Propositional variable: Difference between revisions

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{{short description|A variableVariable that can either be true or false.}}

In [[mathematical logic]], a '''propositional variable''' (also called a '''sentence letter,<ref name=":13">{{Cite book |last=Howson |first=Colin |author-link=Colin Howson |title=Logic with trees: an introduction to symbolic logic |date=1997 |publisher=Routledge |isbn=978-0-415-13342-5 |___location=London; New York |pages=5}}</ref>''' '''sentential variable,''' or '''sentential letter''') is an input [[variable (mathematics)|variable]] (that can either be '''true''' or '''false''') of a [[truth function]]. Propositional variables are the basic building-blocks of [[propositional formula]]s, used in [[propositional logic]] and [[Higherhigher-order logic|higher-order logics]]s.

== Uses ==

Formulas in logic are typically built up recursively from some propositional variables, some number of [[logical connective]]s, and some [[logical quantifier]]s. Propositional variables are the [[atomic formula]]s of propositional logic, and are often denoted using capital [[Latin script|roman letters]] such as <math>P</math>, <math>Q</math> and <math>R</math>.<ref name=":0">{{Cite web|date=2020-04-06|title=Comprehensive List of Logic Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/logic-symbols/|access-date=2020-08-20|website=Math Vault|language=en-US}}</ref><ref>{{Cite web|title=Predicate Logic {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/predicate-logic/|access-date=2020-08-20|website=brilliant.org|language=en-us}}</ref>

;Example

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* 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 [[metavariable]]s, which appear in the [[Propositional_logic#Example_1._Simple_axiom_system|typical axioms of [[propositional calculus]]; the latter effectively range over well-formed formulae, and are often denoted using lower-case greek letters such as <math>\alpha</math>, <math>\beta</math> and <math>\gamma</math>.<ref name=":0" />

== Predicate logic ==

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== See also ==

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* [[Boolean algebra (logic)]]

* [[Boolean datatypedata type]]

* [[Boolean ___domain]]

* [[Boolean function]]

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* [[Logical value]]

* [[Predicate variable]]

* [[Propositional logic]]

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== References ==

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== Bibliography ==

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

{{Mathematical logic}}

[[Category:Propositional calculus]]

[[Category:Concepts in logic]]

[[Category:Logic symbols]]

{{logic-stub}}