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Robert Goldblatt [31]Robert I. Goldblatt [1]
  1. Robert Goldblatt (2012). Topological Proofs of Some Rasiowa-Sikorski Lemmas. Studia Logica 100 (1-2):175-191.
    We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.
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  2. Robert Goldblatt & Tomasz Kowalski (2012). The Power of a Propositional Constant. Journal of Philosophical Logic (1):1-20.
    Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions (...)
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  3. Robert Goldblatt (2011). Grishin Algebras and Cover Systems for Classical Bilinear Logic. Studia Logica 99 (1-3):203-227.
    Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, (...)
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  4. Robert Goldblatt (2011). Quantifiers, Propositions, and Identity: Admissible Semantics for Quantified Modal and Substructural Logics. Cambridge University Press.
    Machine generated contents note: Introduction and overview; 1. Logics with actualist quantifiers; 2. The Barcan formulas; 3. The existence predicate; 4. Propositional functions and predicate substitution; 5. Identity; 6. Cover semantics for relevant logic; References; Index.
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  5. Galym Akishev & Robert Goldblatt (2010). Monadic Bounded Algebras. Studia Logica 96 (1):1 - 40.
    We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of (...)
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  6. Robert Goldblatt (2010). Functional Monadic Bounded Algebras. Studia Logica 96 (1):41 - 48.
    The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ∃ reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions. We show that FMBA is characterised by the disjunction of the equations ∃E = 1 and ∃E = 0. We also define a weaker notion of (...)
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  7. Robert Goldblatt & Michael Kane (2010). An Admissible Semantics for Propositionally Quantified Relevant Logics. Journal of Philosophical Logic 39 (1):73 - 100.
    The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p -instantiations of A . It is also shown that without the admissibility qualification many of the systems considered are (...)
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  8. Robert Goldblatt (2009). Conservativity of Heyting Implication Over Relevant Quantification. Review of Symbolic Logic 2 (2):310-341.
    It is known that propositional relevant logics can be conservatively extended by the addition of a Heyting (intuitionistic) implication connective. We show that this same conservativity holds for a range of first-order relevant logics with strong identity axioms, using an adaptation of Fine’s stratified model theory. For systems without identity, the question of conservatively adding Heyting implication is thereby reduced to the question of conservatively adding the axioms for identity. Some results in this direction are also obtained. The conservative presence (...)
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  9. Robert Goldblatt (2006). Final Coalgebras and the Hennessy–Milner Property. Annals of Pure and Applied Logic 138 (1):77-93.
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  10. Robert Goldblatt (2006). Maps and Monads for Modal Frames. Studia Logica 83 (1-3):309 - 331.
    The category-theoretic nature of general frames for modal logic is explored. A new notion of "modal map" between frames is defined, generalizing the usual notion of bounded morphism/p-morphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategories are equivalent, and are dual to the category of modal algebras and their homomorphisms. An important example of a modal map that (...)
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  11. Edwin D. Mares & Robert Goldblatt (2006). An Alternative Semantics for Quantified Relevant Logic. Journal of Symbolic Logic 71 (1):163-187.
    The quantified relevant logic RQ is given a new semantics in which a formula for all xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of 'extensional confinement': for all x(A V (...)
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  12. Robert Goldblatt, Ian Hodkinson & Yde Venema (2004). Erdős Graphs Resolve Fine's Canonicity Problem. Bulletin of Symbolic Logic 10 (2):186-208.
    We show that there exist 2 ℵ 0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of (...)
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  13. Robert Goldblatt (2003). Observational Ultraproducts of Polynomial Coalgebras. Annals of Pure and Applied Logic 123 (1-3):235-290.
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  14. Robert Goldblatt, Ian Hodkinson & Yde Venema (2003). On Canonical Modal Logics That Are Not Elementarily Determined. Logique Et Analyse 181:77-101.
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  15. Robert Goldblatt (2001). Persistence and Atomic Generation for Varieties of Boolean Algebras with Operators. Studia Logica 68 (2):155-171.
    A variety V of Boolean algebras with operators is singleton-persistent if it contains a complex algebra whenever it contains the subalgebra generated by the singletons. V is atom-canonical if it contains the complex algebra of the atom structure of any of the atomic members of V.This paper explores relationships between these "persistence" properties and questions of whether V is generated by its complex algebras or its atomic members, or is closed under canonical embedding algebras or completions. It also develops a (...)
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  16. Robert Goldblatt (2001). Quasi-Modal Equivalence of Canonical Structures. Journal of Symbolic Logic 66 (2):497-508.
    A first-order sentence is quasi-modal if its class of models is closed under the modal validity preserving constructions of disjoint unions, inner substructures and bounded epimorphic images. It is shown that all members of the proper class of canonical structures of a modal logic Λ have the same quasi-modal first-order theory Ψ Λ . The models of this theory determine a modal logic Λ e which is the largest sublogic of Λ to be determined by an elementary class. The canonical (...)
