15 Nov 2013, 2pm-4pm, Oak Hall 408
Many solutions to the sorties paradox use non-classical logic. But what becomes of the sorties when we turn the tables, and pose the problem using non-classical logics? Is the candidate non-classical logic too weak to be able to formulate the sorites paradox to begin with? There would seem to be something awry if a logic were not strong enough to express the very problems that logic was invoked to address. We will look at some basic tools to demonstrate what a sorties paradox looks like when fully recast in paraconsistent mathematics.
1 Nov 2013, 2pm-4pm, Oak Hall 408
The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the “logical” content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableau-style deductive system for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice. Beyond clarifying the nature of implication in Reverse Mathematics, this work has potential applications in automated theorem proving, such as the Reverse Mathematics Zoo.