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Ryszard Zuber

Towards a Semantic Description of Logical Connectives

Series A: General & Theoretical Papers ISSN 1435-6473 Essen: LAUD 1976 (2nd ed. with divergent page numbering 2013) Paper No. 41

Universität Duisburg-Essen

Ryszard Zuber

Towards a Semantic Description of Logical Connectives

Copyright by the author Reproduced by LAUD 1976 (2nd ed. with divergent page numbering 2013) Linguistic Agency Series A University of Duisburg-Essen General and Theoretical FB Geisteswissenschaften Paper No. 41 Universitätsstr. 12 D- 45117 Essen

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Ryszard Zuber

Towards a Semantic Description of Logical Connectives

In spite of the importance that the notion of plays in the discussions on the and of natural languages, little has been said about the presuppositional properties of the logical connectives of natural language. Prior to Karttunen 1973 no interesting results concerning of complex non-embedded sentences were reported. Karttunen's work on logical connectives and presuppositons can by no means be considered as definitve in this domain and it is very likely not to be viewed as such by Karttunen himself (see for instance his more recent work on presuppositions and context). In this paper I would like to concentrate on the socalled phenomenon of cancellation of presuppositions, the phenomenon which is the basis of the filtering conditions Karttunen proposed for logical connectives. This phenomenon was already discussed before Karttunen (Lakoff 1970, Horn 1972) but Karttunen's paper is the first to give a clear of the problem to which the data give rise and to make an attempt towards solving it, the problem namely, of exactly what it takes to cancel a presupposition. To solve this problem he proposes a special filtering device. I will try to show, however, (1) that this filtering device has some strange consequences and that, strictly speaking, the phenomenon of filtration as conceived by Karttunen does not exist, and (2) that Karttunen's filtering device should be replaced by the principle of neutralisation explained below ((19) - (21) and following comment). In this connection some remarks concerning the (or asymmetry) of logical connectives (predicates) will be made. The discussion of these two problems will lead to certain modifications in the description of presuppositional properties of complex sentences. It will follow from this modified description that some aspects of the meaning of logical connectives which have been considered as pragmatic in nature should be analysed as semantic. I will use the following semantic notion of presupposition: Sentence S presupposes sentence T if S semantically implies T and ¬S semantically implies T (Where ‚¬S’ is the strong (internal) of S, and "S semantically implies T" means that T is true whenever S is true.) When S semantically implies T, we will also say that S entails T or that T is a (semantic) consequence of S.

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It is clear that when S presupposes T then S semantically implies T. Some other consequences of the semantic definition of presupposition, as those examplified in Keenan 1973 for instance, will be assumed. Let me begin by discussing some problems from Karttunen 1973. He proposes a general treatment of complex sentenses that allows for widespread lose of preeuppositions in certain specified circumstances. This lose of presuppositions is done by a special filtering mechanism designed to deal only with the logical connectives of natural languages and, or, and if ... then. Filters, following Karttunen, are predicates which, under certain conditions cancel (filter out) some of the presuppositions of a component sentence. For instance in (1) where filtering conditions are satisfied, the known presupposition of the consequent clause is cancelled: (1) If the king of France exists he is bald. It is precisely the cancellation of (some) presuppositions which is a very strange phenomenon. For it takes place only under some conditions which depend on the semantic relations between the component sentences. This means that we cannot tell from the general logical description of a connective whether it is a filter or not; logical connectives are filters only sometimes, when some semantic conditions between of these connectives are satisfied. Consequently the semantic content of logical connectives of English depends on the semantic content of the arguments of these connectives. Furthermore, following the filtering conditions Karttunen gave, some presuppositions of a component sentence may become presuppositions of the whole sentence, but logical consequences of these presuppositions may be cancelled. For it is possible that a presupposition of the consequent of the conditional, for instance, is not entailed by the antecedent, but that some of thie presupposition is entailed by the antecedent and as such is filtered out. In other words a consequence of a necessary condition (for having -valuee) may not itself be such a necessary condition. For instance, following Karttunen's filtering conditions and usual description of the regret and state, one should consider that (2) presupposes (3) and (4) but does not presuppose (5) and (6) (filtered out because they are consequences of the antecedent): (2) If Sue comes, then Steve will regret that she did not hesitate to come, (3) Steve will know that Sue did not hesitate to come. (4) Sue will not have hesitated to come. (5) Sue will have come. (6) Somebody will have come. This consequence of filtering conditions is not only incompatible with the semantic notion of presupposition but even with our intuition concerning the notion of semantic content in general. For it means that some semantic content can disappear as a whole while logical

