# Definition:Categorical Syllogism

## Definition

A **categorical syllogism** is a logical argument which is structured as follows:

$(1): \quad$ It has exactly two premises and one conclusion.

- The first premise is usually referred to as the major premise.
- The second premise is usually referred to as the minor premise.

$(2): \quad$ It concerns exactly three terms, which are usually denoted:

\(\ds P:\) | the primary term | ||||||||

\(\ds M:\) | the middle term | ||||||||

\(\ds S:\) | the secondary term |

$(3): \quad$ Each of the premises and conclusion is a categorical statement.

## Terms of Syllogism

There are three terms in a categorical syllogism:

### Primary Term of Syllogism

The **primary term** of a categorical syllogism is the term that appears as the second predicate of the conclusion of the syllogism.

It also appears once in one of the premises of the syllogism, traditionally the major premise.

It is usually denoted by $P$.

### Middle Term of Syllogism

The **middle term** of a categorical syllogism is the term that does not appear in the conclusion of the syllogism.

It appears once in each of the premises of the syllogism.

It is usually denoted by $M$.

### Secondary Term of Syllogism

The **secondary term** of a categorical syllogism is the term that appears as the first predicate of the conclusion of the syllogism.

It also appears once in one of the premises of the syllogism, traditionally the minor premise.

It is usually denoted by $S$.

## Premises

There are two premises in a categorical syllogism:

### Major Premise

The **major premise** of a categorical syllogism is conventionally stated first.

It is a categorical statement which expresses the logical relationship between the primary term and the middle term of the syllogism.

### Minor Premise

The **minor premise** of a categorical syllogism is conventionally stated second.

It is a categorical statement which expresses the logical relationship between the secondary term and the middle term of the syllogism.

## Conclusion

The **conclusion** of a categorical syllogism is stated last, conventionally prefaced with the word *Therefore*.

It is a categorical statement which expresses the logical relationship between the primary term and the secondary term of the syllogism.

Furthermore, this categorical statement is *specifically* of the form in which the secondary term occurs *first* and the primary term occurs *second*.

## Shorthand Notation

In order to specify the pattern of a categorical syllogism completely, it is necessary and sufficient to specify:

and

- $(2) \quad$ The types of the three categorical statements that compose the syllogism.

Hence, for example, the following categorical syllogism, which is of the first figure:

- $\begin{array}{r|rl} \text I & & \\ \hline \\ \text{Major Premise}: & \mathbf E & \tuple {M, P} \\ \text{Minor Premise}: & \mathbf I & \tuple {S, M} \\ \hline \\ \text{Conclusion}: & \mathbf O & \tuple {S, P} \\ \end{array}$

is specified completely by:

- $\text I: EIO$

## Also known as

Some sources refer to this as **the syllogism**, narrowing down its field of definition to the precise structure as defined here.

## Also see

- Results about
**categorical syllogisms**can be found**here**.

## Historical Note

*The predicate calculus was undiscovered 100 years ago... For well over 2,000 years before that, some of the same logical material was handled by the theory of the syllogism, which we owe to Aristotle; virtually nothing was added to it in that period. There can today be no doubt that predicate calculus has replaced the syllogism as an instrument for serious logical work; predicate calculus is to syllogism what a precision tool is to a blunt knife... There are no reasons other than historical ones for studying the syllogism; but this theory has been of importance in the history of both logic and philosophy, and perhaps therefore deserves a place in a modern logic course.*- -- 1965: E.J. Lemmon:
*Beginning Logic*: $\S 4.4$: The Syllogism

- -- 1965: E.J. Lemmon:

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**syllogism** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**syllogism**