2 ) are:
(I 3 )
3 :x
3 (x! y )
(I 4 )
3 y
3 (x! y )
There is independent linguistic evidence for the implausibility of inferences rules (I 3 )-(I 4 ) in natural language. Consider for example the following variation on (31a):
(34) Maybe John does not sneeze.? Therefore, maybe if John sneezes then Eric Satie is a German philosopher.
What is disturbing us in the context of contrastive conjunctions is that (I 3 ) and (I 4 ) entail according to definitions ( D 1 )-( D 3 ) the following inferences, respectively:
(I 5 )
3 :p
Θq (p; q) (= pr es(p nvs q))
(I 6 )
3 :q
Θq (p; q) (= pr es(p nvs q))
This is while examples like (33) suggest that in cases where the only data is that 3 :p or 3 :q it is not
automatically guaranteed that the presupposition of contrast for p nvs q is satisfied. Even less digestible are the inference patterns that material implication creates for the non-restrictive but :
(I 7 )
r ^ 3 :r
Θr (p; q) (= pr es(p but q))
This seems quite traumatic: how can evidence on r ^ 3 :r allow automatically any but conjunction, with no
respect to identity of the conjuncts? The problem looks even harder as one notices that according to most
possible world semantics the existence of a proposition r such that r ^ 3 :r holds is something that should
happen in usual linguistic contexts: it only requires the existence of one possible world distinct from the "actual" one! As (I 5 )-(I 7 ) are closely related to the well-known (I 1 )-(I 2 ), we will refer to them as the "paradoxes" of contrast (an alternative name could be ’"paradoxes" of weak implication’ ). Note also the similarity between utterances like (33) and ones like (12b)(1) (section 2). In (12) the problem seemed to be the (missing) "relevance" of r to the context. In (33) the problem is the "relevance"
of the conjuncts p and q to each other. However, (33) is problematic only for treatments of! that allow for
inferences (I 1 ) and (I 2 ) and do not have an explanation for their alleged unsoundness. So, the origins of the problem are the well-discussed "paradoxes" (I 1 ) and (I 2 ) for (material) implication. Basically, there are two main trends in the literature to deal with this kind of problems: one possible strategy claims that (I 1 ) and (I 2 ) are semantic problems for any operator of implication for which they are valid. Therefore, according to this point of view, material implication and many other implication operators are not very relevant for the semantics of implication in natural language (and also in mathematics, if accepting the paradigm of Relevance Logic). Another trend maintains that (I 1 ) and (I 2 ) are to be considered logically valid, and that the question why human beings do not tend to accept these inferences as generally sound should be answered using pragmatic considerations.
Whatever position to be taken, we do not consider possible restrictions on! that are designed to cope
with the "paradoxes" of implication to be part of the semantic interface for CON^2. This should be done by any specific semantic and pragmatic application proposed for the interface. It should be clear, though, that any substantial semantic or pragmatic theory that deals with the problems that (I 1 ) and (I 2 ) raise should explain what is wrong with (I 3 ) and (I 4 ), thus also with (I 5 )-(I 7 ), the "paradoxes" of contrast. In sections 4 and 5 we apply the interface to Veltman’s theory of indicative conditionals, and there it will be our job to explain how these problems are overcome (see 5.3).
A closely related question that comes to mind is whether there are any inherent restrictions on the conjuncts, which should prevent sentences like:
(35)? Isaac won the elections in Jamaica but it is not raining today in Paris.
Generally speaking, when (35) is uttered with no further information it surely sounds like nonsense. After all, what can be the contrast between the results of the elections in Jamaica and the rain in Paris. But consider now the following context:
Interlude - Nesting of contrast operators A subtle empirical question raised by the works of Francez and Meyer & van der Hoek, which is extremely important for a complete formalization of contrast, is the question of nesting of contrast operators. Meyer & van der Hoek consider the following sentence an example for that:
(36) Tweety is a bird but she does not fly but she is high up in the air (in an airplane, for instance).
We are not completely sure that this is an acceptable example in English, but this should not obscure the point here: various kinds of iteration of contrast are quite acceptable in conversation. For example:
(37) A: Did you know it? Tweety is high up in the air! B: How can it be? Tweety is a bird but she cannot fly !? A: But she is in an airplane. Ha Ha!
However, to convince ourselves that unfunny jokes like (37) are indeed a case of nesting we must become sure that it is indeed the contrast in what B says that A tries to contrast in his reaction. Here Meyer & van der Hoek’s arguments do not seem very persuasive: A’s reaction may be equally analyzed as contrasting only the second conjunct ("Tweety cannot fly") in B’s utterance. We were not successful in finding a clear-cut example for a genuine nesting of contrastive operators. Even in the almost artificial (38a) the contrast in the second sentence can be analyzed with respect to the boolean conjunction in the first one, as observed in (38b):
(38) a. It was cloudy but it did not rain. Nevertheless , we were not surprised. b. It was cloudy and it did not rain. Nevertheless , we were not surprised.
