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In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a
truth function In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly o ...
. Propositional variables are the basic building-blocks of
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. A propositional for ...
s, used in
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
and higher-order logics.


Uses

Formulas in logic are typically built up recursively from some propositional variables, some number of
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
s, and some logical quantifiers. Propositional variables are the
atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime 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 subformu ...
s of propositional logic, and are often denoted using capital roman letters such as P, Q and R. ;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 complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
''¬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-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents thi ...
∧),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 \alpha, \beta and \gamma.


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 term A singular term is a paradigmatic referring device in a language. Singular terms are of philosophical importance for philosophers of language, because they ''refer'' to things in the world, and the ability of words to refer calls for scrutiny. Ove ...
s from a
domain of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The doma ...
D), ultimately taking a form such as P''a'', ''a''R''b''.(or with parenthesis, P(11) and R(1, 3)). Propositional logic is sometimes called
zeroth-order logic Zeroth-order logic is first-order logic without variables or quantifiers. Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus,. but an alternative definition extends propositional logic by adding constant ...
due to not considering the internal structure in contrast with
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
which analyzes the internal structure of the atomic sentences.


See also

*
Boolean algebra (logic) In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
*
Boolean datatype In computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted ''true'' and ''false'') which is intended to represent the two truth values of logic and Boolean algebra. It is named ...
*
Boolean domain In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as ...
*
Boolean function In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function ...
*
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, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progra ...
*
Predicate variable In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predi ...
*
Propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...


References


Bibliography

* Smullyan, Raymond M. ''First-Order Logic''. 1968. Dover edition, 1995. Chapter 1.1: Formulas of Propositional Logic. {{logic-stub Propositional calculus Concepts in logic Logic symbols