2 consequences of it, and thus proper parts of it in come sense, may not appear. This fact makes the notion of cancellation still more obscure. Thus my first step will be to present in another light the phenomenon of cancellation. To do this I will apply the idea put torward in Zuber 197: that some presuppositions of one component may combine with the non-presuppositional part of another component of the sentence to give a "new" presupposition of the whole complex sentence. And it may, happen that by such a combination the new presupposition thus obtained expresses a trivial truth (is an analytic sentence). Let me illustrate this point. Usually it is assumed that sentence (7) presupposes (8), (9) and (10): (7) Steve is a bachelor. (8) Steve is a human being. (9) Steve is an adult. (10) Steve is a male. Now, let us substitute the subject NP Steve by another NP which is not a proper noun and which has some semantic feature common with the noun bachelor. We have then (11) for instance, which should presuppose (12), (13) and (14): (11) This king is a bachelor. (12) This king is a human being. (13) This king is male. (14) This king is an adult. Clearly we do not want to eay that the two preeuppositione (presupposed features), those corresponding to human and male, are lost or cancelled; only that they are combined with the features of the noun king to form tautologies with trivial semantic information as in (12) and(13). Similarly with (15): we cannot say that all presuppositions obtained through the supposed features of the NP bachelor were filtered out. Sentence (15) still presupposes (16), (17) and (18), but these presuppositions are not felt because they express analytic : (15) This man is a bachelor. (16) This man is a human being. (17) This man is male. (18) This man is an adult. Now having in mind the analogy between the subject of a simple subject-predicate sentence and the antecedent of a conditional on the one hand, and between the predicate and the consequent clause of the conditional on the other hand, we can posit the following: presuppositional analysis of the : the sentence of the form (19) presupposes (20) and (21): (19) If S, then T

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(20) Pres (S) (21) S implies Pres (T) (where Pres (S) is a of all presuppositions of S and implies has the meaning of the 'horseshoe' taking into the account the third value. 'A implies B' is non-true only in case when A is true and B nontrue. 'Non-true' means either or 'neither true nor false'). This description holds independently of what the semantic relations between the component sentences S and T may be. Thus supposing that presuppositions express truth-preconditions for the truth and falsity of a given sentence, the clause (21) says that for a conditional (of the form (19)) to be considered as (or even to be) true or false, it is necessary that the antecedent of the conditional entails the presupposition of the antecedent clause. For instance (22) (22) If it rains only Steve will come. presupposes that in every in which it is true that it rains, it is also true that Steve will come. Let us now look at the case in which, due to the particular semantic relation taking place between the antecedent and consequent clauses of the conditional, there is a supposed filtering out (or cancellation) of some presuppositions of the consequent. Suppose first that T presupposes S (this is a particular case ot Karttunen is filtering conditions for conditionals). Then following the above description, (19) presupposes (23): (23) S implies S Clearly (23) is a . A more general case of the semantic relation between S and T in which a presupposition of T is supposed to be filtered out is the following: sentence which is a presupposition of T is filtered out, according to Karttunen, it is entailed by the antecedent S. This means that our description and (19), taken to together, presuppose (24): (24) implies what S implies (which we call R) (24) is also obviously a trivial truth. Thus, by (21), the presuppositions of the consequent clause of a conditional amalgamate with the (whole) antecedent clause of the conditional to give a presupposition of the whole conditional. In some cases, when there is a particular semantic relation between the antecedent and the consequent, the amalgamation in yields a tautology, and thereby neutralizes what would otherwise be a presupposition. And in this sense these presuppositions ot the consequent clause are cancelled: psychologically they are not felt as presuppositions because they have become components of semantically empty that is of logical truths -- and logical truths are, trivially "presuppositions" of anything.

