Towards an Algebraic Semantics for Implicatives
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Towards an Algebraic Semantics for Implicatives R. Zuber1
© Springer Nature B.V. 2019
Abstract An algebraic semantics, based on factor algebras, for one-way and two-way implicative verbs is proposed. Implicative verbs denote elements of filters or of ideals generated by identity functions in factor algebras. This semantics explains in particular the problem of implicational equivalence raised by two-way implicative verbs, and shows that the negation necessary to establish the implicativity of these verbs is the negation which preserves the presuppositions of sentences with implicative verbs. In addition, it follows from the proposed semantics that any two implicative verbs denoting in the same algebra but belonging to different categories, are semantically related. Keywords Implicative verb · Factor algebra · Filters and ideals · Implicativity
1 Introduction Implicatives, as understood here, is the class of implicative verbs studied by Karttunen (cf. Karttunen 1971, 1973, 2012) and giving rise to specific inference patterns. Syntactically implicative verbs are functional expressions which take infinitivals or verb phrases as arguments and give verb phrases as result. For simplicity, which in fact does not alter my proposal, I will consider that implicatives are VP modifiers, that is they apply to VP and give VPs as a result. Semantically, roughly speaking, verb phrases formed by implicative verbs entail the finite positive form of their argument or its negative form (i.e. the negation of the argument). Karttunen distinguishes various classes of implicatives according to whether only their positive form, only their negative form or both positive and negative forms give rise to such entailments. One way implicatives are the implicatives which give rise to entailments only when they are taken in positive (affirmative) context or only when they are taken in negative
Thanks to the reviewers for various comments and remarks.
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R. Zuber [email protected] CNRS, Paris, France
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R. Zuber
contexts (when they are negated) and two-way implicatives gives rise to such entailments in positive (affirmative) as well as in negative contexts, that is when they are negated and not negated. In addition there may be agreement or disagrement between the polarity of the implicative and its argument. Thus positive two-way implicatives entail in the positive context the positive form of their argument and in the negative context they entail the negative form of their argument. Similarly, negative two-way implicatives entail in the positive context the negation of their argument and in the negative context they entail positive form of their argument. Positive one-way implicatives entail positive form of their argument only in the positive or only in the negative context. Let me recall some examples. In these examples the implicativity will be illustrated only by implicative verbs which may be called simple implicatives. Karttunen (2012) indicates that there is (at least in English) a large class of mul
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