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.
In a given propositional logic, a formula can be defined 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 can be considered nullary predicates in first order logic, because there are no object variables x, y.., attached to predicate letters Px, xRy,... The internal structure of propositional variables contains predicate letters P, Q,.. in association with individual constants (singular terms) (from a domain of discourse D ) a, b, .. having the form Pa, aRb.
- Smullyan, Raymond M. First-Order Logic. 1968. Dover edition, 1995. Chapter 1.1: Formulas of Propositional Logic.