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It is important to keep in mind that the description given by (19) - (21) is quite general and independent of any semantic relation between the component sentences. The cancellation, in the sense I adopt here (i.e. neutralisation) is only a consequence of the proposed description. One can note that there is an essential difference in the result obtained by the description proposed here and the description given by Karttunen. For Kalttunen, except in the special case in which filtering conditions work, the presuppositions of the consequent clause become presuppositions of the whole sentence. Concerning this subject I will make first a few general comments because, it seems to me, the problem is quite general and independent of the description of just logical connectives. In my description, presuppositions of the complex sentence in the conditional form are always functions of the antecedent clause. Intuitively the expressed by the antecedent of the conditional describes a possible world, usually different from the actual world. Some observations tend to suggest that there is always a "presuppositional link" between a possible non-actual world which we invoke using expressions of natural languages and the actual world. The "normal" situation is such that a sentence which is true in a possible non-actual world has a presupposition which is true in the actual world. But this is not always the case. For example compare (25) Steve's dog will help him. (26) Steve's children will help him. It is actually admitted that, in the case of (25), the existence (in the actual world) of the dog belonging to Steve is presupposed. But in the case of (26) we are not committed to the truth of Steve's actually having children at the moment; we may be thinking about his future children. Why this is so is not at all clear. It depends probably on whether we consider generic/non-generic readings or de re/de dicto interpretations. Returning now to the case of presuppositions of the consequent clause of the conditional, it is worthwhile to note that in this case one encounters problems similar to those discussed above. For many speakers it is not always clear whether the presupposition of the consequent should be regarded as a presupposition of the whole. Karttunen himself, looking at some examples, was obliged to revise the filtering conditions. The effect of these conditions was to make the notion ot the presupposition of a complex sentence relative, with respect to linguistic contexts, to some of sentences whose truth is taken for granted. What is important for us is the fact that the difficulty Karttunen encounters is due to the status of the presupposition of the second clause of a complex sentence.1

1 It seems to me however that the examples Karttunen gives to support the revised conditions are not properly analysed from the semantic point of view. He takes a presupposition for context described by propositions of which the truth is taken for granted mixing up the bath. Thus, says Karttunen, (i) presupposees (ii) in the context in which (iii) is regarded as a fact: (i) If Nixon appoints J. Edgar Hoover to the cabinet, he will regret having appointed a homosexual. 5

Furthermore, and this is far more important, the description here proposed matches very well the fact that in many cases presuppositions of the second clause become presuppositions of the whole. This can be shown in the following way. Suppose a sentence T has two types of presuppositions: first those which are true whenever another given sentence S is true, and second those which are not alwaye true when S is truh. Call them

Pree1(T) and Pres2(T), respectively. Suppose furthermore, that T is the consequent and S the antecedent clause of a conditional of the form (19). Then, if (21) is a presupposition of (19),

Pres2(T) is also a presupposition of (19). For in this case it is evident that (27) is true because the sentence "S implies Pres2 (T)" is false:

(27) The sentence "S implies Pres2 (T)" entails Pree2 (T).

On the other hand Pres1(T) is not entailed by the corresponding instance of (21). In other words (28) is not true:

(28) The sentence "S implies Pres1(T) " entails Pres1(T). Thus according to my description, presuppositions of the consequent clause which are not entailed by the antecedent clause are automatically second-order presuppositions of the whole complex sentence, second-order presuppositions because they are consequences of other presuppositions.2 And this is not true of presuppositions which are entailed by the antecedent clause - they are neutralized. Let me now try to justify the proposed presuppositional description of the connective if ... then. Traditionally, justifications have been based on the intuitions one has about changes (if any) in semantical content when natural negation or modal operators are applied. These are, however, of no help in this case. So I will give some less direct justifications. Let us agree, intuitively, that there exists a synonymy between (a) and (b) sentences in the following set: (29a) Is it the case that if it rains Steve will come? (29b) If it raine is it the case that Steve will come? (30a) I promise you that if it rains I will come. (30b) If it rains I promise you that I will come.

