A Boolean derivation of the Moore-Osgood theorem

1946 ◽  
Vol 11 (3) ◽  
pp. 65-70 ◽  
Author(s):  
Archie Blake
Keyword(s):  
Decision Problem ◽  
Fundamental Problem ◽  
Symbolic Logic ◽  
Logical Function ◽  
Consistent System ◽  
First Order ◽  
Methods Of Calculation ◽  
Truth Value ◽  
Function Calculus ◽  
Logical Calculi

A fundamental problem of symbolic logic is to define logical calculi sufficient to comprise important parts of mathematics, and to develop systematic methods of calculation therein.The possibility of progress in this direction has been severely limited by Gödel's proof that a consistent system sufficient to comprise arithmetic must contain propositions whose truth-value cannot be decided within the system, and by Church's extension of Gödel's method to the result that even in the first order logical function calculus the general decision problem cannot be solved.

A weak completeness theorem for infinite valued first-order logic

1963 ◽  
Vol 28 (1) ◽  
pp. 43-50 ◽  
Author(s):  
L. P. Belluce ◽  
C. C. Chang
Keyword(s):  
Completeness Theorem ◽  
Unit Interval ◽  
Order Logic ◽  
First Order Logic ◽  
Order System ◽  
First Order ◽  
Recursively Enumerable ◽  
Weak Completeness ◽  
Truth Value

This paper contains some results concerning the completeness of a first-order system of infinite valued logicThere are under consideration two distinct notions of completeness corresponding to the two notions of validity (see Definition 3) and strong validity (see Definition 4). Both notions of validity, whether based on the unit interval [0, 1] or based on linearly ordered MV-algebras, use the element 1 as the designated truth value. Originally, it was thought by many investigators in the field that one should be able to prove that the set of valid sentences is recursively enumerable. It was first proved by Rutledge in [9] that the set of valid sentences in the monadic first-order infinite valued logic is recursively enumerable.

Church’s theorem and the decision problem

2018 ◽  
Author(s):  
Rohit Parikh
Keyword(s):  
Decision Problem ◽  
Combinatorial Problem ◽  
Order Logic ◽  
First Order Logic ◽  
Negative Solution ◽  
Great Contribution ◽  
Solvable Problem ◽  
First Order ◽  
Mathematical Definition ◽  
Definition Of

Church’s theorem, published in 1936, states that the set of valid formulas of first-order logic is not effectively decidable: there is no method or algorithm for deciding which formulas of first-order logic are valid. Church’s paper exhibited an undecidable combinatorial problem P and showed that P was representable in first-order logic. If first-order logic were decidable, P would also be decidable. Since P is undecidable, first-order logic must also be undecidable. Church’s theorem is a negative solution to the decision problem (Entscheidungsproblem), the problem of finding a method for deciding whether a given formula of first-order logic is valid, or satisfiable, or neither. The great contribution of Church (and, independently, Turing) was not merely to prove that there is no method but also to propose a mathematical definition of the notion of ‘effectively solvable problem’, that is, a problem solvable by means of a method or algorithm.

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.

On the reduction of the decision problem. First paper. Ackermann prefix, a single binary predicate

1939 ◽  
Vol 4 (1) ◽  
pp. 1-9 ◽  
Author(s):  
László Kalmár
Keyword(s):  
Decision Problem ◽  
Reduction Process ◽  
The Other ◽  
First Order ◽  
Other Hand ◽  
Special Cases ◽  
Binary Predicate ◽  
Ternary Variables ◽  
Order Predicate Calculus ◽  
Predicate Variable

1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form(1) (Ex1)(x2)(Ex3)(x4)…(xm).On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove theTheorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

scholarly journals A decision problem for varieties of commutative semigroups

1993 ◽  
Vol 47 (3) ◽  
pp. 457-464
Author(s):  
W.L. Cao
Keyword(s):  
Decision Problem ◽  
Commutative Semigroups ◽  
First Order ◽  
Finite Set ◽  
Semigroup Identities

For a first order formula P: ∀x1, …, ∀xn, ∃y1, …, ∃ym (u(x1, …, xn, y1, …, ym) ≡ v(x1, …, xn, y1, …, ym)) where u and v are two words on the alphabet {x1, …, xn, y1, …, ym}, and a finite set E of semigroup identities with xy ≡ yx in it, we prove that it is decidable whether P follows from E, that is whether all the semigroups in the variety defined by E satisfy P.

scholarly journals ERRATUM

2008 ◽  
Vol 1 (3) ◽  
pp. 393-393
Keyword(s):  
Modal Logic ◽  
Production Process ◽  
Symbolic Logic ◽  
First Order ◽  
Topological Interpretation ◽  
Final Equation ◽  
First Order Modal Logic

Steve Awodey and Kohei Kishida (2008). Topology and Modality: The Topological Interpretation of First-Order Modal Logic. The Review of Symbolic Logic 1(2): 146-166.On page 148 of this article an error was introduced during the production process. The final equation in the displayed formula 8 lines from the bottom of the page should read,[0, 1) ≠ [0, 1]The publisher regrets this error.

The decision problem for standard classes

1976 ◽  
Vol 41 (2) ◽  
pp. 460-464 ◽  
Author(s):  
Yuri Gurevich
Keyword(s):  
Decision Problem ◽  
Predicate Logic ◽  
Order Theory ◽  
First Order ◽  
First Order Theory

The standard classes of a first-order theory T are certain classes of prenex T-sentences defined by restrictions on prefix, number of monadic, dyadic, etc. predicate variables, and number of monadic, dyadic, etc. operation variables. In [3] it is shown that, for any theory T, (1) the decision problem for any class of prenex T-sentences specified by such restrictions reduces to that for the standard classes, and (2) there are finitely many standard classes K1, …, Kn such that any undecidable standard class contains one of K1, …, Kn. These results give direction to the study of the decision problem.Below T is predicate logic with identity and operation variables. The Main Theorem solves the decision problem for the standard classes admitting at least one operation variable.

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