In [1] Craig introduced a proof procedure for first order classical logic called linear reasoning. In it, a proof of P ⊃ Q consists of a sequence of formulas, each of which implies the next, beginning with P and ending with Q. And one of the formulas in the sequence will be an interpolation formula for P ⊃ Q. Indeed, this was the first proof of the Craig interpolation theorem, some of whose important consequences were demonstrated in a companion paper [2]. In this paper we present systems of linear reasoning for several standard modal logics: K, T, K4, S4, D, D4, and GL. Similar systems can be constructed for several regular, nonnormal modal logics too, though we do not do so here. And just as in the classical case, interpolation theorems are easy consequences. Such theorems are well known for the logics considered here. There is a model theoretic argument in [6], an argument using Gentzen systems in [8], an argument using consistency properties in [4] and [5], and an argument using symmetric Gentzen systems in [5]. This paper presents what seems to be the first modal proof that follows Craig's original methods. We note that if the modal rules given here are dropped, a classical linear reasoning system results, which is related to, but not the same as those in [1] and [10].

Since the basic linear reasoning ideas are fully illustrated by the propositional case, we present that first, to keep the clutter down. Later we show how the techniques can generally be extended to encompass quantifiers. We do not follow [1] in making heavy use of prenex form, since it is not generally available in modal logics. Fortunately, it plays no essential role.

If L is a first-order language, then an L-structure A is called quantifier-eliminable (QE) if every L-formula is equivalent in A to a formula without quantifiers.

The classification problem for QE groups and rings has received attention in work by Berline, Boffa, Cherlin, Feigner, Macintyre, Point, Rose, the present authors, and others. In [1], Berline and Cherlin reduced the problem for rings of prime characteristic p to that for nilrings, but also constructed countable QE nilrings of characteristic p. Likewise, in [3], we constructed countable QE nil-2 groups. Both results can be viewed as “nonstructure theorems”, in that they provide negative evidence for any attempt at classification. In the present paper we show that the situation is equally bad (or rich, depending on one's point of view) for commutative rings:

Theorem 1. For any odd prime p, there existcountable QE commutative nilrings of characteristic p.

This solves a problem posed in [1]. We remark that the examples we produce are uniformly locally finite, hence ℵ0-categorical. A more algebraic description is that each of our rings R is uniformly locally finite (in fact, R3 = 0) and homogeneous, in the sense that any isomorphism of finitely generated subrings extends to an automorphism of R.

Theorem 1 does not cover the case p = 2, and we show that for commutative rings this case is in fact exceptional:

Theorem 2. There exist exactly two nonisomorphic countably infinite QE commutative nilrings of characteristic 2.

In this paper we investigate the connections between categoricity and ranks. We use stability theory to prove some old and new results.

Much of the literature on the model theory of modal logics suffers from two weaknesses. Firstly, there is a lack of generality; theorems are proved piecemeal about this or that modal logic, or at best small classes of logics. Much of the literature, e.g. [1], [2], and [3], confines attention to structures with the expanding domain property (i.e., if wRu then Ā(w) ⊆ Ā(u)); the syntactic counterpart of this restriction is assumption of the converse Barcan scheme, a move which offers (in Russell's phrase) “all the advantages of theft over honest toil”. Secondly, I think there has been a failure to hit on the best ways of extending classical model theoretic notions to modal contexts. This weakness makes the literature boring, since a large part of the interest of modal model theory resides in the way in which classical model theoretic notions extend, and in some cases divide, in the modal setting. (The relation between α-recursion theory and classical recursion theory is analogous to that between modal model theory and classical model theory. Much of the work in α-recursion theory involved finding the right definitions (e.g., of recursive-in) and separating concepts which collapse in the classical case (e.g. of finiteness and boundedness).)

The notion of a well-behaved modal logic is introduced in §3 to make possible rather general results; of course our attention will not be restricted to structures with the expanding domain property. Rather than prove piecemeal that familiar modal logics are well-behaved, in §4 we shall consider a class of “special” modal logics, which obviously includes many familiar logics and which is included in the class of well-behaved modal logics.