(ii) Nixon will have appointed a homosexual. (iii) J. Edgar Hoover is a homosexual. Note first that (i) is amgiguous, the alternative readings depending on whether there was or was not another homosexual appointed before Hoover. On the reading when there was no other homosexual appointed before, (i) presupposes (iii). We obtain easily this presupposition in my description, for according to it (1i presupposes (iv): (iv) If Nixon appoints J. Edgar Hoover to the cabinet, he will have appointed a homosexual. Clearly (iii) is a consequence of (iv) on this reading. 2 It is clear that there is something wrong with the case (27), for these second order presuppositions are obtained only when the first order presuppositions are false. We will see later that this case does not in fact arise in the procees of linguistic coordination. This does not solve however the problem of the presuppositione of the second clause entirely. 6

(31a) It is possible that if it rains Steve will come. (31b) If it rains it is possible that Steve will come. Thus a hole-operator usually serving to detect presuppositions, when applied to a conditional, applies, in effect, only to the consequent clause without changing the structure of the conditional. More precisely we have the following situation. Given a hole-operator O, sentences of the form O (If S then T) are equivalent to sentences of the form If S then O(T), and because of their equivalence both these sentences have to have the same presuppoaitions. Furthermore, because O is a hole, the sentence If S then T has the same presuppositions as the sentence If S then O(T). But because the operator O effectively applies only to the consequent clause, the sentence If S then O(T) and by the same token the simple conditional If S then T have as a presupposition S implies Pres(T). This can be shown in a somewhat different way. The fact that to (29) one can answer by a conditional Yes or by a conditional No can be easily understood by supposing that If S then Quest. (T) presuppoees that in every possible world in which S is true, T is either true or false. And this slast implies (21), given the semantic definition or presupposition. More precisely, if (32) is true then (33) is true as well: (32) If S is true in some possible world then T is either true or false in this world. (33) If S is true in some possible world then Pres(T) is true in this world. For supposing that (33) is not true we would have an in which S is true and Pres (T) not true. But in this case (32) could not be true. A second type of justification will be given later on, but it will first be useful to say a few words about the other logical connectives. As for the conjunction and, we can be guided, on one hand, by the observation of Karttunen that the filtering conditions for and and for if ... then are the same, and, on the other hand, by the existence of the following equivalence: (34) S⋏T = ¬(Sə¬T) = S ⋏(S⊃T) (34) means that to be able to conserve the idea of neutralisation here developed, one should postulate the following presuppositional description of the conjunction and: (35) has (36) and (37) as presuppositions: (35) S and T (36) Pres(S) (37) S implies Pres(T) Similarly with the (propositional) disjunction (either) ... or. In view of the equivalence (38) and because of Karttunen's filtering conditions for disjunction, the following, presuppositional description of (either) ... or seems to be plausible: (39) presupposes (40) and (41):

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(38) S⋎T = S⋎(¬S⊃T) = (¬S⊃T) (39) S or T (40) Pres(S) (41) S implies Pres(T) In both the case of the conjunction and and the disjunction or, the situation with the presupposition of the second clause of the complex sentence is the same as in the case of the conditional if ... then: the pusuppositien of the clause may become automatically a second- order presupposition of the whole sentence. The problem with this possibility is however that in none of the three cases, of if ... then connective, of the conjunction and and of the disjunction or, do we need to use it. More precisely, and this is what I am going to show now, there exist "selection restrictions" imposed by the locical connectives of natural languages which exclude the possibility of connecting two sentenses the second of which has presuppositions not entailed by the first, without the semantic oddness characteristic of false presuppositions. I have tried to show the existence of such restrictions in the case of if ... then in Zuber 1976. The idea there applied is also applicable to the case or other connectives, I believe. Let me first illustrate the point by showing how the relation between a pronoun nnd its antecedent can be used to detect the type of semantic relation between two sentenses of which one contains the grammatical antecedent of a pronoun and the other the pronoun itself. (42) If somebody comes he will be surprised. The proneon he is not a pronoun of lazinese, since it has as its semantic antecedent the noun phrase the onewho comes. Therefore the consequent clause of (42) presupposes that somebody will come, which is exactly what it entailed by the antecedent clause of (42). Now, not every sentence contains a pronoun, and in particular it is not true that in a sentence composed by a logical connective we always have a pronoun in the second sentence referring back to something in the first sentence. My claim is however that in something that might be called a , we do have a kind of such pronoun and that consequently the description of the semantic connection between two component sentences can be based on the pronoun-antecedent relation. I base my claim on the simple observation that every action or event, every physical or mental state, takes place in time and space, and this means that every sentence expressing an action, an event etc. should contain (in its logical form) temporal or spatial referring pronouns. More precisely we can associate to every (non-generic) sentence S a sentence S' equal to ∃t,pSv(t,p) where Sv(t,p) is roughly the sentence S with two free occurring variables t and p running over the set of time points and space points, respectively. Relation between S and S’ is illustrated by the following example where (43) is an instance of S and (44) an instance of S’: (43) Steve will come. 8

