The Terms of ITT

Church described in [30] a logic of sense and denotation based on the simple theory of types [28]; that is, using the terminology of Carnap, a logic of intension and extension. ITT differs from that logic in two ways: First ITT is based on TT and not on the simple theory of types; but of greater importance, ITT identifies the intension of a predicate with its name while the logic of Church treats intensions as separate entities with their own types and notation. The justification for this identification is the belief that in a given context a user discovers the intension of a predicate from its name.

The types of ITT, an intensional type theory, are the types of TT, and the constants and variables of ITT are those of TT. The terms of ITT are an extension of those of TT. Definition 29 of term of TT in §2.1.1 is extended for ITT by a fourth clause that introduces a secondary typing for some of the terms of ITT. Since secondary typing is the main feature of ITT that distinguishes it from TT, it will be motivated in §3.1.1 before being formally expressed in clause (4) of Definition 43 in §3.1.2.

Motivation for secondary typing. The purpose of secondary typing in ITT is to provide a simple but unambiguous way of distinguishing between an occurrence of a predicate name where it is being used and an occurrence where it is being mentioned. The necessity for recognizing this distinction has been stressed many times and a systematic use of quotes is traditionally employed for expressing it; see for example [24] or [120]. But the systematic use of quotes is awkward in a formal logic and subject to abuses as described by Church in footnote 136 of [31]. Secondary typing exploits the typing notation of TT to distinguish between a used predicate name and a mentioned predicate name and is not subject to the abuses cited by Church.