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Wednesday, 27 February 2013

The plurality of logics in UNILOG

Unilog is taking place in Rio in early April. We (Alexandre Costa-Leite and me) hope to make an official début there of our much rehearsed galaxy theory. The abstract of our presentation there gives an idea of what we are after in the paper (currently submitted to Logica Uniersalis):

The availability of multiple logics, although not a novelty, carries on provoking different kinds of puzzlement. From the point of view of those endeavoring to describe and understand parts of the world, it is a pressing issue to understand how different logics coexist – and eventually how to choose between them. For metaphysicians, who often deal in necessity and make frequent use of modal reasoning, the appeal to a logic is also the appeal to a standard to decide what is possible – typically in terms of which worlds are possible (see D. Lewis’ On the plurality of worlds). The use of a single, fixed logic as a standard of possibility is clearly unsatisfactory as it biases all results. Clearly, what is impossible in classical logic is not necessarily so in paraconsistent or intuitionistic logics. Up till now, the use of classical logic as if it were there only logic available was defended on the basis of its entrenchment: in the absence of any reason to pick any other logic, classical logic is best retained once it is deemed sufficiently useful and intuitive in the past. Such a response, nevertheless, has been challenged by the development of tools for a universal logic.

Universal logic engages with multiple logics simultaneously either by comparing them or by combining them. It made it possible to look at the plurality of logics not in order to choose one among them but rather to study relations between them. By considering the space of all logics, universal logic provides a general framework where features and capacities of a logic can be made evident. We have recently sketched a tool for universal logic called galaxy theory. Based on some developments in Kripke’s semantics for modal logic, galaxy theory defines a logic (or rather, a relation of consequence) as a class of possible worlds. Such a class, called galaxy, is itself an element in a topology of galaxies. Typically, modal elements in a logic add to each corresponding galaxy some relations of access, but this can be taken not to affect the underlying galaxy. The emerging image is one where the plurality of logics can be studied as the plurality of galaxies.

In this work we present the framework of galaxies and apply it to the debate about realism concerning different logics – and related issues revolving around dialetheism. We consider galaxy theory together with some concepts developed by Kit Fine (mainly in papers collected in “Modality and Tense”), such as the notion of a inconsistent über-reality that brings together elements in a plurality. We then propose a realism about the different logics that is, at the same time, combined to a form of dialetheism. Galaxy theory paves the way to investigate such issues because it takes each galaxy as a point in a topology. A side aim of this work, nevertheless important, is to show how fruitful the framework of galaxies can be.

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