Willem Conradie, Alessandra Palmigiano, Claudette Robinson, Nachoem Wijnberg
We discuss an ongoing line of research in the relational (non topological) semantics of non-distributive logics. The developments we consider are technically rooted in dual characterization results and insights from unified correspondence theory. However, they also have broader, conceptual ramifications for the intuitive meaning of non-distributive logics, which we explore.
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We define when a ternary term $m$ of an algebraic language $\mathcal{L}$ is called a \textit{distributive nearlattice term} (DN-term) of a sentential logic $\mathcal{S}$. Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a DN-term through the interpretation of the DN-term in the algebras of the algebraic counterpart of the logics. We prove that the canonical class of algebras (under the point of view of Abstract Algebraic Logic) associated with a selfextensional logic with a DN-term is a variet…
Toward the van Benthem Characterization Theorem for Non-Distributive Modal Logic
本文将模拟、互模拟及图像有限模型、模态饱和模型、超滤扩张和超幂扩张等模型论概念推广到非分配模态逻辑的极性语义，并据此证明了Hennessy-Milner定理与van Benthem刻画定理。
A Characterization Result for Non-Distributive Logics
本文将非分配逻辑归约为多类剩余模态逻辑，通过复用现有结果，成功将模态逻辑的 van Benthem 刻定结果推广至非分配逻辑。
Priestley-style duality for filter-distributive congruential logics
本文为带定理的同余、有限且滤子分配逻辑的代数类提出 Priestley 风格对偶，并分析对偶空间性质与逻辑性质（如演绎定理、析取存在性）的对应关系。
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.
Using Sheaf duality theory of Comer for cylindric algebras, we give a representation theorem of of distributive bounded lattices expanded by modalities (functions distributing over joins) as the continuous sections of sheaves. Our representation is defined via a contravariant functor. We give applications to many-valued logics logics and various modifications of first order logic and multi-modal logic, set in an algebraic framework.
Tense distributive lattices: algebra, logic and topology
本文研究了配备Prior时态算子的有界分配格范畴tdlat，建立了其与Kripke框架和Priestley空间的范畴对偶性，刻画了同余格与次直不可约成员，并定义了保持真值度的逻辑。
Compatibility between modal operators in distributive modal logic
本文识别了分配模态逻辑中保证 box 和 diamond 算子源自同一可达关系且不施加额外限制的基本相容性条件，并通过添加局部性公理模块化地获得直觉主义和正模态逻辑。
Non-distributive relatives of ETL and NFL
本文构造了ETL和NFL的非分配相对逻辑，使用非分配格M3作为真值格，提供解析表列系统并证明其正确性与完备性。
In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g. (generalized) orthomodular lattices, and MV-algebras, which admit a natural notion of implication. In fact, it turns out that skew Hilbert algebras play a similar role for (strongly) sectionally pseudocomplemented posets as Hilbert algebras do for relatively pseudocomplemented ones. We will discuss basic properties of closed, dense, and weakly dense elements …
We characterise non-distributive positive logic as the fragment of a single-sorted first-order language that is preserved by a new notion of simulation called a meet-simulation. Meet-simulations distinguish themselves from simulations because they relate pairs of states from one model to single states from another. En route to this result we use a more traditional notion of simulations and prove a Hennessy-Milner style theorem for it, using an analogue of modal saturation called meet-compactness.
