Ideals of statements?

Question

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 $\vee\mapsto \times$. 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?

Answer 1

Yes, of course, the algebraic aspects of logic have been very well studied. There is a lot to say about this, but since I am supposed to be on a voluntary MO hiatus until the end of the semester, I will only mention a few things.

You might want to ask for your "ideals" to be closed under logical equivalence too. Otherwise, statements of length 17 or more form a rather silly ideal. With this change, your "ideals" are called deductively closed theories.

The space you're describing is basically the Stone space of the Lindenbaum algebra of your language. The Lindenbaum algebra is the Boolean algebra which consists of all sentences of the language modulo logical equivalence. This construction also makes sense over a nontrivial base theory T, where logical equivalence is replaced by T-provable equivalence.

Another name for this Stone space is the space of (complete) 0-types over the theory T. The more general space of (complete) n-types is obtained in the same way by using formulas in n fixed variable symbols x₁,...,x_n (instead of sentences, which have no free variables). These spaces of types and their topology are a central concept in model theory. One usually takes T to be a the complete theory of a structure M. Then the space of 0-types has only one point since T is complete and n-types correspond to coherent things that one could say about a n-tuple of elements of M.

Several theorems of model theory have interesting meanings in this context. For example, the Compactness Theorem corresponds to the fact that Stone spaces are compact, and the Omitting Types Theorem corresponds to the Baire Category Theorem (for compact Hausdorff spaces). Some leading model theorists have supported the view that model theory should become more and more algebraic/geometric, which is indeed a current trend.