Propositional calculus

In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. (Compare this to the predicate calculus which is quantificational and whose atomic formulas are propositional functions, and modal logic which may be non-truth-functional.)

Related Topics:
Mathematical logic - Predicate calculus - Modal logic

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A calculus is a logical system which is used to prove valid formulas (i.e. its theorems) and arguments. It is a set of axioms (which may be an empty or countably infinite set) or axiom schemata, and inference rules for deriving valid inferences. A formal grammar (or syntax) recursively defines the expressions and well-formed formulas (wffs) of the language. In addition a semantics is given which defines truth and valuations (or interpretations). It allows us to determine which wffs are valid (i.e. theorems).

Related Topics:
Logical system - Inference rule - Formal grammar - Language - Semantics

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In the propositional calculus the language consists of propositional variables (or placeholders) and sentential operators (or connectives). A wff is any atomic formula or a formula built up from sentential operators.

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In what follows we will outline a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their language (i.e. which operators and variables are part of the language); (2) which (if any) axioms they have; (3) which inference rules are employed.

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