|Fine-grained, logic, philosophy: statements can be be more or less detailed. E.g. the fine-grained statement (x)(y)(x = y > (N(x = x) > N(x = y))) could be displayed in a coarse-grained way as A > B. For the decision of problems the coarse-grained form is often inadequate. See also decidability, possible worlds, situation semantics, possible world semantics, hyperintensionality.|
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Fine-grained/Field: E.g. sets of possible worlds are coarser than possible belief objects: eg the axioms of set theory and the Banach-Tarski theorem (BTT, strongly counter-intuitive) are logically equivalent, i.e. valid in the same worlds - but the BTT is not believed by all the people who believe the axioms of set theory - sentence meaning/Lewis: is fine grained - belief/Lewis: what one believes, is coarse - punch line: so the conviction is always the same: with and without BTT!
FieldVsStalnaker: sets of possible worlds are too coarse to distinguish beliefs - e.g. set theory with and without Banach-Tarski theorem are the same.
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980