Three Essays on Substructural Approaches to Semantic Paradoxes

This thesis consists of three papers on substructural approaches to semantic paradoxes. The first paper introduces a formal system, based on a nontransitive substructural logic, which has exactly the valid and antivalid inferences of classical logic at every level of (meta)inference, but which I argue is still not classical logic. In the second essay, I introduce infinite-premise versions of several semantic paradoxes, and show that noncontractive substructural approaches do not solve these paradoxes. In the third essay, I introduce an infinite metainferential hierarchy of validity curry paradoxes, and argue that providing a uniform solution to the paradoxes in this hierarchy makes substructural approaches less appealing. Together, the three essays in this thesis illustrate a problem for substructural approaches: substructural logics simply do not do everything that we need a logic to do, and so cannot solve semantic paradoxes in every context in which they appear. A new strategy, with a broader conception of what constitutes a uniform solution, is needed

The Logic of Logical Necessity

Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, `it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some views in modal metaphysics. In particular, we present an hitherto unpublished solution to problems 41 and 42 from Friedman's 102 problems, which uses a different method of proof from the solution presented in the paper of Tadeusz Prucnal