Numerical Term Logic
Wallace A. Murphree · Notre Dame Journal of Formal Logic · 1998
This paper is an attempt to show that my work to establish numerically flexible quantifiers for the syllogism can be aptly combined with the term logic advanced by Sommers, Englebretsen, and others. 1 Introduction Sommers, followed by Englebretsen and others, has developed a comprehensive notational and deductive system in categorical logic which has the syllogism as its base (see Sommers [11]) but which extends far beyond the traditional logic.(See Kelley [4], chapter 14, for a textbook presentation.)Indeed, its power rivals that of the first-order predicate calculus and its defenders allege it to be superior to the calculus on various important counts.(For example, see Sommers [10] and Englebretsen [2].)Furthermore, I have developed a numerically expanded scheme of quantification in categorical logic of which the traditional syllogism turns out to be but one of infinitely many numerical instances (see Murphree [5] and [6]).In this paper I propose to show that the two approaches can be aptly combined into a program more comprehensive than either.Specifically, I propose that with only minor adaptations, the symbolic and deductive mechanism developed by Sommers and Englebretsen (called "term logic" or TL) works for the propositions and inferences of my "numerical logic" (NL); and I propose that these latter propositions and inferences, in turn, reveal a vast field of applicability hitherto unavailable to the term logic.The preliminary tasks are those of summarizing the basic features of each system.The numerical logic (NL) is introduced first. Numerical logic (NL)The numerically expanded logic is concerned with quantities between the extremes of "all" and "some".Specifically, it accommodates numerical differences in particular quantifiers and different numerical deviations from universal quantifiers. Propositions of NLThe point of departure allowing the move from traditional quantification to the numerically expanded system is the observation that "all" as in