# 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:

- The Law of Excluded Middle (a statement is either true or false)
- The Rule of Explosion (from a contradiction anything can be deduced)

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 $\psi$.

#### Rule of Addition

- $(1): \quad$ 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 $\phi \implies \psi$.

#### Principle of Non-Contradiction

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

#### Proof by 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.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (next): Preface (except he refers to it as*Johannson's Minimal Calculus*)