MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

62 Robert Goldblattwhether a variety of BAO’s that is closed under canonical embedding algebrasmust be generated by the complex algebras of some elementary class of relationalstructures. This remained a perplexing open problem for three decades, duringwhich time a positive answer was found for all of the canonically closed varieties ofmodal algebras, cylindric algebras and relation algebras that had been investigated.Eventually however it was discovered that the converse of (ii) fails in general, anddoes so as badly as it could. This is shown by Goldblatt, Hodkinson and Venema[2004; 2003], exhibiting 2 ℵ0 different canonical logics that are not characterised byany elementary class. These examples all have the finite model property. Theyinclude logics of every degree of unsolvability, and in particular undecidable logicswith decidable sets of axioms. Some of the examples are based on ideas from theproof of the non-canonicity of the McKinsey axiom, while others use constructionsfrom the theory of graph colouring, and are related to the modal logic KMT studiedby George Hughes [1990]. The validating frames for KMT can be described as thosedirected graphs satisfying the non-elementary condition that the set {y : xRy} ofchildren of any node x has no finite colouring. The logic has an infinite sequenceof axioms whose n-th member rules out colourings that use n colours. But KMT isalso characterised by the elementary class of graphs whose edge relation R satisfies∀x∃y(xRyRy), meaning that every node has a reflexive child. The canonical KMTframesatisfies this condition.Some of the logics that violate the converse of (ii) also have axioms that imposereflexive points on canonical frames. But now a canonical frame is essentially thedisjoint union of a family of directed graphs, and it is only the infinite membersof the family that are required to have a reflexive point to ensure canonicity. Thisis a non-elementary requirement. The proof that the logics are never elementarilycharacterised involves a famous piece of graph theory of Paul Erdős [1959], whoshowed that for each integer n there is a finite graph G n whose chromatic numberand girth are both greater than n, the girth being the length of the shortest cyclein the graph and the chromatic number being the smallest number of coloursneeded to colour it. The essence of the application is that if a certain logic Λ werecharacterised by an elementary class C, and infinitely many of the G n ’s validatedΛ, then by a compactness argument it would follow that C contained an infinitegraph that had no cycles of odd length. But such a graph can be coloured usingonly two colours, a property that invalidates one of the axioms defining Λ. Hencethe existence of C is impossible.7 SOME MATHEMATICAL MODALITIESThe seed of relational semantics sown in the 1950’s has grown into a tree withmany branches. The most notable new dimension of activity beyond that alreadydescribed has been the application of relational modal semantics to a range offormalisms of computational and mathematical interest. This final section willbriefly survey some studies of this kind, providing a sketch of the key ideas and a