In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is a variable which can either be true or false. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher logics.
Formulas in logic are typically built up recursively from some propositional variables, some number of logical connectives, and some logical quantifiers. Propositional variables are the atomic formulas of propositional logic. For example, in a given propositional logic, we might define a formula as follows:
- Every propositional variable is a formula.
- Given a formula X the negation ¬X is a formula.
- Given two formulas X and Y, and a binary connective b (such as the logical conjunction ∧), then (X b Y) is a formula. (Note the parentheses.)
In this way, all of the formulas of propositional logic are built up from propositional variables as a basic unit. Propositional variables should not be confused with the metavariables which appear in the typical axioms of propositional calculus; the latter effectively range over well-formed formulae.
Propositional variables are represented as nullary predicates in first order logic.
... and is a p-morphism of their underlying frames, which satisfies if and only if, for any propositional variable p ... preserve forcing of atomic formulas if w B w’, then if and only if, for any propositional variable p ... (hence also p-morphisms) of models preserve the satisfaction of all formulas, not only propositional variables ...
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