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

Latest revision as of 00:21, 28 January 2019

The "models" symbol, $⊨$ , is used for several purposes in mathematical logic. Roughly, there are two basic purposes:

When the symbol that comes before " $⊨$ " is a structure/interpretation, then it says something about truth in that structure/interpretation.
When the symbol that comes before " $⊨$ " is a sentence or set of sentences, then it says something about semantic consequence (also called logical consequence, logical implication, semantic implication). In this case, we are talking about all possible structures/interpretations.

In either of the above two purposes, we are talking about the semantics (rather than syntax) of a logical system.

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
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] The structure $A$ makes true the formula $ϕ$ .
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 $Γ$ . $ϕ$ is true in every model of $Γ$ .
Nothing	$⊨$	A sentence or formula $ϕ$	$ϕ$ is valid.^[2] $ϕ$ is a tautology (especially in the case of propositional logic).

Notes

Other tricky things:

Some books only use the models symbol for one of the two use cases. E.g. Boolos/Burgess/Jeffrey only uses the symbol for truth in an interpretation. Therefore you might be really confused when you start reading other books and you start seeing stuff like $Γ ⊨ ϕ$ .
In the Boolos/Burgess/Jeffrey book, if a set of sentences comes after $⊨$ , then the sentences are taken disjunctively rather than conjunctively. (see p. 168)
https://math.stackexchange.com/a/2506938/35525

References

↑ ^1.0 ^1.1 Derek Goldrei. Propositional and Predicate Calculus. p. 134.
↑ Boolos; Burgess; Jeffrey. Computability and Logic. p. 168.

[goldrei-1] 1.0 ^1.1 Derek Goldrei. Propositional and Predicate Calculus. p. 134.

[2] Boolos; Burgess; Jeffrey. Computability and Logic. p. 168.

[1]

[2]

@@ Line 18: / Line 18: @@
 ! 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 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>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /><br>The structure <math>\mathfrak A</math> makes true the formula <math>\phi</math>. ||
 |-
-| 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>.<br><br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||
+| 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><math>\Gamma</math> logically implies <math>\phi</math>.<br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>.<br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||
 |-
-| Nothing || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is valid.<ref>Boolos; Burgess; Jeffrey. ''Computability and Logic''. p. 168.</ref><br><br><math>\phi</math> is a tautology (especially in the case of propositional logic). ||
+| Nothing || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is valid.<ref>Boolos; Burgess; Jeffrey. ''Computability and Logic''. p. 168.</ref><br><math>\phi</math> is a tautology (especially in the case of propositional logic). ||
 |}
@@ Line 31: / Line 31: @@
 * Some books only use the models symbol for one of the two use cases. E.g. Boolos/Burgess/Jeffrey only uses the symbol for truth in an interpretation. Therefore you might be really confused when you start reading other books and you start seeing stuff like <math>\Gamma \models \phi</math>.
 * In the Boolos/Burgess/Jeffrey book, if a set of sentences comes after <math>\models</math>, then the sentences are taken disjunctively rather than conjunctively. (see p. 168)
+* https://math.stackexchange.com/a/2506938/35525
 ==See also==