Canonicity Results of Substructural and Lattice-Based Logics

Abstract

In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of ε-join preserving operations, ε-meet preserving operations, ε-additive operations, ε-multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and lattice-based logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach in Dunnet al.(2005) and Gehrkeet al.(2005).

Coalgebraic completeness-via-canonicity for distributive substructural logics

“…As an example, consider modal substructural logics; for instance, the 'relevant modal logic' of [Suz11] or the modal resource logics of [CMP12,CG13,CG15]. The language of positive modal logic ( [Dun95]) is given by the functor…”

Reasoning with Incomplete Information in Generalized Galois Logics Without Distribution: The Case of Negation and Modal Operators

“…Dropping distribution may present a number of new technical difficulties in the semantic treatment of the related calculi [witness the difficulties in the semantics of necessity and possibility in any of Kamide (2002), Gehrke (2006), Conradie and Palmigiano (2015), Suzuki (2010Suzuki ( , 2012Suzuki ( , 2014, Järvinen and Orlowska (2005)], but it does also present itself as the natural approach to reasoning with incomplete information. Information sites (worlds, points of the underlying set of the Kripke frame) possess only partial knowledge of the world.…”

“…The approaches to the semantics of modal extensions of non-distributive logics developed over the last decade or so Kamide (2002), Gehrke (2006), Suzuki (2010Suzuki ( , 2012Suzuki ( , 2014, Conradie and Palmigiano (2015), Düntsch et al (2004), invariably depart from the standard Kripke semantics for the modal operators in important ways. First and because of lack of distribution distinct accessibility relations for each of the necessity and possibility operators seem to be forced.…”

“…We also complement our study of negation as impossibility by introducing and studying a possibility operator, where we provide standard relational semantics, despite the lack of distribution, thus significantly improving over the results of Conradie and Palmigiano (2015), which follows the approach of Gehrke (2006) of so-called generalized Kripke frames, or of Suzuki (2010Suzuki ( , 2012Suzuki ( , 2014, as well as of Järvinen and Orlowska (2005). The systems we study are based on non-distributive lattice logic, variously extending it with negation operators (minimal negation, De Morgan or intuitionistic negation, negation as impossibility, etc.)…”

Completeness via Canonicity for Distributive Substructural Logics: A Coalgebraic Perspective

“…Finally, we may wish to add modal operators to the language (see the 'relevant modal logic' in [33]), for example ♦. In this case, we can in the same way add the syntax constructor for modal logic, namely,…”