Willem Conradie, Alessandra Palmigiano, Claudette Robinson, Apostolos Tzimoulis, Nachoem M. Wijnberg
We introduce a complete many-valued semantics for basic normal lattice-based modal logic. This relational semantics is grounded on many-valued formal contexts from Formal Concept Analysis. We discuss an interpretation and possible applications of this logical framework for categorization theory to the formal analysis of multi-market competition.
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We begin the study of categorical logic for continuous model theory. In particular, we 1. introduce the notions of metric logical categories and functors as categorical equivalents of a metric theory and interpretations, 2. prove a continuous version of conceptual completeness showing that $T^\eq$ is the maximal conservative expansion of $T$, and 3. define the concept of a metric pre-topos.
It is shown that the proof theory for sketches and forms provided in Part I of this paper (see http://www.cwru.edu/1/class/mans/math/pub/wells) is strong enough to produce all the theorems of the entailment system for multisorted equational logic provided in Goguen and Meseguer's article ``Completeness of many-sorted equational logic'' (Technical Report CSL-135, SRI International Computer Science Laboratory, 333 Ravenswood Ave., Menlo Park, CA 94025, USA, 1982).
Pawlak, Belnap and the magical number seven
本文通过Pawlak-Brouwer-Zadeh格区分模糊性与歧义性，揭示其衍生的一般七值逻辑可推导出Belnap四值逻辑，并关联至公元前四世纪耆那教推理系统。
Moisil logic, having as algebraic counterpart Łukasiewicz-Moisil algebras, provide an alternative way to reason about vague information based on the following principle: a many-valued event is characterized by a family of Boolean events. However, using the original definition of Łukasiewicz-Moisil algebra, the principle does not apply for subalgebras. In this paper we identify an alternative and equivalent definition for the $n$-valued Łukasiewicz-Moisil algebras, in which the determination principle is also saved for arbitrary subalgebras, which are characterized by a Boolean algebra and a fa…
First-Order Modal Logic via Logical Categories
本文将逻辑范畴框架扩展至一阶模态逻辑，证明Joyal风格表示定理并引入模态预拓扑os概念，最后展示如何从一阶模态理论构建语法范畴。
In this paper we develop an almost general process to switch from abstract logics in the sense of Brown and Suszko to lattices. With this method we can establish dualities between some categories of abstract logics to the correspondent topological space categories. In more detail we will explain the duality between the category of abstract intuitionistic logics with intuitionistic morphisms and the category of Esakia spaces with the Esakia morphisms.
Fibered Universal Algebra for First-Order Logics
本文扩展 Lawvere-Pitts prop-范畴，为非经典一阶逻辑构建代数语义，并刻画两类对应 Birkhoff 及 Blok-Jónsson 定理的一阶闭包算子。
Unification of Modal Logic via Topological Categories
本文通过拓扑范畴框架统一描述模态逻辑的不同语义，建立模态逻辑语法扩展（如模态依赖、群体智能体结构和逻辑动态）与拓扑范畴语义结构间的精确对应，深化了对模态逻辑抽象结构的理解。
The primary goal of this paper is to recast the semantics of modal logic, and dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We first review the category of relations and categories of Kripke frames, with particular emphasis on the duality between relations and adjoint homomorphisms. Using these categories, we then reformulate the semantics of DEL in a more categorical and algebraic form. Several virtues of the new formulation will be demonstrated: The DEL idea of updating a model into another is captured naturally by the categorical perspective -- which emphasizes a…
In this paper we show how the theory of monads can be used to deduce in a uniform manner several duality theorems involving categories of relations on one side and categories of algebras with homomorphisms preserving only some operations on the other. Furthermore, we investigate the monoidal structure induced by Cartesian product on the relational side and show that in some cases the corresponding operation on the algebraic side represents bimorphisms.
