SYLLOGISM
In the Aristotelian world, the theory of the syllogism encompasses all reasoning, whether in science, dialectic or rhetoric. In science, that 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 consisting of three propositions, where “certain things being established” are the premises of the syllogism, and “something other than these necessarily follows from them” is the conclusion.
Syllogistic inference involves two premises, while immediate inference is based upon one premise, see proposition.
The logic of the analyzed propositions concerns the conditions of validity of the syllogism. A valid syllogism is one such that, if its premises are true, its conclusion is necessarily true. The premises of a syllogism cannot be true and its conclusion false.
The conclusion of a syllogism need not be a necessary truth, it’s a truth that necessarily follows from the premises.
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 minor 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 position (subject or predicate) of the middle term in the major and minor premises. There are 4 possibilities, which form the 4 figures of the syllogism, see figures. For example, a syllogism in which 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 Syllogism
– The mode of the syllogism depends on the quantity of the three propositions that make up the syllogism. A proposition can be universal or particular, affirmative or negative, giving a total of four possibilities.
– Each of these four possibilities for the major premise can be combined with a minor premise, that also admits four possibilities, to give a conclusion that also admits four possibilities, for a total of 4 x 4 x 4 = 64 forms.
– Moreover, each of these forms admits the 4 figures, making a total of 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 is derived from two universal premises. This deduction corresponds to the valid mode:
Major Premise All humans are rational
Minor Premise All Greeks are human beings
Conclusion All Greeks are rational
This mode is known as Barbara, where the three occurrences of the vowel a mark that the major, minor and conclusion are universal, see proposition.
4. Example: The conclusive modes of the first figure
Syllogistic reasoning is clearly expressed in the language of set theory.
— Two (non-empty) sets are disjoint if their intersection is empty; they have no elements in common.
— Two sets intersect if they have some elements in common.
— One set is contained in the other if all the elements of the first set also belong to the second set.
In the following, M will be read 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 basic forms of reasoning are used in classification, §3, Syllogistic reasoning about natural taxonomies.
5. Evaluation of syllogisms
See Paralogisms
6. Syllogisms with premise(s) having a concrete subject
The above definitions correspond to the traditional (Aristotelian) categorical syllogism, using quantified variables.
The word syllogism is also used to refer to a form of reasoning in which a premise has a concrete subject. A concrete subject is a subject that refers to a unique single individual, by means of various expressions such as this, this being, Peter, the N who.
Syllogisms that instantiate a universal proposition are examples of such syllogisms. These assign to an individual the properties of the class to which it belongs:
| the x-s are B | human beings have a right to respect |
| this is an x | this man is a human being |
| this is B | this man deserves respect |
The following type of reasoning is based on two concrete propositions. It can also be called, rather metaphorically, “syllogistic”. It disproves universal propositions like as “all swans are white”, see proof by fact.
This is a swan the proposition refers to a concrete individual
This is black the proposition attaches a property to the same individual
Applied to the same subject, “to be black” and “to be white” are opposite predicates
Therefore the claim “all the swans are white” is false.
7. Syllogistic Forms
A polysyllogism, is « a series of syllogisms connected in such a way that the conclusion of one serves as a premise for the next. » (Chenique 1975, p. 255). The polysyllogism is also called a logical sorite.
The term sorite can also applies to an abbreviated polysyllogism “in which the conclusion of each syllogism is not expressed, except in the last one” (Chenique 1975, pp. 256-257).
In the polysyllogism stricto sensu, the rules of syllogism apply at each step; the previous conclusion enters as a premise in the following one, and a new premise is introduced, that allows the reasoning to continue. The transmission of the truth is flawless, from the first argument to the final conclusion.
A chain of propositions whose syntactic form and mode of linking more or less imitate those of a syllogism may also be called a syllogism, with more or less justification; see expression; linked argument; epicheirema
The famous syllogism “Everything rare is expensive, a cheap horse is a rare thing, so a cheap horse is expensive” is based on two contradictory premises, so it is natural that its conclusion is absurd.