model theory - completeness of a theory

Question

For an $L$-structure $\mathcal{A}$, the $L$-theory of $\mathcal{A}$ is the set of $L$-sentences: $$ \mathrm{Th}_L(\mathcal A) = \{\sigma : \mathcal A \models \sigma\} $$

Prove that $\mathrm{Th}_L(\mathcal A)$ is complete.

Whys is this true? Why can't there be an $L$-sentence such that it is neither true or false?

Answer 1

Assume there's an $L$-sentence $\sigma$ that's neither true nor false. Then we have that $\mathcal{A}\nvDash\sigma$ and $\mathcal{A}\nvDash\neg\sigma$. But this is a contradiction since $\mathcal{A}\vDash\sigma$ if and only if $\mathcal{A}\nvDash\neg\sigma$, and vice versa, by definition of what a structure is.