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# Syllogism

In traditional logic, a syllogism is an inference in which one proposition (the conclusion) follows of necessity from two others (known as premises). The definition is traditional, but is derived loosely from Aristotle's Prior Analytics, Book I, c. 1. The Greek "sullogismos" means "deduction".

Syllogisms consist of three things: major, minor (the premises) and conclusion, which follows logically from the major and the minor. A major is a general principle. A minor is a specific statement. Logically, the conclusion follows from applying the major to the minor.

For example, this is the classic "Barbara" syllogism, given by Aristotle:

If all humans (B's) are mortal (A), (major)
and all Greeks (C's) are humans (B's), (minor)
then all Greeks (C's) are mortal (A). (conclusion)

That is,

Men die. (general principle)
Socrates is a man. (specific statement)
Socrates will die. (application of major to minor)

A metaphor, in contrast, resembles a form of syllogism called affirming the consequent, which is a logical fallacy:

Grass (B) dies (A).
Men (C's) die (A).
Men (C's) are grass (B).

A Barbara syllogism involves grammar and logical types; it has a subject (e.g. Socrates) and a predicate (mortal). Affirming the Consequent, the basis of metaphor, is grammatically symmetrical: it equates two predicates. This form of syllogism is logically invalid.

Syllogisms may also be invalid if they have four terms or the middle term is not distributed.

Epagoge are weak syllogisms that rely on inductive reasoning.

By the definition of conditional and biconditional the consequences of the principle of the syllogism may be stated in the following formulas:

$(a \Rightarrow b) \wedge (b \Rightarrow c) \Rightarrow (a \Rightarrow c)$
$(a \Leftrightarrow b) \wedge (b \Leftrightarrow c) \Rightarrow (a \Leftrightarrow c)$

The conclusion is a biconditional only when all premises are biconditionals. This statement is of great practical value. In a succession of deductions we must pay close attention to see if the transition from one proposition to the other takes place by means of a biconditional or only of a conditional. There is no equivalence between two extreme propositions unless all intermediate deductions are equivalences; in other words, if there is one single implication in the chain, the relation of the two extreme propositions is only that of implication.

Last updated: 10-17-2005 13:13:51

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