Definition:Johansson's Minimal Logic

Johansson's Minimal Calculus is a system of propositional logic of the intuitionist school.

The philosophy behind it is that it may not necessarily be the case that either the Law of the Excluded Middle or the principle of non-contradiction may not hold.

That is, the rules that say:


 * A statement must be either true or false, and
 * A statement may not be both true and not true at the same time

are not accepted.

Thus this philosophical school allows for the foundation of the study of fuzzy logic.

Axioms of Johansson's Minimal 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) If we can conclude $$p \land q$$, then we may infer $$p$$.
 * 2) If we can conclude $$p \land q$$, then we may infer $$q$$.


 * The Rule of Addition:
 * 1) If we can conclude $$p$$, then we may infer $$p \lor q$$.
 * 2) 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) By making the assumption $$p$$, we can conclude $$r$$;
 * 2) By making the assumption $$q$$, we can conclude $$r$$;

then we may infer $$r$$.


 * Modus Ponendo Ponens: If we can conclude $$p \Longrightarrow q$$, 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 $$p \Longrightarrow q$$.


 * The Rule of Not-Elimination: If we can conclude both $$p$$ and $$\lnot p$$, 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 $$\lnot p$$.