On a proportionality analysis of syllogistic private reasoning

Synthese 146 (1-2):129 - 138 (2005)
.  Syllogisms like Barbara, “If all S is M and all M is P, then all S is P”, are here analyzed not in terms of the truth of their categorical constituents, “all S is M”, etc., but rather in terms of the corresponding proportions, e.g., of Ss that are Ms. This allows us to consider the inferences’ approximate validity, and whether the fact that most Ss are Ms and most Ms are Ps guarantees that most Ss are Ps. It turns out that no standard syllogism is universally valid in this sense, but special ‘default rules’ govern approximate reasoning of this kind. Special attention is paid to inferences involving existential propositions of the “Some S is M” form, where it is does not make sense to say “Almost some S is M”, but where it is important that in everyday speech, “Some” does not mean “At least one”, but rather “A not insignificant number”
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