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24.5 What Are Syllogisms?

Syllogism: Any two-premise argument.

A syllogism is a two-premise argument.

For example, this form of argument is called the disjunctive syllogism, because it’s a nifty valid argument using a disjunction:

Disjunctive Syllogism: from PvQ and ~P we can infer Q.

1. Pia or Quinn is guilty.
2. Pia is not guilty.
3. Quinn is guilty.

Or here’s one of the most famous syllogisms ever:

1. All humans are mortal.
2. Socrates is human.
3. Socrates is mortal.

Notice that the first premise is one of the Aristotelian forms.

The second premise of the argument, though, “Socrates is human”, is not identical to an Aristotelian form. But it could be re-phrased like this: “All objects identical with Socrates are human.” That paraphrase now has the “All Ps are Q” form.

Our new argument, then, would be this:

1. All humans are mortal.
2. All objects identical to Socrates are human.
3. All objects identical to Socrates are mortal.

Categorical Syllogism: a syllogism with three terms in which the premises and conclusion are Aristotelian forms.

A syllogism like this, where the premises and conclusion are Aristotelian forms, we will call a categorical syllogism.

It’s called “categorical” because the terms like “human” and “mortal” pick out categories or collections of things. “Objects identical to Socrates” also picks out a category which happens to have only one member (Socrates himself).

A categorical syllogism must also have exactly three terms or categories, which occur in a certain pattern: each term appears twice total in two of the different sentences.

Not every categorical syllogism is valid. For example, in the argument about Socrates, if we exchanged the conclusion with one of the premises, we would have another categorical syllogism.

1. All objects identical to Socrates are human.
2. All objects identical to Socrates are mortal.
3. All humans are mortal.

That argument has this form:

1. All Ps are Q.
2. All Ps are R.
3. All Qs are R.

That is clearly invalid: all cats are animals, and all cats are mammals, but not all animals are mammals.

The goal of Aristotle’s logic was to figure out and systematize all the valid categorical syllogisms.

The first thing Aristotle noticed is that the conclusion has two different terms: in the example above, the conclusion “All Qs are R” has Q as subject term and R as the predicate term.

Minor Term: the subject of the conclusion.
Major term: the predicate of the conclusion.
Middle term: the term in both premises.

He called the subject of the conclusion the “minor term” and the predicate of the conclusion the “major term.”

A categorical syllogism also must have a third term, which only occurs in the two premises. Aristotle dubbed that the “middle term.”

Let’s practice all this terminology.

Standard Form: 1. Major Premise, 2. Minor Premise, 3. Conclusion.

The order of premises does not matter to the validity of an argument, so all we need is a convention for presenting arguments in the same way.

We will follow tradition and list the major premise first, the minor premise second. This is called standard form.

1. Major Premise
2. Minor Premise
3. Conclusion

In the next section we’ll see how Aristotle used this setup to identify all the valid categorical syllogisms.

Before that, here’s some more practice.