Some Ways of Constructing a Propositional Calculus of Any Required Degree of Unsolvability

Gladstone, M. D.

doi:10.2307/1993954

Cited by 3 publications

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“…THEOREM 1. For each recursively enumerable degree of unsolvability D there exists a class of partial propositional calculi {P"}D> where the axioms of P n consist of a fixed recursive set P So together with a single formula V n which depends uniformly effectively on n, such that the problem to determine of an arbitrary n whether or not P n is finitely axiomatizable is of degree D. 2 Let S 0 be a set of ordered pairs of positive integers 3 having the properties (1) there is an effective procedure to determine of any ordered pair of positive integers, (m, ri), whether or not (m, ri) is a member of S 0 , (2) the problem to determine for an arbitrary positive integer n whether or not there is an m such that (m, ri) is a member of S 0 is of degree D, and (3) the set of second members of the ordered pairs of S 0 is infinite.…”

Section: Corollary 1 For a Positive Integer N ~~W N Isa Theorem Ofpmentioning

confidence: 99%

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A note on finite axiomatization of partial propositional calculi

Singletary

1967

J. symb. log.

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At the Princeton Bicentennial in 1946 Tarski posed the question as to whether or not certain problems connected with partial propositional calculi were recursively solvable. Since that time the specific problems mentioned by Tarski as well as a number of related problems have been shown to be recursively unsolvable. Such results are due to Post and Linial [5], Yntema [9], Gladstone [2], Ihrig [3] and Singletary [7], [8].

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Section: Corollary 1 For a Positive Integer N ~~W N Isa Theorem Ofpmentioning

confidence: 99%

“…Since that time the specific problems mentioned by Tarski as well as a number of related problems have been shown to be recursively unsolvable. Such results are due to Post and Linial [5], Yntema [9], Gladstone [2], Ihrig [3] and Singletary [7], [8].…”

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A note on finite axiomatization of partial propositional calculi

Singletary

1967

J. symb. log.

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“…During recent months, several independent detailed proofs of the existence of undecidable fragments of classical propositional calculus have been announced. Among the people concerned with these proofs were Singletary [50], [51], Ihrig [27] and Yntema [54] in the United States and Gladstone [14], [15], [16] and the present author [22], [23], [24] in the United Kingdom. The present position seems to be that Gladstone, Ihrig and Singletary have shown the existence of undecidable fragments of classical propositional calculus of any recursively enumerable degree of unsolvability, using, in the cases of Gladstone and Singletary, only the connective -^.…”

Section: Finite Model Propertymentioning

confidence: 99%

Some structure results for propositional calculi

Harrop

1965

J. symb. log.

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1. Introduction. Since this paper is a written version of one presented as a survey lecture at a meeting of the Association for Symbolic Logic, its form and content have been essentially determined by the form and content of that lecture, and these latter were themselves considerably influenced by a deliberate attempt to avoid the lecture consisting essentially either of a catalogue of results in some fairly wide field or of the presentation of detailed proofs of a few particular results in a very limited field, with, in either case, a consequential serious limitation of the time available for discussion of the motivation for the results and of the connections between them. There is thus no attempt to cover, even in outline, the full range of topics which could reasonably fall within the scope of its title.The paper consists mainly of a general discussion of some results concerned with finite models of axiomatically defined calculi of propositional form, with particular reference to the occurrence of such models in proofs that certain propositional calculi are decidable. In addition, there is an account of some results concerned with the existence of undecidable propositional calculi and of others connected with the decidability or undecidability of certain structure problems which relate either to sets of propositional calculi or to different presentations of the same calculus.

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A Single‐Axiom Impligational Calculus of Given Unsolvability

Gladstone

1968

Mathematical Logic Qtrly

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in Bristol (England) Q 1. Introduction Erst a paragraph of terminology.A propositional calculus, as the phrase is used here, consists of the following three things :(1) A set of coianectives, adequate to express implication (a connective conaists of a ( 2 ) A particular espression of implication in terms of the given connectives;(3) A set of tautologies, to be kiaown as "atuioms", in which no connectives appearThe wff of a propositional calculus are built up in the usual way from variables (throughout this paper '. variable " means "propositional variable "), the given connectives, and, possibly (according to the convention used), brackets and coinmas. The theorems are those of the wff which can be derived from the axioms, using as rules of inferencc,We are interested in propositional calculi which are uwolvable; that is, there is no effective procedure for deciding whether an arbitrary wff is a theorem.The purpose of this paper is to show that even a, propositional calculus of the Sirrrplwt possiblc structure, may be unsolvable. More precisely, we can satisfy the following three conditions simultaneously :( I ) The calculus is of any given recursively enumerable degree of unsolvability.(2) There is just one axiom.(3) Thc only connective is one representing implication. Say this connective is 3 ; then the designated expression for " A implies U" is " ( A 3 B ) " (without this stipulation, thc designated cxpression could be, for instancc, " ( A 3 ( A 3 B ) ) " ) . Tn other wolds, we have an irnplicational calculus.The strongest result of this kind hitherto obtained, as far as I know, is as above but without the single-axiom condition; it is given in my own paper [2], also by HARKWP in (41. For an extensive survcy of the literature, the reader is referred t o H ZKKC)P')R paper 131.symbol and an assignment to that symbol of LL classical truth-function) ; other than the given. one$.15 X h h r f iiiritli Locik 13*

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Some Ways of Constructing a Propositional Calculus of Any Required Degree of Unsolvability

Cited by 3 publications

References 3 publications

A note on finite axiomatization of partial propositional calculi

A note on finite axiomatization of partial propositional calculi

Some structure results for propositional calculi

A Single‐Axiom Impligational Calculus of Given Unsolvability

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