# Definition:Johansson's Minimal Logic/Axioms

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## Definition

### 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 $\phi$.