In recent years, the compositional distributional approach in computational linguistics has opened the way for an integration of the \emph{lexical} aspects of meaning into Lambek's type-logical grammar program. This approach is based on the observation that a sound semantics for the associative, commutative and unital Lambek calculus can be based on vector spaces by interpreting fusion as the tensor product of vector spaces.
In this paper, we build on this observation and extend it to a `vector space semantics' for the \emph{general} Lambek calculus, based on \emph{algebras over a field} $\mathbb{K}$ (or $\mathbb{K}$-algebras), i.e. vector spaces endowed with a bilinear binary product. Such structures are well known in algebraic geometry and algebraic topology, since they are important instances of Lie algebras and Hopf algebras. Applying results and insights from duality and representation theory for the algebraic semantics of nonclassical logics, we regard $\mathbb{K}$-algebras as `Kripke frames' the complex algebras of which are complete residuated lattices.
This perspective makes it possible to establish a systematic connection between vector space semantics and the standard Routley-Meyer semantics of (modal) substructural logics.
Kripke frames (and models) provide a suitable semantics for sub-classical logics, for example Intuitionistic Logic (of Brouwer and Heyting) axiomatizes the reflexive and transitive Kripke frames (with persistent satisfaction relations), and the Basic Logic (of Visser) axiomatizes transitive Kripke frames (with persistent satisfaction relations). Here, we investigate whether Kripke frames/models could provide a semantics for fuzzy logics. For each axiom of the Basic Fuzzy Logic, necessary and sufficient conditions are sought for Kripke frames/models which satisfy them. It turns out that the onl…
We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On the one hand, it is well-known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e., a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on `natural' propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplet…
We study the algorithmic properties of first-order monomodal logics of frames $\langle \mathbb{N}, \leq \rangle$, $\langle \mathbb{N}, < \rangle$, $\langle \mathbb{Q}, \leq \rangle$, $\langle \mathbb{Q}, < \rangle$, $\langle \mathbb{R}, \leq \rangle$, $\langle \mathbb{R}, < \rangle$, as well as some related logics, in languages with restrictions on the number of individual variables as well as the number and arity of predicate letters. We show that the logics of frames based on $\mathbb{N}$ are $Π^1_1$-hard -- thus, not recursively enumerable -- in languages with two individual variables, one …
Kripke Completeness of Strictly Positive Modal Logics over Meet-semilattices with Operators
University of Oxford（牛津大学）；Birkbeck, University of London（伦敦大学伯贝克学院）；Institute for Information Transmission Problems（信息传输问题研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；King’s College London（伦敦国王学院）；Kyushu Sangyo University（九州产业大学）；University of Liverpool（利物浦大学）
We give a procedure for translating geometric Kripke frame axioms into structural hypersequent rules for the corresponding intermediate logics in Int^*/Geo that admit weakening, contraction and in some cases, cut. We give a procedure for translating labelled sequents in the corresponding logic to hypersequents that share the same linear models (which correspond to Gödel-Dummett logic). We prove that labelled proofs Int^*/Geo can be translated into hypersequent proofs that may use the linearity rule, which corresponds to the well-known communication rule for Gödel-Dummett logic.
Japan Advanced Institute of Science and Technology（日本北陆先端科学技术大学院大学）
This paper establishes a dual theory about knowledge and argumentation. Our idea is rooted at both epistemic logic and argumentation theory, and we aim to merge these two fields, not just in a superficial way but to thoroughly disclose the intrinsic relevance between knowledge and argumentation. Specifically, we define epistemic Kripke models and argument Kripke models as a dual pair, and then work out a two-way generation method between these two types of Kripke models. Such generation is rigorously justified by a duality theorem on modal formulae's invariance. We also provide realistic examp…
We introduce a class of neighbourhood frames for graded modal logic embedding Kripke frames into neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation with respect to Kripke frames by modifying the definition of monotonic bisimulation.
Center for Information and Bubble Studies, University of Copenhagen（哥本哈根大学信息与泡沫研究中心）
The literature on awareness modeling includes both syntax-free and syntax-based frameworks. Heifetz, Meier \& Schipper (HMS) propose a lattice model of awareness that is syntax-free. While their lattice approach is elegant and intuitive, it precludes the simple option of relying on formal language to induce lattices, and does not explicitly distinguish uncertainty from unawareness. Contra this, the most prominent syntax-based solution, the Fagin-Halpern (FH) model, accounts for this distinction and offers a simple representation of awareness, but lacks the intuitiveness of the lattice structur…
This paper is focused on the study of modal logics defined from valued Kripke frames, and particularly, on computability and expressibility questions of modal logics of transitive Kripke frames evaluated over certain residuated lattices. It is shown that a large family of those logics -- including the ones arising from the standard MV and Product algebras -- yields an undecidable consequence relation. Later on, the behaviour of transitive modal Lukasiewicz logic is compared with that of its non transitive counterpart, exhibiting some particulars concerning computability and equivalence with ot…
In this work we study the decidability of a class of global modal logics arising from Kripke frames evaluated over certain residuated lattices, known in the literature as modal many-valued logics. We exhibit a large family of these modal logics which are undecidable, in contrast with classical modal logic and propositional logics defined over the same classes of algebras. This family includes the global modal logics arising from Kripke frames evaluated over the standard Lukasiewicz and Product algebras. We later refine the previous result, and prove that global modal Lukasiewicz and Product lo…
We propose a new definition of the representation theorem for many-valued logics, with modal operators as well, and define the stronger relationship between algebraic models of a given logic and relational structures used to define the Kripke possible-world semantics for it. Such a new framework offers a new semantics for many-valued logics based on the truth-invariance entailment. Consequently, it is substantially different from current definitions based on a matrix with a designated subset of logic values, used for the satisfaction relation, often difficult to fix. In the case when the many-…
We work primarily with the Kripke frame consisting of two-dimensional Minkowski spacetime with the irreflexive accessibility relation 'can reach with a slower-than-light signal'. We show that in the basic temporal language, the set of validities over this frame is decidable. We then refine this to PSPACE-complete. In both cases the same result for the corresponding reflexive frame follows immediately. With a little more work we obtain PSPACE-completeness for the validities of the Halpern-Shoham logic of intervals on the real line with two different combinations of modalities.
The present paper establishes systematic connections among the first-order correspondents of Sahlqvist modal reduction principles in various relational semantic settings which include crisp and many-valued Kripke frames, and crisp and many-valued polarity-based frames (aka enriched formal contexts). Building on unified correspondence theory, we aim at introducing a theoretical environment which makes it possible to: (a) compare and inter-relate the various frame correspondents (in different relational settings) of any given Sahlqvist modal reduction principle; (b) recognize when first-order se…
The modal systems S1--S3 were introduced by C. I. Lewis as logics for strict implication. While there are Kripke semantics for S2 and S3, there is no known natural semantics for S1. We extend S1 by a Substitution Principle SP which generalizes a reference rule of S1. In system S1+SP, the relation of strict equivalence $φ\equivψ$ satisfies the identity axioms of R. Suszko's non-Fregean logic adapted to the language of modal logic (we call these axioms the axioms of propositional identity). This enables us to develop a framework of algebraic semantics which captures S1+SP as well as the Lewis sy…
