Real Analysis in Paraconsistent Logic

Abstract

This paper begins an analysis of the real line using an inconsistencytolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistencyreliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.This paper has two aims. The first is to show that, by weakening the logic within which we work, so as to allow for the possibility of non-trivial inconsistency, we are still able to do everyday mathematics. We see why such practice is possible: We take classical proofs, note the obstacles from a paraconsistent viewpoint, and then reconstruct the arguments, using only inferences that are valid in nontrivial inconsistent contexts. This 'breakdown' of analysis enables a fuller understanding of the fine structure of proofs. The strategy is to reverse engineer classical results (cf. [19, sec.16.1]), eventually deriving a paraconsistent basis for analysis, and this paper is a step in that direction. A fundamental result is the Heine-Borel theorem, characterising compactness of bounded closed intervals.The second aim of the paper is to explain, in part, a basic historical fact: the original calculus of Newton and Leibniz was inconsistent-in the sense that the original definition of derivative involves a number ε which is at some points non-zero (to permit division), and at other points zero (to allow 'infinitessimal' quantities to vanish); see [6], [18].1 Yet, despite the inconsistency, early practitioners were able to draw meaningful and useful conclusions. Despite contradictions, they got the right answers. Accounting for how this is possible provides one motivation for working in a paraconsistent logic-reconstructing structures in which inconsistent reasoning actually did take place, and yet did so without falling into incoherence. 1]), again involving a different approach. By contrast, this paper is not a model-theoretic development, and so does not include similar claims to non-triviality. 1Our interest here is in compactness properties. It turns out that, to understand the Heine-Borel theorem in a paraconsistent system, detours through ideas from constructive analysis are needed; e.g. Brouwer's fan theorem plays a key role [5, §5.3]. This is something of a surprise, since paraconsistency has often been taken as a dual or opposite to constructive, paracomplete theories [16, ch. 11], [7].2 It has been quipped of constructive mathematicians that they perform mathematics with one hand tied behind their back, working without universal application of the law of excluded middle. Here we tie the other hand behind our backs, and work without disjunctive syllogism or general forms of reductio. Constructive mathematicians have developed powerful tools to overcome truth value gaps; here, by restoring excluded middle, but considering the possibility of truth-value gluts, th...

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