Dialetheism and the Impossibility of the World

Ben Martin

doi:10.1080/00048402.2014.956768

Abstract

This paper first offers a standard modal extension of dialetheic logics that respect the normal semantics for negation and conjunction, in an attempt to adequately model absolutism, the thesis that there are true contradictions at metaphysically possible worlds. It is shown, however, that the modal extension has unsavoury consequences for both absolutism and dialetheism. While the logic commits the absolutist to dialetheism, it commits the dialetheist to the impossibility of the actual world. A new modal logic AV is then proposed which avoids these unsavoury consequences by invalidating the interdefinability rules for the modal operators with the use of two valuation relations. However, while using AV carries no significant cost for the absolutist, the same isn't true for the dialetheist. Although using AV allows her to avoid the consequence that the actual world is an impossible world, it does so only on the condition that the dialetheist admits that she cannot give a dialetheic solution to all self-referential semantic paradoxes. Thus, unless there are any further available modal logics that don't commit her to the impossibility of the actual world, the dialetheist is faced with a dilemma. Either admit that the actual world is an impossible world, or admit that her research programme cannot give a comprehensive solution to the self-referential paradoxes.

Highlights

This paper first offers a standard modal extension of dialetheic logics that respect the normal semantics for negation and conjunction, in an attempt to adequately model absolutism, the thesis that there are true contradictions at metaphysically possible worlds
Using AV allows her to avoid the consequence that the actual world is an impossible world, it does so only on the condition that the dialetheist admits that she cannot give a dialetheic solution to all self-referential semantic paradoxes
W is a set of possible worlds, wa is a distinguished member of our domain known as the actual world, R is a binary relation between sets of worlds known as the accessibility relation, and e is a valuation relation assigning truth-values to world-indexed propositions

Summary

Absolutism

Assuming that contradictions are formalized as ‘A ^ ~A’, absolutism requires a special type of paraconsistent logic that invalidates both the unconjoined {A, ~A} ‘ B and the conjoined {A ^ ~A} ‘ B forms of explosion, thereby blocking triviality, and allows contradictions to be assigned the truth-value true. We have three apparently conceptually distinct positions, Dialetheism: There are true contradictions at the actual world Absolutism: There are true contradictions at a metaphysically possible world Paraconsistency: Explosion is invalid, we don’t yet have logical evidence for the non-equivalence of absolutism and dialetheism. Even if we possess no good reasons at present to believe that there are true contradictions at non-actual possible worlds but not at the actual world, absolutism and dialetheism seem conceptually distinct positions. Having shown that the logics resulting from these standard semantics have unsavoury consequences for both absolutism and dialetheism, we will secondly, propose a new modal logic AV that avoids these consequences

A Standard Modal Semantics

Consequence One

Consequence Two

Possible Solution to the Problem

Stipulating Exclusivity and a Dilemma for the Dialetheist

Conclusion