Propositional calculus(2)

A logical calculus in which the derivable objects are propositional formulas (cf. Propositional formula). Every propositional calculus is given by a set of axioms (particular propositional formulas) and derivation rules (cf. 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 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). The rule of substitution then becomes superfluous.

The classical propositional calculus is given by the following axioms:

1) $ p \supset ( q \supset p ) $;

2) $ ( p \supset ( q \supset r ) ) \supset ( ( p \supset q ) \supset ( p \supset r ) ) $;

3) $ ( p \& q ) \supset p $;

4) $ ( p \& q ) \supset q $;

5) $ p \supset ( q \supset ( p \& q ) ) $;

6) $ p \supset ( p \lor q ) $;

7) $ q \supset ( p \lor q ) $;

8) $ ( p \supset r ) \supset ( ( q \supset r ) \supset ( ( p \lor q ) \supset r ) ) $;

9) $ ( p \supset q ) \supset ( ( p \supset \neg q ) \supset \neg p ) $;

10) $ \neg \neg p \supset p $.

In this propositional calculus the propositional connectives $ \& , \lor , \supset , \neg $( cf. Propositional connective) are not independent. It can also be given by 1), 2), 9), and 10), taking the connectives $ \supset , \neg $ as primitive ones. The connectives $ \& $ and $ \lor $ are then abbreviations:

$$ A \& B \iff \neg ( A \supset \neg B ) ,\ \ A \lor B \iff \neg A \supset B , $$

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) $ \neg p \supset ( p \supset q ) $.

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.

Other examples of propositional calculi are: the implicative propositional calculus; the minimal propositional calculus; and the positive propositional calculus.

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

$$ \mathfrak M = < M , D ; \& ^ {*} ,\ \lor ^ {*} , \supset ^ {*} , \neg ^ {*} > . $$

Here $ M $ is the set of truth values (cf. Truth value), $ D $ is the set of distinguished truth values, $ D \subset M $, and $ \& ^ {*} , \lor ^ {*} , \supset ^ {*} , \neg ^ {*} $ are the operations on $ M $ corresponding to the connectives $ \& , \lor , \supset , \neg $. The set $ D $ must satisfy the following condition: For any $ a , b \in D $, if $ a \in D $ and $ ( a \supset ^ {*} b ) \in D $, then $ b \in D $( compatibility with the rule of modus ponens). A formula is said to be universally valid on $ \mathfrak M $ if it takes values in $ D $ for every interpretation of its variables as elements of $ M $. The simplest matrix is the matrix $ \mathfrak M _ {2} $, consisting of two elements $ 0 $ and $ 1 $( "false" , "true" ) and the single distinguished value $ 1 $; the operations $ \& ^ {*} , \lor ^ {*} , \supset ^ {*} , \neg ^ {*} $ are defined in the usual way (cf. Algebra of logic). A propositional formula that is universally valid on $ \mathfrak M _ {2} $ is called a tautology. A formula is a tautology if and only if it is a theorem of the classical propositional calculus.

References

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

Comments

Sometimes the propositional calculus is also called the sentential calculus, [a3].

References

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