Intuitionistic Logic with A Definitely Operator, Research Report 97.05.pdf

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2017-04-06 发布于江苏
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Intuitionistic Logic with A Definitely Operator, Research Report 97.05.pdf

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Intuitionistic Logic with A Definitely Operator, Research Report 97.05

- 1 - Intuitionistic Logic with a “Definitely” Operator Peter Mott School of Computer Studies University of Leeds email: pmott@scs.leeds.ac.uk Abstract This paper introduces a logic ILED derived from standard intuitionistic sentence logic by adding two operators D? for “Definitely ?” and ~? for “Experience rejects ?”. A further negation ?? = def (?→⊥) ∨ ~? , which we call real negation, is introduced. Real negation is like intuitionistic negation when there are no D-operators but deviates when there are. We see that D? ? ? is valid but ?D? → ?? is not and hence that contraposition fails for real negation. We give a semantics for this logic, axiomatise it and prove the axiomatisation complete. Finally we show that real negation behaves as standard intuitionistic negation within D-free contexts. The logic ILED is proposed as an extension of intuitionistic logic apt for use as a general logic. - 2 - Introduction The use of intuitionistic logic as a general logic is made difficult by the usual interpretation of intuitionistic negation: to assert not-? is to assert that ? derives absurdity. This is constrained by the definition of ?? by ?→⊥ and the usual understanding of intuitionistc negation that goes with it (Dummett [3], Troelstra Van Dalen [14]). Yet just saying “It’s not raining” does not appear to commit us to derivations of absurdity. Why might we want to use intuitionistic logic as a general logic? One reason is that the Sorites Paradox has a simple solution in the framework of intuitionistic logic (Putnam [10], Mott [8]). But using intuitionistic logic to solve the Sorites requires its use as a general logic. There is found in the literature (Williamson [16], Wright [17], Read Wright [12]) an interpretation of intuitionistic negation Williamson states thus: “Now, intuitionistically, what proves ?B is what proves there could not be a proof of B” (p. 136). This semantic principle can be supported by the following sort of argument. Suppose that an int