View source for Propositional calculus(2)

A [[Logical calculus|logical calculus]] in which the derivable objects are propositional formulas (cf. [[Propositional formula|Propositional formula]]). Every propositional calculus is given by a set of axioms (particular propositional formulas) and derivation rules (cf. [[Derivation rule|Derivation rule]]). A formula that is derivable in a given propositional calculus is called a theorem of this propositional calculus. One usually takes the rules of [[Modus ponens|modus ponens]] and substitution (of arbitrary propositional formulas for variables) as the derivation rules. Sometimes a propositional calculus is given not by axioms, but by axiom schemes (cf. [[Axiom scheme|Axiom scheme]]). The rule of substitution then becomes superfluous.

The classical propositional calculus is given by the following axioms:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p0754801.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p0754802.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p0754803.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p0754804.png" />;

5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p0754805.png" />;

6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p0754806.png" />;

7) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p0754807.png" />;

8) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p0754808.png" />;

9) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p0754809.png" />;

10) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548010.png" />.

In this propositional calculus the propositional connectives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548011.png" /> (cf. [[Propositional connective|Propositional connective]]) are not independent. It can also be given by 1), 2), 9), and 10), taking the connectives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548012.png" /> as primitive ones. The connectives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548014.png" /> are then abbreviations:

<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/p/p075/p075480/p07548015.png" /></td> </tr></table>

while 3)–8) become theorems. The classical propositional calculus is also called the complete propositional calculus, since the addition of any formula not derivable in it as a further axiom leads to a contradictory propositional calculus, i.e. one in which all propositional formulas are derivable. The classical propositional calculus is often simply called propositional calculus.

The intuitionistic (constructive) propositional calculus can be obtained from the classical propositional calculus by replacing 10) by the weaker axiom

11) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548016.png" />.

A propositional calculus that is obtained from the intuitionistic propositional calculus by the addition of a finite (or recursive) set of axioms is called intermediate, superintuitionistic or superconstructive. See also [[Intermediate logic|Intermediate logic]].

Other examples of propositional calculi are: the [[Implicative propositional calculus|implicative propositional calculus]]; the [[Minimal propositional calculus|minimal propositional calculus]]; and the [[Positive propositional calculus|positive propositional calculus]].

An interpretation of a propositional calculus is given by an algebra (matrix) of the form

<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/p/p075/p075480/p07548017.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548018.png" /> is the set of truth values (cf. [[Truth value|Truth value]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548019.png" /> is the set of distinguished truth values, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548021.png" /> are the operations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548022.png" /> corresponding to the connectives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548023.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548024.png" /> must satisfy the following condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548025.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548028.png" /> (compatibility with the rule of modus ponens). A formula is said to be universally valid on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548029.png" /> if it takes values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548030.png" /> for every interpretation of its variables as elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548031.png" />. The simplest matrix is the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548032.png" />, consisting of two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548034.png" /> ( "false" ,  "true" ) and the single distinguished value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548035.png" />; the operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548036.png" /> are defined in the usual way (cf. [[Algebra of logic|Algebra of logic]]). A propositional formula that is universally valid on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075480/p07548037.png" /> is called a tautology. A formula is a tautology if and only if it is a theorem of the classical propositional calculus.

====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>



====Comments====
Sometimes the propositional calculus is also called the sentential calculus, [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Bell,   M. Machover,   "A course in mathematical logic" , North-Holland  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Wójcicki,   "Theory of logical calculi" , Kluwer  (1988)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Tarski,   "Introduction to logic and to the methodology of deductive sciences" , Oxford Univ. Press  (1946)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.C. Kleene,   "Introduction to metamathematics" , North-Holland  (1959)  pp. Chapts. IX; XI, §54</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.F. Strawson,   "Introduction to logical theory" , Methuen  (1952)</TD></TR></table>