Some Compactness Results for Modal Logic

A modal logic L is said to be compact if every L-consistent set of formulas has a model on a frame for L. Some large classes of compact (noncompact) logics are identified, and it is shown that there are uncountably many compact (noncompact) logics. A modal logic L is compact if every L-consistent set of formulas has a model on a frame for L, classically compact if a set of formulas fails to have a model on a frame for L only if some finite subset fails to have a model on a frame for L, canonical if determined by its canonical frame, and complete if determined by some class of frames. These four properties are related in an obvious way: CANONICALCY => COMPACTNESS => COMPLETENESS CLASSICAL COMPACTNESS Given the amount of attention that has been lavished upon canonicalcy and completeness, it is therefore mildly surprising that compactness has enjoyed relatively little press. The first explicit mention of it would seem to be found in Corcoran and Weaver [1]. However, they were working with a different concept of a model than the now standard one used here, and as a result the nice connection between canonicalcy and compactness is lost. (See [2], in which they show that the canonical logics Γand B are, on their account, noncompact.) Fine raises the issue of compactness at the end of [5], but his important work on this notion did not appear in print until more than a decade later with the publication of [6]. Even then, although he was undoubtedly aware that his argument could be generalized, only one example of a familiar, noncompact logic is actually mentioned. Hughes and Cress well [10] go a bit further, giving several examples. Unfortunately, their proofs (and suggested proofs) contain a seductive error, and *An earlier version of this paper was presented to the Annual Meeting of the Association for Symbolic Logic held in Chicago, Illinois, April 26-27, 1985. With the exception of Theorem 4, these results were obtained before I saw [6], [10], and [11]. Received August 31, 1987 286 GEORGE F. SCHUMM correct proofs appear only in [11]. Finally, Fitting [8] and van Fraassen [9] show how to obtain compactness without going through canonicalcy, though all of the logics they consider are in fact canonical. For complete logics compactness and classical compactness come to the same thing. But in general the two concepts pull apart. In [13] it is shown that there exists an incomplete logic L with the same frames as 54.3. Given the canonicalcy, and hence classical compactness, of the latter, it follows that L is also classically compact. Classical compactness therefore implies neither completeness, compactness, nor canonicalcy. Perhaps this helps explain why it is compactness rather than classical compactness which is generally discussed in modal contexts. As we shall see, completeness fails to imply compactness or even classical compactness —and fails with a vengeance. The tougher question, apparently still open, is whether compactness implies canonicalcy. It is becoming increasingly clear, however, that if there is a counterexample, it is not likely to be simple. Fine (in [6] and [7]) has shown that the two notions are equivalent over two broad and important classes of logics (the complete elementary and "subframe" logics), and other results of this sort are also to be found in the literature. Here I shall identify some additional classes, running across and up through the lattice of modal logics, over which compactness and canonicalcy stand or fall together. Let us say that a logic L is finitely accessible if no frame for L contains a world having infinitely many worlds accessible from it. Theorem 1 IfL is finitely accessible, then L is compact if and only ifL is canonical. Proof: It is enough to show that if L is finitely accessible and compact, then L is canonical. By compactness, every L-consistent set Γ has a model 2lr based upon a frame for L. Form the "union" of these models. To be precise, where {(Wh Ri>Φi)]i is a class of pairwise disjoint isomorphic copies of these Sir's, put W = \J Wh R = (J Rh and φ = \J φh and set 21 = (W,R,φ). Define a / / i function/from Wto the domain of %L = (WL,RL,φL), the canonical model forZ,, by f(u) = [A\(%u) )rA}. Since g = (W,R) is a frame for L, to see that %L — (WL>RL) is also a frame for L one needs only verify that %L is ap-morphic image of § under /. ( l)/is onto. (2) If uRυ, then f(u)RLf(v). (3) Supposing thatf(u)RLa, we need to show that α =f(v) for some v, where uRυ. Since (8lL,/(w)) \= 0 τ , (81,«) f= 0 τ . So u has at least one world accessible from it under R. Let υu . , vn be all such worlds, and suppose for a reductio that (*) {VAHWiVj) M ~ ( a L , α ) ¥A) for no Vi. Then, for each υh there is some At such that (21, Vj) N At and (2ίL,α) ψ Ah But now ( 8 , « ) ^ U i v. vA,), COMPACTNESS RESULTS 287