# Definition:Aristotelian Logic

## Definition

**Aristotelian logic** is a system of logic which is based upon the philosophy of Aristotle.

It forms the cornerstone of the entirety of classical logic.

The school of **Aristotelian logic** consists almost entirely of the relationships between the various categorical syllogisms.

This school of philosophy forms the basis of mainstream mathematics, although, for example, mathematicians of the intuitionistic school do not accept the Law of the Excluded middle value.

It was Aristotle who, in particular, introduced the following two axioms of logic:

### Law of Excluded Middle (LEM)

The **law of (the) excluded middle** can be expressed in natural language as:

This is one of the Aristotelian principles upon which rests the whole of classical logic, and the majority of mainstream mathematics.

The **LEM** is rejected by the intuitionistic school, which rejects the existence of an object unless it can be constructed within an axiomatic framework which does not include the LEM.

### Principle of Non-Contradiction (PNC)

The **Principle of Non-Contradiction** can be expressed in natural language as follows:

This means: if we have managed to deduce that a statement is both true and false, then the sequence of deductions show that the pool of assumptions upon which the sequent rests contains assumptions which are mutually contradictory.

Thus it provides a means of eliminating a logical not from a sequent.

## Also see

- Results about
**Aristotelian logic**can be found**here**.

## Source of Name

This entry was named for Aristotle.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Aristotelian logic** - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Aristotelian logic**