• README ⤓ .txt 2026-07-17T121634.146 000000000000792
  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?
• E-Connections ⤓ .md 2026-07-17T120407.600 000000000000840
  Oliver Kutz, Carsten Lutz, Frank Wolter, and Michael Zakharyaschev, "E-connections of abstract description systems" (Artificial Intelligence 156, 2004), with the earlier "E-connections of description systems" and Kutz's thesis (2004). Built for a stated purpose: fusion is too weak to say anything and products are undecidable, so the field needed a construction in the gap, and this is the one that was found.
• Feferman-Vaught Theorem ⤓ .md 2026-07-17T120407.600 000000000000920
  Solomon Feferman and Robert Vaught, "The first order properties of products of algebraic systems" (Fundamenta Mathematicae 47, 1959), generalizing Mostowski's "On direct products of theories" (JSL 17, 1952), which had proved the case Skolem arithmetic needs. Läuchli, Shelah, and Gurevich extended the method to monadic second-order logic, where it is called the composition method. Makowsky's "Algorithmic uses of the Feferman–Vaught theorem" (APAL 126, 2004) is the survey and the argument for its computational reading.
• Fibring ⤓ .md 2026-07-17T120407.600 000000000000792
  Dov Gabbay, "Fibred semantics and the weaving of logics" (Journal of Symbolic Logic, 1996) and Fibring Logics (Oxford Logic Guides 38, 1999). Developed algebraically by Sernadas, Sernadas, and Caleiro (1999), whose categorical account made the collapsing problem visible. The most general of the combination methods: it joins arbitrary logics, not just modal ones.
• Fusion and Products ⤓ .md 2026-07-15T233630.000 000000000035624
  Fusion (independent join) was studied by Thomason (1980) and Fine and Schurz (1996); the transfer theorems are theirs and Kracht and Wolter's (1991). Products were introduced by Shehtman (1978) and Segerberg (1973), and the subject was systematized in Gabbay, Kurucz, Wolter, and Zakharyaschev, Many-Dimensional Modal Logics (2003). The two constructions look similar and behave completely differently, which is the point.
• Nelson-Oppen Combination ⤓ .md 2026-07-17T120407.600 000000000000952
  Greg Nelson and Derek Oppen, "Simplification by cooperating decision procedures" (ACM TOPLAS 1, 1979). Robert Shostak's alternative method (1984) covered a narrower class more efficiently and its original correctness proof was wrong — Rueß and Shankar (2001) gave the first correct version, and Shostak's method is now understood as an instance of Nelson–Oppen. Tinelli and Harandi (1996) gave the clean model-theoretic proof; Ghilardi (2004) extended the result to non-disjoint signatures under model-completeness conditions.
• Temporalization and Parameterization ⤓ .md 2026-07-17T120407.600 000000000001024
  Marcello Finger and Dov Gabbay, "Adding a temporal dimension to a logic system" (Journal of Logic, Language and Information 1, 1992), with "Combining temporal logic systems" (Notre Dame Journal of Formal Logic, 1996) extending the method. Caleiro, Carnielli, Coniglio, Sernadas, and Sernadas (1999) generalized it to parameterization, and showed that parameterization is a particular case of constrained fibring — which places the method inside the fibring picture rather than beside it.
• CRITERIA ⤓ .txt 2026-07-17T120407.600 000000000000808
  Not sufficient: A logic that happens to have two kinds of operator (that belongs where its apparatus places it—epistemic temporal logic is a modal logic, not a combination method, however it was built). A translation or embedding of one logic into another, which relates two given logics rather than constructing a third. A framework for defining logics uniformly (Metatheory/Categorical for the categorical ones, Model-Theory for institutions). A theory that is a union of others but is studied for its models rather than for what the union preserves (Theories).