We therefore tend to think that members of CON^2 cannot be used "meta-linguistically"– to convey contrast to implications from linguistic conditions on an utterance (e.g. conditions on contrast between elements within one of the conjuncts). It seems that a similar conclusion can be drawn from a remark in the appendix in Horn(1985). However, the empirical status of "genuine" nesting of contrast is not completely clear to us. From a logical point of view, this is not to say that the problem of nesting of contrast operators is uninteresting, on the contrary. Still, a full analysis of the formal aspects of nesting would have dictated an entirely different approach to this work, and we must defer it to another occasion.
4 Contrastive Data Logic – an incorporation of the connectives of contrast into
Veltman’s data semantics for conditionals
In this section a possible application of the "semantic interface" proposed in the previous section is inves- tigated. The connectives but and nvs are incorporated into the syntax and semantics of the formalism of Data Logic proposed in Veltman(1986). The resulting extended logic will be called Contrastive Data Logic (CDL). We chose to use Veltman’s data semantics as a case study because of its formal elegance and the interesting pragmatic part of his analysis for indicative conditionals. Another interesting aspect of Veltman’s data semantics is its "dynamic" conception of knowledge, which dictates some recapitulation in the proposed static definition for contrast ( D 1 ). For the convenience of the reader we first briefly summarize the main ideas and definitions in Velt- man(1986) that are needed for our purposes. However, in order to understand also the linguistic and philosophical motivations behind Veltman’s theory, familiarity with his brief and stimulating work is rec- ommended.
4.1 Data semantics - the basic definitions
Let the logical language L have a vocabulary that consists of a set P of atomic propositions, parentheses,
the one-place operators :, may and must , and the two-place connectives ^; _ and !. The syntax of L is
defined using the standard formation rules for these operators. The operators may and must stand for the epistemic readings of the corresponding modalities in English,
! stands for the indicative conditional if...then.
A model for L is a partially ordered set S of information states and a valuation function V per state for the atomic propositions in P. Each state represents information (possibly partial) on the truth values of
the atomic propositions. Growth of information is modeled by requiring that for any two states s 1 s 2 ,
Vs 1 Vs 2. It is also assumed that each state can grow into a complete information state and therefore it is
required that for each chain of ordered states there is a maximal state s, which is assigned a total valuation
function Vs. Formally:
An information model is a triple < S; ; V > with the following properties:
(i) S 6 =
(ii) is a partial ordering of S. Each chain in < S; > contains a maximal element.
(iii) V is a function with a domain S:
(a) for each s 2 S, Vs is a partial function P! f 0 ; 1 g.
(b) if s s^0 then Vs Vs^0
(c) if s is a maximal element of < S; > then Vs is total and s is called a complete state
of information.
The semantics for L is defined in quite a natural way, where j= and =j are the satisfaction/falsification
relations between an information state in an information model and propositions in L:
Let M =< S; ; V > be an information model and s 2 S. For every proposition A is L:
If A is atomic then
(M,s) j= A iff Vs (A) = 1
(M,s) =j A iff Vs (A) = 0
Otherwise, recursively:
(M,s) j= :A iff (M,s) =j A
(M,s) =j :A iff (M,s) j= A
(M,s) j= may A iff for some information state s^0 s, (M,s^0 ) j= A
(M,s) =j may A iff for no information state s^0 s, (M,s^0 ) j= A
(M,s) j= mustA iff for no information state s^0 s, (M,s^0 ) =j A
(M,s) =j mustA iff for some information state s^0 s, (M,s^0 ) =j A
(M,s) j= A ^ B iff (M,s) j= A and (M,s) j= B
(M,s) =j A ^ B iff (M,s) =j A or (M,s) =j B
(M,s) j= A _ B iff (M,s) j= A or (M,s) j= B
(M,s) =j A _ B iff (M,s) =j A and (M,s) =j B
(M,s) j= A! B iff for no information state s^0 s, (M,s^0 ) j= A and (M,s^0 ) =j B
(M,s) =j A! B iff for some information state s^0 s, (M,s^0 ) j= A and (M,s^0 ) =j B
4.2 Incorporation of but and nvs – Contrastive Data Logic
We add now the two-place connectives but and nvs to the vocabulary of L, with the expected formation rules. In order to define the semantics of these two connectives in data semantics we should adopt a "dynamic" notion of contrast, which is more appropriate here. This requires a slight modification in the straightforward
