In the Aristotelian world, the theory of the syllogism covers all reasoning, in science, dialectic or rhetoric. In science, this is in logic, the syllogism is defined as

an argument in which, certain things being laid down, something other than these necessarily comes about through them (Aristotle, Top., I, 1).

The classical syllogism is a discourse composed of three propositions, the “things being laid down” are the premises of the syllogism, and “something other than these necessarily comes about through them” is the conclusion.

Syllogistic inference involves two premises, while immediate inference is based upon one premise, S. Proposition.

The logic of the analyzed propositions concerns the conditions of validity of the syllogism. A valid syllogism is a syllogism such that, if its premises are true, necessarily its conclusion is true (the conclusion of a syllogism does not have to be a necessary truth, it’s a truth that follows necessarily from the premises). The premises of the syllogism cannot be true and its conclusion false.

In Aristotle’s words, the syllogism, is a “demonstration” “when the premises from which the reasoning starts are true and primary, or are such that our knowledge of them has originally come through premises which are primary and true” (ibid.).

1. Terms of the syllogism

The syllogism articulates three terms, the major term T, the minor term t and the middle term M:

— The major term T is the predicate of the conclusion.
The premise containing the great term T is called the major premise.

— The minor term t is the subject of the conclusion.
The premise containing the small term t is called the minor premise.

— The middle term M connects the major and the minor terms, and consequently disappears in the conclusion, which is of the form < t is T >.

2. Figures of the syllogism

The form of the syllogism varies according to the subject or predicate position of the middle term in the major and minor premise. There are four (maybe only three) possibilities, which constitute the four figures of the syllogism, S. Figures. For example, a syllogism where the middle term is subject in the major premise and predicate in the minor premise is a syllogism of the first figure:

Major Premise          M – T              man-reasonable
Minor Premise         t – M                horse-man
Conclusion               t – T                 horse-reasonable

3. Modes of the syllogism

The mode of the syllogism depends on the quantity of the three propositions which constitute the syllogism. A proposition may be universal or particular, affirmative or negative, giving a total of four possibilities.

Each of these four possibilities for the major premise may combine with a minor premise, also admitting four possibilities, to give a conclusion that also admits four possibilities, totaling 4 x 4 x 4 = 64 forms.

Moreover, each of these forms admits the four figures, making all together 256 modes. Some of these modes are valid, others are not.

For example, the first figure of the syllogism corresponds to the case where, a universal conclusion derives from two universal premises. This deduction corresponds to the valid mode:

Major Premise             All human are reasonable
Minor Premise             All Greeks are human
Conclusion                   All Greeks are reasonable

This mode is known as Barbara, where the vowel a marks that the major, minor and conclusion are universal, S. Proposition.

4. Example: the conclusive modes of the first figure

Syllogistic deductions are clearly exposed in the language of set theory.

— Two (non-empty) sets are disjointed if their intersection is empty; they have no elements in common.
— The two sets intersect if they have some elements in common.
— One set is included in the other if all the elements of the first set also belong to the second set.

In what follows, M reads as “set M”, similarly for P and S. The first figure of the syllogism admits four conclusive modes.

A – A – A syllogism

A          every M is P
A          all S are M
A          hence all S are P.

A – I – I syllogism

A          every M is P
I           some S are M
I           hence some S are P

E – A – E syllogism

E          no M is P
A          all S are M
E          therefore no S is P

E – I – O syllogism

E          no M is P
I           some S is M
O          therefore some S is not P

These forms of basic reasoning used in categorization.

5. Evaluation of syllogisms

S. Paralogisms

6. Syllogisms with premise(s) having a concrete subject

The preceding definitions correspond to the traditional (Aristotelian) categorical syllogism, bearing on quantified variables. The word syllogism is also used to refer to a form of reasoning where one premise has a concrete subject. A concrete subject is a subject referring to a unique single individual, by means of various expressions such as this, this being, Peter, the N who.

Syllogisms instantiating a universal proposition are examples of such syllogisms. These assign to an individual the properties of the class to which he or she belongs:

the x are B
this is an x
this is B

The following type of reasoning is based on two concrete propositions can also be called, rather metaphorically, “syllogistic”. It refutes universal propositions such as “all swans are white”:

This is a swan      the proposition refers to a concrete individual
This is black         the proposition attach a property to the same individual
Applied to the same subject, “being black” and “being white” are opposite predicates
Therefore the claim “all the swans are white” is false

7. Syllogistic forms with more than two premises

By extension, one also calls a syllogism an argument based on several arguments either linked, convergent, or again having the form of an epicheirema.

A chain of propositions where the syntactic form and mode of linking more or less mimic those of a syllogism may also be called a syllogism, S. Expression.

The famous syllogism “everything rare is expensive, a cheap horse is a rare thing, so a cheap horse is expensive” is developed on the basis of two contradictory premises, so it is normal that its conclusion be absurd.