¹ For a more detailed account of propositional calculi with s designated truth-values, see the present authors' Axiom schemes for m-valued propositional calculi, this Journal, vol. 10, no. 3, pp. 61–82.

² Introduction to a general theory of elementary propositions, American journal of mathematics, vol. 43 (1921), pp. 163–185.

³ The functional incompleteness of the Łukasiewicz-Tarski calculi based on CPQ and NP was first proved by Slupecki, Jerzy in his Der volle dreiwertige Aussagenkalkül, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 29 (1936), pp. 9–11.Google Scholar

⁴ Op. cit.

⁵ Op. cit.

⁶ J_κ(P) may be read “P takes the truth-value κ.”

⁷ See p. 178 of the present paper.

⁸ See Rosser and Turquette, op. cit., p. 78.

⁹ A formalization is deductively complete when all its formulas taking designated truth-values exclusively are provable in the formalization.

¹⁰ For a definition of plausibility for m-valued prepositional calculi see Rosser and Turquette, op. cit., p. 68.

¹¹ The operators ·, v, and ⊃ satisfy standard conditions when the following conditions are satisfied: (1) (P·Q) takes a designated truth-value if and only if both P and Q take designated truth-values. (2) (P v Q) takes an undesignated truth-value if and only if both P and Q take undesignated truth-values. (3) (P ⊃ Q) takes an undesignated truth-value if and only if P takes a designated truth-value and Q takes an undesignated truth-value.

¹² See p. 178 of the present paper.

¹³ That CPQ does not satisfy standard conditions may be seen from the fact that CPQ takes the undesignated truth-value 2 when P and Q take the undesignated truth-values 2 and 3 respectively.

¹⁴ For a more detailed account of the procedures involved in the present illustration, see Rosser and Turquette, op. cit., pp. 61–82.

¹⁵ For a definition of plausibility for m-valued functional calculi of first order see p. 189 of the present paper.

¹⁶ A system is quantificationally complete if all possible quantifiers are definable in terms of the basic quantifiers of the system.

¹⁷ Henceforth, we shall let (ai) denote our i^th assumption.

¹⁸ One could give the following recursive definition of well-formed formula and free and bound occurrence of a variable:

(α)1. A propositional variable which occurs alone is a free occurrence and is well-formed. 2. If G is a functional variable and X ₁, …, X _n are individual variables, then G(X ₁, …, X _n) is well-formed and G, X ₁, …, X _n are free occurrences of functional and individual variables respectively.

(β)1. If P ₁, …, P _αi, are well-formed, then F _i(P ₁, …, P _αi) is well-formed where F_i(—₁, …, —_αi) is a basic operator of the calculus. Variables occurring in F_i(P ₁, …, P _αi) are free or bound according as they occur free or bound in P ₁, …, P _αi. 2. If P is well-formed, (Π_i,Χ₁, …, Χφi)P is well-formed and all occurrences of Χ₁, …, Χφi, are bound occurrences in (Π_i,Χ₁, …, Χφi)P. Occurrences of any other variables than Χ₁, …, Χφi, in (Π_i,Χ₁, …, Χφi)P are free or bound according as they occur free or bound in P.

In what follows we will use the term “formula” to mean “well-formed formula” as here defined.

¹⁹ See pp. 178–181, of the present paper.

²⁰ See footnote 6, p. 178, of the present paper.

²¹ An m-valued ∼ satisfies standard conditions if ∼P takes a designated truth-value when and only when P takes an undesignated truth-value. P satisfies standard conditions. See p. 180 of the present paper.

²² See p. 187 of the present paper.

²³ For a more precise statement of such a definition see pp. 187–8 of the present paper.

²⁴ See Alonzo Church's Introduction to mathematical logic, part I, sec. 2.5.

²⁵ Ibid.

²⁶ See p. 184 of the present paper.

²⁷ See footnote 18, p. 182, of the present paper.

²⁸ Determining appropriately chosen prepositional variables to be associated with G(Χ₁, …, Χ_n) in the present case is strictly aralogous to the 2-valued case. In particular, expressions such as g(x ₁, …, x_n) and g(y ₁, …, y_n) are to be treated as the same numerical variables while expressions such as g(x ₁, …, x_n) and h(x ₁, …, x_n) are to be treated as different numerical variables. See Hilbert, and Ackermann, , Grundzüge der theoretischen Logik, 2nd revised editionGoogle Scholar, section 9, and Alonzo Church, op. cit.

²⁹ For the definition of F_i(x ₁, …, x _αi) see p. 180 of the present paper.

³⁰ (Χ)P is our m-valued universal quantifier which satisfies standard conditions. See p. 184 of the present paper.

³¹ See p. 184 of the present paper.

³² See p. 180 of the present paper.

³³ See p. 183 of the present paper.

³⁴ For a definition of J_κ(P) in terms of the Łukasiewicz-Tarski operators С and N, see Rosser and Turquette, op. cit., sec. 2.

³⁵ See p. 186 of the present paper.

³⁶ See p. 186 of the present paper.

³⁷ For a definition of standard conditions for ⊃, ·, and v, see footnote 11, p. 180, of the present paper. For a definition of standard conditions for ∼, see footnote 21, p. 184, of the present paper. J _κ(P) satisfies standard conditions if it takes a designated truth-value when and only when P takes the truth-value κ.

³⁸ See p. 181 of the present paper.

³⁹ See p. 179 of the present paper.

⁴⁰ See p. 184 of the present paper.

⁴¹ See p. 183 of the present paper.

⁴² See p. 181 of the present paper.

⁴³ See Rosser and Turquette, op. cit., p. 68.

⁴⁴ See p. 188 of the present paper.

⁴⁵ See p. 186 of the present paper.

⁴⁶ See p. 185 of the present paper.

⁴⁷ Note that the present conception of an m-valued quantifier-free formula is strictly analogous to the 2-valued notion of an associated quantifier-free formula F* of F. See p. 185 of the present paper.

⁴⁸ See pp. 185, 188, of the present paper.

⁴⁹ See footnote 37, p. 188, of the present paper.

⁵⁰ The present definition is an extension of the definition of a yields sign given in Rosser and Turquette, op. cit., p. 70.

⁵¹ Our proof of plausibility for the m-valued case is closely analogous to proof of consistency for the 2-valued case (see Alonzo Church and Hilbert and Ackermann, op. cit.). As we have indicated elsewhere, however, (see Rosser and Turquette, op. cit., p. 68) we do not use the term “consistency” in the m-valued case since it is usually defined in such a manner as to make consistency independent of a choice of designated truth-values.

⁵² See p. 186 of the present paper.

⁵³ See Theorem 1, p. 185, of the present paper.

⁵⁴ See Alonzo Church, op. cit., p. 44.