A Brief Intro to Syllogistic Logic

Syllogistic or Aristotelian logic takes a middle ground between informal logic (which concerns itself with distinguishing good and bad arguments in everyday discourse) and formal logic (which concerns itself with the form of arguments, and its validity).

Disclaimer: This page is intended only to provide a little background to syllogistic logic as an aid to understanding any formal fallacies on this site. Consider it a work in progress: by no means should it be considered a comprehensive source of information about intentional logic. I welcome suggestions or corrections from experienced logicians.

Propositions

According to Aristotle, a proposition is a sentence that affirms or denies a predicate of a subject. In other words, a proposition makes a truth claim about something or someone. The subject and predicate of a proposition are also known as terms.

There are only four kinds of propositions:

• the universal affirmative (or A proposition): All S is P.
• the particular affirmative (I proposition): Some S are P.
• the universal negative (E proposition): No S is P.
• the particular negative (O proposition): Some S are not P.

The nicknames A, I, E, and O come from the vowels in the Latin words affirmo ("I affirm") and nego ("I deny").

Distribution

A term is said to be distributed if the proposition gives information about every member of its class.

• In an A proposition, the subject is distributed.
• In an I proposition, neither subject nor predicate is distributed.
• In an E proposition, both subject and predicate are distributed.
• In an O proposition, the predicate is distributed.

Syllogisms

A syllogism is a deductive argument comprising three propositions. The first two are are premises. In a valid syllogism, if the two premises are true, then the third proposition, the conclusion, must also be true.

The classic syllogism is as follows:

Premise 1. All men are mortal.
Premise 2. Socrates is a man.
Conclusion. Therefore, Socrates is mortal.

Every valid syllogism contains no more or less than three terms:

• The minor term (in blue) is the subject of the conclusion, and common to the conclusion and the minor premise.
• The major term (in red) is the predicate of the conclusion, and common to the conclusion and the major premise.
• The middle term (in green) is common to the two premises, and does not appear in the conclusion.

For a syllogism to be valid, the following must be true:

• There must be no more than three terms.
• The middle term must be distributed at least once.
• Any term distributed in the premises must be distributed in the conclusion.
• A particular premise cannot support a universal conclusion.
• A negative premise cannot support an affirmative conclusion.
• Two negative premises do not support any conclusion.

If a syllogism follows these rules, then it is valid: the conclusion necessarily follows from the premises. Of course, it is possible to draw a false conclusion from a valid syllogism, if one or both premises are false:

All elephants are Communists.
Stephen Harper is an elephant.
Therefore, Stephen Harper is a Communist.

If a syllogism is valid and its premises are true, then it is sound.