Laurent De Rudder, Georges Hansoul, Valentine Stetenfeld
The aim of this paper is to show that even if the natural algebraic semantic for modal (normal) logic is modal algebra, the more general class of subordination algebras (roughly speaking, the non symmetric contact algebras) is adequate too - so leading to completeness results. This motivates for an algebraic (in the sense of universal algebra) study of those relational structures that are subordinate algebras.
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In the present paper, we study the correspondence and canonicity theory of modal subordination algebras and their dual Stone space with two relations, generalizing correspondence results for subordination algebras in \cite{dR20,dRHaSt20,dRPa21,Sa16}. Due to the fact that the language of modal subordination algebras involves a binary subordination relation, we will find it convenient to use the so-called quasi-inequalities and $Π_2$-statements. We use an algorithm to transform (restricted) inductive quasi-inequalities and (restricted) inductive $Π_2$-statements to equivalent first-order corresp…
We establish a novel connection between two research areas in non-classical logics which have been developed independently of each other so far: on the one hand, input/output logic, introduced within a research program developing logical formalizations of normative reasoning in philosophical logic and AI; on the other hand, subordination algebras, investigated in the context of a research program integrating topological, algebraic, and duality-theoretic techniques in the study of the semantics of modal logic. Specifically, we propose that the basic framework of input/output logic, as well as i…
Correspondence and Inverse Correspondence for Input/Output Logic and Region-Based Theories of Space
本文发展输入/输出逻辑的代数语义，给出闭开解析不等式的模态刻画、一阶对应与逆对应算法，并将对偶表示推广到预接触和对偶预接触代数。
Normative implications
本文沿从属代数进路研究I/O逻辑，以斜（余）Heyting代数刻画带从属关系的分配格，建立算法对应与逆对应，并给出具道义意义的公理化扩张及示例。
Priestley Representation of Distributive Precontact Lattices
研究带预接触关系的有界分配格（预接触格）的Priestley表示，刻画预接触关系条件对应的对偶一阶性质，并描述子结构和同余。
We prove several representation theorems for infinitary predicate modal logic
The aim of the paper is to study the topological modal logic of $T_0$ spaces, with the difference modality (for $T_n$, where $n\geq1 $ the corresponding logics were known). We consider propositional modal logic with two modal operators $\square$ and $[\ne]$. $\square$ is interpreted as an interior operator and $[\ne]$ corresponds to the inequality relation. We introduce the logic $S4DT_0$ and show that $S4DT_0$ is the logic of all $T_0$ spaces and has the finite model property.
We show that the standard Gödel modal logics, as initially introduced by Caicedo and Rodriguez in \cite{CR2009,CR2010}, are not realized by the basic Gödel justification logics although being related by the forgetful projection.
In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic ${\sf KD}$. Secondly, we introduce a new normal modal logic ${\sf KDR}$ which is a proper extension of ${\sf KD}$, and prove that there exists a Rosser provability predicate whose provability logic includes ${\sf KDR}$.
The Modal Logic of Provability and Forcing
本文研究可证性与力迫在模态逻辑中的交互，提出双模态逻辑 PF 及 PF^ω，并分别刻画一般情形与集合论 ω-模型中的对应逻辑。
Modal dependence logics are modal logics defined on the basis of team semantics and have the downward closure property. In this paper, we introduce sound and complete deduction systems for the major modal dependence logics, especially those with intuitionistic connectives in their languages. We also establish a concrete connection between team semantics and single-world semantics, and show that modal dependence logics can be interpreted as variants of intuitionistic modal logics.
A sceptical examination of the motivations for strong modal plural logics.
Intuitionistic and Constructive Modal Logics for Classical Modal Logicians
本文面向经典模态逻辑研究者，综述直觉主义框架下的两类命题模态逻辑，研究介于两者间的新逻辑$\textbf{KI}$并证其完备性，还指出$\textbf{CK}$语义缺陷并证明$\textbf{CK}$、$\textbf{WK}$的Lyndon插值定理。
This paper involves generalizing the Goldblatt-Thomason and the Lindström characterization theorems to first-order modal logic.
