# An Alternative Semantics for Modal Predicate-Logic

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## Abstract

A N A L T E R N A T I V E S E M A N T I C S F O R M O D A L P R E D I C A T E - L O G I C (A) In this paper I present a realistic semantics for first-order modal predicate-logic that differs in two respects from the standard Tarski- Kripke approach. Firstly, the background theory is not axiomatic set theory supple- mented by the concept possible world, but a theory of basic intensional entities: propositions, properties and relations;1 this theory will here be developed only as far as it is necessary for the semantics of an elementary language, that is, the only properties and relations con- sidered are properties of, and relations between individuals. In standard (extensional) ontology intensional entities are reduced to sets and pos- sible worlds (and individuals; they are complex sets involving in their specification possible worlds and/or individuals);2 in intensional on- tology, however, sets (of individuals) are reduced to properties (of individuals), and possible worlds to propositions. The latter reduction is more natural than the former, and the inveterate skepticism against intensional entities as basic ("What, after all, are these things?? When are they identical??") can be dispelled. For intensional ontology is completely on a par with set theory. All we really know about sets is stated in the axioms of set theory; all we really know about properties, relations and propositions is stated in the axioms of intensional on- tology; the latter axioms are no less precise than the former, and identity conditions for intensional entities are not left unclear. Secondly, the basic semantic concept is different; expressions do not have extensions relative to possible worlds in some interpretation, but absolutely intend intensions. Sentences intend propositions, monadic predicates intend properties, polyadic predicates intend relations, indi- vidual-constants intend individuals. To intend is not to mean; we can, however, think of the intension of an expression as an

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