Syllogism

A syllogism (Greek: συλλογισμός — "conclusion", "inference"), more correctly a categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises). 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." (24b18–20) Despite this very general definition, however, he limits himself almost entirely to categorical syllogisms, as discussed in this article.

Basic structure

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 predicate of the conclusion; in the case of the minor premise it is the minor term, the subject of the conclusion. For example:

Major premise: All men are mortal.

Minor premise: Socrates is a man.

Conclusion: Socrates is mortal.

"Being mortal" is the major term and "Socrates" the minor term; the connection between them is made by 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:

Major premise: All mortal things die.

Minor premise: All men are mortal.

Conclusion: All men die.

Here, the major term is "die", the minor term is "all men", and the middle term is "being mortal".

Types of syllogism

There are four kinds of propositions involved in syllogisms:

A. universal affirmatives (e.g., "all humans are mortal")

I. particular affirmatives (e.g., "some humans are healthy")

E. universal negatives (e.g., "no humans are perfect")

O. particular negatives (e.g., "some humans are not clever")

From this comes one classification of all syllogisms into different types depending upon the nature of the premises and conclusion (see Square of opposition).

Another classification is arrived at using the position of the middle term in the premises — the figure; the four figures are:

	Figure 1	Figure 2	Figure 3	Figure 4
Major premise:	M–P	P–M	M–P	P–M
Minor premise:	S–M	S–M	M–S	M–S
Conclusion:	S–P	S–P	S–P	S–P

"M" is the middle term, "S" is the subject, and "P" the predicate. Each of these figures has a variety of forms depending on the types of proposition involved. The letters standing for the types of proposition (A, E, I, O) have been used since the mediæval Schools to form mnemonic names for the forms:

Figure 1	Figure 2	Figure 3	Figure 4
Barbara	Cesare	Darapti	Bramantip
Celarent	Camestres	Disamis	Camenes
Darii	Festino	Datisi	Dimaris
Ferio	Baroco	Felapton	Fesapo
		Bocardo	Fresison
		Ferison

Forms can be converted to other forms, following certain rules, and all forms can be converted into one of the first-figure forms.

The place of the syllogism in logic

Logic was dominated by syllogistic reasoning until the 19th century, though it was very limited in its application, being able to deal with a small number of types of valid argument (partly because of its being traditionally restricted to categorical syllogisms), yet cumbersome and complex. Although attempts have occasionally been made to resuscitate and expand it, syllogistic reasoning has been replaced by the simpler and more powerful predicate logic and quantification theory.

Validity

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

For example:

Grass (B) dies (A).

Men (C) die (A).

Men (C) are grass (B).

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.

Errors in syllogism

People have difficulty with categorical syllogisms. For instance, given the following parameters: some A are B, some B are C, people tend to come to a definitive conclusion that therefore some A are C. However, this is logically incorrect. As a more concrete example, assume that some people from New York attend the University of Pennsylvania and some people from the University of Pennsylvania will work for IBM. Many people might jump to the conclusion that some people from New York must work for IBM. However, this is not a conclusion we can make for certain. The people from New York (attending the University of Pennsylvania) could be working for Miscrosoft or Google, etc. and the people working at IBM from the University of Pennsylvania could be people from Boston, Chicago etc. In general, a syllogism with two "some"s yields no conclusion.

This sort of mistake is rarely made by most adults, although children will make it. In general, errors of this sort result from failure to consider alternative mental models of the situation, aside from the first model to come to mind. A model, in this case, is a way of putting the premises together. This theory, proposed by Philip Johnson-Laird, has received wide support, but it was preceded by other theories.

Sources and reading

Aristotle, Prior Analytics. transl. Robin Smith (Hackett, 1989) ISBN 0-87220-064-7
Simon Blackburn, Oxford Dictionary of Philosophy (Oxford University Press, 1996) ISBN 0-19-283134-8
Alexander Broadie, Introduction to Medieval Logic (Oxford University Press, 1993) ISBN 0-19-824026-0

External links

Abbreviatio Montana article by Prof. R. J. Kilcullen of Macquarie University on the medieval classification of syllogisms.
The Figures of the Syllogism is a brief table listing the forms of the syllogism.
Stanford Encyclopedia of Philosophy entry on Medieval Theories of Syllogisms

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