Definition:Intuitionistic Propositional Logic

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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.