translation of ( D 1 ): instead of saying that p "weakly implies" :r and q implies r on the same state of
information, we use a dynamic notion: if in a state of information s^0 there is a possibility that p implies
:r , and s^0 grows into an information state s s^0 where q implies r, then the contrast between p and q is
established in s. Contrast then is understood as a transition in implications. This agrees with the intuition
that in a contrastive conjunction there is some "cancellation" effect of what p might have implicated, if not for the presence of q. Formally:
(M,s) j= Θr (p; q) (read: r establishes contrast between p and q) iff
(M,s) j= q! r and there is some state of information s^0 s, s.t. (M,s^0 ) j= may (p! :r )
This definition can be illustrated as follows (an arrow from a state s 1 to a state s 2 represents the relation
s 1 s 2 ):
s^0
B
B
p! :r BN
s
q! r
Figure 1: Satisfaction of Θr (p; q) in a state s
Contrast is un-established in the complementary case:
(M,s) =j Θr (p; q) iff
(M,s) 6 j= q! r or there is no state of information s^0 s s.t. (M,s^0 ) j= may (p! :r )
Presuppositions can be incorporated into data semantics by using the undefined truth value also to represent a case of presupposition failure. We use here a common semantic definition for presuppositions in multi-valued semantics:
B = pr es(A) iff A ) B and :A ) B
Therefore, if (M,s) 6 j= A and (M,s) 6 =j A (A’s truth value is undefined in s) then we can say that either (as in
Veltman’s original framework) there is no direct evidence in s for A, or that a presupposition of A is false
in s. The difference between the two cases is that in the case of presupposition failure, even in a complete
state of information A’s truth value can remain undefined.
Now we can define the truth-functional content of p but q and p nvs q as p ^ q, and their
presuppositional content as Θr (p; q) and Θq (p; q), respectively:
(M,s) j= p but q iff (M,s) j= p ^ q and there is a proposition r s.t. (M,s) j= Θr (p; q).
(M,s) =j p but q iff (M,s) =j p ^ q and there is a proposition r s.t. (M,s) j= Θr (p; q).
(M,s) j= p nvs q iff (M,s) j= p ^ q and (M,s) j= Θq (p; q).
(M,s) =j p nvs q iff (M,s) =j p ^ q and (M,s) j= Θq (p; q).
This definition for the semantics of the formal connectives but and nvs guarantees that the presupposition
for contrast Θr (p; q) is preserved under negation, which agrees with the linguistic intuitions presented in
section 2. We do not try here to cope with other linguistic problems of presupposition accommodation, cancellation or modification. Our goal here in this respect is only to incorporate presuppositions into Data Logic with minimal technical complications. A better notion of presupposition in Data Logic can be achieved using orthogonal mechanisms of truth and presupposition-satisfaction assignments (four truth values) or using van Eijck’s novel dynamic error state semantics (see van Eijck(1993)).
4.3 Satisfaction of (M 1 )–(M 4 ) within CDL
In order to count CDL as an admissible application of the interface we presented, we should verify that the restrictions ( M 1 )–( M 4 ) are satisfied in CDL.
Proposition 1: For descriptive propositions p,q, and r, ( M 1 ) is satisfied:
Θr(p; q) ^ p ^ q ) :(p! :r)
Proof: By definition: assume by negation that (M,s) j= Θr(p; q) ^ p ^ q ^ (p! :r).
Then (M,s) j= p ^ q ^ (p! :r) ^ (q! r). So in any complete state s^0 s, (M,s^0 ) j= p ^ q (p and
q are descriptions, therefore stable). r’s truth value must be defined in s^0 (it is a description), and
any of the possibilities (M,s^0 ) j= :r(^q) and (M,s^0 ) j= r(^p) is contradictory to the assumption, by
definition of !.
( M 1 ) does not always hold in CDL when p, q or r are not descriptive. Consider for example the model
M as illustrated in figure 2 (s < s 1 ; s < s 2 ; s 1 6 < s 2 ; s 2 6 < s 1 ), with A a description with an undefined truth
value in s, and (M,s 1 ) j= :A, (M,s 2 ) j= A:
s
B
B
BN
s 1
:A
s 2
A
Figure 2: A model M for CDL
If we denote now p = may :A, q = may A and r = A then it is easy to see that (M,s) j= (p!
:r ) ^ (q! r ) and therefore (M,s) j= Θr (p; q) and obviously (M,s) j= p ^ q. Still, (M,s) j= p! :r does
not agree with ( M 1 ). This is a problem for the linguistic adequacy of CDL that is to be discussed in sub-section 5.4, when the pragmatic principles are introduced.
Proposition 2: ( M 2 ):
a. If (M,s) j= Θq(p; q) then there is a proposition r s.t. (M,s) j= Θr(p; q).
