Definition:Johansson's Minimal Logic

Definition

Johansson's minimal logic is a subbranch of propositional logic of the intuitionist school.

The philosophical position behind it is that it may be the case that the proof rules:

do not hold.

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

Axioms of Johansson's Minimal Logic

Rule of Assumption

An assumption may be introduced at any stage of an argument.

Rule of Conjunction

If we can conclude both $\phi$ and $\psi$, we may infer the compound statement $\phi \land \psi$.

Rule of Simplification

$(1): \quad$ If we can conclude $\phi \land \psi$, then we may infer $\phi$.
$(2): \quad$ If we can conclude $\phi \land \psi$, then we may infer $\psi$.

$(1): \quad$ If we can conclude $\phi$, then we may infer $\phi \lor \psi$.
$(2): \quad$ If we can conclude $\psi$, then we may infer $\phi \lor \psi$.

Proof by Cases

If we can conclude $\phi \lor \psi$, and:
$(1): \quad$ By making the assumption $\phi$, we can conclude $\chi$
$(2): \quad$ By making the assumption $\psi$, we can conclude $\chi$
then we may infer $\chi$.

Modus Ponendo Ponens

If we can conclude $\phi \implies \psi$, and we can also conclude $\phi$, then we may infer $\psi$.

Rule of Implication

If, by making an assumption $\phi$, we can conclude $\psi$ as a consequence, we may infer $\phi \implies \psi$.

If we can conclude both $\phi$ and $\neg \phi$, we may infer a contradiction.

If, by making an assumption $\phi$, we can infer a contradiction as a consequence, then we may infer $\neg \phi$.
The conclusion does not depend upon the assumption $\phi$.

Also known as

This system is also known as Johansson's minimal calculus.

Source of Name

This entry was named for Ingebrigt Johansson.