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 $\phi$.