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Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of ∀p and ∃p*

Published online by Cambridge University Press:  12 March 2014

Philip Kremer
Philip Kremer*
Affiliation:
Department of Philosophy, University of Pittsburgh, Pittsburg, Pennsylvania 15260, E-mail: PHIL@CIS.PITT.EDU
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Extract

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take a model to be one of these structures, α, together with some function or relation which associates with every formula A a subset of α. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take a proposition of α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers the primary interpretation associated with the semantics.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 1 , March 1993 , pp. 334 - 349
Copyright
Copyright © Association for Symbolic Logic 1993

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Footnotes

*

I am much indebted to Nuel Belnap for his constant help and encouragement, and, not least of all, for rather closely inspecting the proofs. I thank Aldo Antonelli for asking me whether the systems studied here fail to be arithmetical (in the recursion theoretic sense of Hinman [1978], Odifreddi [1989] and others) as well as recursively enumerable. Not only are the systems here nonarithmetical, they are recursively isomorphic to full second-order logic. Finally I thank a referee and Richard Shore for pointing in the right direction toward proving this stronger result.

References

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