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Propositional connective

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A symbol in a formal language used for denoting a logical operation by means of which a new statement can be obtained from given statements. The most important propositional connectives are: the conjunction (or ), the disjunction , the implication (or , or ), the negation (or ), and equivalence (or , or ). These propositional connectives correspond in the English language to the expressions "and" , "or" , "implies" , "it is not true that" , and "is equivalent with" . Sometimes one considers other propositional connectives; for example, the Sheffer stroke.

The symbol is usually not introduced as an independent propositional connective, but as an abbreviation:

(1)

If a language contains the propositional constant , denoting "untruth" , then negation can be regarded as an abbreviation: .

The propositional connectives , and are not independent in classical logic, since the following equivalences hold:

(2)
(3)
(4)

Thus, each of the propositional connectives can be expressed in terms of and one of the others. Therefore, in formulating the classical propositional calculus of expressions, one can choose two propositional connectives as primitive: and one of ; the others are regarded as abbreviations, according to (1)–(4). In intuitionistic logic, , and are independent.


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References

[a1] J.L. Bell, M. Machover, "A course in mathematical logic" , North-Holland (1977)
How to Cite This Entry:
Propositional connective. S.K. Sobolev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Propositional_connective&oldid=15926
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098