Classifying material implications over minimal logic
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Mathematical Logic
Classifying material implications over minimal logic Hannes Diener1 · Maarten McKubre-Jordens1 Received: 16 May 2016 / Accepted: 14 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. Keywords Reverse mathematics · Minimal logic · Ex falso quodlibet · Implication · Paraconsistent logic · Peirce’s principle Mathematics Subject Classification 03B20 · 03C98
1 Introduction The project of constructive reverse mathematics [10] has given rise to a wide literature where various theorems of mathematics and principles of logic have been classified over intuitionistic logic. What is less well-known is that the subtle difference that arises when the principle of explosion, ex falso quodlibet, is dropped from intuitionistic
This research was carried out while McKubre-Jordens was partially supported on Marsden Grant UOC1205, funded by the Royal Society of New Zealand; and Diener and McKubre-Jordens were partially funded by the Royal Society of New Zealand’s Counterpart Funding Initiative PIRE for the FP7 Marie Curie project CORCON.
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Maarten McKubre-Jordens [email protected] Hannes Diener [email protected]
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School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
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H. Diener, M. McKubre-Jordens
logic (thus giving (Johansson’s) minimal logic) enables the distinction of many more principles. The focus of the present paper are a range of principles known collectively (but not exhaustively) as the paradoxes of material implication; paradoxes because they illustrate that the usual interpretation of formal statements of the form “. . . → . . .” as informal statements of the form “if…, then…” produces counter-intuitive results. These paradoxes have motivated the development of many non-classical logics over the years [2–4,7,18]. Here we present a carefully worked-out chart, classifying a number of such principles over minimal logic. These principles hold classically, and intuitionistically either hold or are equivalent to one of three well-known principles (see Sect. 6). As it turns out, over minimal logic these principles divide cleanly into a small number of distinct categories. We hasten to add that the principles we classify here are considered as formula schemas, and
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