ABSTRACT

This chapter provides the Aristotelian concept of distribution by expanding Robert Carnes' application of it. It explains distribution to k-quantity "fractional" syllogistic systems. The chapter considers unrestricted "proportional" syllogisms and sorites. It describes proportional systems with an infinite number of quantities. The chapter shows how the new understanding of distribution survives Geach's criticisms. Every valid syllogism is such that the algebraic formulae for the premises together with the formula for the denial of the conclusion are inconsistent. When a syllogism is invalid a counter-example can be formed. Failed attempts to demonstrate validity supply reduced formulae for hypothesizing counter examples. The quantifier-theoretic approach need not be reduced to some species of relational-quantifiers or restricted quantifiers in the predicate calculus. It is an Aristotelean theory of quantifiers based on propositional quantity. I-quantity system appears to be equivalent to Johnson's syllogistic with "fractional quantifiers".