Syntactical and semantical properties of simple type theory

Schütte, Kurt

doi:10.2307/2963525

Cited by 83 publications

(40 citation statements)

References 7 publications

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“…The definition of Hintikka sequents is close to that of the "Hintikka sets" in Smullyan [27] and Fitting [9,10], but is also analogous to that of Schütte's [26] semi-valuations. A Hintikka sequent Γ in L is said to be complete if L : ϕ ∈ Γ or R : ϕ ∈ Γ, for each sentence ϕ of L.…”

Section: Model Existencementioning

confidence: 95%

Intensional models for the theory of types

Muskens¹

2007

J. symb. log.

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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...

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Section: Model Existencementioning

confidence: 95%

Intensional models for the theory of types

Muskens¹

2007

J. symb. log.

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“…The goal, and the difficulty in the Simple Type Theory case -or higherorder logic -as identified by Schütte [14], resides in the extension of V into a model interpretation. The answer, given independently by Takahashi [16] and Prawitz [13], is to construct a new interpretation domain, called V -complexes, as follows.…”

Section: Principles Of the Proof With V -Complexesmentioning

confidence: 99%

“…We assume given a semi-valuation V [14] compatible with this sequent i.e. a partial interpretation function from the propositions into {0, 1} such that V (Γ) = 1 and V (∆) = 0.…”

Section: Principles Of the Proof With V -Complexesmentioning

confidence: 99%

A Simple Proof that Super-Consistency Implies Cut Elimination

Dowek¹,

Hermant²

2012

Notre Dame J. Formal Logic

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We give a simple and direct proof that super-consistency implies the cut elimination property in deduction modulo. This proof can be seen as a simplification of the proof that super-consistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cut-free calculus. As an application, we compare our work with the cut elimination theorems in higher-order logic that involve V-complexes.

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“…Unlike IL, which is based on hierarchies of functions, the logic MTT will be based on hierarchies of relations (Orey [42], Schütte [47]), as relational models are pleasant to work with. Some definitions therefore must be changed and we shall start with the definition of types.…”

Section: Types and Termsmentioning

confidence: 99%

“…The latter also play an important role in Andrews' [1] proof that his (non-extensional) resolution calculus corresponds to the first six axioms of Church [14]. The structures considered by these authors are proof-generated and are defined on the basis of a purely syntactic notion (Schütte's [47] semivaluations), but recently purely semantic, stand-alone, generalizations of such models have been offered in Fitting [19] ('generalized Henkin models') and in Benzmüller et al [9] ('Σ-models'). Fitting's models involve a non-standard interpretation of abstraction, while the models of Benzmüller et al have a non-standard form of application, but these complications seem unnecessary, as our intensional models will do without them.…”

mentioning

confidence: 99%

10 Higher order modal logic

Muskens¹

2007

Handbook of Modal Logic

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Syntactical and semantical properties of simple type theory

Cited by 83 publications

References 7 publications

Intensional models for the theory of types

Intensional models for the theory of types

A Simple Proof that Super-Consistency Implies Cut Elimination

10 Higher order modal logic

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