Logic

Aristotle (384–322 B.C.E.) was the first person to formulate an explicit theory of correct reasoning, as he himself claimed in Sophistical Refutations. He owed a good deal to the exploration into forms of argument carried out in the course of argument contests, such as those illustrated by Plato in some of his Socratic dialogues. Book 8 of his own Topics reads like a handbook for contestants, and the Topics as a whole is designed to teach its readers how to construct "dialectical" arguments: arguments that, in keeping with the idea of a real contest, use generally accepted premises that will be granted by the interlocutor. In an argument, says Aristotle, "when certain things have been laid down, something other than what has been laid down necessarily results from them." This definition captures the idea of logical consequence, and in his Prior Analytics Aristotle develops his "syllogistic," a formal theory of logical consequence, which he applies to "demonstrations," arguments in which the premises must not be merely accepted, but true.

Syllogisms (in the narrow sense considered in the Prior Analytics) consist of three assertoric sentences, two of them premises, from which the third, the conclusion, follows. In an assertoric sentence, something is "predicated" of a subject, and a predicate can stand in one of just five relations to a subject: it may be its definition, its genus ("Man is an animal"), its differentia (the element of the definition that differentiates things of one species from another: "Man is rational"), an accident (a characteristic the particular thing happens to have: "Socrates is curly-haired"), or its characteristic property (a feature that all and only things of the subject's species have, but is not part of its definition: "Man is able to laugh").

The two premises of a syllogism share a common ("middle") term, and they have "quantity" (universal/particular) and "quality" (affirmative/negative). They may be, then, universal affirmative (A-sentences: "Y belongs to every X"), universal negative (E-sentences: "Y belongs to no X"), particular affirmative (I-sentences: "Y belongs to some X"), or particular negative (O-sentences: "Y does not belong to some X"). Depending on the position of the middle term—predicate of both premises ("third figure"), subject of both premises ("second figure"), or subject of the first, predicate of the second ("first figure")—from some combinations of two A, E, I, and O sentences as premises, there follows an A, E, I, or O sentence as a conclusion—and this conclusion follows purely in virtue of the form of the argument. (Although ancient logicians rarely used false premises as their examples, they too made their conclusions follow logically.) So, for example, in the first figure, the patterns AAA, EAE, AII, and EIO are valid arguments. First-figure syllogisms were held by Aristotle to be self-evident: for example, Mortal belongs to every man (every man is mortal); man belongs to every philosopher; thus mortal belongs to every philosopher. Aristotle also shows how second-and third-figure syllogisms can be reduced to first-figure ones, using a set of conversion rules.

Aristotle's other logical works both fill in the discussions in the Topics and the Prior Analytics and introduce new philosophical dimensions. On Interpretation discusses assertoric statements and their relations (such as contradiction and contrariety). It also proposes a basic semantics, in which sentences are signs for thoughts, and thoughts for things, and it ventures into difficult questions of possibility and necessity. If it is true that there will be a sea battle tomorrow, then how can we avoid the unpalatable conclusion that it is a matter of necessity that the battle will take place tomorrow? The Posterior Analytics uses the theory of demonstration as a basis for a theory of scientific knowledge. The Sophistical Refutations explore fallacious but apparently valid arguments. The Categories has, in part, the aspect of a preface to the Topics, but it is in part a work of metaphysics—a forerunner of Aristotle's treatise of that name.