Hauptsatz for higher order logic

1968 ◽  
Vol 33 (3) ◽  
pp. 452-457 ◽  
Author(s):  
Dag Prawitz
Keyword(s):  
Finite Type ◽  
Type Theory ◽  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Simple Type ◽  
Higher Order Logic ◽  
Simple Type Theory ◽  
Second Order Logic

I shall prove in this paper that Gentzen's Hauptsatz is extendible to simple type theory, i.e., to the predicate logic obtained by admitting quantification over predicates of arbitrary finite type and generalizing the second order quantification rules to cover quantifiers of other types. (That Gentzen's Hauptsatz is extendible to this logic has been known as Takeuti's conjecture; see [4].) Gentzen's Hauptsatz has been extended to second order logic in a recent paper by Tait [3]. However, as remarked by Tait, his proof seems not to be extendible to higher orders. The present proof is rather an extension of a different proof of the Hauptsatz for second order logic that I have presented in [1].

scholarly journals POLYMORPHISM AND THE OBSTINATE CIRCULARITY OF SECOND ORDER LOGIC: A VICTIMS’ TALE

2018 ◽  
Vol 24 (1) ◽  
pp. 1-52
Author(s):  
PAOLO PISTONE
Keyword(s):  
Type Theory ◽  
Order Type ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Related Notion ◽  
Foundational Research ◽  
The One ◽  
Second Order Logic

AbstractThe investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher-order logic. However, the epistemological significance of such investigations has not received much attention in the contemporary foundational debate.We discuss Girard’s normalization proof for second order type theory or System F and compare it with two faulty consistency arguments: the one given by Frege for the logical system of the Grundgesetze (shown inconsistent by Russell’s paradox) and the one given by Martin-Löf for the intuitionistic type theory with a type of all types (shown inconsistent by Girard’s paradox).The comparison suggests that the question of the circularity of second order logic cannot be reduced to Russell’s and Poincaré’s 1906 “vicious circle” diagnosis. Rather, it reveals a bunch of mathematical and logical ideas hidden behind the hazardous idea of impredicative quantification, constituting a vast (and largely unexplored) domain for foundational research.

On nonstandard models in higher order logic

1984 ◽  
Vol 49 (1) ◽  
pp. 204-219
Author(s):  
Christian Hort ◽  
Horst Osswald
Keyword(s):  
Model Theory ◽  
Type Theory ◽  
Order Logic ◽  
Elementary Embedding ◽  
Simple Type ◽  
Order Model ◽  
Higher Order Logic ◽  
First Order ◽  
Simple Type Theory ◽  
Nonstandard Models

There are two concepts of standard/nonstandard models in simple type theory.The first concept—we might call it the pragmatical one—interprets type theory as a first order logic with countably many sorts of variables: the variables for the urelements of type 0,…, the n-ary relational variables of type (τ1, …, τn) with arguments of type (τ1,…,τn), respectively. If A ≠ ∅ then 〈Aτ〉 is called a model of type logic, if A0 = A and . 〈Aτ〉 is called full if, for every τ = (τ1,…,τn), . The variables for the urelements range over the elements of A and the variables of type (τ1,…, τn) range over those subsets of which are elements of . The theory Th(〈Aτ〉) is the set of all closed formulas in the language which hold in 〈Aτ〉 under natural interpretation of the constants. If 〈Bτ〉 is a model of Th(〈Aτ〉), then there exists a sequence 〈fτ〉 of functions fτ: Aτ → Bτ such that 〈fτ〉 is an elementary embedding from 〈Aτ〉 into 〈Bτ〉. 〈Bτ〉 is called a nonstandard model of 〈Aτ〉, if f0 is not surjective. Otherwise 〈Bτ〉 is called a standard model of 〈Aτ〉.This first concept of model theory in type logic seems to be preferable for applications in model theory, for example in nonstandard analysis, since all nice properties of first order model theory (completeness, compactness, and so on) are preserved.

scholarly journals A Resolution Calculus for Second-order Logic with Eager Unification

10.29007/zpg2 ◽  
2018 ◽  
Author(s):  
Alexander Leitsch ◽  
Tomer Libal
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Unification Algorithm ◽  
Efficient Search ◽  
First Order ◽  
Unification Algorithms ◽  
Second Order Logic ◽  
Resolution Calculus

The efficiency of the first-order resolution calculus is impaired when lifting it to higher-order logic. The main reason for that is the semi-decidability and infinitary natureof higher-order unification algorithms, which requires the integration of unification within the calculus and results in a non-efficient search for refutations.We present a modification of the constrained resolution calculus (Huet'72) which uses an eager unification algorithm while retaining completeness. Thealgorithm is complete with regard to bounded unification only, which for many cases, does not pose a problem in practice.