In this paper we study frame definability in finitely-valued modal logics and establish two main results via suitable translations: (1) in finitely-valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf.~\citep[Thm.~8]{tho}), and (2) a large family of finitely-valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and \MV-algebras, or even \BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt--Thomason theorem applies imm…
Deciding some displayable modal logics
本文利用显示演算证明了正规模态逻辑K及其若干扩张的可判定性。
We consider modal logics of products of neighborhood frames and prove that for any pair $L$ and $L'$ of logics from set $\{S4, D4, D, T\}$ modal logic of products of $L$-neighborhood frames and $L'$-neighborhood frames is the fusion of $L$ and $L'$.
Non-iterative normal modal logics are defined by axioms of modal degree 1. In this paper we use calculations with normal forms to determine the set of all possible non-iterative normal modal logics, unimodal propositional extensions of K. To characterize them, we construct the complete set (lattice) of such logics and we provide the generic specification of their axioms and their semantics.
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。
We consider quantified pretransitive Horn modal logic. It is known that such logics are complete with respect to predicate Kripke frames with expanding domains. In this paper we prove that they are also complete with respect to neighbourhood frames with constant domains.
In this chapter we study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, $T_1$-spaces, dense-in-themselves spaces, a zero-dimensional dense-in-itself separable metric space, $R^n$ $(n\ge 2)$. We also discuss the correlation between languages with different combinations of the topological, the derivational, the universal and the difference modality in terms of definability.
Axiomatizing modal inclusion logic and its variants
本文为模态包含逻辑及其两个表达能力等价的变体（分别含可能算子及其单世界变体）建立了完全公理化，并通过范式定理和自然演绎系统完成完全性证明。
We analyze a class of modal logics rendered insensitive to reflexivity by way of a modification to the semantic definition of the modal operator. We explore the extent to which these logics can be characterized, and prove a general completeness theorem on the basis of a translation between normal modal logics and their reflexive-insensitive counterparts. Lastly, we provide a sufficient semantic condition describing when a similarly general soundness result is also available.
Obligations and permissions, algebraically
本研究拓展输入/输出逻辑的代数研究路径，提出预接触代数可作为否定许可的代数语义环境，并通过模态语言刻画多类许可的性质及其与规范系统的互动，丰富了道义逻辑的代数语义研究。
Affine modal propositional logic
研究仿射模态命题逻辑，通过用下半连续包络算子取代内部算子，证明了该逻辑的完备性和仿射紧致性定理。
Complexity of the variable-free fragments of non-normal modal logics (extended version)
本文研究了非正规模态逻辑的无变元片段的复杂性，证明了包含经典命题逻辑且包含于弱Grzegorczyk逻辑的模态逻辑的可满足性问题是NP难的，并指出E、EM、EN和EMN的无变元片段是coNP完全的。
We study public announcement operators in the context of standard Gödel modal logic as introduced by Caicedo and Rodriguez. Over that base logic, admitting a natural semantics over [0,1]-valued generalizations of modal Kripke models, we exhibit three possible semantic interpretations of the public announcement operator, all equivalent in a classical setting, and show that these logics are all different in terms of expressive strength. We provide partial completeness results for these logics via Hilbert-style calculi.
Closure Semantics for Weak Modal Logics: Completeness and Representations
本文为弱模态逻辑引入闭包语义，证明其可靠性与强完备性，并给出与邻域、FMT及克里普克语义间的保真翻译。
A model category for modal logic
本文在Kripke模型树展开的预层拓扑斯上定义Quillen模型结构，为模态逻辑建立同伦理论，并重新审视模态保持定理与Hennessy-Milner性质。
We present a rooted hypersequent calculus for modal propositional logic S5. We show that all rules of this calculus are invertible and that the rules of weakening, contraction, and cut are admissible. Soundness and completeness are established as well.
We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal{M}$, or an axiomatization $S$ thereof, we find a modal logic $M$ such that a modal sentence $φ$ is a theorem of $M$ if and only if the sentence $φ^*$ obtained by translating the modal operator with the truth predicate is true in $\mathcal{M}$ or a theorem of $S$ under all such translations. To this end, we introduce a novel version of possible worlds semantics featuring both classical and nonclassical worlds and es…
We propose a new modal logic endowed with a simple deductive system to interpret Aristotle's theory of the modal syllogism. While being inspired by standard propositional modal logic it is also a logic of terms that admits a (sound) extensional semantics involving possible states-of-affairs in a given world. Applied to the analysis of Aristotle's modal syllogistic as found in the \emph{Prior Analytics} A8-22 it sheds light on various fine-grained distinctions which when made allow us to clarify some ambiguities and obtain a completely consistent system and prove all of the modal syllogisms con…
We give a purely syntactical proof of the fixed point theorem for Sacchetti's modal logics ${\bf K} + \Box(\Box^n p \to p) \to \Box p$ ($n \geq 2$) of provability. From our proof, an effective procedure for constructing fixed points in these logics is obtained. We also show the existence of simple fixed points for particular modal formulas.
