Intermediate quantifiers versus percentages.
Robert D. Carnes, Philip L. Peterson · Notre Dame Journal of Formal Logic · 1991
In his 1986 paper {Notre Dame Journal of Formal Logic) Thompson offers rules for determining validity and invalidity of so-called "statistical syllogisms" (syllogisms with percentages replacing the traditional quantities of universal and particular) which are both wnsound and mcomplete.As a result, his claim that the genuine 5-quantity syllogistic (the traditional syllogistic with the three "intermediate" quantities added, expressible by "few", "many", and "most") is included in his system is trivial, // true at all.It turns out not to be even true, as revealed by detailed examination of distribution, Thompson's rules, and his claims for equivalences.In his "Syllogisms with Statistical Quantifiers" [12], Thompson has used and abused our unpublished manuscript "The Compleat Syllogistic: Including Rules and Venn Diagrams for Five Quantities".He used it as a basis for developing his own system of so-called "statistical syllogisms" and he abused it through insufficient acknowledgement, by misrepresenting our extension of the classical syllogistic, and by claiming that our extension was included in his inadequate system.Our aim herein is not to present and discuss our results, for that is what is done in our manuscript. 1Rather, 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.In Section 3, we give the underlying reasons for Thompson's basic misapprehensions of the relation between his "statistical" quantifiers-viz., percentages -and genuinely intermediate quantifiers. 1Thompson offers some rules for determining the validity of syllogistic-like argument forms -forms which are just like Aristotelian syllogisms with regard