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  17. Hajnal Andréka, Robert Goldblatt & István Németi (1998). Relativised Quantification: Some Canonical Varieties of Sequence-Set Algebras. Journal of Symbolic Logic 63 (1):163-184.
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  18. Anita Feferman, Solomon Feferman, Robert Goldblatt, Yuri Gurevich, Klaus Grue, Sven Ove Hansson, Lauri Hella, Robert K. Meyer & Petri Mäenpää (1997). Stål Anderaa (Oslo), A Traktenbrot Inseparability Theorem for Groups. Peter Dybjer (G Öteborg), Normalization by Yoneda Embedding (Joint Work with D. Cubric and PJ Scott). Abbas Edalat (Imperial College), Dynamical Systems, Measures, Fractals, and Exact Real Number Arithmetic Via Domain Theory. [REVIEW] Bulletin of Symbolic Logic 3 (4).
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  19. Robert Goldblatt (1992). Logics of Time and Computation. Csli.
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  20. Robert Goldblatt (1992). Parallel Action: Concurrent Dynamic Logic with Independent Modalities. Studia Logica 51 (3-4):551 - 578.
    Regular dynamic logic is extended by the program construct, meaning and executed in parallel. In a semantics due to Peleg, each command is interpreted as a set of pairs (s,T), withT being the set of states reachable froms by a single execution of, possibly involving several processes acting in parallel. The modalities ] are given the interpretations>A is true ats iff there existsT withsRT andA true throughoutT, and.
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  21. Robert Goldblatt (1991). The McKinsey Axiom is Not Canonical. Journal of Symbolic Logic 56 (2):554-562.
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  22. Robert Goldblatt (1989). First-Order Spacetime Geometry. In Jens Erik Fenstad, Ivan Timofeevich Frolov & Risto Hilpinen (eds.), Logic, Methodology, and Philosophy of Science Viii: Proceedings of the Eighth International Congress of Logic, Methodology, and Philosophy of Science, Moscow, 1987. Sole Distributors for the U.S.A. And Canada, Elsevier Science.
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  23. Robert Goldblatt (1989). Varieties of Complex Algebras. Annals of Pure and Applied Logic 44 (3):173-242.
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  24. Robert Goldblatt (1986). [Omnibus Review]. Journal of Symbolic Logic 51 (1):225-227.
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  25. Robert Goldblatt (1985). An Algebraic Study of Well-Foundedness. Studia Logica 44 (4):423 - 437.
    A foundational algebra ( , f, ) consists of a hemimorphism f on a Boolean algebra with a greatest solution to the condition f(x). The quasi-variety of foundational algebras has a decidable equational theory, and generates the same variety as the complex algebras of structures (X, R), where f is given by R-images and is the non-wellfounded part of binary relation R.The corresponding results hold for algebras satisfying =0, with respect to complex algebras of wellfounded binary relations. These algebras, however, (...)
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  26. Robert Goldblatt (1985). On the Role of the Baire Category Theorem and Dependent Choice in the Foundations of Logic. Journal of Symbolic Logic 50 (2):412-422.
    The Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Čech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the "Henkin method" of proving deductive completeness of logical systems. The Rasiowa-Sikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces.
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  27. Robert Goldblatt (1984). An Abstract Setting for Henkin Proofs. Topoi 3 (1):37-41.
    A general result is proved about the existence of maximally consistent theories satisfying prescribed closure conditions. The principle is then used to give streamlined proofs of completeness and omitting-types theorems, in which inductive Henkin-style constructions are replaced by a demonstration that a certain theory respects a certain class of inference rules.
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  28. Robert Goldblatt (1984). Orthomodularity is Not Elementary. Journal of Symbolic Logic 49 (2):401-404.
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  29. Robert Goldblatt (1982). The Semantics of HOARE's Iteration Rule. Studia Logica 41 (2-3):141 - 158.
    Hoare's Iteration Rule is a principle of reasoning that is used to derive correctness assertions about the effects of implementing a while-command. We show that the propositional modal logic of this type of command is axiomatised by Hoare's rule in conjunction with two additional axioms. The proof also establishes decidability of the logic. The paper concludes with a discussion of the relationship between the logic of while and Segerberg's axiomatisation of propositional dynamic logic.
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  30. Robert Goldblatt (1981). “Locally-at” as a Topological Quantifier-Former. In. In U. Mönnich (ed.), Aspects of Philosophical Logic. Dordrecht. 119--127.
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  31. Robert I. Goldblatt (1981). Grothendieck Topology as Geometric Modality. Mathematical Logic Quarterly 27 (31‐35):495-529.
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  32. Robert Goldblatt (1980). Diodorean Modality in Minkowski Spacetime. Studia Logica 39 (2-3):219 - 236.
    The Diodorean interpretation of modality reads the operator as it is now and always will be the case that. In this paper time is modelled by the four-dimensional Minkowskian geometry that forms the basis of Einstein's special theory of relativity, with event y coming after event x just in case a signal can be sent from x to y at a speed at most that of the speed of light (so that y is in the causal future of x).It is (...)
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