Henkin, L.: Completeness in the theory of types. Journal of Symbolic Logic 15 (1950) 81--91  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rst order sentence . By Tarski s theorem, the left hand side is not arithmetically de nable, then the set of second order validities cannot be arithmetical either. Thus, there is no e ective and complete axiomatization of second order validity (cf. 14] On the other hand, Henkin proves in [5] that it is possible to give an axiomatization of hol sound and complete w.r.t. a wider class of models, called general models, in which types of the form ( 1 2 ) are interpreted as subsets of the set of maps from (the interpretation of) 1 to (the interpretation of) 2 . The trick consists ....

....a sequent calculus for local set theory is not surprising: The proof methods for hol one can found in the literature are, in general, expressed as sequent calculus or natural deduction systems. On the other hand, the Hilbert calculus presentations of hol contain complicated rules of inference (cf. [5, 1]) The goal of this article is to introduce two very simple and natural Hilbertstyle axiomatizations of hol, which are sound and complete w.r.t. topos semantics. The rst one is obtained by adapting the sequent calculus for local set theory mentioned above, de ned in a language with power types ....

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L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15:81-91, 1950.

....[Harrison, 1996a] which has been successfully employed for industrial verification tasks of floating point arithmetic. See also [Gordon, 2000] for further background information on HOL and its relatives. The HOL logic is based on a version of Church s Simple Theory of Types [Church, 1940] [Henkin, 1950] [Andrews, 1986] which has been extended by schematic polymorphism, first order type constructors, and a semantic type definition scheme [Gordon, 1985a] Gordon, 1985b] Pitts, 1993] The HOL methodology emphasizes a strictly definitional discipline of theory development; arbitrary ....

....to note that this incomplete view on HOL is merely an e#ect of the particular set theoretic interpretation required by [Pitts, 1993] in order to make the typedef primitive appear as a definitional concept. The very higher order nature of HOL does not make it apt to incompleteness. In fact, [Henkin, 1950] shows completeness of the original formulation of [Church, 1940] The proof may use essentially the same techniques as for propositional or first order logic, cf. the detailed exposition in [Andrews, 1986] 8.6.2 Derived rules of type definitions The primitive axioms of HOL type definitions are ....

L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15(2):81--91, 1950.

....in formal derivations. 2.2 Model Theory To show consistency and to provide a reference point for the proof theory, we define simple term models. These models correspond to Gilmore s models for NaDSyL [Gil97] in the same way that standard models correspond to nonstandard, Henkin style models [Hen50]: they do not require the model to select denotations for use variables from a single given set. This relaxation simplifies the semantics. We use the symbols T; F to denote the truth values true and false , respectively. A model consists of a total function v and a countably infinite sequence ....

Leon Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15:81--91, 1950.

....as follows: fair = def (s:sched) 8p:proc:infinitely often( s :sched) who(s ) p) s) Expanding fair(start) gives a single formula expressing the required property. The language defined above is a trimmed version of the classical theory of simple types as introduced by Henkin in [Hen50]. Henkin considers nonstandard models for which a natural Gentzen style proof system is sound and complete. A good reference is also Chapter 4 of Schutte s monograph [Sch77] where cut elimination for this system is established. 3 Semantics of higher order logic Let Sigma = hB; Ci be a ....

....fact (Example 3.34) that Nat = for as given in Example 3.16 gives Term = j= Phi ctr . Then Theorem 3.35 tells us that Phi ctr , which is what we concluded in Example 3.31. And vice versa. We believe that the above development would go through, mutatis mutandis, for Henkin models [Hen50] as well as in a constructive framework like that of topos theory [Pho92] see Section 3.9 of [TvD88] In the absence of the axiom of choice, e.g. in topos theory, one must replace the function in the proof of Theorem 3.35 by a relation which is functional up to . 18 4 Behavioural equivalence ....

L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic 15:81--91 (1950).

....B of course has a similar description. Before going into details, let us mention that in fact there are many different spaces which will do the job, depending on various parameters that one is free to choose. We exhibit here just one such choice, intended to be illuminating. To begin, recall from [9] that classical higher order logic is complete with respect to general models, nowadays called Henkin models. The basic feature of a Henkin model M of a theory T is that a function type Z (or power type 2 ) is interpreted by a subset (Z ) M ae (ZM ) of the set of all functions from YM ....

.... way to a left exact, continuous functor BT Sh(XT ) that preserves exponentials, inducing the covering map OE : Sh(XT ) Sh(BT ) Finally, the stalk x Phi of the c model Phi at a point x = M; ff) of XT is just the Henkin model M itself, which gives the relationship between our results and [9]. 16 ....

