Paraconsistency in Mathematics

Abstract

Paraconsistent logic makes it possible to study inconsistent theories in a coherent way. From its modern start in the mid-20th century, paraconsistency was intended for use in mathematics, providing a rigorous framework for describing abstract objects and structures where some contradictions are allowed, without collapse into incoherence. Over the past decades, this initiative has evolved into an area of non-classical mathematics known as inconsistent or paraconsistent mathematics. This Element provides a selective introductory survey of this research program, distinguishing between `moderate' and `radical' approaches. The emphasis is on philosophical issues and future challenges.

True, Untrue, Valid, Invalid, Provable, Unprovable

“…If a proof is a sequence of propositions arrived at by step-by-step application of valid rules, it is perhaps harder than truth or validity to see how derivations might be inconsistent (Shairo, 2002). Nevertheless, there is reason to think the proof relation is inconsistent (Priest, 1979;Routley, 1979;Berto, 2007;Weber, 2022). If G = 'this sentence is unprovable' is both true and false there is a sentence which is both provable and unprovable, which means that there is both a proof of it and there is no proof of it.…”

“…There are many approaches to paraconsistency (see Beall and Restall, 2005, p. 80;Weber, 2022). Some are very moderate, aiming to extend standard classical theories with a non-explosive consequence relation in certain places, analogous to the way the transfinite extends but does not alter the finite natural numbers (Carnielli and Coniglio, 2016, p. x).…”

Routes to relevance: Philosophies of relevant logics

“…The study of naive set theory is an important motivation for the rejection of contraction principles. For more about naive set theory, see, for example,Routley (2019) orWeber (2010Weber ( , 2021Weber ( , 2022.14 In this section, we will talk about proofs involving assumptions, although the interest is in the theorems obtainable after discharging all assumptions.15 Anderson and Belnap (1975, p. 18). Emphasis in the original.…”

“…The study of naive set theory is an important motivation for the rejection of contraction principles. For more about naive set theory, see, for example, Routley (2019) or Weber (2010, 2021, 2022).…”

The Liberation Argument for Inconsistent Mathematics

“…This is a controversial claim -see[Vic13] for a rebuttal of many of the usual examples -but either way, historical examples are for the most part very different from the contemporary investigations, as the latter do not seem to treat contradictions as a mere accident or temporary step towards future consistentization. Since my interest is in inconsistent mathematics as a particular field of study distinct from mainstream mathematics, I will stick to recent work.5 For a proper first introduction to the field, see[Web22]. For a more detailed discussion, see[Man23].Australasian Journal of Logic (20:2) 2023, Article no.…”