Provability as a deontic notion

Theory and Decision 2 (1):1-15 (1971)
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Abstract

The purpose of this paper is to mark a significant difference between classical and several non-classical prepositional calculi. The argument presupposes familiarity with Kripke/Hintikka semantics for modal logic. The non-classical systems are Hintikka's logic of belief and alethic modal systems which have Kripke/Hintikka semantics. The difference is marked by showing that the semantic validity operator in classical logic behaves as a normal alethic necessity-operator while the non-classical semantic validity operators behave as normal deontic ought-operators. The crucial step is showing that a formula, valid by non-classical semantics, can be falsified. I show that the negation of a non-classical thesis can be added to a consistent set of formulae without making the set inconsistent or any other set inconsistent. This is shown by observing that consistent sets of formulae do not need to be related to other consistent sets by any of the alternativeness relations of Kripke/Hintikka model structures for non-classical systems. The deontic behavior of non-classical semantical validity operators is interpreted as showing that being a thesis of a non-classical system means, not that the thesis is a logical truth, but that the thesis is the content of a norm on how we ought to use, crucial terms such as ‘believe’ and ‘necessity’.

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References found in this work

Knowledge and belief.Jaakko Hintikka - 1962 - Ithaca, N.Y.,: Cornell University Press.
An Introduction to Modal Logic.George Edward Hughes & M. J. Cresswell - 1968 - London, England: Methuen. Edited by M. J. Cresswell.
A completeness theorem in modal logic.Saul Kripke - 1959 - Journal of Symbolic Logic 24 (1):1-14.
Semantical Analysis of Modal Logic I. Normal Propositional Calculi.Saul A. Kripke - 1963 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 9 (5‐6):67-96.

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