Propositional variable: Difference between revisions

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{{short description|Variable 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 aan input [[Variablevariable (mathematics)|variable]] which(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 [[higher-order logicslogic]]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 Forare example,often indenoted ausing givencapital propositional[[Latin logicscript|roman letters]] such as <math>P</math>, we<math>Q</math> mightand define<math>R</math>.<ref>{{Cite aweb|title=Predicate formulaLogic as{{!}} followsBrilliant Math & Science Wiki|url=https://brilliant.org/wiki/predicate-logic/|access-date=2020-08-20|website=brilliant.org|language=en-us}}</ref>

;Example

*Every propositional variable is a formula.

In a given propositional logic, a formula can be defined as follows:

*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]] ∧), then ''(X b Y)'' is a formula. (Note the parentheses.)

* Every propositional variable is a formula.

In this way, all of the formulas of propositional logic are built up from propositional variables as a basic unit.

* 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]] ∧), thenthe 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 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>.

Propositional variables are represented as nullary [[predicates]] in [[first order logic]].

==See alsoPredicate 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.

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The internal structure of propositional variables contains [[predicate symbol|predicate letters]] such as P and Q, in association with [[bound variable|bound]] individual variables (e.g., x, ''y''), individual constants such as ''a'' and ''b'' ([[singular term]]s from a [[___domain of discourse]] D), ultimately taking a form such as P''a'', ''a''R''b''.(or with parenthesis, <math>P(11)</math> and <math>R(1, 3)</math>).<ref>{{Cite web|date=2015-06-24|title=Mathematics {{!}} Predicates and Quantifiers {{!}} Set 1|url=https://www.geeksforgeeks.org/mathematic-logic-predicates-quantifiers/|access-date=2020-08-20|website=GeeksforGeeks|language=en-US}}</ref>

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 ==

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* [[Boolean algebra (logic)]]

* [[Boolean datatypedata type]]

* [[Boolean ___domain]]

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* [[Boolean function]]

* [[Logical value]]

* [[Predicate variable]]

* [[Propositional logic]]

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== References ==

{{reflist}}

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

== Bibliography ==

[[Category:Propositional calculus]]

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

[[Category:Concepts in logic]]

{{Mathematical logic}}

[[Category:Propositional calculus]]

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[[Category:Concepts in logic]]

[[Category:Logic symbols]]

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[[pl:Zmienna zdaniowa]]

[[zh:命题变量]]