Second- and higher-order logics

2018 ◽  
Author(s):  
Shaughan Lavine
Keyword(s):  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Third Order ◽  
First Order ◽  
Proof Systems ◽  
The Universe ◽  
Second Order Logic

In first-order predicate logic there are symbols for fixed individuals, relations and functions on a given universe of individuals and there are variables ranging over the individuals, with associated quantifiers. Second-order logic adds variables ranging over relations and functions on the universe of individuals, and associated quantifiers, which are called second-order variables and quantifiers. Sometimes one also adds symbols for fixed higher-order relations and functions among and on the relations, functions and individuals of the original universe. One can add third-order variables ranging over relations and functions among and on the relations, functions and individuals on the universe, with associated quantifiers, and so on, to yield logics of even higher order. It is usual to use proof systems for higher-order logics (that is, logics beyond first-order) that include analogues of the first-order quantifier rules for all quantifiers. An extensional n-ary relation variable in effect ranges over arbitrary sets of n-tuples of members of the universe. (Functions are omitted here for simplicity: remarks about them parallel those for relations.) If the set of sets of n-tuples of members of a universe is fully determined once the universe itself is given, then the truth-values of sentences involving second-order quantifiers are determined in a structure like the ones used for first-order logic. However, if the notion of the set of all sets of n-tuples of members of a universe is specified in terms of some theory about sets or relations, then the universe of a structure must be supplemented by specifications of the domains of the various higher-order variables. No matter what theory one adopts, there are infinitely many choices for such domains compatible with the theory over any infinite universe. This casts doubt on the apparent clarity of the notion of ‘all n-ary relations on a domain’: since the notion cannot be defined categorically in terms of the domain using any theory whatsoever, how could it be well-determined?

Higher‐order Logic Reconsidered

2009 ◽  
Author(s):  
Ignacio Jané
Keyword(s):  
Set Theory ◽  
Logical Consequence ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Consequence Relation ◽  
Fourth Section ◽  
Second Order Logic

This article discusses canonical (i.e., full, or standard) second-order consequence and argues against it being a case of logical consequence. The discussion is divided into three parts. The first part comprises the first three sections. After stating the problem in Section 1, Sections 2 and 3 examine the role that the consequence relation is expected to play in axiomatic theories. This leads to put forward two requirements on logical consequence, which are called “formality” and “noninterference.” It is this last requirement that canonical second-order consequence violates, as the article sets out to substantiate. The fourth section argues that canonical second-order logic is inadequate for axiomatizing set theory, on the grounds that it codes a significant amount of set-theoretical content.

GLIVENKO AND KURODA FOR SIMPLE TYPE THEORY

2014 ◽  
Vol 79 (2) ◽  
pp. 485-495 ◽  
Author(s):  
CHAD E. BROWN ◽  
CHRISTINE RIZKALLAH
Keyword(s):  
Type Theory ◽  
Predicate Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Simple Type ◽  
Universal Quantifier ◽  
First Order ◽  
Simple Type Theory ◽  
Glivenko’S Theorem ◽  
First Order Predicate Logic

AbstractGlivenko’s theorem states that an arbitrary propositional formula is classically provable if and only if its double negation is intuitionistically provable. The result does not extend to full first-order predicate logic, but does extend to first-order predicate logic without the universal quantifier. A recent paper by Zdanowski shows that Glivenko’s theorem also holds for second-order propositional logic without the universal quantifier. We prove that Glivenko’s theorem extends to some versions of simple type theory without the universal quantifier. Moreover, we prove that Kuroda’s negative translation, which is known to embed classical first-order logic into intuitionistic first-order logic, extends to the same versions of simple type theory. We also prove that the Glivenko property fails for simple type theory once a weak form of functional extensionality is included.

Higher‐order Logic

2009 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Philosophy Of Mathematics ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Deductive Systems ◽  
Second Order Logic

The philosophical literature contains numerous claims on behalf of and numerous claims against higher-order logic. Virtually all of the issues apply to second-order logic (vis-à-vis first-order logic), so this article focuses on that. It develops the syntax of second-order languages and present typical deductive systems and model-theoretic semantics for them. This will help to explain the role of higher-order logic in the philosophy of mathematics. It is assumed that the reader has at least a passing familiarity with the theory and metatheory of first-order logic.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

scholarly journals Definability theorem for the intuitionistic predicate logic with equality

1982 ◽  
Vol 87 ◽  
pp. 1-15
Author(s):  
Chiharu Mizutani
Keyword(s):  
Predicate Logic ◽  
Second Order ◽  
Order Logic ◽  
Infinitary Logic ◽  
Second Order Logic

Svenonius’ definability theorem and its generalizations to the infinitary logic Lω1ω or to a second order logic with countable conjunctions and disjunctions have been studied by Kochen [1], Motohashi [2], [3] or Harnik and Makkai [4], independently. In this paper, we consider a (Svenonius-type) definability theorem for the intuitionistic predicate logic IL with equality.

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