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Diagram (mathematical logic)

Definition[edit]

Let ${\displaystyle {\mathcal {L}}}$ be a first-order language and ${\displaystyle T}$ be a theory over ${\displaystyle {\mathcal {L}}.}$ For a model ${\displaystyle {\mathfrak {A}}}$of${\displaystyle T}$ one expands ${\displaystyle {\mathcal {L}}}$ to a new language

${\displaystyle {\mathcal {L}}_{A}:={\mathcal {L}}\cup \{c_{a}:a\in A\}}$

by adding a new constant symbol ${\displaystyle c_{a}}$ for each element ${\displaystyle a}$in${\displaystyle A,}$ where ${\displaystyle A}$ is a subset of the domain of ${\displaystyle {\mathfrak {A}}.}$ Now one may expand ${\displaystyle {\mathfrak {A}}}$ to the model

${\displaystyle {\mathfrak {A}}_{A}:=({\mathfrak {A}},a)_{a\in A}.}$

The positive diagram of ${\displaystyle {\mathfrak {A}}}$, sometimes denoted ${\displaystyle D^{+}({\mathfrak {A}})}$, is the set of all those atomic sentences which hold in ${\displaystyle {\mathfrak {A}}}$ while the negative diagram, denoted ${\displaystyle D^{-}({\mathfrak {A}}),}$ thereof is the set of all those atomic sentences which do not hold in ${\displaystyle {\mathfrak {A}}}$.

The diagram ${\displaystyle D({\mathfrak {A}})}$of${\displaystyle {\mathfrak {A}}}$ is the set of all atomic sentences and negations of atomic sentences of ${\displaystyle {\mathcal {L}}_{A}}$ that hold in ${\displaystyle {\mathfrak {A}}_{A}.}$^[1][2] Symbolically, ${\displaystyle D({\mathfrak {A}})=D^{+}({\mathfrak {A}})\cup \neg D^{-}({\mathfrak {A}})}$.

See also[edit]

• Elementary diagram

References[edit]

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