Syllogistic

syllogistic

Formal analysis of the syllogism. Developed in its original form by Aristotle in his Prior Analytics c. 350 BC, syllogistic represents the earliest branch of formal logic. Syllogistic comprises two domains of investigation. Categorical syllogistic confines itself to categorical propositions and their variation with respect to modalities. Noncategorical syllogistic is a form of logical inference using whole propositions as its units, an approach traceable to the Stoics but only fully developed by John Neville Keynes (1852–1949).

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Syllogistic 

a theory of logical deduction that studies inferences consisting of categorical statements (judgments): the universal affirmative (every S is P), universal negative (no S is P), particular affirmative (some S are P), and particular negative (some S are not P). Syllogistic examines the deduction of a conclusion from one premise (direct inferences) and complex and compound syllogisms, or polysyllogisms, which have at least three premises. However, syllogistic emphasizes primarily the theory of the categorical syllogism, which has only two premises and one conclusion of an abovementioned type.

Aristotle, the founder of logic as a science, devised a system of classifying and validating the forms (moods) of syllogisms. Subsequently, syllogistic was refined by various schools of classical and medieval logicians, including the Peripatetics and the Stoics. Although F. Bacon, R. Descartes, J. S. Mill, and other scholars noted that syllogistic was of limited applicability, it was long an integral, traditional element of classical education in the humanities. Thus, it is often called traditional logic. With the establishment of the calculi of mathematical logic, the role of syllogistic became very modest. It was proved that by using the one-place predicate calculus, a fragment of the predicate calculus, it is possible to obtain almost the entire content of syllogistic—all deductions not dependent on the typical syllogistic assumption of an empty object field. A number of axiomatic statements of syllogistic have also been obtained in the terms of modern mathematical logic (J.Łukasiewicz, 1939).

REFERENCES

Aristotle. Analiliki, pervaia i vtoraia. Leningrad, 1952. (Translated from Greek.)
Bacon, F. Novyi organon. Leningrad, 1935. (Translated from English.)
Descartes, R. Izbrannyeproizvedeniia. Moscow, 1950. (Translated from French.)
Hilbert, D., and W. Ackermann. Osnovy teoreticheskoi logiki. Moscow, 1947. Chapter II, sect. 3. (Translated from German.)
Łukasiewicz, J. Aristotelevskaia sillogistika s tochki zreniia sovremennoi formal’noi logiki. Moscow, 1959. (Translated from English.)
Bourbaki, N. Ocherki po istorii matematiki. Moscow, 1963. (Translated from French.)
Culbertson. J. T. Matematika i logika tsifrovykh ustroistv. Moscow, 1965. Chapter 5. (Translated from English.)
Subbotin, A. L. Teoriia sillogistiki v sovremennoi formal’noi logike. Moscow, 1965.
Subbotin, A. L. Traditsionnaia i sovremennaia formal’naia logika. Moscow, 1969.