Sources of Delusion in Analytica Posteriora 1.5

Abstract

Aristotle's philosophically most explicit and sophisticated account of the concept of a (primary-)universal proof is found, not in Analytica Posteriora 1.4, where he introduces the notion, but in 1.5. In 1.4 Aristotle merely says that a universal proof must be of something arbitrary as well as of something primary and seems to explain primacy in extensional terms, as concerning the largest possible domain. In 1.5 Aristotle improves upon this account after considering three ways in which we may delude ourselves into thinking we have a primary-universal proof. These three sources of delusion are shown to concern situations in which our arguments do establish the desired conclusion for the largest possible domain, but still fail to be real primary-universal proofs. Presupposing the concept of what may be called an immediate proof, in which something is proved of an arbitrary individual, Aristotle in response now demands that a proof be immediate of the primary thing itself and goes on to sketch a framework in which an intensional criterion for primacy can be formulated.For the most part this article is a comprehensive and detailed commentary on Aristotle's very concise exposition in 1.5. One important result is that the famous passage 74a17-25 referring to two ways of proving the alternation of proportions cannot be used as evidence for the development of pre-Euclidean mathematics.

Some remarks against non-epistemic accounts of immediate premises in Aristotle’s Posterior Analytics

“…However, if we "remove" the attribute of being a triangle (or more general ones, like being a 20 See, for instance, Lennox (1987, 92); Kosman (1973, 375-376); McKirahan (1992, 95-102); Charles (2000, 206-209); Hasper (2006); Angioni (2012, 50-52;2018, 179, n.44); Bronstein (2016, 43-44). 21 This is the common reading, anyway.…”

“…Here it becomes clear that condition (iii) for primary-universality cannot be understood in purely extensional terms. 23 In this chapter, Aristotle analyses situations in which we mistakenly believe to be demonstrating a primary-universal predication, when in fact we are not (APo I 5, 74a4-6; see Hasper 2006). For instance, in a situation in which all triangles happen to be isosceles, we could wrongly assume that 2R is predicated of them for the very fact that they are isosceles, which would make us believe that isosceles is the kind to which 2R belongs in a primary-universal way.…”

Aristóteles e a necessidade do conhecimento científico

“…Praticamente se dá o mesmo com Bronstein 2016 (que, no entanto, diz algo sobre 76a4-6 na nota 21, p. 56). 22 Ver Angioni, 2016, p. 95-102;Ferejohn, 2013, p. 81-90;Hasper, 2006;Zuppolini, 2018a, p. 129-132. 23 Sobre esse ponto, ver Angioni, 2012a, p. 42-52.…”

Syllogisms and Existence in Aristotle’s Posterior Analytics

“…322), though he presents the sequence as decreasing. See also Hasper (2006). Admittedly, the appearance of bronze in the sequence is perplexing, but Aristotle is probably conceiving of it as a specific (accidental) determination of isosceles.…”