No abstract

Lindenbaum's construction of a matrix for a propositional calculus, in which the wffs themselves are taken as elements and the theorems as the designated elements, immediately establishes two general results: that every prepositional calculus is many-valued, and that every many-valued propositional calculus is also ℵ0-valued. These results are however concerned exclusively with theoremhood, the inferential structure of the calculus being relevant only incidentally, in that it may serve to determine the set of theorems. We therefore ask what happens when deducibility is taken into consideration on a par with theoremhood. The answer is that in general the Lindenbaum construction is no longer adequate and both results fail.

A semicycle is said to turn at a point a if the arcs incident to a are both to it or both from it. We prove that if a nonempty set of points of a finite directed graph contains a turning point of each semicycle, then one of its members is a turning point of every semicycle to which it belongs; and we indicate the application of this result to mathematical logic through the modeling of arguments by graphs.We follow the terminology of Harary [l], and we assume in particular that a graph is finite and without loops or multiple arcs. We say also that a point a is a turning-point of a semiwalk. . . ,E, a, E', . . . if E and E' are both to a or both from a. Thus a walk is a semiwalk without turningpoints; and similarly with paths and cycles. Theorem 1. If a nonempty set S of points of a directed graph contains a turning point of each semicycle, then some member of S is a turning point of every semicycle to which it belongs.