Syllogism (Greek: συλλογισμός, meaning "conclusion" or "inference"), more correctly categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of certain forms. In his Prior Analytics, Aristotle defines syllogism as "a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so." Despite this very general definition, he initially limits himself to categorical syllogisms (and later to modal syllogisms). The study of logical structures of syllogisms have been the main issue in the area of logic until the nineteenth century, when first-order logic emerged. Frege greatly shaped this new logic, which was the first radical transformation of logic since Aristotle. Aristotelian logic has since then been subsumed by modern logic.
A syllogism consists of three parts—a major premise, a minor premise, and a conclusion. Each of the premises has one term in common with the conclusion. In the case of the major premise this is the major term, or the predicate of the conclusion; in the case of the minor premise it is the minor term, the subject of the conclusion. For example:
"Being mortal" is the major term and "Socrates" the minor term. The premises also have one term in common with each other, which is known as the middle term, in this case "being a man." Here the major premise is general and the minor particular, but this needn't be the case. For example:
Here, the major term is "die," the minor term is "all men," and the middle term is "being mortal."
Syllogisms are categorized into logically distinct types. For instance, both of the syllogisms above share the same abstract form:
The premises and conclusion of a syllogism can be any of four types, which are labeled by letters as follows. For instance, in the syllogisms above, only universal affirmatives (A) are used.
(See Square of opposition for a discussion of the logical relationships between these types of propositions.)
Let us denote the subject of the conclusion as S, the predicate of the conclusion as P, the middle term as M. Then the major premise links M with P and the minor premise links M with S. However, M can be either the subject or the predicate of each premise that it appears in. This gives rise to another classification of syllogisms known as the figure. The four figures are:
|Figure 1||Figure 2||Figure 3||Figure 4|
Putting it all together, there are 256 possible types of syllogism. Each premise and the conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the number for the figure. For example, the syllogisms above are AAA-1.
Of course, the vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from the premises). The table below shows the valid forms of syllogism. The letters standing for the types of proposition (A, E, I, O) have been used since the Medieval Schools to form mnemonic names for the forms:
|Figure 1||Figure 2||Figure 3||Figure 4|
An example syllogism of each type follows.
Forms can be converted to other forms, following certain rules, and all forms can be converted into one of the first-figure forms.
Logic was dominated by syllogistic reasoning until the nineteenth century. Modifications were incorporated to deal with disjunctive ("A or B") and conditional statements ("if A then B"). Kant infamously claimed that logic was the one completed science, and that Aristotle had more or less discovered everything about it there was to know.
Still, it was cumbersome and very limited in its ability to reveal the logical structure of complex sentences. For example, it was unable to express the claim that the real line is a dense order. In the late ninteenth century, Frege's invention of first order logic revolutionized the field and the Aristotelian system has since been left to introductory materials and historical studies of logic.
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