Alan Rose

Strong Completeness of Fragments of the Propositional Calculus

There has recently been developed a method of formalising any fragment of the propositional calculus, subject only to the condition that material implication is a primitive function of the fragmentary system considered. Tarski has stated, without proof, that when implication is the only primitive function a formulation which is weakly complete (i.e., has as theorems all expressible tautologies) is also strongly complete (i.e., provides for the deduction of any expressible formula from any which is not a tautology). The methods used by Henkin suggest the following proof of the Theorem. If in a fragment of the propositional calculus material implication can be defined in terms of the primitive functions, then any weakly complete formalisation of the fragmentary system which has for rules of procedure the substitution rule and modus ponens is also strongly complete.

Extensions of some theorems of Anderson and Belnap

The object of this note is to extend to Posts m -valued propositional calculus 1 the axiomatization by Anderson and Belnap 2 of the 2-valued propositional calculus. Notation and definitions. We shall use the symbols ∼ and v for the primitives of Post and shall otherwise, except as stated below, use the same notation and definitions as in the previous paper.

An Extension of a Theorem of Margaris

Margaris has shown 1 that for every triple 〈 s,t,m 〉 of integers such that 1 ≦ s t m it is possible to construct a formalisation of an m -valued propositional calculus satisfying the following conditions: I. Every statement which takes only truth-values belonging to the set {1, …, s } is provable. II. Every provable statement takes only truth-values belonging to the set {1, …, t }. III. There exist statements P k , Q k which take only truth-values belonging to the set {1, …, k ) and neither of which takes only truth-values belonging to the set {1, …, k —1} such that P k is provable and Q k is unprovable ( k = s +1, …, t ). The systems of Margaris are all functionally incomplete and he appears to suggest 2 that it is impossible to construct functionally complete systems having the required properties.

Formalisations of further ℵ 0 -valued Łukasiewicz propositional calculi

It has been shown that, for all rational numbers r such that 0≤ r ≤ 1, the ℵ 0 -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 D αβ such that if P, D αβ take the truth-values x , d αβ ( x ) respectively, α, β are relatively prime integers and r = α/β then It follows at once from a theorem of McNaughton that the functor D αβ is definable in terms of C and N in an effective way. If r = 0 we make the definition We note first that if x ≤ α/β then d αβ ( x )≤(β + 1)α/β − α = α/β. Hence Let us now define the functions d n αβ ( x ) ( n = 0,1,…) by Since it follows easily that and that Thus, 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.

The

The 2-valued calculus of non-contradiction of Dexter has been extended to 3-valued logic. The methods used were, however, too complicated to be capable of generalisation to m -valued logics. The object of the present paper is to give an alternative method of generalising Dexters work to m -valued logics with one designated truth-value. The rule of procedure is generalised in the same way as before, but the deductive completeness of the system is proved by applying results of Rosser and Turquette. The system has an infinite set of primitive functions, written n ( P 1 , P 2 , …, P r ) ( r = 1,2, …). With the notation of Post, n ( P 1 , P 2 , …, P r ) has the same truth-value as ~( P 1 & P 2 & … & P r ). Thus n ( P ) is Posts primitive ~ P , and we can define & by We use n 2 ( P 1 , P 2 , …, P r ) as an abbreviation for n ( n ( P 1 , P 2 , …, P r )); similarly for higher powers of n . But if we set up the 1-1 correspondence of truth-values i ↔ m − i +1, then & corresponds to ∨ and ~ m−1 corresponds to ~. Hence the functional completeness of our system follows from a theorem of Post. We define the functions N ( P ), N ( P, Q ) by Thus the truth-value of N ( P ) is undesignated if and only if the truth-value of P is designated, and the truth-value of N ( P, Q ) is undesignated if and only if the truth-values of P and Q are both designated.