In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is a input variable (that can either be true or false) of a truth function. Propositional variables are the basic buildingblocks of propositional formulas, used in propositional logic and higherorder logics.
Uses
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, and are often denoted using capital roman letters such as , and .^{[1]}^{[2]}
 Example
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 ∧),the expression (X b Y) is a formula. (Note the parentheses.)
Through this construction, all of the formulas of propositional logic can be 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 wellformed formulae, and are often denoted using lowercase greek letters such as , and .^{[1]}
Predicate logic
Propositional variables with no object variables such as x and y attached to predicate letters such as Px and xRy, having instead individual constants a, b, ..attached to predicate letters are propositional constants Pa, aRb. These propositional constants are atomic propositions, not containing propositional operators.
The internal structure of propositional variables contains predicate letters such as P and Q, in association with bound individual variables (e.g., x, y), individual constants such as a and b (singular terms from a domain of discourse D), ultimately taking a form such as Pa, aRb.(or with parenthesis, and ).^{[3]}
Propositional logic is sometimes called zerothorder logic due to not considering the internal structure in contrast with firstorder logic which analyzes the internal structure of the atomic sentences.
See also


References
 ^ ^{a} ^{b} "Comprehensive List of Logic Symbols". Math Vault. 20200406. Retrieved 20200820.
 ^ "Predicate Logic  Brilliant Math & Science Wiki". brilliant.org. Retrieved 20200820.
 ^ "Mathematics  Predicates and Quantifiers  Set 1". GeeksforGeeks. 20150624. Retrieved 20200820.
Bibliography
 Smullyan, Raymond M. FirstOrder Logic. 1968. Dover edition, 1995. Chapter 1.1: Formulas of Propositional Logic.