The History of Logic

Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and negation in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical Refutations—goes far beyond any of these. Aristotle holds that a proposition is a complex involving two terms, a subject and a predicate, each of which is represented grammatically with a noun. The logical form of a proposition is determined by its quantity (universal or particular) and by its quality (affirmative or negative). Aristotle investigates the relation between two propositions containing the same terms in his theories of opposition and conversion. The former describes relations of contradictoriness and contrariety, the latter equipollences and entailments. The analysis of logical form, opposition, and conversion are combined in syllogistic, Aristotle’s greatest invention in logic. A syllogism consists of three propositions. The first two, the premisses, share exactly one term, and they logically entail the third proposition, the conclusion, which contains the two non-shared terms of the premisses. The term common to the two premisses may occur as subject in one and predicate in the other (called the ‘first figure’), predicate in both (‘second figure’), or subject in both (‘third figure’). A given configuration of premisses and conclusions is called a ‘mood’. In the scholastic period, mnemonic names for the validmoods canvassed in the Prior Analytics were devised. Two first-figure valid moods were considered perfect and not in need of any further validation: BARBARA (consisting entirely in universal affirmatives) and CELARENT (consisting in a universal negative and a universal affirmative, concluding in a universal negative). For the validation of the rest, Aristotle used three techniques: reduction, in which a given mood is transformed through conversions into BARBARA or CELARENT; reductio ad absurdum; and âkqesij, which proceeds by selection of an arbitrary individual. He regularly describes moods by using variables in place of terms. To reject a proposed inference he typically gives a list of terms that, when substituted as values of the term-variables, produce truepremisseswith false conclusions. This is similar to themodern