L. Henkin, Completeness in the theory of types, Journal of Symbolic

....P n ( L 0 ; Q ; Q n ) this also gives us a leftmost derivation of P from L of length n. 2 4 Classical Type Logic Mainly in order to x notation, we give a short exposition of classical type logic, Church s formulation of Russell s Simple Theory of Types. For more extensive treatments see [2, 8, 9, 4, 25, 1]. In classical type logic (higher order logic) every expression comes with a type. Types are either basic or complex. The type t of truth values should be among the basic types, but there may be other basic types as well. In this paper we x the set of basic types as ft; e; sg, where e is the ....

L. Henkin. Completeness in the Theory of Types. Journal of Symbolic Logic, 15:81-91, 1950.

....by Russell in 1908 as a solution to paradoxes in set theory (see [19] and was reformulated by Church in [4] By G odel s second theorem, it is immediate to show that there is no (reasonable) proof system complete for the standard (set) semantics of hol. On the other hand, Henkin proves in [8] that it is possible to give an axiomatization of hol sound and complete w.r.t. a wider class of models, called general models, in which types of the form ) are interpreted as subsets of the set of maps from (the carrier of) to (the carrier of) From the works of Lawvere (see for example ....

L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15:81-91, 1950.

....and results. The basic result in this paper is the existence of a sound and complete deduction system for full R logics. This is not in contradiction to the general incompleteness of higher order predicate logics, since we use a particular nonstandard model notion, similar to Henkin models [9]. Furthermore, this category of models together with standard order sorted homomorphisms contains an initial object for each presentation, since we use a Horn clause fragment and all operations are deterministic. 3 Illustration of R Let us give some examples of specifications in R ....

L. Henkin. Completeness in the theory of types. The Journal of Symbolic Logic, 15(2):81--91, June 1950.

....m is a surjective geometric morphism. The corollary follows since models in Sh(X) correspond to OX valued models, see [8] for details. 2 What are the points of the topological space X Classical second order logic is complete with respect to models which are called nowadays Henkin models, see [10]. Combining Henkin s proof and the standard proof of Heyting valued completeness for first order intuitionistic logic one shows that our logic L (but in fact, full intuitionistic second order logic) is complete with respect to Heyting valued Henkin models. Fixing a set of enough Heytingvalued ....

L. Henkin. Completeness in the theory of types. J. Symbolic Logic, 15:81--91, 1950.

.... predicate calculus in that it has variables and quanti ers not only for individuals but also for subsets of the universe (sometimes variables for n ary relations as well, but this is not important in this context) The deductive calculus DED 2 of secondorder logic is based on rules and axioms ([Hen50]) which guarantee that the I am grateful to Juliette Kennedy for many helpful discussions while developing the ideas of this paper. y Research partially supported by grant 40734 of the Academy of Finland. 1 quanti ers range at least over de nable subsets. As to the semantics, there are two ....

....of second order logic the full semantics and second order logic with the full semantics the full second order logic. 1 . The following facts are the main features of second order logic: The Completeness Theorem: A sentence is provable in DED 2 if and only if it holds in all Henkin models ([Hen50]) The L owenheim Skolem Theorem: A sentence with an in nite Henkin model has a countable Henkin model. The Compactness Theorem: A set of sentences, every nite subset of which has a Henkin model, has itself a Henkin model. The Incompleteness Theorem: Neither DED 2 nor any other ....

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Leon Henkin. Completeness in the theory of types. J. Symbolic Logic, 15:81-91, 1950.

....for use as a basis for declarative programming languages and learning systems. Historically, this logic can be traced back to Church s simple theory of types [Chu40] which I refer to as type theory in the following. The appropriate model theory for type theory was given subsequently by Henkin in [Hen50]. More recent accounts of type theory appear in [And86] and [Wol93] Accounts of intuitionistic versions of the logic are in [Bel88] and [LS86] In fact, the logic presented here extends type theory in that it is polymorphic and admits product types. The polymorphism introduced is a simple form of ....

....function, there is a set of axioms contained in the definition for the function. The definitions for some typical functions are also given in Appendix A. 2. 9 Model Theory Next I turn to the semantics of the logic, which is derived from the semantics for type theory originally given by Henkin [Hen50]. The main concept is that of an interpretation which provides a meaning for the symbols used to model an application. Definition. A type substitution # is closed if, for each binding a # in #, # is closed. Definition. For each # # S c , a domain D# corresponding to # is a non empty set ....

L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15(2):81-- 91, 1950.

