Alan Rose. Systems of logic whose truth-values form lattices. Mathematische Annalen, vol. 123 (1951), pp. 152–165. - Alan Rose. A lattice-theoretic characterisation of the ℵ0-valued Propositional Calculus. Mathematische Annalen, vol. 123 (1951), pp. 285–287. - Alan Rose. The degree of completeness of some Łukasiewicz-Tarski propositional calculi. The journal of the London Mathematical Society, vol. 26 (1951), pp. 47–49.

1952 ◽  
Vol 17 (2) ◽  
pp. 147-148
Author(s):  
A. R. Turquette
Keyword(s):  
Propositional Calculus ◽  
London Mathematical Society ◽  
Mathematical Society ◽  
Truth Values ◽  
Propositional Calculi

Formalisations of further ℵ0-valued Łukasiewicz propositional calculi

1978 ◽  
Vol 43 (2) ◽  
pp. 207-210 ◽  
Author(s):  
Alan Rose
Keyword(s):  
Propositional Calculus ◽  
Rational Numbers ◽  
Modus Ponens ◽  
Truth Values ◽  
Rules Of Procedure ◽  
C And N ◽  
Image Position ◽  
Propositional Calculi

It has been shown that, for all rational numbers r such that 0≤ r ≤ 1, the ℵ0-valued Łukasiewicz propositional calculus whose designated truth-values are those truth-values x such that r ≤ x ≤ 1 may be formalised completely by means of finitely many axiom schemes and primitive rules of procedure. We shall consider now the case where r is rational, 0≥r≤1 and the designated truth-values are those truth-values x such that r≤x≤1.We note that, in the subcase of the previous case where r = 1, a complete formalisation is given by the following four axiom schemes together with the rule of modus ponens (with respect to C),the functor A being defined in the usual way. The functors B, K, L will also be considered to be defined in the usual way. Let us consider now the functor Dαβ such that if P, Dαβ take the truth-values x, dαβ(x) respectively, α, β are relatively prime integers and r = α/β thenIt follows at once from a theorem of McNaughton that the functor Dαβ is definable in terms of C and N in an effective way. If r = 0 we make the definitionWe note first that if x ≤ α/β then dαβ(x)≤(β + 1)α/β − α = α/β. HenceLet us now define the functions dnαβ(x) (n = 0,1,…) bySinceit follows easily thatand thatThus, if x is designated, x − α/β > 0 and, if n > − log(x − α/β)/log(β + 1), then (β + 1)n(x−α/β) > 1.

Axiom schemes for m-valued propositions calculi

1945 ◽  
Vol 10 (3) ◽  
pp. 61-82 ◽  
Author(s):  
J. B. Rosser ◽  
A. R. Turquette
Keyword(s):  
Propositional Calculus ◽  
Value Functions ◽  
Truth Values ◽  
Truth Value ◽  
Propositional Calculi

In an m-valued propositional calculus, or a formalization of such a calculus, truth-value functions are allowed to take any truth-value t where 1 ≦ t ≦ m and m ≧ 2. In working with such calculi, or formalizations thereof, it has been decided to distinguish those truth-values which it is desirable for provable formulas to have from those which it is not desirable for provable formulas to have. The first class of truth-values is called designated and the second undesignated. This specification of certain of the m truth-values as designated and the remainder as undesignated is one of the distinguishing characteristics of m-valued propositional calculi, and it should be observed at the outset that two m-valued propositional calculi will be considered to differ even if they differ only in respect to the number of truth-values which are taken as designated.

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