Coherence with respect to Kelly-Mac Lane graphs is proved for categories that correspond to the multiplicative fragment without constant propositions of classical linear first-order predicate logic without or with mix. To obtain this result, coherence is first established for categories that correspond to the multiplicative conjunction-disjunction fragment with first-order quantifiers of classical linear logic, a fragment lacking negation. These results extend results published in previous two books by the authors, where coherence was established for categories of the corresponding fragments o…
On the lack of colimits in various categories arising in pointfree topology and algebraic logic
研究证明McKinsey-Tarski代数范畴不等价于代数品种，反驳了Peter Jipsen的问题。各类BAO、Heyting代数及框架的范畴不完整，因此不等价于预品种。
This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, and orthomodularity. For instance, the Sasaki hook and and-then connectives are obtained, as adjoints…
First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Löf type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved for DFOL. The semantics is functorial in the …
This is the author's Ph.D. Thesis. It contains results from four years of research into realizability and categorical logic. The main subjects are the axiomatisation of realizable propositions, and a characterization of realizability categories as pseudoinitial objects. Realizability is a collection of techniques in the study of constructive logic. Some forms of realizability induce realizability categories, which are Heyting categories and therefore have a first order intuitionistic logic as internal language. The axiomatisation chapter of the thesis explains how and to what extend we can axi…
In flowchart languages, predicates play an interesting double role. In the textual representation, they are often presented as conditions, i.e., expressions which are easily combined with other conditions (often via Boolean combinators) to form new conditions, though they only play a supporting role in aiding branching statements choose a branch to follow. On the other hand, in the graphical representation they are typically presented as decisions, intrinsically capable of directing control flow yet mostly oblivious to Boolean combination. While categorical treatments of flowchart languages ar…
In the framework of propositional Łukasiewicz logic, a suitable notion of implicit definability, tailored to the intended real-valued semantics and referring to the elements of its domain, is introduced. Several variants of implicitly defining each of the rational elements in the standard semantics are explored, and based on that, a faithful interpretation of theories in Rational Pavelka logic in theories in Łukasiewicz logic is obtained. Some of these results were already presented in Hájek's "Metamathematics of Fuzzy Logic" as technical statements. A connection to the lack of (deductive) Bet…
We introduce the framework of AECats (abstract elementary categories), generalising both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory ("cat", as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat. The Kim-Pillay theorem for first-order logic characterises simple theories by the properties dividing independence has. We prove a version of the Kim-Pillay theorem for …
We give a new Esakia-style duality for the category of Sugihara monoids based on the Davey-Werner natural duality for lattices with involution, and use this duality to greatly simplify a construction due to Galatos-Raftery of Sugihara monoids from certain enrichments of their negative cones. Our method of obtaining this simplification is to transport the functors of the Galatos-Raftery construction across our duality, obtaining a vastly more transparent presentation on duals. Because our duality extends Dunn's relational semantics for the logic R-mingle to a categorical equivalence, this also …
Semi-Substructural Logics à la Lambek
本文以非结合 Lambek 演算为视角，构造了左斜幺半闭范畴与斜幺半双闭范畴的无切割矢列演算，证明其与既有演算等价、对关系模型可靠完备，并建立框架条件与结构律的对应关系。
Bicategories, Biequivalence, and Bi-Interpretability
本文通过构造相干理论双范畴与相干范畴双范畴之间的双等价，给出双可解释性的范畴刻画，并利用正合完备化得到两个理论双可解释的充要条件，推广到一阶逻辑常见片段。
Monoidal bicategories, differential linear logic, and analytic functors
研究通过引入线性指数余单子和余废弃变换的双范畴对应物发展幺半双范畴理论，将若亚尔解析函子微计算扩展到预层范畴间的解析函子，如同普通微积分的变量扩展。
In the present work, a natural sequel to \cite{MaPi1}, we further discuss the existence of adjunctions between categories of institutions and of $π$-institutions. This is done at both a foundational and an applied level. Firstly, we reformulate and conceptually clarify such adjunctions in terms of the $2$-categorical data involved in the construction of categories of institution-like structures. More precisely, we remark that the process used for passing from rooms to institutions (\cite{Diac2}) can be extended, due to its $2$-functoriality, to more general room-like and institution-like struc…
From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2. In the present work, we generalize the entire arrangement from propositional to first-order logic. Boole…
Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof rules of regular logic in terms of the free regular category $\mathsf{FRg}(\mathrm{T})$ on a set $\mathrm{T}$. From this point of view, regular theories are certain monoidal 2-functors from a suitable 2-category of contexts---the 2-category of relations in $\mathsf{FRg}(\mathrm{T})$---to the 2-category of posets. Such functors assign to each context the set of formulas in that context, ordered by ent…
We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of promonoidal category. Since the theory of promonoidal categories has not been developed very thoroughly, at least in the published literature, we need to develop it here. The most natural way to do this - and the simplest, once the (substantial) groundwork has been laid - is to consider promonoidal categories as an instance of the general theory of pseudomonoids in a monoidal bicategory. Accordingly, we describe and explain the notions of monoi…
Braided monoidal categories arise naturally as centres of monoidal categories and have been the focus of much recent attention in both mathematics and physics. By suitably restricting the use of the exchange rule, we obtain a sequent calculus whose categorical semantics may be seen as freely constructing the centre of a monoidal category. This calculus is shown to admit a strongly normalising and confluent cut elimination procedure. The resulting logic fits neatly into the landscape of noncommutative logics and is distinguished by possessing a particularly perspicuous semantics.
We seize the opportunity of the publication of selected papers from the \emph{Logic, categories, semantics} workshop in the \emph{Journal of Applied Logic} to survey some current trends in logic, namely intuitionistic and linear type theories, that interweave categorical, geometrical and computational considerations. We thereafter present how these rich logical frameworks can model the way language conveys meaning.
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