Validity Rules for Proportionally Quantified Syllogisms

Abstract

Since the time, about a century ago, when DeMorgan (8), Boole (1) and Jevons (12), inaugurated the study of the logic of numerically definite reasoning, no one has been concerned to establish the validity rules for a very general type of numerically definite inference which is a strong analogue of the classical syllogism. The reader will readily agree that the traditional rules of syllogistic inference cannot even begin to decide whether the following proportionally quantified syllogism is a valid argument: at most 4/7 p are at most 2/3 not-m and at least 3/5 s are precisely 7/8 m, hence, at most 1/3 s are p. Likely reasons, of broad methodological interest, why this readily definable type of formal inference has been neglected will presently be proposed. Among the masters of twentieth century logic, Hilbert and Bernays (11) have, to be sure, meticulously shown how such numerically definite propositions as “at least 7 individuals have the property p” can be expressed in the predicate calculus with the use of the identity relation. They do not, since their concern with such propositions is indirect, proceed to establish a systematic calculus of classes whose “size” or extent is numerically definite. As we shall always notice, traditional logic and even ordinary Boolean algebra prefer to generalize about the laws governing the interrelationship of classes in inference systems without making any hypotheses concerning class “size” or extent except by the classical quantifier “some”, which quantifier admits the measure or extent of a class only in terms of the proposition that a given class contains or does not contain at least one member. Such a restricted admission of the relevance for inference of class size is among the fundamental reasons for the neglect of proportional quantifiers, i.e., quantifiers where numerical value is any rational number r such that 0/100 ≤ r ≤ 100/100. It can, of course, also be pointed out that the traditional syllogism and ordinary Boolean algebra do not i.e. measure, numerically define, the degree of inclusion of one class in another. Inclusion is taken either as total or partial, with no discrimination of the degree of partial inclusion.

Mental probability logic

“…Generalized quantifiers have been developed in mathematics and logics (Mostowski 1957, Lindström 1966, Väänänen 2004, artificial intelligence and computer science (Schwartz 1997, Liu & Kerre 1998, Novák 2001, and linguistics (Barwise & Cooper 1981, van Bentham & ter Meulen 1985, Gärdenfors 1987, van der Does & van Eijck 1991, Keenan & Westerståhl 1997). Peterson's work is an improvement on Finch (1957) and belongs to the philosophy of language tradition.…”

Syllogisms with fractional quantifiers

“…See De Morgan [3]. And, unlike Finch [4], we shall not assign true to any sentences of the form 'more than 2/3 of the A are B' if A denotes an infinite set. (But in the system below such a sentence may be true if B denotes an infinite set.…”

Intermediate quantifiers versus percentages.

“…Our aim herein is not to present and discuss our results, for that is what is done in our manuscript. 1 Rather, our aim is to expose the inadequacy of Thompson's allegedly syllogistic system for evaluating certain arguments-those arguments that resemble genuine syllogisms but which contain percentages instead of ordinary quantifiers. In Section 1, we demonstrate the unsoundness and incompleteness of Thompson's system and give rejoinders to likely replies by Thompson. In Section 2, we present important simplifications of Thompson's notations and rules, discuss distribution, and reveal further difficulties.…”

“…to number of premises (two), number of terms (three), and quality (affirmative, negative), but which are different with regard to quantity. In place of the two quantities of traditional categorical propositions, Thompson permits indefinitely many quantities by incorporating quantifier expressions (quantifier plus subject-term) of the following sorts: (1) n<Ίo of S Many more than n°7o of 5 (for 0 < n < 100; Almost n°7o of S More than n°7o of S fractions expressed Less than n°/o of S decimally).…”