S. Ú. Maslov, G. É. Minc, and V. P. Orévkov. Nérazréšimost′ ν konstruktivnom isčislénii prédikatov nékotoryh klassov formul, sodéržaščih tol′ko odnoméstnyé prédikatnyé péréménnyé. Doklady Akadémii Nauk, vol. 163 (1965), pp. 295–297. - S. Ju. Maslov, G. E. Minc, and V. P. Orevkov. Unsolvability in the constructive predicate calculus of certain classes of formulas containing only monadic predicate variables. Translation of the preceding by E. Mendelson. Soviet mathematics, vol. 6 (1965), pp. 918–920.

The paper presents a logic which is an algorithmic extension of the classical predicate calculus and is based on the ideas given by F. Kröger. The programs and the effects of their execution are the formulas of this logic which are considered at any time scale. There are many interesting properties of the logic which are connected with the notion of time scale. These properties are examined in the paper. Moreover the problem of the formulas normalization is presented. Our logic is compared with the algorithmic logic introduced by A. Salwicki. Next, the usefulness of a new logic in the theory of programs is shown.

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Resolution systems and their applications I

Fundamenta Informaticae ◽

10.3233/fi-1980-3209 ◽

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.

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S. C. Kleene. Permutability of inferences in Gentzen's calculi LK and LJ. Two papers on the predicate calculus, by S. C. Kleene (Memoirs of the American Mathematical Society, no. 10), lithographed, Providence1952, pp. 1–26. - S. C. Kleene. Finite axiomatizability of theories in the predicate calculus using additional predicate symbols. Two papers on the predicate calculus, by S. C. Kleene (Memoirs of the American Mathematical Society, no. 10), lithographed, Providence1952, pp. 27–66. - S. C. Kleene. Bibliography. Two papers on the predicate calculus, by S. C. Kleene (Memoirs of the American Mathematical Society, no. 10), lithographed, Providence1952, pp. 67–68. - William Craig. On axiomatizability within a system. The journal of symbolic logic, vol. 18 (1953), pp. 30–32.

Journal of Symbolic Logic ◽

10.2307/2267666 ◽

1954 ◽

Vol 19 (1) ◽

pp. 62-63

Author(s):

Robert McNaughton

Keyword(s):

American Mathematical Society ◽

Predicate Calculus ◽

Mathematical Society ◽

Symbolic Logic ◽

Finite Axiomatizability

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Model theory for the higher order predicate calculus

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1959-0107600-8 ◽

1959 ◽

Vol 92 (1) ◽

pp. 72-72 ◽

Cited By ~ 20

Author(s):

Steven Orey

Keyword(s):

Model Theory ◽

Higher Order ◽

Predicate Calculus ◽

Order Predicate Calculus

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Preface

Mathematical Structures in Computer Science ◽

10.1017/s0960129511000235 ◽

2011 ◽

Vol 21 (4) ◽

pp. 671-677 ◽

Cited By ~ 1

Author(s):

GÉRARD HUET

Keyword(s):

Mathematical Logic ◽

Computer Science ◽

20Th Century ◽

Theorem Proving ◽

Propositional Calculus ◽

Interactive Theorem Proving ◽

Predicate Calculus ◽

Special Issue ◽

Mathematical Structures ◽

Infinite Domains

This special issue of Mathematical Structures in Computer Science is devoted to the theme of ‘Interactive theorem proving and the formalisation of mathematics’.The formalisation of mathematics started at the turn of the 20th century when mathematical logic emerged from the work of Frege and his contemporaries with the invention of the formal notation for mathematical statements called predicate calculus. This notation allowed the formulation of abstract general statements over possibly infinite domains in a uniform way, and thus went well beyond propositional calculus, which goes back to Aristotle and only allowed tautologies over unquantified statements.

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