This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following Trafford :22–40, 2015). It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their relation such (...) that classical logic is simulated where proofs and refutations are conclusive. (shrink)

We show that numerous distinctive concepts of constructive mathematics arise automatically from an “antithesis” translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically “constructivize” classical definitions, handling the resulting bookkeeping automatically.

This article takes inspiration from Kittler’s claim that philosophy has neglected the means used for its production. Kittler’s argument for media ontology will be compared to the post-Kantian project of re-inventing philosophy through the medium of thought. The article discusses these views in the context of the automation of logical thinking where procedures, tasks, and functions are part of the instrumental processing of new ends evolving a new mode of reasoning. In particular, the article suggests that in constructivist logic and (...) information theory, the temporal gap between truth and proof, between input and output, can be taken to argue that the means of thought expose the indetermination or the incomputability of proof. The automation of reasoning in logical processing coincides not with mindless correlations of data, replacing axioms with data, truths with self-validating proofs. Instead, the problem of the indeterminacy of proof within automated logic re-habilitates techne or instrumentality, and the relation between means and ends away from classical idealism and analytic realism. By following John Dewey’s argument for instrumentality, it will be argued that the task of thinking today needs to re-invent a logic of techne away from the teleological view of ends or the crisis of finality. If the post-Kantian preoccupations about the task of thinking already announced that the medium of thought could offer possibilities for a non-human philosophy, this article envisions a machine philosophy originating from within computational media. (shrink)