\begin{definition}[Definition:Intuitionistic Propositional Logic]
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".
{{refactor|The below needs its own place (it's "Natural Deduction for Intuitionistic Propositional Logic")|level = medium}}
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".
{{transclude:Definition:Intuitionistic Propositional Logic/Axioms
  |section  = definition
  |increase = 1
  |title    = Axioms of Intuitionistic Propositional Logic
  |header   = 2
  |link     = true
}}
\end{definition}