Logic 3 Deduction

A deductive argument is one in which the premisses provide conclusive grounds for the truth of the conclusion. A deductive argument may be valid or invalid – valid if it is not possible for the premisses to be true without the conclusion being true also, invalid if this is not so.

Categorical Propositions

Aristotle began his study of deduction by examining arguments that involved categorical propositions - that is, propositions about categories or classes. For example, all men are bipeds is a proposition about a class, and no men are tripeds is another. Such propositions either affirm or deny that one class is included in another.

Categorical Propositions and Classes

There are four standard forms of categorical proposition and they are illustrated by the following examples (and I thank Copi for these):

All politicians are liars.
No politicians are liars.
Some politicians are liars.
Some politicians are not liars.

The first is called a universal affirmative by logicians and is known as Type A. Schematically, it is written as:

All S is P.

Where S stands for subject and P for predicate.

The second is called a universal negativeand is known as Type E. Schematically, it is written as:

No S is P.

The third is called a particular affirmative and is known as Type I. It is written as:

Some S is P.

The fourth is called a particular negative and is known as Type O. It is written as:

Some S is not P.

Whether a proposition is affirmative or negative is known as its quality. Whether it is universal or particular is known as its quantity. The words all, no, or some are known as quantifiers. If we have time to look at predicate calculus, we will see quantifiers used to very powerful effect.

The Square of Opposition

Aristotle is generally credited with the analysis of categorical propositions that we have just seen and also for expressing the relations between them in the diagrammatical form of a square – the square of opposition.

The Square of Opposition

A (All S is P)E (No S is P)

SuperalternSuperaltern

I (Some S is P)O (Some S is not P)

SubalternSubaltern

Two propositions contradict if one denies the other – they cannot both be true and they cannot both be false. Two propositions are contraries if they cannot both be true but they could both be false. For example, all artists are lazy and no artists are lazy cannot both be true but they could both be false. On the other hand, two propositions are subcontraries if they can both be true but they cannot both be false. For example, some politicians are good and some politicians are not good are clearly subcontraries. When two propositions with the same subject and predicate terms agree in quality but differ in quantity, they exhibit subalternation. In such a case, one proposition is universal and the other particular; the universal proposition is known as the superaltern (and sometimes the superalternate) and the particular proposition is known as the subaltern (and sometimes the subalternate). Notice that the superaltern implies the subaltern but the subaltern does not imply the superaltern.

The Syllogism

Aristotle noticed that, in an argument composed of three statements, if the subject of the first statement is the predicate of the second, and the third statement is a combination of the remaining subject and predicate, the truth of the third statement is guaranteed by the truth of the first two statements. We call the first statement the major premiss, the second the minor premiss, and the third the conclusion. And we call the three-statement system a syllogism; in fact, we call it an Aristotelian syllogismin recognition of its originator. A much-quoted example is:

All men are mortal.

Socrates is a man.

Therefore Socrates is mortal.

John Gibbs