Basic Logic Symbols

Question

I’m assuming that T means a given theory...I’m wondering if this is correct. Additionally, I’m wondering what the capital phi and lowercase phi with the brackets are?

Answer 1

The definition means just what it says. $L$ is a set of formulas, and $$T\cup \Phi\cup \{\varphi\}\subseteq L.$$

Unpacking a bit: this is just the same as saying that $T$ is a subset of $L$, $\Phi$ is a subset of $L$, and $\varphi$ is an element of $L$. That's it!

If you define an $L$-formula to be an element of the set $L$ and an $L$-theory to be a set of $L$-formulas, then $T$ and $\Phi$ are $L$-theories and $\varphi$ is an $L$-formula.