....declarative metaprogramming. These and many other gains are discussed extensively in [14] Semantics: Declarative and Operational Over the past decade a series of papers have provided semantics for certain classical or intuitionistic fragments of HOHH (see [22, 3] based on work by Henkin [13] and Andrews [1] on the semantics of Church s classical Type theory [6] Nadathur s dissertation [20] provides a notion of term model for the HOHH fragment. Miller de nes a Kripke like bottom up semantics for a rst order fragment [15] in which the syntax of programs is built into the notion of ....

Leon Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15:81-91, 1950.

....y, z of appropriate types. ii) A (weak) PTS A is total, or is a (weak) type structure, if each ## is a total function. iii) A (weak) PTS A is extensional if for all #, # and all f, g # A ### we have (#x # A # .fx # gx) # f = g. Extensional (total) type structures were introduced in [Hen50] and so are often called Henkin models (see e.g. Mit96, Section 4.5] Total type structures are called combinatory type structures in [Bar84, Appendix A] and weak total type structures are called typed applicative structures in [Mit96] A weak PTS A is called a partial type frame (cf. Mit96, ....

L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15(2):81--91, 1950.

....course has a similar description. Before going into details, let us mention that in fact there are many different spaces which will do the job, depending on various parameters that one is free to choose. We exhibit here just one such choice, intended to be illuminating. 16 To begin, recall from [9] that classical higher order logic is complete with respect to general models , nowadays called Henkin models . The basic feature of a Henkin model M of a theory T is that a function type Z Y (or power type 2 Y ) is interpreted by a subset (Z Y ) M ae (ZM ) Y M ) of the set of all ....

.... way to a left exact, continuous functor BT Sh(XT ) that preserves exponentials, inducing the covering map OE : Sh(XT ) Sh(BT ) Finally, the stalk x Phi of the c model Phi at a point x = M; ff) of XT is just the Henkin model M itself, which gives the relationship between our results and [9]. ....

L. Henkin, Completeness in the theory of types, Journal of Symbolic Logic 15 (1950), 81--91. 18

....Z Gamma . There is also a more or less obvious proof theoretic statement of this situation. We conclude by indicating how to pass from the strong compleness theorem to the classical higher order completeness theorem using nonstandard models in the single topos Sets, in the style of Henkin [10]. First, observe that any well pointed topos W has a canonical faithful functor into Sets, namely the global sections functor Gamma = W(1; Gamma) W Sets: Lemma 14. For any well pointed topos W, the global sections functor Gamma has the following properties: i) Gamma is left exact and ....

.... X ) ae Gamma2 GammaY = 2 GammaY = P ( GammaX ) 26) for any object X, its power object PX, and the powerset P ( GammaX ) Now, given a model M of a classical theory T in a well pointed topos W, the image of M under Gamma is a Henkin model of T (a general model in the sense of Henkin [10]; cf. 1] for a recent treatment) More precisely, recall that such a Henkin model M consists of sets XM ; interpreting the basic types of T) plus subsets (PZ)M P (Z M ) for each type Z (interpreting the power types of T) plus distinguished elements c M ; of these sets (interpreting ....

L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15:81--91, 1950. 16

.... oe v 1 z PROJ1 REF z : oe Theta h ; oe i v h 2 z; 1 zi PAIR REF z : oe Theta Thetaoe v h 2 z; 1 zi TRANS REF z : oe Theta : Thetaoe v z : oe Theta :h 2 z; 1 zi ABS REF 4 Models We can interpret the calculus using a simple generalisation of Henkin models [3]. A Henkin model (with products) is an applicative structure, that is, a tuple of families of sets and type indexed mappings hfA oe g; fProj oe; 1 g; fProj oe; 2 g; fApp oe; g; Consti which is also extensional and satisfies the environment model condition. See [6] for details. It is ....

L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15(2), 1950.

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Henkin, L.: Completeness in the theory of types. Journal of Symbolic Logic 15 (1950) 81--91

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L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15:81--91, 1950.

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Leon Henkin. Completeness in the Theory of Types. Journal of Symbolic Logic, 15:81--91, 1950.

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L. Henkin, Completeness in the Theory of Types, Journal of Symbolic Logic, 15, 2, 1950, pp. 81-91. 16

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L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15:81--91, 1950. 8

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Henkin, Leon. (1950). Completeness in the theory of types. The Journal of Symbolic Logic, 15:81--91.

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L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15:81-91, 1950.

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Leon Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15(2):81-91, 1950.

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L. Henkin. Completeness in the theory of types. Journal of Symbolic Logic 15:8191, 1950.

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