CORCORAN ON ARISTOTLE’S AXIOMATIC BEGINNINGS. AC ABSTRACT Aristotle’s axiomatic beginnings. Bulletin of Symbolic Logic. 21 (2015) 441–442.► JOHN CORCORAN, Aristotle’s axiomatic beginnings. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USAE-mail:
[email protected] Aristotle’s Analytics uses the noun archē (“beginning”, plural archai)—source of archaic, archeology, archetype, etc.—for foundational items [3, pp. 511ff, 538f]. Archē translates “element”, “origin”, “principle”, etc. It corresponds somewhat to our noun primitive in logic-related uses. However, primitive carries pejorative connotations—naïve, crude, undisciplined, ignorant—and excludes honorific connotations—clear, firm, reliable, basic, etc. For archē, it is almost the reverse. Current arithmetic-foundations literature mentions primitive entities (numbers, operations, relations, etc.), primitive concepts (“number”, “zero”, “successor”, etc.), primitive propositions (axioms, postulates), and primitive rules of inference. Aristotle’s archai, likewise inhomogeneous, included entities, concepts, and propositions: rules were conspicuously, though not explicitly, excluded. Aristotle recognized primitive rules without calling them archai. Applications of certain two-premise primitive rules were called teleioi sullogismoi (immediately-complete deductions) without using a Greek expression translating primitive rule [2, pp. 7ff]. We discuss Ross’ view of Aristotle’s propositional archai, primitive propositions, or axiomatic beginnings—propositions basic to demonstrative science as treated in Analytics [3]. Ross has three non-overlapping classes of axiomatic beginnings: “special beginnings” (idiai archai), “general beginnings” (koinai archai), and “enabling beginnings” (axiomata). This classification applies separately to each science; examples include geometry, arithmetic, and astronomy. Consider geometry. Geometrical special beginnings correspond to Euclid’s “postulates” and “definitions”. Aristotle’s only example is “the definition of line”. Geometrical general beginnings correspond to Euclid’s common notions: “metric” axioms governing geometrical magnitudes. Enabling beginnings correspond to principles of “formal ontology” [1, pp. 18ff]. Aristotle explicitly says his enabling beginnings don’t occur as premises in geometrical demonstrations: they are propositions that enable knowledge. Without them demonstration is impossible: examples resemble “laws of thought” [3, pp. 511, 531]. [1] JOHN CORCORAN, Founding of logic, Ancient Philosophy, vol. 14 (1994), pp. 9–24. [2] JOHN CORCORAN, Aristotle's Demonstrative Logic, History and Philosophy of Logic, vol. 30 (2009), pp. 1–20. [3] DAVID ROSS (editor), Aristotle’s Analytics, Oxford UP, Oxford, 1965.