Non-distributive description logic
本研究提出基于非分配格命题逻辑的描述逻辑LE-ALC，在FCA形式背景的关系结构上解释语义，给出检验带无环TBox知识库一致性的tableaux算法并证明其性质，且该场景下一致性检验复杂度低于经典ALC。
In the study of algebras related to non-classical logics, (distributive) semilattices are always present in the background. For example, the algebraic semantic of the $\{\rightarrow,\wedge,\top\}$-fragment of intuitionistic logic is the variety of implicative meet-semilattices \cite{CelaniImplicative} \cite{ChajdaHalasKuhr}. In this paper we introduce and study the class of distributive meet-semilattices endowed with a monotonic modal operator $m$. We study the representation theory of these algebras using the theory of canonical extensions and we give a topological duality for them. Also, we …
We extend the theory of unified correspondence to a very broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as `lattices with operators'. Specifically, we introduce a very general syntactic definition of the class of Sahlqvist formulas and inequalities, which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. Together with this, we introduce a variant of the algorithm ALBA, specific to the setting of LEs, which effectively…
We continue work of our earlier paper (Lewitzka and Brunner: Minimally generated abstract logics, Logica Universalis 3(2), 2009), where abstract logics and particularly intuitionistic abstract logics are studied. Abstract logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theoretic machinery usually applied to represent logics. Motivated by that point of view, we define in this paper the cate…
A non-distributive two-sorted hypersequent calculus \textbf{PDBL} and its modal extension \textbf{MPDBL} are proposed for the classes of pure double Boolean algebras and pure double Boolean algebras with operators respectively. A relational semantics for \textbf{PDBL} is next proposed, where any formula is interpreted as a semiconcept of a context. For \textbf{MPDBL}, the relational semantics is based on Kripke contexts, and a formula is interpreted as a semiconcept of the underlying context. The systems are shown to be sound and complete with respect to the relational semantics. Adding approp…
Superamalgamation for modal lattices via non-distributive dualities
本文通过非分配对偶性证明模态格的超拼接性质，并由此得出弱正模态逻辑具有Craig插值性质，同时扩展到其他弱正模态逻辑。
In this paper, we study logics of bounded distributive residuated lattices with modal operators considering $\Box$ and $\Diamond$ in a noncommutative setting. We introduce relational semantics for such substructural modal logics. We prove that any canonical logic is Kripke complete via discrete duality and canonical extensions. That is, we show that a modal extension of the distributive full Lambek calculus is the logic of its frames if its variety is closed under canonical extensions. After that, we establish a Priestley-style duality between residuated distributive modal algebras and topolog…
We extend unified correspondence theory to Kripke frames with impossible worlds and their associated regular modal logics. These are logics the modal connectives of which are not required to be normal: only the weaker properties of additivity and multiplicativity are required. Conceptually, it has been argued that their lacking necessitation makes regular modal logics better suited than normal modal logics at the formalization of epistemic and deontic settings. From a technical viewpoint, regularity proves to be very natural and adequate for the treatment of algebraic canonicity Jónsson-style.…
This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means of finite Hilbert calculi. On the side of negative results, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely axiomatizable; likewise, we establish that the order-preserving logic of the variety of all …
A categorical equivalence for monadic algebras of first-order substructural logics motivated by Kalman's construction
本文通过卡曼构造，研究单子余分配格的范畴等价性，并扩展了相关领域的研究成果。
Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics
本文引入子结构逻辑FL_ew的Frobenius-Galois扩张，证明其可代数化，建立Kalman等价与正锥语义，统一了代数、范畴与逻辑框架。
In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalizing the well-known Gödel-McKinsey-Tarski translation of intuitionistic logic into {\bf S4}. Došen's translation worked well for (variants of) {\bf BCI} and stronger systems ({\bf BCW}, {\bf BCK}), but not for systems below {\bf BCI}. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (…
