In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, a propositional variable (also called a sentential variable or sentential letter) is a variable

Variable (mathematics)

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 which can either be true or false. Propositional variables are the basic building-blocks of propositional formula

Propositional formula

In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value...

s, used in propositional logic and higher logics.

Formulas in logic are typically built up recursively from some propositional variables, some number of logical connective

Logical connective

In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

s, and some logical quantifiers. Propositional variables are the atomic formula

Atomic formula

In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...

s 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
  Negation
  In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
  
   ¬X is a formula.
• Given two formulas X and Y, and a binary connective b (such as the logical conjunction
  Logical conjunction
  In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....
  
   ∧), 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 are represented as nullary predicates in first order logic.

See also

• Boolean algebra (logic)
• Boolean datatype
  Boolean datatype
  In computer science, the Boolean or logical data type is a data type, having two values , intended to represent the truth values of logic and Boolean algebra...
  
  
• Boolean domain
  Boolean domain
  In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true...
  
  

• Boolean function
• Logical value
  Logical value
  In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.In classical logic, with its intended semantics, the truth values are true and false; that is, classical logic is a two-valued logic...
  
  
• Propositional logic