¹ Throughout, I use the notation variable-slant-value and formula-slant-value as abbreviation for “takes on the value” (usually “truth-value”).

² In a 1938 paper, O. V. Zich discussed a truth-value system of prepositional logic whose statements can be true or false under either of two different interpretations or fundamental presuppositions. (See the review by K. Reach, this Journal, vol. 4 (1939), pp. 165–166.) Here there are four basic truth-values T₁ and F₁ for truth and falsity relative to the first interpretation, and T₂ and F₂ for truth and falsity relative to the second. Thus a particular proposition can have any one of four complex truth-values T₁–T₂, T₁–F₂, F₁–T₂, F₁–F₂. The reviewer rightly observes that this is simply a notationally different mode of presentation for a four-valued prepositional logic (of the strictly truth-functional type). The possibility of identifying the two types of truth-values, thus arriving at a system which embodies truth-functions which are not single-valued, is not envisaged by Zich.

³ It should be noted that certain two-valued tautologies — for example p⊃p and (p⊃q)⊃(∼q⊃∼p) — are not tautologies in Q. We could, however, at the cost of added complication, consider a system in which the truth-value definition of “⊃” is given by conditional rules of the type: To determine the truth-value p⊃q, use truth-table I when p and q are identical formulas, and truth-table II otherwise.

In this system, the outcome of evaluating the truth-value of a formula will always be consistent with the result obtained in Q, but may be more specific in giving T (or F) in a case in which Q gives the noncommittal result (T, F). But in any event, the “unnatural” character of Q should not be considered an obstacle, since we are not concerned here to advocate Q, but to examine a whole family of systems of which Q serves merely as an example. All of these systems are fragments of propositional calculus, and of course in any fragment (proper), some “plausible” (two-valued) tautologies will inevitably be missing.

⁴ Note that this system (and the others to be considered below) is of the following general type: between T and F are inserted a series of “intermediate” truth-values such that the ranking T, I₁, I₂, …, I_n, F may be viewed as a ranking in increasing order of “falseness” in the sense that (i) the truth-table column for negation is obtained by inverting this truth-value series, (ii) the truth-table entry for alternation is always the truth-value of the “truer” member, and (iii) the truth-table entry for conjunction is always the truth-value of the “falser” member.

⁵ It is customary to characterize a truth-value as designated when tautologies are permitted to take on this truth-value (i.e., when a formula is a tautology when it takes on only these designated truth-values for any assignment of truth-values to its variables). Extending this terminology, we may characterize as anti-designated those truth-values which contradictions are permitted to take on. It is usual to identify the set of anti-designated truth-values with the set of non-designated ones, but this requirement — violated in option (D) — is not a necessary one in general. It is, however, clearly undesirable to accord such double-status to truth-values which some formula can take on identically (i.e., for all values of its variables), for such a formula would then be both a tautology and a contradiction.

⁶ Difficulties of this nature have led Andrzej Mostowski to observe (this Journal, vol. 15 (1950), p. 223) that he does not have “any hope that it will ever be possible to find a reasonable interpretation of the three-valued logic of Łukasiewicz in terms of everyday language.”

⁷ Note that N, like M, is built on the principles enunciated in footnote 4 above, with respect to the mode of many-valued generalization of ∼, v, and ·.

⁸ A many-valued system, in the sense of this discussion, is a set of truth-tables (with specified designated and anti-designated truth-values) for the usual connectives of the propositional calculus.

⁹ In this usage, the term “characteristic” is analogous with the usual sense in which a system of truth-tables is called characteristic of a system of prepositional calculus if the tautologies according to these truth-tables are the same as the original tautologies (i.e., theorems) of this system. Any characteristic extension of a regular system (i.e. one for which modus ponens applies) must be regular. Church's, Alonzo paper Non-normal truth-tables for the prepositional calculus (Boletin de la Sociedad Matematica Mexicana, vol. 10 (1953), pp. 41–52Google Scholar) contains an excellent survey of all the customary terminology relating to the logic of truth-tables.

¹⁰ To accommodate n-ary quasi-truth-functional connectives (with n > 2), we may reduce these to binary connectives. This calls for only minor and obvious modifications in the standard reduction procedure (see W. V. Quine, Mathematical logic, §8).

¹¹ The upper left-hand 2×2 submatrix of the 4×4 matrix agrees completely with the two-valued matrix, except in those places where the latter has (T, F). Now in collapsing rows (and columns) 1 and 3, and 2 and 4, of the 4×4 matrix, by setting I₁ = T and I₂ = F, the T's and F's must always agree pairwise, except where rule R3 has led to opposites, which yield (T, F), as is appropriate.

¹² Thus in the case of a three-valued (T, F, I) quasi-truth-functional system we would need seven truth-values, to represent: T. F, I, (T, F), (T, I), (F, I), (T, F, I).

¹³ This Journal, vol. 14 (1949), pp. 95–97.

¹⁴ This was remarked by A. Church and N. Rescher, ibid., vol. 15 (1950), pp. 69–70.