# propositional calculus

## propositional calculus

*noun*

## Definition of *PROPOSITIONAL CALCULUS*

**:**the branch of symbolic logic that uses symbols for unanalyzed propositions and logical connectives only —called also

*sentential calculus*— compare predicate calculus

## First Known Use of *PROPOSITIONAL CALCULUS*

## propositional calculus

*noun*

*(Concise Encyclopedia)*

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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