Propositional variable

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 ${\displaystyle P}$, ${\displaystyle Q}$ and ${\displaystyle R}$.[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 well-formed formulae, and are often denoted using lower-case greek letters such as ${\displaystyle \alpha }$, ${\displaystyle \beta }$ and ${\displaystyle \gamma }$.[1]

Propositional variables can be considered nullary predicates in first order logic, because there are no object variables such as x and y attached to predicate letters such as Px and xRy. The internal structure of propositional variables contains predicate letters such as P and Q, in association with 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, ${\displaystyle P(11)}$ and ${\displaystyle R(1,3)}$).[3]

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