COMBINATION
The methods by which logics, theories, and structures are built out of others, and the theorems that say what survives the construction. Fusion joins two modal logics over a shared classical base; products let their accessibility relations commute; fibring joins arbitrary logics by gluing their languages at a shared category of formulas; temporalization refuses to glue them at all and layers one on top of the other; E-connections keep their domains disjoint and run relations between them. Nelson–Oppen combines first-order theories over disjoint signatures; Feferman–Vaught combines structures into generalized products. Each is a construction, and each comes with a transfer question: given that the components are complete, decidable, or finitely axiomatizable, is the combination?
The subject is the relation between logics rather than any logic in particular, which is why it lives in Metatheory. The combined systems themselves — the products, the fusions, the many-dimensional modal logics — are cross-listed with the divisions their components come from.
The negative results are the interesting half, and taken together the positive and negative ones say the same thing. Fusion transfers almost everything and adds no interaction. Temporalization transfers everything unconditionally and forbids interaction by construction. Feferman–Vaught reduces a product's whole first-order theory to its factors' and the index set's, because coordinates in a product do not interact. Nelson–Oppen combines decision procedures across disjoint signatures, where the components can see nothing of each other but equalities on finitely many shared variables. Products force one commutation axiom and lose decidability. Unconstrained fibring permits arbitrary interaction and collapses, so that the combination proves each component's theorems in the other's vocabulary.
The pattern is that transfer is bought with interaction, and the exchange rate is sharp rather than vague: what a combination cannot afford is enough interaction to encode a grid. Products become undecidable because two commuting relations address a plane and tilings reduce into it. E-connections keep decidability because disjoint domains leave no plane to address — and strengthening the link language with Booleans, inverses, or counting returns undecidability exactly when it returns the ability to force a grid. Nelson–Oppen's impoverished interface is the same restriction imposed on theories rather than frames. What a construction preserves is not obvious in advance, is rarely everything, and is decided by whether the components can build a plane between them.