In
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, a propositional variable (also called a sentence letter,
sentential variable, or sentential letter) is an input
variable (that can either be true or false) of a
truth function. Propositional variables are the basic building-blocks of
propositional formulas, used in
propositional logic and
higher-order 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
.
;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
In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...
''¬X'' is a formula.
* Given two formulas ''X'' and ''Y'', and a
binary connective ''b'' (such as the
logical conjunction
In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or ...
∧), 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 propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
; the latter effectively range over well-formed formulae, and are often denoted using lower-case greek letters such as
,
and
.
Predicate logic
Propositional variables with no object variables such as ''x'' and ''y'' attached to predicate letters such as P''x'' and ''x''R''y'', having instead individual constants ''a'', ''b'', ..attached to predicate letters are propositional constants P''a'', ''a''R''b''. 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 P''a'', ''a''R''b''.(or with parenthesis,
and
).
Propositional logic is sometimes called
zeroth-order logic due to not considering the internal structure in contrast with
first-order logic which analyzes the internal structure of the atomic sentences.
See also
*
Boolean algebra (logic)
*
Boolean data type
*
Boolean domain
*
Boolean function
*
Logical value
*
Predicate variable
*
Propositional logic
References
Bibliography
* 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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