Substitution contradiction, its resolution and the Church-Rosser Theorem in TIL

Abstract

I present an analysis according to which the current state of the definition of substitution leads to a contradiction in the system of Transparent Intensional Logic (TIL). I entail the contradiction using only the basic definitions of TIL and standard results. I then analyse the roots of the contradiction and motivate the path I take in resolving the contradiction. I provide a new amended definition of collision-less substitution which blocks the contradiction in a non-ad hoc way. I elaborate on the consequences of the amended definition, namely the invalidity of the Church-Rosser theorem (the so-called diamond property). I present a counterexample to the validity of the theorem in TIL with an amended definition of substitution.