scholarly journals Completeness of two systems of illative combinatory logic for first-order propositional and predicate calculus

1998 ◽  
Vol 37 (5-6) ◽  
pp. 327-341 ◽  
Author(s):  
Wil Dekkers ◽  
Martin Bunder ◽  
Henk Barendregt
Keyword(s):  
Predicate Calculus ◽  
Combinatory Logic ◽  
First Order ◽  
Two Systems

scholarly journals Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic

1998 ◽  
Vol 63 (3) ◽  
pp. 869-890 ◽  
Author(s):  
Wil Dekkers ◽  
Martin Bunder ◽  
Henk Barendregt
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Lambda Calculus ◽  
Preceding Paper ◽  
Predicate Calculus ◽  
Combinatory Logic ◽  
First Order ◽  
Two Systems

AbstractIllative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions.

scholarly journals Systems of illative combinatory logic complete for first-order propositional and predicate calculus

1993 ◽  
Vol 58 (3) ◽  
pp. 769-788 ◽  
Author(s):  
Henk Barendregt ◽  
Martin Bunder ◽  
Wil Dekkers
Keyword(s):  
Lambda Calculus ◽  
Predicate Calculus ◽  
Combinatory Logic ◽  
Consistent System ◽  
First Order ◽  
Base Logic

AbstractIllative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).

scholarly journals On the Definitional Embeddability of the Combinatory Logic Theory into the First-Order Predicate Calculus

2015 ◽  
Vol 21 (2) ◽  
pp. 9-14
Author(s):  
В. И. Шалак
Keyword(s):  
Predicate Logic ◽  
Computability Theory ◽  
Predicate Calculus ◽  
Combinatory Logic ◽  
Applied Theory ◽  
First Order ◽  
Computable Functions ◽  
Order Predicate Calculus ◽  
Classical Predicate

In this article we prove a theorem on the definitional embeddability of the combinatory logic into the first-order predicate calculus without equality. Since all efficiently computable functions can be represented in the combinatory logic, it immediately follows that they can be represented in the first-order classical predicate logic. So far mathematicians studied the computability theory as some applied theory. From our theorem it follows that the notion of computability is purely logical. This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.

Resolution systems and their applications I

1980 ◽  
Vol 3 (2) ◽  
pp. 235-268
Author(s):  
Ewa Orłowska
Keyword(s):  
Theorem Proving ◽  
Sufficient Conditions ◽  
Predicate Calculus ◽  
Resolution Method ◽  
First Order ◽  
Deductive Systems ◽  
Resolution System ◽  
Resolution Principle ◽  
Necessary And Sufficient ◽  
Classical Predicate

The central method employed today for theorem-proving is the resolution method introduced by J. A. Robinson in 1965 for the classical predicate calculus. Since then many improvements of the resolution method have been made. On the other hand, treatment of automated theorem-proving techniques for non-classical logics has been started, in connection with applications of these logics in computer science. In this paper a generalization of a notion of the resolution principle is introduced and discussed. A certain class of first order logics is considered and deductive systems of these logics with a resolution principle as an inference rule are investigated. The necessary and sufficient conditions for the so-called resolution completeness of such systems are given. A generalized Herbrand property for a logic is defined and its connections with the resolution-completeness are presented. A class of binary resolution systems is investigated and a kind of a normal form for derivations in such systems is given. On the ground of the methods developed the resolution system for the classical predicate calculus is described and the resolution systems for some non-classical logics are outlined. A method of program synthesis based on the resolution system for the classical predicate calculus is presented. A notion of a resolution-interpretability of a logic L in another logic L ′ is introduced. The method of resolution-interpretability consists in establishing a relation between formulas of the logic L and some sets of formulas of the logic L ′ with the intention of using the resolution system for L ′ to prove theorems of L. It is shown how the method of resolution-interpretability can be used to prove decidability of sets of unsatisfiable formulas of a given logic.

scholarly journals Higher-Order Illative Combinatory Logic

2013 ◽  
Vol 78 (3) ◽  
pp. 837-872 ◽  
Author(s):  
Łukasz Czajka
Keyword(s):  
Strong Consistency ◽  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Model Construction ◽  
Partial Answer ◽  
Combinatory Logic ◽  
First Order ◽  
Propositional Quantifiers

AbstractWe show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

A reduction class containing formulas with one monadic predicate and one binary function symbol

1976 ◽  
Vol 41 (1) ◽  
pp. 45-49
Author(s):  
Charles E. Hughes
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Conjunctive Normal Form ◽  
Function Symbol ◽  
Predicate Calculus ◽  
Binary Function ◽  
First Order ◽  
The Matrix ◽  
Reduction Class ◽  
Order Predicate Calculus

AbstractA new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

On the strong semantical completeness of the intuitionistic predicate calculus

1968 ◽  
Vol 33 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Richmond H. Thomason
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
First Order ◽  
Semantic Tableaux ◽  
Weak Completeness ◽  
Skolem Theorem ◽  
Completeness Theorems

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

scholarly journals On a Propositional Calculus Whose Decision Problem is Recursively Unsolvable

1970 ◽  
Vol 38 ◽  
pp. 145-152
Author(s):  
Akira Nakamura
Keyword(s):  
Decision Problem ◽  
Propositional Calculus ◽  
Predicate Calculus ◽  
Reduction Theorem ◽  
List Type ◽  
First Order ◽  
Order Predicate Calculus

The purpose of this paper is to present a propositional calculus whose decision problem is recursively unsolvable. The paper is based on the following ideas: (1) Using Löwenheim-Skolem’s Theorem and Surányi’s Reduction Theorem, we will construct an infinitely many-valued propositional calculus corresponding to the first-order predicate calculus.(2) It is well known that the decision problem of the first-order predicate calculus is recursively unsolvable.(3) Thus it will be shown that the decision problem of the infinitely many-valued propositional calculus is recursively unsolvable.

Translating Advanced Integrity Checking Technology to SQL

2002 ◽  
pp. 203-249 ◽  
Author(s):  
Hendrik Decker
Keyword(s):  
Natural Language ◽  
Order Logic ◽  
Predicate Calculus ◽  
Integrity Constraints ◽  
First Order Logic ◽  
First Order ◽  
Integrity Checking ◽  
Natural Language Interface ◽  
User Friendly ◽  
Declarative Specifications

The main goal of this chapter is to arrive at a coherent technology for deriving efficient SQL triggers from declarative specifications of arbitrary integrity constraints. The user may specify integrity constraints declaratively as closed queries in predicate calculus syntax (i.e., sentences in the language of first-order logic, abbr. FOL), as datalog denials, as query conditions in SQL WHERE clauses, or in some other, possibly more user-friendly manner (e.g., via a dialog-driven graphical or natural language interface which internally translates to equivalent WHERE clause conditions). As we are going to see, the triggers derived from such specifications behave such that whenever some update event would violate any of the integrity constraints, one or several of the triggers derived from that constraint are activated in order to enforce the constraint. That is, the violation is either prevented by rolling back the update or repaired instantly by subsequent further updates.

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