Minimal propositional calculus

minimal calculus of expressions

The logical calculus obtained from the positive propositional calculus $\Pi$ by the addition of a new connective $\neg$ (negation) and the axiom scheme

$$(A\supset B)\supset((A\supset\neg B)\supset\neg A),$$

which is called the law of reductio ad absurdum.

The minimal propositional calculus is distinguished by the fact that in it not every formula is deducible from "false", that is, from a formula of the form $\neg A\&A$. The minimal propositional calculus can be obtained from the calculus $\Pi$ in another way by adding to the language instead of the connective $\neg$ a new propositional constant $\bot$ (falsehood) without the addition of new axiom schemes. Here the formula $A\supset\bot$ serves as the negation $\neg A$ of a formula $A$.

References

[1]	A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956)