# Definition:Intuitionistic Propositional Logic

## Definition

The **intuitionist** school of mathematics is one which adopts the following philosophical position:

- "Although we may know that it is not the case that a statement $p$ is (provably) false, we don't necessarily know that it is (provably) true either."

Thus the intuitionist school rejects the Law of the Excluded Middle.

The classical school, by affirming that if a statement is not true it must be false, and if it is not false it must be true, accepts as an axiom that "not not-true" must mean "true".

The system of **intuitionistic propositional logic** is, in consequence, based on the same axioms as that of classical propositional logic except for that disputed Law of the Excluded Middle.

It is worth mentioning that fuzzy logic is a branch of logic in which truth values are selected from a far wider range than just "true" and "false".

## Axioms of Intuitionistic Propositional Logic

## Also known as

Some prefer to address this logic by its practitioners, calling it **intuitionist (propositional) logic**.

This field of logic is also (more intuitively) (no pun intended) known as **constructive logic** or **constructivist logic**.

That is, a mathematical object can not be deduced to exist based on the Law of Excluded Middle — it specifically needs to be **constructed**.

## Sources

- 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*... (previous) ... (next): Chapter $1$: Introduction: $\S 1.4$: Non-standard logics - 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