Context recognition for a hyperintensional inference machine

Abstract

The goal of this paper is to introduce the algorithm of context recognition in the functional programming language TIL-Script, which is a necessary condition for the implementation of the TIL-Script inference machine. The TIL-Script language is an operationally isomorphic syntactic variant of Tichý’s Transparent Intensional Logic (TIL). From the formal point of view, TIL is a hyperintensional, partial, typed λ-calculus with procedural semantics. Hyperintensional, because TIL λ-terms denote procedures (defined as TIL constructions) producing set-theoretic functions rather than the functions themselves; partial, because TIL is a logic of partial functions; and typed, because all the entities of TIL ontology, including constructions, receive a type within a ramified hierarchy of types. These features make it possible to distinguish three levels of abstraction at which TIL constructions operate. At the highest hyperintensional level the object to operate on is a construction (though a higher-order construction is needed to present this lower-order construction as an object of predication). At the middle intensional level the object to operate on is the function presented, or constructed, by a construction, while at the lowest extensional level the object to operate on is the value (if any) of the presented function. Thus a necessary condition for the development of an inference machine for the TIL-Script language is recognizing a context in which a construction occurs, namely extensional, intensional and hyperintensional context, in order to determine the type of an argument at which a given inference rule can be properly applied. As a result, our logic does not flout logical rules of extensional logic, which makes it possible to develop a hyperintensional inference machine for the TIL-Script language.