INCLUSION CRITERIA
An entry belongs in this subdivision if and only if its subject is a construction that builds one logic, theory, or structure from others, or a theorem about what such a construction preserves.
Required: At least one of the following:
- A combination operation on logics (fusion, product, fibring, dovetailing, temporalization, E-connection) defined on their languages, semantics, or consequence relations
- A combination operation on theories or structures (union of theories over disjoint signatures, generalized product) whose components are combined without being merged
- A transfer theorem stating that a property of the components (completeness, decidability, finite axiomatizability, interpolation, the finite model property) does or does not hold of the combination
- A collapse or non-conservativity result about a combination
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).
Boundary: The combined systems live where their apparatus places them and are cross-listed here—products and fusions of modal logics are Modal, many-dimensional modal logics are Modal, and the combination method is what lives here. Institutions and logical frameworks describe what a logic is and are not combination operations; they are cross-listed from Model-Theory and Applications/Computation where a combination result depends on them.
Boundary: The second Required clause was added when Feferman–Vaught and Nelson–Oppen were catalogued. Combination of logics, of theories, and of structures are three constructions with one shape—components, an operation, and a question about what survives—and the original criteria admitted only the first because they were written with the modal and fibring literature in view. The test is whether the entry's subject is the operation and its transfer question. A generalized product studied as a model-theoretic tool is cross-listed with Model-Theory; a union of theories studied as a decision procedure is cross-listed with Applications/Verification; the transfer theorem itself is here.