Deduction System for TIL-2010

The goal of this paper is to introduce a deductive system for Tichý’s Transparent Intensional Logic (TIL). Tichý defined a sequent calculus for pre-1988 TIL, that is TIL based on the simple theory of types. Thus we first briefly recapitulate the rules of this simple-type calculus. Then we describe the adjustments of the calculus so that it be applicable to hyperintensions within the ramified hierarchy of types. TIL operates with a single procedural semantics for all kinds of logical-semantic context, be it extensional, intensional or hyperintensional. We show that operating in a hyperintensional context is far from being technically trivial. Yet it is feasible. To this end we introduce a substitution method that operates on hyperintensions. The syntax of TIL is the typed lambda calculus. Its semantics is based on a procedural redefinition of, inter alia, functional abstraction and application. The only two non-standard features are a hyperintension (called Trivialization) that presents objects, including hyperintensions, and a four-place substitution function (called Sub) defined over hyperintensions.