Definition:Reduction to First Figure

Definition

Reduction to the first figure is a method for determining the validity of a categorical syllogism, as follows:

$(1): \quad$ Certain "self-evident" patterns are identified in the first figure of the categorical syllogism.

$(2): \quad$ Using various rules of categorical statements, the valid patterns of other figures of the categorical syllogism are deduced.

There are two forms this procedure takes:

Direct Reduction

Direct reduction to the first figure is a form of the reduction to the first figure method for determining the validity of a categorical syllogism.

This form of the method works forward from the first figure using various rules of categorical statements in order to derive valid patterns of other figures.

Indirect Reduction

Indirect reduction to the first figure is a form of the reduction to the first figure method for determining the validity of a categorical syllogism.

This form of the method works backward by supposing that a pattern is not valid, and then deriving a contradiction against a known valid pattern in the first figure.

Hence by Reductio ad Absurdum the pattern is deduced to be valid.

Historical Note

This technique for analysis of categorical syllogisms was devised by Aristotle.

Reduction is of great historical interest, as being perhaps the earliest known attempt to derive conclusions systematically from given assumptions. Though Aristotle's presentation is crude and informal by modern standards of rigour, it is possible to follow the outlines of his programme and derive the valid syllogisms as conclusions, by purely propositional calculus reasoning, from a very small set of syllogistic assumptions.

— 1965: E.J. Lemmon: Beginning Logic: $\S 4.4$: The Syllogism

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism