﻿

# propositional calculus

[1900-05]

* * *

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.

* * *

logic
also called  Sentential Calculus,

in logic, symbolic system of treating compound and complex propositions and their logical relationships. As opposed to the predicate calculus, the propositional calculus employs simple, unanalyzed propositions rather than terms or noun expressions as its atomic units; and, as opposed to the functional calculus, it treats only propositions that do not contain variables. Simple (atomic) propositions are denoted by letters, 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 A and B) AB is provable if and only if B is always a logical consequence of A. The propositional calculus is consistent in that there exists no formula 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 predicate calculus; thought, laws of.

* * *

Universalium. 2010.

### Look at other dictionaries:

• Propositional calculus — In mathematical logic, a propositional calculus or logic (also called sentential calculus or sentential logic) is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules… …   Wikipedia

• propositional calculus — The logical calculus whose expressions are letters representing sentences or propositions, and constants representing operations on those propositions, to produce others of higher complexity. The operations include conjunction, disjunction,… …   Philosophy dictionary

• propositional calculus — noun Date: 1903 the branch of symbolic logic that uses symbols for unanalyzed propositions and logical connectives only called also sentential calculus compare predicate calculus …   New Collegiate Dictionary

• propositional calculus — noun propositional logic …   Wiktionary

• propositional calculus — noun a branch of symbolic logic dealing with propositions as units and with their combinations and the connectives that relate them • Syn: ↑propositional logic • Hypernyms: ↑symbolic logic, ↑mathematical logic, ↑formal logic …   Useful english dictionary

• propositional calculus — proposi′tional cal′culus n. math. pho sentential calculus • Etymology: 1900–05 …   From formal English to slang

• propositional calculus — /prɒpəˌzɪʃənəl ˈkælkjələs/ (say propuh.zishuhnuhl kalkyuhluhs) noun that part of modern logic which systematises the relations between unanalysed propositions …   Australian English dictionary

• Implicational propositional calculus — In mathematical logic, the implicational propositional calculus is a version of classical (two valued) propositional calculus which uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by… …   Wikipedia

• Frege's propositional calculus — In mathematical logic Frege s propositional calculus was the first axiomatization of propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus, in 1879 as part of his second order predicate calculus (although… …   Wikipedia

• implicational propositional calculus — noun A minimalist version of propositional calculus which uses only the logical connectives ( implies ) and ( false ) …   Wiktionary