# The meaning of Th(E) in default logic?

Hello I couldn't find an answer for that question anywhere. what is the meaning of Th() ? and also the meaning of the more specific Th(E)=E from Reiter's paper

• In first order logic,given a language $\mathcal{L}$ and a $\mathcal{L}$-structure $M$, $Th(M)$ is the set of $\mathcal{L}$-sentences $\phi$ such that $M\models \phi$. I am not sure if this is useful for you. Could you include a link of "Reiter's" paper? Mar 22, 2016 at 12:14
• Here is the link: cs.nott.ac.uk/~psznza/G52HPA/articles/Reiter:80a.pdf It was very helpful thank you. Mar 22, 2016 at 13:41

The notation $\operatorname{Th}(\cdot)$ is commonly used in logic for two different meanings:
• $\operatorname{Th}(\mathcal M)$, where $\mathcal M$ is a structure, is the theory of $\mathcal M$, defined as the set of all formulas (or sometimes only sentences) that are true in $\mathcal M$: $$\operatorname{Th}(\mathcal M) = \{ \varphi \mid \mathcal M\Vdash \varphi \}$$
• $\operatorname{Th}(T)$, where $T$ is a set of formulas, is the set of theorems of the theory $T$: $$\operatorname{Th}(T) = \{ \varphi \mid T\vdash \varphi \}$$
If the context is $\operatorname{Th}(E)=E$, only the second of these makes sense -- and in that case the equation $\operatorname{Th}(E)=E$ simply asserts that $E$ is a set of formulas that is closed under logical entailment.