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Switch to: Citations
References in:
How to Ekman a Crabbé-Tennant
Synthese 199 (Suppl 3):617-639 (2018)
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Recently Ekman gave a derivation in natural deduction such that it either contains a substantial redundant part or else is not normal. It is shown that this problem is caused by a non-normality inherent in the usual modus ponens rule. |
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In the present paper, the Fregean conception of proof-theoretic semantics that I developed elsewhere will be revised so as to better reflect the different roles played by open and closed derivations. I will argue that such a conception can deliver a semantic analysis of languages containing paradoxical expressions provided some of its basic tenets are liberalized. In particular, the notion of function underlying the Brouwer–Heyting–Kolmogorov explanation of implication should be understood as admitting functions to be partial. As argued in previous (...) |
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The system of natural deduction that originated with Gentzen , and for which Prawitz proved a normalization theorem, is re-cast so that all elimination rules are in parallel form. This enables one to prove a very exigent normalization theorem. The normal forms that it provides have all disjunction-eliminations as low as possible, and have no major premisses for eliminations standing as conclusions of any rules. Normal natural deductions are isomorphic to cut-free, weakening-free sequent proofs. This form of normalization theorem renders (...) |
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This is a reply to the considerations advanced by Schroeder-Heister and Tranchini as prima facie problematic for the proof-theoretic criterion of paradoxicality, as originally presented in Tennant and subsequently amended in Tennant. Countering these considerations lends new importance to the parallelized forms of elimination rules in natural deduction. |
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From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as a crucial (...) |
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The interpretation of implications as rules motivates a different left-introduction schema for implication in the sequent calculus, which is conceptually more basic than the implication-left schema proposed by Gentzen. Corresponding to results obtained for systems with higher-level rules, it enjoys the subformula property and cut elimination in a weak form. |
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Prawitz observed that Russell’s paradox in naive set theory yields a derivation of absurdity whose reduction sequence loops. Building on this observation, and based on numerous examples, Tennant claimed that this looping feature, or more generally, the fact that derivations of absurdity do not normalize, is characteristic of the paradoxes. Striking results by Ekman show that looping reduction sequences are already obtained in minimal propositional logic, when certain reduction steps, which are prima facie plausible, are considered in addition to the (...) |
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Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox. |
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This paper presents and motivates a new philosophical and logical approach to truth and semantic paradox. It begins from an inferentialist, and particularly bilateralist, theory of meaning---one which takes meaning to be constituted by assertibility and deniability conditions---and shows how the usual multiple-conclusion sequent calculus for classical logic can be given an inferentialist motivation, leaving classical model theory as of only derivative importance. The paper then uses this theory of meaning to present and motivate a logical system---ST---that conservatively extends classical (...) |
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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 this paper it is shown that addition of certain reductions to the standard cut removing reductions of deductions in prepositional logic makes prepositional logic non-normalizable. From this follows that some provable propositions in prepositional logic has no direct proof. |
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Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of such systems from logic (...) |
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In this paper we show how Dummett-Prawitz-style proof-theoretic semantics has to be modified in order to cope with paradoxical phenomena. It will turn out that one of its basic tenets has to be given up, namely the definition of the correctness of an inference as validity preservation. As a result, the notions of an argument being valid and of an argument being constituted by correct inference rules will no more coincide. The gap between the two notions is accounted for by (...) |
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