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Switch to: Citations
References in:
Natural deduction and Curry's paradox
Journal of Philosophical Logic 36 (2):155 - 179 (2007)
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This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics. |
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In his paper [6], Greg Restall conjectured that a logic supports a naïve comprehension scheme if and only if it is robustly contraction free, that is, if and only if no contracting connective is definable in terms of the primitive connectives of the logic. In this paper, we present infinitely many counterexamples to Restall''s conjecture, in the form of purely implicational logics which are robustly contraction free, but which trivialize naïve comprehension. |
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It is known that a number of inference principles can be used to trivialise the axioms of naïve comprehension - the axioms underlying the naïve theory of sets. In this paper we systematise and extend these known results, to provide a number of general classes of axioms responsible for trivialising naïve comprehension. |
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A logic is said to be contraction free if the rule from A→(A→B) to A→B is not truth preserving. It is well known that a logic has to be contraction free for it to support a non-trivial naïve theory of sets or of truth. What is not so well known is that if there is another contracting implication expressible in the language, the logic still cannot support such a naïve theory. A logic is said to be robustly contraction free if (...) |
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Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will encounter, originally as an example of (...) |
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We propose contractionless constructive logic which is obtained from Nelson's constructive logic by deleting contractions. We discuss the consistency of a naive set theory based on the proposed logic in relation to Curry's paradox. The philosophical significance of contractionless constructive logic is also argued in comparison with Fitch's and Prawitz's systems. |
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