This allows for rank-theoretic theories of belief revision and of nonmonotonic reasoning, which are the topic of the final two sections.
(If we take necessity measures as primitive rather than letting them be induced by possibility measures, and if we continue to follow the rank-theoretic policy of adopting a well-ordered range, we can obviously also define countably and completely minimitive necessity measures.)
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The set-theoretic model can be criticized on the grounds that it models the type of functions as the set of all set-theoretic functions, in spite of the fact that a function in type theory is always computable, whereas a set-theoretic function may not be.
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But model-theoretic validity (or derivability) might be theoretically adequate in some way even if some possible meaning-assignments are not modeled straightforwardly by (actual) set-theoretic structures.
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Although both methods are property-theoretic, the first method yields a precursor of set-theoretic versions of Curry’s paradox, while the second yields a precursor of truth-theoretic versions.
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Proof-theoretic semantics is an alternative to truth-condition semantics.
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The proof-theoretic and model-theoretic perspectives have been considered as providing rival accounts of logical consequence.
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Taking the topic of intensionality seriously may reshape many fields of proof-theoretic semantics.
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As an immediate application of this graph-theoretic setting, the preservation of the finite-model property by graph-theoretic fibring was proved in Coniglio et al. 2011.
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Fortunately, even a rough description of the model-theoretic interpretation is enough to show that no such “close ties” exist.
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As mentioned, the game-theoretic semantics for dependence logic is a variant of the imperfect-information semantics for independence-friendly logic, which is itself an adaptation of the game-theoretic semantics of first-order logic.
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Besides the model-theoretic side, axiomatic investigations of truth and related paradoxes have become increasingly important since the seminal papers of Friedman and Sheard 1987, Feferman 1991.
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The logic of negation may be presented in quite different ways, by considering various styles of proof systems (axiom systems, sequent calculi, systems of natural deduction, tableaux, etc.) or different kinds of semantics (algebraic, model-theoretic, proof-theoretic, game-theoretic, etc.).
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These calculi of reasoning also apply to games, where they can capture aspects of specialized game-theoretic argumentation.
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For example in Boole’s case the set-theoretic consequences that he relies on are all easily provable by formal proofs in first-order logic, not even using any set-theoretic axioms; and by the completeness theorem (see the entry on classical logic) the same is true for first-order logic.
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Insofar as they follow Boole, modern textbooks of logic establish that English arguments are valid by reducing them to model-theoretic consequences.
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In proof-theoretic semantics, proof systems of a suitable form are seen as providing a meaning theory, see the entry proof-theoretic semantics.
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Importantly, the set-theoretic objects are found in more than one of the axioms of the theory, and Newton’s calculus is reconstructed in a novel, set-theoretic form.
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Within philosophy, proof-theoretic semantics has mostly figured under the heading “theory of meaning”.
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In recent years, the most widely discussed version of Skolem's Paradox has come come in (one version of) Hilary Putnam's so-called “model-theoretic argument against realism.”
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