(44) Steve will come at some time somewhere. Note that, intuitively3, S entails S’. Now, in a compound sentence one can observe that the value of the temporal (and spatial but from now on I will oparate only with temporal pronouns) pronoun in the second constituent sentence is a of the time point determined by the first constituent sentence; it is determined relative to the value of the time point of the first sentence - it is a relative temporal pronoun. This can be described more precisely in the followint way: sentence pattern (45) becomes (46): (45) S ⊛ T (46) S at some time ⊛ T at the time t = f (time at which S) where '⊛' symbolises any connective and f is a senuine mathematical function of the time point at which S. The function f can mean for instance: at the same time as, before when, after when, etc. What is important for us is the fact that the function f applies to an containing a relative pronoun and as such presupposing the existence of the time point at which S happens and consequently presupposing the truth of S’.4 Let us take some examples. Note first that in the case of if ... then connective we do have such a relative pronoun which is precisely then (Zuber 1976 can be consulted for more details). Compare: (47) becomes (48): (47) If it rains then Steve will come. (48) If it rains at some time then in this situation Steve will come. Consequently the second clause presupposes the first one. Similarly with (49) and (50): (49) It was raining and Steve was singing. (50) It was raining at some time and Steve was singing at this time. What is interesting here is the fact that the two sentences in conjunction refer to the same time point, even though in isolation they can refer to different time points. Thus here also the second clause presupposes a sentence entailed by the first clause. As far as I can see this

3 This is a very imprecise statement, for from what we have said it follows that S' is a (kind of) logical form of S. Usually this means that S and S’ do not belong to the same languages and thus it is difficult to talk about any semantic relation between them. It is however clear that we do have an entailment relation between (43) and (44) and for convenience we can consider that the logical form of a sentence is expressed in the same language as the sentence itself. Note also that in this case probably there exists not only a one-sided entailment but mutual entailment (). However I do not need this stronger relation for my analysis. 4 All this might be written in more using in particular the definite description operator. What is in point however is the fact that I use here a pre-suppositional analysis of restrictive relative clauses: the x which is P is O presupposes some x is P. Note furthermore that this property of complex sentences just mentioned mentioned reflects the fact that natural languages do not incorporate "modern" way of counting time; I rather in their underlying time system time values are determined relative to some event. This seems to be quite natural. 9 relation is independent of the type of the connective used: it is at the basis of the phenomenon of linguistic coordination.5 Note that it is the case that any presupposition of the second clause of a complex sentence is a consequence of the first clause, for every sentence expressing such a presupposition contains (somewhere in the logical structure) a time-referring pronoun which refers to the point in time of the event described in the first clause. And this achieves justification of (21): to say that a complex sentence has aspresuppositions the sentence expressing the sentential selection restrictions of the type described above imposed by the connectives, is to say that a complex sentence presupposes that any presupposition of its second clause is entailed by the first clause. At least one point remains to be explained in this context. If in any complex sentence every presupposition of the second clause of this sentence is entailed by the first clause, then according to my description we should not find a complex sentence which has as a presupposition some (non-trivial) presupposition of the second clause. This apparently seems not to correspond to the reality. Most speakers agree that (51) presupposes (52): (51) If it rains only Steve will come. (52) Steve will come. My answer to this is that (51), if acceptable, is an elliptic expression for a larger expression in which a part was deleted. Thus (51) stands for a larger sentence (53): (53) If it rains only Steve will come but if it does not rain not only Steve will come. (52) seems to be a presupposition of (51) because of the relation of (51) to (53) but (52) is not in fact a presupposition of (51) taken by itself. The plausibility of this solution by ellipsis is reinforced by the fact that come complex sentences are semantically very odd, because the whole expression of which they are ellipses contains contradictory constituent; (54) is odd bacause (55) is impossible: (54) If there is a water in the pool only Steve swims (in this pool). (55) If there is a water in the pool only Steve swims but if there is no water in the pool not only Steve swims. Note also that (54) is an example showing that it is not true that any two sentences can be connected (even when there are no syntactic restrictions).

