analyse categorical statements and syllogisms, including the four standard forms and the rules for valid syllogistic reasoning

What this dot point is asking

QCAA wants you to handle Aristotelian categorical logic: the logic of statements about classes of things. You need to classify statements into the four standard forms, identify the terms, understand distribution, lay a syllogism out in standard form, and apply the rules of the syllogism to decide validity. This is examined in IA1 and the external exam, often by asking you to test a given syllogism.

The answer

The four standard categorical forms

A categorical statement relates two classes (a subject term S and a predicate term P). There are four standard forms, traditionally labelled by vowels:

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

The vowels come from the Latin affirmo (I affirm, giving A and I) and nego (I deny, giving E and O). "Some" means "at least one".

Distribution of terms

A term is distributed when the statement says something about every member of that class. Distribution drives the validity rules, so learn this table:

• A (All S are P): S is distributed, P is not.
• E (No S are P): both S and P are distributed.
• I (Some S are P): neither is distributed.
• O (Some S are not P): P is distributed, S is not.

A useful pattern: universals (A, E) distribute the subject; negatives (E, O) distribute the predicate.

The categorical syllogism

A categorical syllogism has exactly two premises and a conclusion, built from exactly three terms:

• the major term (P) is the predicate of the conclusion;
• the minor term (S) is the subject of the conclusion;
• the middle term (M) appears in both premises but not the conclusion.

In standard form the major premise comes first. Example:

• Major premise: All mammals (M) are warm-blooded (P).
• Minor premise: All whales (S) are mammals (M).
• Conclusion: Therefore all whales (S) are warm-blooded (P).

Rules for a valid syllogism

A standard-form syllogism is valid if and only if it breaks none of these rules:

1. The middle term must be distributed at least once. If it is not, the premises fail to connect S and P. Violating this is the fallacy of the undistributed middle.
2. Any term distributed in the conclusion must be distributed in the premise where it appears. Violating this is an illicit major or illicit minor.
3. No syllogism can have two negative premises. Two negatives (E or O) leave the classes disconnected.
4. If either premise is negative, the conclusion must be negative, and if the conclusion is negative, exactly one premise must be negative.
5. From two universal premises you cannot validly draw a particular conclusion (the existential rule, on the modern reading that universals do not assert existence).

Working an example

Test: "All cats are mammals; some pets are cats; therefore some pets are mammals."

• Terms: middle = cats, major (P) = mammals, minor (S) = pets.
• Premise 1 is A (All cats are mammals): cats distributed.
• Premise 2 is I (Some pets are cats): nothing distributed.
• The middle term (cats) is distributed in premise 1, satisfying Rule 1.
• The conclusion is I, distributing nothing, so no term is distributed in the conclusion: Rule 2 cannot be broken.
• No negative premises, so Rules 3 and 4 hold; conclusion is particular from a particular premise, so Rule 5 holds.

All rules pass: the syllogism is valid.

Try this

Q1. State the four standard categorical forms and give the distribution of terms for each. [4 marks]

• Cue. A distributes S only; E distributes both; I distributes neither; O distributes P only.

Q2. Test for validity: "All birds are animals; some pets are birds; therefore some pets are animals." [3 marks]

• Cue. Middle (birds) distributed in premise 1; conclusion distributes nothing; valid.

Q3. Name the fallacy in "All dogs are animals; all cats are animals; therefore all cats are dogs." [2 marks]

• Cue. Undistributed middle; the middle term "animals" is never distributed.