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 strange correspondence $T\mapsto 0$, $F\mapsto 1$, $\wedge\mapsto +$ 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 ideals of $R$ and 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.

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 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?