Intensional models for the theory of types

Abstract

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up. IntroductionThe axiom scheme of Extensionality states that whenever two predicates or relations are coextensive they must have the same properties:Historically Extensionality has always been problematic, the main problem being that in many areas of application, though not perhaps in the foundations of mathematics, the statement is simply false. This was recognized by Whitehead and Russell in Principia Mathematica [32], where intensional functions such as 'A believes that p' or 'it is a strange coincidence that p' are discussed at length. However, in the introduction to the second edition (1927) of the Principia Whitehead and Russell (influenced by Wittgenstein's Tractatus) already entertain the possibility that "all functions of functions are extensional". Thirteen years later, in Church's [6] canonical formulation of the Theory of Types, it is observed that axioms of Extensionality should be adopted "[i]n order to obtain classical real number theory (analysis)", a wording that does not seem to rule out the option of not adopting them. Church's formulation of type theory was completely syntactic and axioms could be adopted or dropped at will, * The Journal of Symbolic Logic, to appear.1 but in Henkin's [12] classical proof of generalized completeness the models that are considered, both the "standard" models and the "general" ones, simply validate Extensionality. Although Henkin's text still allows giving up the axiom 1 the formal set-up now effectively rules out intensional predicates and functions. This poses problems for those areas of application of the logic where it is important to distinguish between predicates that are coextensive and where propositions that determine the same set of possible worlds should be kept apart nevertheless. Linguistic semantics and Artificial Intelligence are such applications and the problem has been dubbed one of "logical omniscience" there, for it is with propositional attitudes like knowledge and belief that predicates of predicates and predicates of propositions most naturally arise. Is there a...

Automation of Higher-Order Logic

“…Boolean extensionality, however, is independent of any of these principles. A whole landscape of respective notions of models structures for ETT between Andrews's v-complexes and Henkin semantics that further illustrate and clarify the above connections is developed in Benzmüller et al (2004);Brown (2004);Benzmüller (1999a), and an alternative development and discussion has been contributed by Muskens (2007).…”

Free Choice in Deontic Inquisitive Semantics (DIS)

“…1 There are now several approaches to type theory that manage to avoid making (2) valid. Fitting (2002) and Benzmüller et al (2004) are two of them, but since both of these papers interpret the central machinery of type logic in some nonstandard way, 2 the logic used here will be the ITL of Muskens (2007). In this logic all operators have standard interpretations and in fact the interpretation of the logic is a rather straightforward generalisation of that of Henkin (1950), making (2) invalid but retaining all classical rules for logical operators.…”

“…These last clauses constrain extensions to behave as in the usual Tarski value definition. For the treatment of abstraction and application in a relational setting, see also Muskens (1995).…”

“…Proofs The Gentzen calculus in Table 1 is generalised complete for the semantic notion of entailment just defined (see Muskens (2007) for a proof). Table 2 gives derived rules for some operators defined from the two primitives ⊂ and ⊥.…”

Leo-III Version 1.1 (System description)

“…The semantics of monomorphic HOL can be found in the literature [6,7,22]. HOL augmented with quantification over types and type operators was studied in [20].…”