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Paradox and Logical Revision



  Time: 2 April, 2011 - 3 April, 2011
  Location: School II

Workshop Report

Can a logic--a theory of correct inference--be defective in various ways, and so open to revision? Some have argued that this is, indeed, the case by appeal to the shortcomings of standard logical theory with respect to the semantic and set-theoretic paradoxes. But is there a strong case to be made for revising logic on these grounds? And how does revision of logical theory feed back into the practice that it tries to codify, or impact our understanding of speech acts, cognitive states, and rationality? This workshop brought together top researchers in the field to answer these and other questions about the relation between semantic paradox and the revision of logic.

The first day of talks started with Julien Murzi (Munich) arguing that anyone who rejects the inference of conditional contraction in order to avoid Curry paradoxes involving the conditional should, analogously, reject the rule of structural contraction to avoid Curry paradoxes involving the validity predicate. Next up, Aaron Cotnoir (Aberdeen) argued for a parity between 'truth-value-gap' and 'truth-value-glut' interpretations of the paradoxes. He used this to motivate a three valued logic for truth which is at the intersection of K3 and LP. Zach Weber (Melbourne) discussed the reconstruction of number theory in paraconsistent (naive) set theory. Amongst other things he showed that, for the paraconsistentist, there is a difference between implying something false and implying something absurd and that '0=1' is just as absurd paraconsistently as it is classically. Roy Sorensen (Washington University) closed out the first day of talks, drawing a distinction between two stances toward logical revision: a 'naturalist' stance and a 'non-existence' stance. He argued that the only view which can avoid 'the deviant logicians dilemma' is the revisionist who denies the existence of the competing, classical operators.

On the second day, Toby Meadows (Arch) drew connections between two kinds of Kripkean truth definitions and some tree proof methods from recursion theory. He used this connection to motivate an interpretation of partial, Kripkean truth predicates as generalized proof predicates. Stewart Shapiro (Ohio State) discussed the open-texture of concepts and the possibility of precisification. He concluded that if all the technical work in non-classical truth theory at best serves to explicate one ('naive') way of sharpening our truth concept, then perhaps it is not worth the cost. Volker Halbach (Oxford) presented some results about the relative proof-theoretic strength of axiomatizations of 'external' (KFS) vs. 'internal' (PKF) readings of Kripke's truth definition on the Strong Kleene scheme. He showed that PKF is significantly proof-theoretically weaker than KFS, attaching a definite, mathematical cost to the revision of logic. Ole Hjortland (Munich) argued that we can approximate the logic of Field's theory of truth using some results from substructural logic. Field's logic has a lot in common with contraction-free substructural logics, and the similarities raise the prospect that contraction is in some way essential to the paradoxes. Dave Ripley (Melbourne) closed out the workshop with a talk about non-transitive logic for truth. This was motivated by a novel conception of logic on which he regards his logic as an extension of classical logic in much the way quantifier theory extends the propositional fragment of the logic.

The workshop had about 30 participants, and the discussion was extremely fruitful.