5 It is also true of "non-logical" connectives such as but, because, etc. (when as in the case of logical connectives the conjoined members are not generic sentences). One may object that in the case of the disjunction we do not have this relation. It seems to me however that most disjunctions in natural languages are elliptic expressions and in fact S or T means S or (if non-S then T). It is important to keep in mind that what is said here means that a compound sentence presupposes that its second member presupposes something entailed by the first member (or when there are no personal pronouns that the second clause presupposes the first one). It does not mean, as it was mistakenly suggested in Zuber 1972, that a compound sentence presupposes the truth of ist first sentential clause. 10

The last thing I am going to discuss briefly here is the problem of the symmetry of logical connectives in natural languages and the "asymmetrical" presuppositional analysis here proposed for conjunction and disjunction. In some sense my analysis of and and or is based on equivalences (34) and (38). Both equivalences contain a non-symmetrical part in the form of the conditional. Now, the logical impossibilities in these conditional parts may serve to establish the order of component sentences. Por instance even if S ⋎ T is equivalent either to (56) or to (57), the semantic relation between S and T render one of the conditionals S ⊃ T or T ⊃ S nessesarily false (or indeterminante): (56) S ⋎( ¬ S ⊃ T) (57) T ⋎ ( ¬ T ⊃ S) To illustrate this, take example (58) which is apparently equivalent to (59): (58) Either the victim was a bachelor or no one got killed at all. (59) Either no one got killed at all or the victim was a bachelor. If we give to (58) and (59) the form indicated by (56) and (57) then the sentence corresponding to (58) will contain (60) as the conditional clause and the sentence corresponding to (59) will contain (61): (60) If the victim was not a bachelor then no one got killed at all. (61) If somebody got killed then the victim was a bachelor. Clearly (60) is semantically not acceptable (has a false presupposition) and consequently we should consider (59) and not (58) as having a „proper“ order of disjuncts. This ends my paper. I hope that its results show in a different light various problems concerning the semantic properties of complex sentences and the status of presuppositions.6

6 I forced Newton Carver to help me with the form of this paper. He also caused some changes in the content. 11

References: Horn, L. (1972) On the Semantic Properties of Logical Operators in English. Unpublished doctoral dissertation, UCLA, Los Angeles. Karttunen, L. (1973) "Presuppositions of Compound Sentences", Linguistic Inquiry IV, 169 - 193. Keenan, E. L. (1973) "Presupposition in Natural Logic", The Monist, Vol. 57, 343 - 364. Lakoff, G. (1970) "Linguistics and Natural Logic", Synthese 22, 151 - 271. Zuber, R. (1972) Structure presuppositionnelle du langage, Dunod, Paris. Zuber, R. (1975) "On the semantics of Complex Sentences", distributed by Indiana University Linguistics Club. Zuber, R. (1976) "Conditionnelle: sémantique ou pragmatique?", in Modèles Logiques et Niveaux d'Analyse Linguistique, Actes du Colloque puliés par J. David et R. Martin, Etudes publiées par le Centre d'Analyse syntaxique de l'Université de Metz.

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