The unified correspondence theory for distributive lattice expansion logics (DLE-logics) is specialized to strict implication logics. As a consequence of a general semantic consevativity result, a wide range of strict implication logics can be conservatively extended to Lambek Calculi over the bounded distributive full non-associative Lambek calculus (BDFNL). Many strict implication sequents can be transformed into analytic rules employing one of the main tools of unified correspondence theory, namely (a suitably modified version of) the Ackermann lemma based algorithm $\msf{ALBA}$. Gentzen-st…
Recovery operators in quasi-Nelson logic: the prelinear case
研究准Nelson逻辑中的一致性和确定性算子，在预线性条件下推广了LFIs/LFUs的结果，无需对合性假设。
This paper deals with join-semilattices whose sections, i.e. principal filters, are pseudocomplemented lattices. The pseudocomplement of a\vee b in the section [b,1] is denoted by a\rightarrow b and can be considered as the connective implication in a certain kind of intuitionistic logic. Contrary to the case of Brouwerian semilattices, sections need not be distributive lattices. This essentially allows possible applications in non-classical logics. We present a connection of the semilattices mentioned in the beginning with so-called non-classical implication semilattices which can be converte…
Topological Duality for Distributive Lattices: Theory and Applications
本书系统讲解Stone-Priestley对偶性理论，旨在快速提供全面的对偶工具，并将其应用于命题逻辑语义、逻辑形式的域理论及profinite monoids理论等三大标志性领域。
This paper is motivated by the desire to study package management using the toolkit of the semantics of functional languages. As it transpires, this is deeply related to the semantics of concurrent computation. The models we produce are not solely of theoretical interest, but amenable to analysis and computation. This work makes a number of related contributions. First, it relates the specification of branching dependency structures, which exist in fields from knowledge-representation to package management, to the specification of semantics of concurrent computation. Second, it relates depende…
Reconciliation of Approaches to the Semantics of Logics without Distribution
本文澄清并完成了一种针对非分配逻辑的关系语义方法，通过使用更简单的分类框架，在放弃 RS-frames 额外限制的同时捕获了其主要语义结果，从而统一了该领域并补充了 Dunn 的 gaggle 理论项目。
In this article, we define general normal forms for any logic that has propositional part and whose non-propositional connectives distribute over the finite disjunctions. We do not require the non-propositional connectives to be closed on the set of formulas, so our normal forms cover logics with partial connectives too. We also show that most of the known normal forms in the literature are in fact particular cases of our general forms. These general normal forms are natural improvement of the distributive normal forms of J. Hintikka and their modal analogues, e.g. [Anderson] and [Fine].
Game semantics for lattice-based modal μ-calculus
本研究将模态μ-演算拓展为基于格的非分配版本并用于分类场景形式化，构建了对应的博弈语义并证明其恰当性，还定义了极性互模拟并验证了该演算在其下的不变性。
We consider a proper propositional quantum logic and show that it has multiple disjoint lattice models, only one of which is an orthomodular lattice (algebra) underlying Hilbert (quantum) space. We give an equivalent proof for the classical logic which turns out to have disjoint distributive and non-distributive ortholattices as its models. In particular, we prove that both classical and quantum logics are sound and complete with respect to each of these lattices. We also show that there is one common non-orthomodular lattice that is a model of both quantum and classical logics. In technical t…
A vector logic for extensional formal semantics
本文提出一种向量逻辑，证明扩展式形式语义与分布向量空间语义的同构关系，展示二者在结构上的兼容性。
By exploiting the algebraic and order theoretic mechanisms behind Sahlqvist correspondence, the theory of unified correspondence provides powerful tools for correspondence and canonicity across different semantics and signatures, covering all the logics whose algebraic semantics are given by normal (distributive) lattice expansions (referred to as (D)LEs). In particular, the algorithm ALBA, parametric in each (D)LE, effectively computes the first order correspondents of (D)LE-inductive formulas. We present an algorithm that makes use of ALBA's rules and algebraic language to invert its steps i…
