Bernard Bolzano on Non-Syllogistic Reasoning

Abstract

Abstract Bernard Bolzano, in his Theory of Science, discusses the ground-consequence relation and within that context he analyzes forms of valid reasoning which he argues cannot be reduced to syllogisms. In section 262 of the Theory of Science, titled “The syllogism in the received logic,” Bolzano provides a rigorous argument aimed at showing that proofs (including mathematical proofs) cannot all be captured by chains of syllogistic inference rules. Bolzano expresses astonishment that Aristotle, having started with a broad definition of syllogism, restricted its range quite narrowly. A possible justification for the restriction consists in two claims, made by Aristotle at I.25 and I.23 of Prior Analytics, to the effect that every inference can be reduced to a sequence of syllogistic arguments, each of which consisting of two premises and one conclusion. Bolzano objects that it is questionable whether Aristotle ever gave such a proof. The chapter reconstructs Aristotle’s argument and Bolzano’s objections to it.