View source for Minimal propositional calculus

''minimal calculus of expressions''

The logical calculus obtained from the [[Positive propositional calculus|positive propositional calculus]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m0638801.png" /> by the addition of a new connective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m0638802.png" /> (negation) and the axiom scheme

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m0638803.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m0638804.png" />. The minimal propositional calculus can be obtained from the calculus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m0638805.png" /> in another way by adding to the language instead of the connective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m0638806.png" /> a new propositional constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m0638807.png" /> (falsehood) without the addition of new axiom schemes. Here the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m0638808.png" /> serves as the negation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m0638809.png" /> of a formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063880/m06388010.png" />.

====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>