In an immediate inference, we draw a conclusion directly from a single A, E, I, or O proposition. Moreover, when two categorical propositions are contradictories, the falsity of one can be validly inferred from the truth of the other, and the truth of one can be validly inferred from the falsity of the other. All these forms of argument contain only one premise. The next step in understanding categorical propositions is to consider arguments with two premises.

An important group of such arguments is called categorical syllogisms. The basic idea behind these arguments is commonsensical. Suppose you wish to prove that all squares have four sides. A proof should present some link or con- nection between squares and four-sided figures. This link can be provided by some intermediate class, such as rectangles. You can then argue that, because the set of squares is a subset of the set of rectangles and rectangles are a subset of four-sided figures, squares must also be a subset of four-sided figures.

Of course, there are many other ways to link two terms by means of a third term. All such arguments with categorical propositions are called categorical syllogisms. More precisely, a categorical syllogism is any argument such that:

1. The argument has exactly two premises and one conclusion;

2. The argument contains only basic A, E, I, and O propositions;

3. Exactly one premise contains the predicate term;

4. Exactly one premise contains the subject term; and

5. Each premise contains the middle term.

The predicate term is simply the term in the predicate of the conclusion. It is also called the major term, and the premise that contains the predicate term is called the major premise. The subject term is the term in the subject of the conclusion. It is called the minor term, and the premise that contains the sub- ject term is called the minor premise. It is traditional to state the major prem- ise first, the minor premise second.

Our first example of a categorical syllogism then looks like this:

All rectangles are things with four sides. (Major premise) All squares are rectangles. (Minor premise)

All squares are things with four sides. (Conclusion)

Subject term = “Squares”

Predicate term = “Things with four sides”

Middle term = “Rectangles”

To get the form of this syllogism, we replace the terms with variables:

All M is P. All S is M.

All S is P.

Of course, many other arguments fit the definition of a categorical syllogism. Here is one with a negative premise:

No ellipses are things with sides. All circles are ellipses.

No circles are things with sides.

The next categorical syllogism has a particular premise:

All squares are things with equal sides. Some squares are rectangles.

Some rectangles are things with equal sides.