Propositional calculus, many-valued
Propositional calculus, many-valued
The truth-table method for the classical (two-valued) propositional calculus is explained in the article logic, formal, 1. It depends on assigning truth-tables to the fundamental connectives, with the result that every formula — of the pure propositional calculus, to which we here restrict ourselves for the sake of simplicity — has one of the two truth-values for each possible assignment of truth-values to the variables appearing. A formula is called a tautology if it has the truth-value truth for every possible assignment of truth-values to the variables; and the calculus is so constructed that a formula is a theorem if and only if it is a tautology.
This may be generalized by arbitrarily taking n different truth-values, t1, t2, . . . , tm, f1, f2, . . . , fn-m, of which the first m are called designated values — and then setting up truth tables (in terms of these n truth-values) for a set of connectives, which usually includes connectives notationally the game as the fundamental connectives of the classical calculus, and may also include others. A formula constructed out of these connectives and variables is then called a tautology if it has a designated value for each possible assignment of truth-values to the variables, and the theorems of the n-valued propositional calculus are to coincide with the tautologies.
In 1920, Lukasiewicz introduced a three-valued propositional calculus, with one designated value (interpreted as true) and two non-designated values (interpreted as problematical and false respectively). Later lie generalized this to n-valued propositional calculi with one designated value (first published in 1929). Post introduced n-valued propositional calculi with an arbitrary number of designated values in 1921. Also due to Post (1921) is the notion of symbolic completeness — an n-valued propositional calculus is symbolically complete if every possible truth-function is expressible by means of the fundamental connectives.
The case of infinitely many truth-values was first considered by Lukasiewicz. — A.C.
J. Lukasiewicz, O logice trojwartosciowej, Ruch Fifozoficzny, vol. 5 (1920), pp. 169-171.
E. L. Post, Introduction to a general theory of elementary propositions, American Journal of Mathematics, vol. 43 (1921), pp. 163-185.
Lukasiewicz and Tarski, Untersuchungen ber den Aussagenkalkl, Comptes Rendus des Seances de la Societe des Sciences et des Lettres de Varsovie, Classe III, vol. 23 (1930), pp. 30-50.
J. Lukasiewicz, Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalkls, ibid , pp 51-77.
Lewis and Langford, Symbolic Logic, New York and London, 1932.