# 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.