This paper from 2008 is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the foundations are laid for later results. These foundations consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic ${\textbf{IL}}$. Next, the method is applied to obtain new results: the modal completeness of the logic ${\textbf{IL}}{\sf M_0}$, and modal completeness of ${\textbf{IL}…
Product update is an operation on models introduced into epistemic logic in order to represent a broad class of informational events. If adding modalities representing product update to a language does not alter its expressive power then we say that the language is "closed for product update." The basic modal language is known to be closed for product update. We establish that monadic second order logic is closed for product update (Theorem 5). Our technique is to pass via an intermediate language with what we call "action nominals." We obtain as corollaries that propositionally quantified mod…
We study propositional logical systems arising from the language of Johansson's minimal logic and obtained by weakening the requirements for the negation operator. We present their semantics as a variant of neighbourhood semantics. We use duality and completeness results to show that there are uncountably many subminimal logics. We also give model-theoretic and algebraic definitions of filtration for minimal logic and show that they are dual to each other. These constructions ensure that the propositional minimal logic has the finite model property. Finally, we define and investigate bi-modal …
We present a sequent calculus for the modal Grzegorczyk logic Grz allowing non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs.
Profiniteness, Monadicity and Universal Models in Modal Logic
本文证明投射有限模态代数在集合范畴上具有单子性，并通过分析单子函子重新得到模态逻辑中用于刻画自由模态代数与典范模型子框架的普遍模型构造。
On algorithmic expressivity of finite-variable fragments of intuitionistic modal logics
本文证明了直觉主义模态逻辑FS和MIPC可多项式时间嵌入其正单变量片段，揭示了其算法表达力。
More on modal logics and deduction
本文研究了在共轭模态逻辑的代数语义中，可加性与推理定理之间的关系，提出强非可加变种，并证明其在保持局部推理-脱落定理的情况下，可将正常模态逻辑转换为强非可加逻辑，同时显示相关类不是初等的。
In 1933, Gödel introduced a provability interpretation of the propositional intuitionistic logic to establish a formalization for the BHK interpretation. He used the modal system, $\mathbf{S4}$, as a formalization of the intuitive concept of provability and then translated $\mathbf{IPC}$ to $\mathbf{S4}$. His work suggested the problem to find a concrete provability interpretation of the modal logic $\mathbf{S4}$. In this paper, we will try to answer this problem. In fact, we will generalize Solovay's provability interpretation of the modal logic $\mathbf{GL}$ to capture other modal logics suc…
We present a sequent calculus for the Grzegorczyk modal logic Grz allowing cyclic and other non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs. As an application, we establish the Lyndon interpolation property for the logic Grz proof-theoretically.
Ivlev's pioneering work started in the 1970's showed a new and promissory way in the study of modal logic from the perspective of many-valued logics. Continuing our previous work on Ivlev-like non-normal modal logics with non-deterministic semantics, we present in this paper tableau systems for Tm, S4m and S5m, the non-normal versions of T, S4 and S5, respectively, as well as for their corresponding first-order extensions Tm*, S4m* and S5m*.
The {\em tetravalent modal logic} ($\cal TML$) is one of the two logics defined by Font and Rius (\cite{FR2}) (the other is the {\em normal tetravalent modal logic} ${\cal TML}^N$) in connection with Monteiro's tetravalent modal algebras. These logics are expansions of the well--known {\em Belnap--Dunn's four--valued logic} that combine a many-valued character (tetravalence) with a modal character. In fact, $\cal TML$ is the logic that preserve degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $\cal TML$ and the algebras is…
In this paper we analyze the propositional extensions of the minimal classical modal logic system E, which form a lattice denoted as CExtE. Our method of analysis uses algebraic calculations with canonical forms, which are a generalization of the normal forms applicable to normal modal logics. As an application, we identify a group of automorphisms of CExtE that is isomorphic to the symmetric group S4.
