A Pragmatic Defense of Logical Pluralism
Cian Guilfoyle Chartier
Abstract:
The thesis characterises logic as a formal presentation of a guide to undertaking a rational practice, a guide which is itself constituted by epistemic norms and their consequences. There in general may be more than one "good" presentation, more than one "good" practice, and more than one way to conceive of the practice. This is a pragmatic conception of logical pluralism we call thoroughgoing logical pluralism. The thesis consists in a defence of thoroughgoing logical pluralism, and a case for how such a characterisation of logic is helpful in addressing problems in logical revision, semantic paradoxes, and the incommensurability of logical theories.
The first chapter of the thesis outlines the aforementioned view of logic (clarifying what is meant by "formal" and "epistemic norms"), and defends it against the usual objections to logical pluralism and treats of a few rival approaches to logical pluralism. The second chapter of the thesis defends the view that logical revision is a form of narrow reflective equilibrium with respect to the norms which constitute the practice that the logic is formalising, and the formalisation itself.
The third chapter of the thesis outlines a family of pragmatic solutions to the liar paradox along the lines of the practitioner identifying the norms they wish to keep, and changing their practice or formalisation in accordance with either keeping a truth predicate in her logical language or not. Either way, paradoxical statements do not appear in the extension of a truth predicate in this language. This comes about through identifying the existence of paraconsistent (dual) Kripke fixed point-defined truth predicate extensions of a family of languages of arithmetic, starting from simply that of arithmetic and stretching into a hierarchy of predicates: the truth predicate, and a proper class of ordinal-indexed "paradoxicality" predicates. Such a language is defined and the existence of fixed points for the extensions of a hierarchy of predicates for paraconsistent models of arithmetic is proven.
The fourth chapter of the thesis outlines how, through a modification of Craige Roberts' Questions Under Discussion framework, one can provide an account of how practitioners of different logics may understand and learn from each-other's proofsâ€”to the extent that a formal translation sufficient to do this is allowed, given linguistic constraints. This is managed on the basis of a characterisation of the potential ability of the formal mathematician to make the relevant inferences.
This account of logic provides the beginning of an answer to the question of how logic is normative for thought. A terse response to the question in light of this account is that logics are normative for thought insofar as they outrun the mental capacities of practitioners to think through the consequences of their norm-following. Further suggestions of how to elaborate on this response, particularly in how to amend it for applications of logics outside of descriptive languages, are provided at the end of the thesis.