Inmathematical logic, apropositional variable (also called asentence letter,^[1]sentential variable, orsentential letter) is an inputvariable (that can either betrue orfalse) of atruth function. Propositional variables are the basic building-blocks ofpropositional formulas, used inpropositional logic andhigher-order logics.

Formulas in logic are typically built up recursively from some propositional variables, some number oflogical connectives, and somelogical quantifiers. Propositional variables are theatomic formulas of propositional logic, and are often denoted using capitalroman letters such as${\displaystyle P}$,${\displaystyle Q}$ and${\displaystyle R}$.^[2]

Example

In a given propositional logic, a formula can be defined as follows:

• Every propositional variable is a formula.
• Given a formulaX, thenegation¬X is a formula.
• Given two formulasX andY, and abinary connectiveb (such as thelogical 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 themetavariables, which appear in the typical axioms ofpropositional calculus; the latter effectively range over well-formed formulae, and are often denoted using lower-case greek letters such as${\displaystyle \alpha }$,${\displaystyle \beta }$ and${\displaystyle \gamma }$.

Propositional variables with no object variables such asx andy attached to predicate letters such as Px andxRy, having instead individual constantsa,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 containspredicate letters such as P and Q, in association withbound individual variables (e.g., x,y), individual constants such asa andb (singular terms from adomain of discourse D), ultimately taking a form such as Pa,aRb.(or with parenthesis,${\displaystyle P(11)}$ and${\displaystyle R(1,3)}$).^[3]

Propositional logic is sometimes calledzeroth-order logic due to not considering the internal structure in contrast withfirst-order logic which analyzes the internal structure of the atomic sentences.

• Smullyan, Raymond M.First-Order Logic. 1968. Dover edition, 1995. Chapter 1.1: Formulas of Propositional Logic.

• Propositional calculus
• Concepts in logic
• Logic symbols

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