User:IssaRice/Computability and logic/Models symbol: Difference between revisions

Revision as of 21:10, 27 January 2019

The "models" symbol, $⊨$ , is used for several purposes in mathematical logic.

If $A$ is a structure/interpretation and $ϕ$ is a sentence, then $A ⊨ ϕ$ means ...
If $A$ is a structure/interpretation and $Γ$ is a set of sentences, then $A ⊨ Γ$ means ...
If $T$ is a theory and $ϕ$ is a sentence, then $T ⊨ ϕ$ means ...
If $T$ is a theory and $Γ$ is a set of sentences, then $T ⊨ Γ$ means ...
If $Σ$ is a set of axioms for a theory $T$ , and $ϕ$ is a sentence, then $Σ ⊨ ϕ$ means ...
If $Σ$ is a set of axioms for a theory $T$ , and $Γ$ is a set of sentences, then ...
if $ϕ$ is a formula (or wff), then ...
also the variant without anything in front, e.g., $⊨ ϕ$

Part before " $⊨$ "	$⊨$	Part after $⊨$	Possible pronunciations	Meaning
A structure/interpretation $A$	$⊨$	A sentence or formula $ϕ$	The structure $A$ satisfies the formula $ϕ$ .^[1] The formula $ϕ$ is true in $A$ .^[1]
A set of sentences or formulas $Γ$	$⊨$	A sentence or formula $ϕ$	$ϕ$ is a logical consequence of $Γ$ . $Γ$ logically implies $ϕ$ . $ϕ$ is a semantic consequence of $Γ$ .

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-! Part before "<math>\models</math>" !! <math>\models</math> !! Part after <math>\models</math> !! Pronunciation !! Meaning
+! Part before "<math>\models</math>" !! <math>\models</math> !! Part after <math>\models</math> !! Possible pronunciations !! Meaning
 |-
 | A structure/interpretation <math>\mathfrak A</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || The structure <math>\mathfrak A</math> satisfies the formula <math>\phi</math>.<ref name="goldrei">Derek Goldrei. ''Propositional and Predicate Calculus''. p. 134.</ref><br><br>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /> ||
+|-
+| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><br><math>\Gamma</math> logically implies <math>\phi</math>.<br><br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>. ||
 |}