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:

1. "True" is in $I(V)$,
2. If $i$ and $j$ are in $I(V)$ then $i\wedge j$ is in $I(V)$, and
3. 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 $\vee\mapsto \times$.

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?