[CG] Re: Functional Logic • Inquiry and Analogy

Functional Logic • Inquiry and Analogy • 4 (Part 1) • https://inquiryintoinquiry.com/2023/06/24/functional-logic-inquiry-and-anal… Inquiry and Analogy • Aristotle’s “Apagogy” • Abductive Reasoning • https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Aristo… Peirce's notion of abductive reasoning is derived from Aristotle's treatment of it in the “Prior Analytics”. Aristotle's discussion begins with an example which may seem incidental but the question and its analysis are echoes of the investigation pursued in one of Plato's Dialogue, the “Meno”. It concerns nothing less than the possibility of knowledge and the relationship between knowledge and virtue, or between their objects, the true and the good. It is not just because it forms a recurring question in philosophy, but because it preserves a close correspondence between its form and its content, that we shall find this example increasingly relevant to our study. <QUOTE Aristotle:> We have Reduction (απαγωγη, abduction): (1) when it is obvious that the first term applies to the middle, but that the middle applies to the last term is not obvious, yet nevertheless is more probable or not less probable than the conclusion; or (2) if there are not many intermediate terms between the last and the middle; for in all such cases the effect is to bring us nearer to knowledge. (1) E.g., let A stand for “that which can be taught”, B for “knowledge”, and C for “morality”. Then that knowledge can be taught is evident; but whether virtue is knowledge is not clear. Then if BC is not less probable or is more probable than AC, we have reduction; for we are nearer to knowledge for having introduced an additional term, whereas before we had no knowledge that AC is true. (2) Or again we have reduction if there are not many intermediate terms between B and C; for in this case too we are brought nearer to knowledge. E.g., suppose that D is “to square”, E “rectilinear figure”, and F “circle”. Assuming that between E and F there is only one intermediate term — that the circle becomes equal to a rectilinear figure by means of lunules — we should approximate to knowledge. Aristotle, “Prior Analytics” 2.25, Hugh Tredennick (trans.) </QUOTE> A few notes on the reading may be helpful. The Greek text seems to imply a geometric diagram, in which directed line segments AB, BC, AC indicate logical relations between pairs of terms taken from A, B, C. We have two options for reading the line labels, either as implications or as subsumptions, as in the following two paradigms for interpretation. Table of Implications • https://inquiryintoinquiry.files.wordpress.com/2022/04/table-of-implication… Table of Subsumptions • https://inquiryintoinquiry.files.wordpress.com/2022/04/table-of-subsumption… In the latter case, P ⩾ Q is read as “P subsumes Q”, that is, “P applies to all Q”, or “P is predicated of all Q”. Regards, Jon