Difference between revisions of "Minimal propositional calculus"
From Encyclopedia of Mathematics
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''minimal calculus of expressions'' | ''minimal calculus of expressions'' | ||
| − | The logical calculus obtained from the [[Positive propositional calculus|positive propositional calculus]] | + | The logical calculus obtained from the [[Positive propositional calculus|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. | 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 | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Church, "Introduction to mathematical logic" , '''1''' , Princeton Univ. Press (1956)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Church, "Introduction to mathematical logic" , '''1''' , Princeton Univ. Press (1956)</TD></TR></table> | ||
Latest revision as of 08:08, 12 August 2014
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) |
How to Cite This Entry:
Minimal propositional calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_propositional_calculus&oldid=32858
Minimal propositional calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_propositional_calculus&oldid=32858
This article was adapted from an original article by S.K. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article