Using the following table to you convert between propositional logic and arithmetic of multivariate polynomials over $\mathbb{F}_2$: $$ \mbox{TRUE} \leftrightarrow 1 $$ $$ \mbox{ FALSE} \leftrightarrow 0 $$ $$ X \mbox{ or } Y \leftrightarrow xy+x+y$$ $$ X \mbox{ and } Y \leftrightarrow xy$$ $$ !X \leftrightarrow x+1 $$ So a proposition $P(X_1,X_2,\ldots, X_n)$ can be satisfied if and only if the corresponding polynomial equation $p(x_1,x_2,\ldots,x_n)=1$ has a solution. For example, the proposition $$X \mbox{ and } !X$$ is not satisfiable. This corresponds to the fact the polynomial $x(x+1)=1$ or $x^2 + x +1=0$ has no solutions over $\mathbb F_2$.

We now should do in logic as we do in algebra. Since this proposition isn't satisfiable over our standard logic we create an algebraic extension of logic where truth values now live in
$$ \mathbb F_2[x]/(x^2 + x + 1)!$$

I don't know how to extend these ideas to first order logic.