Reforming Logic and the Philosophy of Mathematics

The usual formulation of the basic part of our logic, known as first-order logic, can be enriched by adding to it means of expressing different patterns of dependence and independence between variables. This task of enrichment is already being implemented in the form of the refurnishing of logic know as IF logic (it was created earlier by the PI). The projected research carries this work further toward greater generality. It will be shown that a suitable extension of IF first-order logic can capture the force of entire second-order logic and hence practically all mathematical reasoning, unlike the currently used first-order logic. It thus makes set theory dispensable for foundational purposes. The project was initially led by Jaakko Hitnikka. I became the manager of the project in October, 2015.

Various lines of development in recent history of logic and philosophy mathematics are re-evaluated in the light of historical evidence and recent logical innovations, especially those relating to IF logic. IF logic is applied to various key problems in related fields.