The following is a somewhat vague question concerning logic, but with ideas from algebraic geometry (see in particular the example at the end). The vagueness is in the notion of "language".

Let $A$ be a set and fix some language in which to write propositions which assign a value of true or false to each element of A. Let $R$ be the set of such propositions about elements of $A$, and suppose that $R$ contains "True" and "False" and is closed under "and" ($\wedge$) and "or" ($\vee$).

Now let $V\subseteq A$ be a subset and consider the set $I(V)\subseteq R$ such that $r\in I(V)$ iff $r$ holds of every element $v\in V$. This set $I(V)$ has the following properties:

- "True" is in $I(V)$,
- If $i$ and $j$ are in $I(V)$ then $i\wedge j$ is in $I(V)$, and
- If $i$ is in $I(V)$ and $r\in R$ then $r\vee i$ is in $I(V)$.

Thus $I(V)$ satisfies the axioms of an "ideal" in the "ring" $R$ if we take the following "strange correspondence" between statements and algebraic operations: $T\mapsto 0$, $F\mapsto 1$, $\wedge\mapsto +$ and $\vee\mapsto \times$. Note that in fact $R$ is a commutative "Rig" (ring without negatives) under this "strange correspondence." The correspondence is "strange" because we usually think of "or" as $+$, "and" as $\times$, "true" as $1$, and "false" as $0$. But we can find intuition for this correspondence via Example 2 below.

Suppose we define an ideal in $R$ to be a subset $I\subseteq R$ satisfying conditions 1,2,3. Given an ideal $I$ we can consider the set $Z(I)\subseteq A$ of all elements $a\in A$ such that $i$ holds of $a$ for each $i\in I$. Call such subsets "closed".

Observation 1: There is an (order-reversing) correspondence between the ideals of $R$ and the closed subsets of $A$.

Example 2: Suppose we take the language of commutative rings. Say $A$ is the set ${\mathbb R}^n$ and $R$ is the set of statements of the form $f(x_1,\ldots,x_n)=0$ where $f$ is a polynomial with real coefficients. The set of such statements forms a ring, by operating on the polynomials; that is $R\cong {\mathbb R}[x_1,\ldots,x_n]$.

Given a subset $V\subseteq A$ we can consider those equations that are true of all points in $V$. Such a subset will satisfy conditions 1,2,3 above (where "True" is 0=0). We see that the "strange correspondence" discussed above does make sense: if $f=0$ and $g=0$ hold of every point in $V$ then so does $f+g=0$; for any equation $r=0$ in $R$, if $f=0$ holds of every point in $V$ then so does $rf=0$. The empty subset of $A$ is satisfied by $1=0$ (or "false"). In fact, Observation 1 is the basic observation of algebraic geometry in this context.

Question 3: Has anyone looked at this correspondence? Can it be made more rigorous? Can algebro-geometric notions (like schemes) be applied to other "languages" in an interesting way?

deductively closed theories. The space you're describing is basically the Stone space of the Lindenbaum algebra, or thespace of types, which is widely studied in model theory. The Compactness Theorem takes a very nice form when stated in this language. – François G. Dorais♦ Apr 28 '10 at 8:18