Propositional Calculus

propositional calculus

Formal system of propositions and their logical relationships. As opposed to the predicate calculus, the propositional calculus employs simple, unanalyzed propositions rather than predicates as its atomic units. Simple (atomic) propositions are denoted by lowercase Roman letters (e.g., p, q), and compound (molecular) propositions are formed using the standard symbols ∧ for “and,” ∨ for “or,” ⊃ for “if . . . then,” and ¬ for “not.” As a formal system, the propositional calculus is concerned with determining which formulas (compound proposition forms) are provable from the axioms. Valid inferences among propositions are reflected by the provable formulas, because (for any formulas A and B) A ⊃ B is provable if and only if B is a logical consequence of A. The propositional calculus is consistent in that there exists no formula A in it such that both A and ¬ A are provable. It is also complete in the sense that the addition of any unprovable formula as a new axiom would introduce a contradiction. Further, there exists an effective procedure for deciding whether a given formula is provable in the system. See also logic, predicate calculus, laws of thought.

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propositional calculus [‚präp·ə′zish·ən·əl ′kal·kyə·ləs]

(mathematics)

The mathematical study of logical connectives between propositions and deductive inference. Also known as sentential calculus.

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propositional calculus - propositional logic

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Propositional Calculus 

a branch of mathematical logic in which the formal axiomatic method is used to study complex (compound) propositions, which are put together from simple (elementary, unanalyzable) propositions with the help of the logical connectives “and,” “or,” “if… then,” and “not.” Moreover, the goal is set of determining propositional forms of general significance in one sense or another, that is, those formulas that upon any substitution of propositions in place of the variables give propositions that are true in the appropriate sense.

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Propositional Calculus 

a branch of mathematical logic that studies the logical forms of compound propositions formed from simpler propositions by means of such connectives as “and”; “or”; “if …, then …”; and “not” (negation).