I'm a recent graduate who majored in computer science and minored in applied mathematics at the University of Washington. My interests span several theoretical and applied disciplines, including computational complexity theory, logic and programming languages, and formal methods.

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Theorem (Law of the Excluded Middle, Model Theoretic Version). For any predicate $A$ and structure $M$ on a language $\mathcal{L}$, $$M \vDash (A(x) \lor \neg A(x))$$

Proof. Let $s$ be a variable assignment. Note that $|M| = A^M \cup (|M| - A^M)$, $A$ is modelled by some subset of $|M|$. Suppose we have $\text{Val}_s^M(x) = x_s^M$ ($x$ is interpreted as $x_s^M$ in the meta-language). Then there are two cases:

1. $x_s^M \in A^M$. By definition, we have that $M,s \vDash A(x)$.
2. $x_s^M \in |M| - A^M$. Then, $x_s^M \notin A^M$, and so we have $M, s \nvDash A(x)$. By definition, $M,s \vDash \neg A(x)$.

So it is the case that $M, s \vDash (A(x) \lor \neg A(x))$. Since $s$ was arbitrary, we may disregard it, giving us the claim. $\qquad \square$