One of the greatest problems in philosophy is that of meaning. The turning point in thinking on meaning was Tarski's definition of truth, and the rapid development of logical semantics and model theory was a consequence of this achievement. Perhaps less well-known among classical logicians and philosophers is that it is category theory that provides adequate mathematical tools to study the relationship between the syntax of formalized theories and their semantics. The aim of this article is to change this situation and make a preliminary philosophical analysis of the results obtained so far. T…
It is shown that propositional calculuses of both quantum and classical logics are non-categorical. We find that quantum logic is in addition to an orthomodular lattice also modeled by a weakly orthomodular lattice and that classical logic is in addition to a Boolean algebra also modeled by a weakly distributive lattice. Both new models turn out to be non-orthomodular. We prove the soundness and completeness of the calculuses for the models. We also prove that all the operations in an orthomodular lattice are five-fold defined. In the end we discuss possible repercussions of our results to qua…
Natural Term Logic
本文提出自然项逻辑（NTL），通过定义归约系统证明每个NTL项可归约为唯一范式，从而捕捉自然语言中保持核心逻辑意义的语法变换。
Notion of validity for the bilateral classical logic
本文通过引入Prawitz的有效性概念，为双边经典逻辑（BCL）提供全面辩护，利用不动点构造解决RAA循环性问题，证明经典逻辑的辩护无需依赖排中律。
Absolute Logic: an alternative framework for formal logic
本文提出绝对逻辑框架，通过统一术语与公式的概念，建立自包含语义学，解决传统一阶逻辑的限制，并整合递归理论约束以平衡表达力与有效性。
We study the correspondence theory of intuitionistic modal logic in modal Fairtlough-Mendler semantics (modal FM semantics) \cite{FaMe97}, which is the intuitionistic modal version of possibility semantics \cite{Ho16}. We identify the fragment of inductive formulas \cite{GorankoV06} in this language and give the algorithm $\mathsf{ALBA}$ \cite{CoPa12} in this semantic setting. There are two major features in the paper: one is that in the expanded modal language, the nominal variables, which are interpreted as atoms in perfect Boolean algebras, complete join-prime elements in perfect distributi…
This paper proposes a way to compute the meanings associated with sentences with generic noun phrases corresponding to the generalized quantifier most. We call these generics specimens and they resemble stereotypes or prototypes in lexical semantics. The meanings are viewed as logical formulae that can thereafter be interpreted in your favourite models. To do so, we depart significantly from the dominant Fregean view with a single untyped universe. Indeed, our proposal adopts type theory with some hints from Hilbert ε-calculus (Hilbert, 1922; Avigad and Zach, 2008) and from medieval philosophy…
In quantum logic there is well-known arbitrariness in choosing a binary operation for conditional. Currently, we have at least three candidates, called the Sasaki conditional, the contrapositive Sasaki conditional, and the relevance conditional. A fundamental problem is to show how the form of the conditional follows from an analysis of operational concepts in quantum theory. Here, we attempt such an analysis through quantum set theory (QST). In this paper, we develop quantum set theory based on quantum logics with those three conditionals, each of which defines different quantum logical truth…
Constructing a fuzzy subset logic L with Boolean properties is notoriously difficult because under a handful of "reasonable" conditions, we have the following three debilitating constraints: (1) Bellman and Giertz in 1973 showed that if L is distributive, then it must be idempotent. (2) Dubois and Padre in 1980 showed that if L has the excluded middle or the non-contradiction property or both, then it must be non-idempotent. (3) Bellman and Giertz also demonstrated in 1973 that even if L is idempotent, then the only choice available for the (meet,join) logic operator pair is the (min,max) oper…
An ecumenical view of proof-theoretic semantics
本文将逻辑普世主义与证明论语义学相结合，构建了一个经典逻辑与直觉主义逻辑及其证明概念能够共存的统一框架，为解决两者的哲学争议提供了新视角。
The Distributional Compositional Categorical (DisCoCat) model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validate…
A Proof-Theoretic Approach to the Semantics of Classical Linear Logic
本文提出通过基础扩展语义（BeS）和基础支持的概念，从证明论角度研究经典线性逻辑的乘法-加法片段（MALL）语义学。
Proof-theoretic Semantics for Second-order Logic
本文为二阶逻辑发展了基于基础扩展语义的证明论语义，通过改变原子系统类别获得经典与直觉主义版本，证明其可靠性与完备性等价于 Henkin 解释，将二阶量化重构为系统替换，为高阶逻辑提供了哲学轻量且富有表达力的语义基础。
Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use the operational view of the latter as provided by uniform proof-sear…
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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