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.

On the formalisation of indirect discourse

1958 ◽  
Vol 23 (4) ◽  
pp. 417-419 ◽  
Author(s):  
R. L. Goodstein
Keyword(s):  
Predicate Logic ◽  
Logical Truth ◽  
Predicate Calculus ◽  
Inference Rules ◽  
First Order ◽  
Indirect Discourse ◽  
Image Position ◽  
Order Predicate Calculus

Mr. L. J. Cohen's interesting example of a logical truth of indirect discourse appears to be capable of a simple formalisation and proof in a variant of first order predicate calculus. His example has the form:If A says that anything which B says is false, and B says that something which A says is true, then something which A says is false and something which B says is true.Let ‘A says x’ be formalised by ‘A(x)’ and let assertions of truth and falsehood be formalised as in the following table.We treat both variables x and predicates A (x) as sentences and add to the familiar axioms and inference rules of predicate logic a rule permitting the inference of A(p) from (x)A(x), where p is a closed sentence.We have to prove that from

Embedding first order predicate logic in fragments of intuitionistic logic

1976 ◽  
Vol 41 (4) ◽  
pp. 705-718 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Propositional Calculus ◽  
Expressive Power ◽  
Predicate Calculus ◽  
First Order ◽  
Intuitionistic Propositional Calculus ◽  
Individual Variables ◽  
The One ◽  
Order Predicate Calculus

Some syntactically simple fragments of intuitionistic logic possess considerable expressive power compared with their classical counterparts.In particular, we consider in this paper intuitionistic second order propositional logic (ISPL) a formalisation of which may be obtained by adding to the intuitionistic propositional calculus quantifiers binding propositional variables together with the usual quantifier rules and the axiom scheme (Ex), where is a formula not containing x.The main purpose of this paper is to show that the classical first order predicate calculus with identity can be (isomorphically) embedded in ISPL.It turns out an immediate consequence of this that the classical first order predicate calculus with identity can also be embedded in the fragment (PLA) of the intuitionistic first order predicate calculus whose only logical symbols are → and (.) (universal quantifier) and the only nonlogical symbol (apart from individual variables and parentheses) a single monadic predicate letter.Another consequence is that the classical first order predicate calculus can be embedded in the theory of Heyting algebras.The undecidability of the formal systems under consideration evidently follows immediately from the present results.We shall indicate how the methods employed may be extended to show also that the intuitionistic first order predicate calculus with identity can be embedded in both ISPL and PLA.For the purpose of the present paper it will be convenient to use the following formalisation (S) of ISPL based on [3], rather than the one given above.

Extensional interpretations of modal logics

1966 ◽  
Vol 31 (1) ◽  
pp. 23-45 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Predicate Logic ◽  
Modal Logics ◽  
Predicate Calculus ◽  
First Order ◽  
Image Position ◽  
Order Predicate Calculus ◽  
First Order Predicate Logic

By ΡL we shall mean the first order predicate logic based on S4. More explicitly: Let Ρ0 stand for the first order predicate calculus. The formalisation of Ρ0 used in the present paper will be given later. ΡL is obtained from Ρ0 by adding the rules the propositional constant □ and

scholarly journals On the Definitional Embeddability of Some Elementary Algebraic Theories into the First-Order Predicate Calculus

2015 ◽  
Vol 21 (2) ◽  
pp. 15-20
Author(s):  
В. И. Шалак
Keyword(s):  
Group Theory ◽  
Abelian Groups ◽  
Predicate Logic ◽  
Predicate Calculus ◽  
General Algebra ◽  
First Order ◽  
Algebraic Theories ◽  
Logical Content ◽  
Order Predicate Calculus ◽  
First Order Predicate Logic

In this article we prove a theorem on the definitional embeddability into first-order predicate logic without equality of such well-known mathematical theories as group theory and the theory of Abelian groups. This result may seem surprising, since it is generally believed that these theories have a non-logical content. It turns out that the central theory of general algebra are purely logical. Could this be the reason that we find them in many branches of mathematics? 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.

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.

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.

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.

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