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。
Topological product of modal logics with the McKinsey axiom
本文研究带McKinsey公理的S4.1与S4模态逻辑的拓扑积，证明其等价于两逻辑融合加一条额外公理且可判定，给出了介于融合与扩张积之间的拓扑积实例。
We study modal completeness and incompleteness of several sublogics of the interpretability logic $\mathbf{IL}$. We introduce the sublogic $\mathbf{IL}^-$, and prove that $\mathbf{IL}^-$ is sound and complete with respect to Veltman prestructures which are introduced by Visser. Moreover, we prove the modal completeness of twelve logics between $\mathbf{IL}^-$ and $\mathbf{IL}$ with respect to Veltman prestructures. On the other hand, we prove that eight natural sublogics of $\mathbf{IL}$ are modally incomplete. Finally, we prove that these incomplete logics are complete with respect to general…
In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form $\forall x_0 \exists x_1 \dots \exists x_n \bigwedge x_i R_λx_j$. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
In the first part of this paper we analyzed finite non-deterministic matrix semantics for propositional non-normal modal logics as an alternative to the standard Kripke's possible world semantics. This kind of modal systems characterized by finite non-deterministic matrices was originally proposed by Ju. Ivlev in the 70's. The aim of this second paper is to introduce a formal non-deterministic semantical framework for the quantified versions of some Ivlev-like non-normal modal logics. It will be shown that several well-known controversial issues of quantified modal logics, relative to the iden…
Cover Systems for the Modalities of Linear Logic
该研究定义了结合Beth-Kripke-Joyal语义与Girard对!模态解释的模态FL-覆盖系统，证明任意模态FL-代数都能表示为这类系统的命题代数，为线性逻辑模态提供了新的语义模型。
We prove completeness, interpolation, decidability and an omitting types theorem for certain multi dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing (varieties generated by) complex algebras of Kripke semantics for such logic. Those algebras, whose elements are sets of states are common reducts of cylindric and polyadic algebras
Uncountably many maximally consistent neighborhood complete congruential modal logics
该研究解决了Peter Fritz提出的开放问题，通过代数方法构造不可数极小模态代数簇，证明存在不可数多个C-Post完全且邻域完全的同态模态逻辑。
We study the modal logic of the closure algebra $P_2$, generated by the set of all polygons in the Euclidean plane $\mathbb{R}^2$. We show that this logic is finitely axiomatizable, is complete with respect to the class of frames we call "crown" frames, is not first order definable, does not have the Craig interpolation property, and its validity problem is PSPACE-complete.
Supervenience is an important philosophical concept. In this paper, inspired by the supervenience-determined consequence relation and the semantics of agreement operator, we introduce a modal logic of supervenience, which has a dyadic operator of supervenience as a sole modality. The semantics of supervenience modality is very natural to correspond to the supervenience-determined consequence relation, in a quite similar way that the strict implication corresponds to the inference-determined consequence relation. We show that this new logic is more expressive than the modal logic of agreement, …
We present the system G3S5, a Gentzen-style sequent calculus system for the modal propositional logic S5, which in a sense has the subformula property. We formulate the rules of G3 S5 in the system G3S5; which has the subformula property and prove the admissibility of the weakening, contraction and cut rules for it.
Neighborhood and algebraic models for predicate modal logics with $ω$-rules
本文研究了带有ω规则的谓词模态逻辑的邻域与代数模型，建立了一类逻辑在具有常域邻域模型下完备性的充分条件，并扩展了Tanaka和Arló-Costa等人之前的研究结果。
Given a class $\mathcal C$ of models, a binary relation ${\mathcal R}$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $\mathcal C$ in $L$ where the modal operator is interpreted via $\mathcal R$. We discuss how modal theories of $\mathcal C$ and ${\mathcal R}$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside $L$. We calculate such theories for the submodel and the quotient relations. We prove a downward Löwenheim--Skolem theorem for first-order language expanded w…
Tho Modal Logic of Minimal Upper Bounds
本文提出最小上界的模态逻辑MIN，证明其与Van Benthem的模态信息逻辑MIL在模态语言上不可区分，并基于已有工作给出MIN的公理化与可判定性结果。
Completeness theorems for modal logic in second-order arithmetic
本文在二阶算术框架下研究模态命题逻辑完备性定理的逻辑强度，证明弱完备性定理在 $\mathrm{RCA}_0$ 中可证，且强完备性定理分别等价于 $\mathrm{ACA}_0$ 和 $\mathrm{WKL}_0$。
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