The Strong Theory of Relative Identity

This dissertation considers a theory of numerical identity, first presented by P.T. Geach (1962). I label this theory `the strong theory of relative identity'. I suggest that the strong theory of relative identity involves three theses, which I name `GT', `RI', and `SRI'. I argue that each of these theses is logically independent. I consider arguments for and against each of these theses in turn. I conclude that none of the arguments for GT, RI, or SRI are conclusive. However, I also argue that the arguments against GT, RI and SRI are unsuccessful. I argue, further, that the strong theory of relative identity, and GT in particular, is incompatible with classical semantics and classical first-order logic with identity. I consider alternative non-classical logical systems and semantics which might be compatible with the strong theory of relative identity. Finally, I consider the philosophical applications of the strong theory of relative identity. I focus on one area, specifically philosophical theology, and I argue, with respect to the logical problem of the Trinity, that either GT is true or orthodoxy is false.