I was reading through some lecture notes about a theory in propositional calculus. I was wondering if someone could help me understand an ambiguous statement.
"if $\alpha \in Th(A)$ then for any $\beta \in \mathscr{F}^0$, $\alpha \lor \beta$, $\beta \implies \alpha \in Th(A)$."
I think this is saying if $\alpha \in Th(A)$ then for any $\beta \in \mathscr{F}^0$, $(\alpha \lor \beta) \in Th(A)$ and $(\beta \implies \alpha) \in Th(A)$.
Can anyone confirm this, or give the correct interpretation?
Thank you.
Your conclusion is basically right, also if the details depend on the correct interpretation of the symbols.
Usually, in model theory, with $\text {Th}(\mathfrak A)$ we mean the set of all sentences true in the structure $\mathfrak A$.
See : Katrin Tent & Martin Ziegler, A Course in Model Theory, Springer (2012), page 15 :
Thus, we may apply it to the case of propositional logic, assuming that $A$ is a truth assignment.
Thus, the statement asserts that, if formula $\alpha$ is TRUE in $A$, then also formulas $(\alpha \lor \beta)$ and $(\beta \to \alpha)$ are TRUE in A.
And this is consistent with the truth tables for the two conenctives.
For a different reading of the symbols, see the comments below.