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  1. Softness of hypercoherences and full completeness.Richard Blute, Masahiro Hamano & Philip Scott - 2005 - Annals of Pure and Applied Logic 131 (1-3):1-63.
    We prove a full completeness theorem for multiplicative–additive linear logic using a double gluing construction applied to Ehrhard’s *-autonomous category of hypercoherences. This is the first non-game-theoretic full completeness theorem for this fragment. Our main result is that every dinatural transformation between definable functors arises from the denotation of a cut-free proof. Our proof consists of three steps. We show:• Dinatural transformations on this category satisfy Joyal’s softness property for products and coproducts.• Softness, together with multiplicative full completeness, guarantees that (...)
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  • Modeling linear logic with implicit functions.Sergey Slavnov - 2014 - Annals of Pure and Applied Logic 165 (1):357-370.
    Just as intuitionistic proofs can be modeled by functions, linear logic proofs, being symmetric in the inputs and outputs, can be modeled by relations . However generic relations do not establish any functional dependence between the arguments, and therefore it is questionable whether they can be thought as reasonable generalizations of functions. On the other hand, in some situations one can speak in some precise sense about an implicit functional dependence defined by a relation. It turns out that it is (...)
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  • Coherent phase spaces. Semiclassical semantics.Sergey Slavnov - 2005 - Annals of Pure and Applied Logic 131 (1-3):177-225.
    The category of coherent phase spaces introduced by the author is a refinement of the symplectic “category” of A. Weinstein. This category is *-autonomous and thus provides a denotational model for Multiplicative Linear Logic. Coherent phase spaces are symplectic manifolds equipped with a certain extra structure of “coherence”. They may be thought of as “infinitesimal” analogues of familiar coherent spaces of Linear Logic. The role of cliques is played by Lagrangian submanifolds of ambient spaces. Physically, a symplectic manifold is the (...)
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  • A really fuzzy approach to the sorites paradox.Francesco Paoli - 2003 - Synthese 134 (3):363 - 387.
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  • Some intuitions behind realizability semantics for constructive logic: Tableaux and Läuchli countermodels.James Lipton & Michael J. O'Donnell - 1996 - Annals of Pure and Applied Logic 81 (1-3):187-239.
    We use formal semantic analysis based on new constructions to study abstract realizability, introduced by Läuchli in 1970, and expose its algebraic content. We claim realizability so conceived generates semantics-based intuitive confidence that the Heyting Calculus is an appropriate system of deduction for constructive reasoning.Well-known semantic formalisms have been defined by Kripke and Beth, but these have no formal concepts corresponding to constructions, and shed little intuitive light on the meanings of formulae. In particular, the completeness proofs for these semantics (...)
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  • Z-modules and full completeness of multiplicative linear logic.Masahiro Hamano - 2001 - Annals of Pure and Applied Logic 107 (1-3):165-191.
    We prove that the full completeness theorem for MLL+Mix holds by the simple interpretation via formulas as objects and proofs as Z-invariant morphisms in the *-autonomous category of topologized vector spaces. We do this by generalizing the recent work of Blute and Scott 101–142) where they used the semantical framework of dinatural transformation introduced by Girard–Scedrov–Scott , Logic from Computer Science, vol. 21, Springer, Berlin, 1992, pp. 217–241). By omitting the use of dinatural transformation, our semantics evidently allows the interpretation (...)
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  • A categorical semantics for polarized MALL.Masahiro Hamano & Philip Scott - 2007 - Annals of Pure and Applied Logic 145 (3):276-313.
    In this paper, we present a categorical model for Multiplicative Additive Polarized Linear Logic , which is the linear fragment of Olivier Laurent’s Polarized Linear Logic. Our model is based on an adjunction between reflective/coreflective full subcategories / of an ambient *-autonomous category . Similar structures were first introduced by M. Barr in the late 1970’s in abstract duality theory and more recently in work on game semantics for linear logic. The paper has two goals: to discuss concrete models and (...)
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  • The shuffle Hopf algebra and noncommutative full completeness.R. F. Blute & P. J. Scott - 1998 - Journal of Symbolic Logic 63 (4):1413-1436.
    We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffie algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in (...)
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