Category:Intuitionist Propositional Calculus

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 intuitionist propositional calculus is, in consequence, based on the same axioms as that of classical propositional calculus 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 Intuitionist Propositional Calculus

• The Rule of Assumption: An assumption may be introduced at any stage of an argument.

• The Rule of Conjunction: If we can conclude both $p$ and $q$, we may infer the compound statement $p \land q$.

• The Rule of Simplification:

$(1): \quad$ If we can conclude $p \land q$, then we may infer $p$.

$(2): \quad$ If we can conclude $p \land q$, then we may infer $q$.

• The Rule of Addition:

$(1): \quad$ If we can conclude $p$, then we may infer $p \lor q$.

$(2): \quad$ If we can conclude $p$, then we may infer $q \lor p$.

• The Rule of Or-Elimination: If we can conclude $p \lor q$, and:

$(1): \quad$ By making the assumption $p$, we can conclude $r$

$(2): \quad$ By making the assumption $q$, we can conclude $r$

then we may infer $r$.

• Modus Ponendo Ponens: If we can conclude $p \implies q$, and we can also conclude $p$, then we may infer $q$.

• The Rule of Implication: If, by making an assumption $p$, we can conclude $q$ as a consequence, we may infer $p \implies q$.

• The Rule of Not-Elimination: If we can conclude both $p$ and $\neg p$, we may infer a contradiction.

• The Rule of Proof By Contradiction: If, by making an assumption $p$, we can infer a contradiction as a consequence, then we may infer $\neg p$.

• The Rule of Bottom-Elimination: If we can conclude a contradiction, we may infer any statement.