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The Degrees of Completeness of Dummetts Lc AND THOMAS's LC n
Author(s) -
Anderson J. G.
Publication year - 1971
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-3.3.558
Subject(s) - completeness (order theory) , citation , library science , computer science , sociology , mathematics , mathematical analysis
The purpose of this paper is to establish the degree of completeness of certain superconstructive propositional calculi, namely Dummett's LC and Thomas's sequence {LC n }. The next four paragraphs summarize background knowledge needed to read the paper. A sentence is formed in the usual way from propositional variables and the connectives-• , A , V , H. The closure of a set S of sentences, S, is the intersection of all sets of sentences containing S and closed under modus ponens and substitution. A superconstructive propositional calculus (S.P.C.) is formed from the intuitionistic propositional calculus (IC) by the addition of a finite number of extra axiom schemes. Aside from the trivial, inconsistent, S.P.C. of which every sentence is a theorem, every S.P.C. is intermediate between the classical propositional calculus and IC in the sense that its set of theorems includes all theorems of IC and is included in the set of tautologies. Two S.P.C.'s are called equivalent if they have identical sets of theorems. The degree of completeness of a set S of sentences, deg. S, is defined to be the smallest ordinal number <5 # 0 for which there exists no increasing sequence of sets of sentences {S a } a < 6 of type S satisfying 1. S a is consistent for a < 8 (i.e. S a # F) 2. S 0 = S 3. S. g S ^ c i ? and S a # S p whenever a < /? < 8, where F is the set of all sentences. For example it is easy to see that deg.F = 1, and the degree of completeness of the set of all tautologies is 2. The degree of completeness of an S.P.C. is the degree of its set of theorems. This definition is taken from an early paper of Tarski, translated in [4]. We need the notion of a model (or matrix) for a propositional calculus. Our usage will be conventional, we refer to [3] for definition of model and related terminology. Every model for an S.P.C. is, a fortiori, a model for IC, and hence can be associated in a standard way with a pseud o complemented lattice with smallest element, alternatively called a pseudo-Boolean algebra or Brouwerian algebra. It is most convenient to define the models we consider by means of their associated lattice rather than by giving their " truth tables " …