Carsten Butz, Ieke Moerdijk
For any first order theory T we construct a Boolean valued model M, in which precisely the T--provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a first order formula. Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory.
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We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.
This paper involves generalizing the Goldblatt-Thomason and the Lindström characterization theorems to first-order modal logic.
The Beth Definability Property holds for an algebraizable logic if and only if every epimorphism in the corresponding category of algebras is surjective. Using this technique, Urquhart in 1999 showed that the Beth Definability Property fails for a wide class of relevant logics, including T, E, and R. However, the counterexample for those logics does not extend to the algebraic counterpart of the super relevant logic KR, the so-called Boolean monoids. Following a suggestion of Urquhart, we use modular lattices constructed by Freese to show that epimorphisms need not be surjective in a wide clas…
Both propositional dependence logic and inquisitive logic are expressively complete. As a consequence, every formula with intuitionistic disjunction or intuitionistic implication can be translated equivalently into a formula in the language of propositional dependence logic without these two connectives. We show that although such a (non-compositional) translation exists, neither intuitionistic disjunction nor intuitionistic implication is uniformly definable in propositional dependence logic.
We prove that n-variable logics do not have the weak Beth definability property, for all n greater than 2. This was known for n=3 (Ildikó Sain and András Simon), and for n greater than 4 (Ian Hodkinson). Neither of the previous proofs works for n=4. In this paper we settle the case of n=4, and we give a uniform, simpler proof for all n greater than 2. The case for n=2 is still open.
Let ML(U^+) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We characterize the relative definability of ML(U^+) relative to finite transitive frames in the spirit of the well-known Goldblatt-Thomason theorem. We show that a class F of Kripke frames is definable in ML(U^+) relative to finite transitive frames if and only if F is closed under taking generated subframes and bounded morphic images. In addition, we study modal definability in team-based logics. We study (extended) modal dependence logic, (extended) mod…
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…
The dual character of invariance under transformations and definability by some operations has been used in classical work by for example Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this paper, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in $\La_{\infty\infty}$ so as to cover the cases (quantifiers, logics without equality) that are of interest in the logicality debates, getting McGee's theorem about quantifiers invarian…
Modal definability in Euclidean modal logics
本文研究欧几里得模态逻辑框架类中模态可定义性问题的可计算性，刻画了哪些欧几里得模态逻辑会导致该问题不可判定。
The use of Extended Logics to replace ordinary second order definability in Kleene's {\em Ramified Analytical Hierarchy} is investigated. This mirrors a similar investigation of Kennedy, Magidor and Väänänen \cite{KeMaVa2016} where Gödel's universe $L$ of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain {\em minimal correct} models of analysis. A wide spectrum …
We study model and frame definability of various modal logics. Let ML(A+) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We show that a class of Kripke models is definable in ML(A+) if and only if the class is elementary and closed under disjoint unions and surjective bisimulations. We also characterise the definability of ML(A+) in the spirit of the well-known Goldblatt--Thomason theorem. We show that an elementary class F of Kripke frames is definable in ML(A+) if and only if F is closed under taking generated s…
Definability in affine continuous logic
该文研究仿射连续逻辑的可定义集，证明了仿射理论在有一阶模型和紧模型两种情形下的可定义集性质，并以概率代数理论为例验证一维可定义集为区间[a,b]。
Model theoretic results such as Characterization and Definability give important information about different logics. It is well known that the proofs of those results for several modal logics have, somehow, the same 'taste'. A general proof for most modal logics below first order is still too ambitious. In this thesis we plan to isolate sufficient conditions for the characterization and definability theorems to hold in a wide range of logics. Along with these conditions we will prove that, whichever logic that meets them, satisfies both theorems. Therefore, one could give an unifying proof for…
We deal with various splitting methods in algebraic logic. The word `splitting' refers to splitting some of the atoms in a given relation or cylindric algebra each into one or more subatoms obtaining a bigger algebra, where the number of subatoms obtained after splitting is adjusted for a certain combinatorial purpose. This number (of subatoms) can be an infinite cardinal. The idea originates with Leon Henkin. Splitting methods existing in a scattered form in the literature, possibly under different names, proved useful in obtaining (negative) results on non-atom canonicity, non-finite axiomat…
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…
In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt-Thomason definability theorem in terms of the well established first-order topological language $L_t$.
We reformulate Hrushovski's definability patterns from the setting of first order logic to the setting of positive logic. Given an h-universal theory T we put two structures on the type spaces of models of T in two languages, \mathcal{L} and \mathcal{L}_π. It turns out that for sufficiently saturated models, the corresponding h-universal theories \mathcal{T} and \mathcal{T}_π are independent of the model. We show that there is a canonical model \mathcal{J} of \mathcal{T}, and in many interesting cases there is an analogous canonical model \mathcal{J}_π of \mathcal{T}_π, both of which embed int…
This paper has three parts. First, we study and characterize amenable and extremely amenable topological semigroups in terms of invariant measures using integral logic. We prove definability of some properties of a topological semigroup such as amenability and the fixed point on compacta property. Second, we define types and develop local stability in the framework of integral logic. For a stable formula $ϕ$, we prove definability of all complete $ϕ$-types over models and deduce from this the fundamental theorem of stability. Third, we study an important property in measure theory, Talagrand's…
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey's Theorem for colorings of pairs could help in simplifying the definitions of some $Δ^0_2$ sets, while some others could not. We also investigate some consequences of Ramsey's Theorem for colorings of longer tuples. These results of definability strength have some interesting consequences in reverse mathemat…
Frame definability in second-order arithmetic
该文研究模态逻辑框架可定义性的反数学强度，证明估值扩张引理等相关原则与二阶算术子系统$\text{ACA}^{+}_0}$等价，还考察了CTL、LTL对应原则的强度。
Generalizing a theorem of Campercholi, we characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Isbell, as follows: in any prevariety having at most s non-logical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m + s + aleph0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable "bridge theorems", matching the surjectivity of all epim…
Probability logic has contributed to significant developments in belief types for game-theoretical economics. We present a new probability logic for Harsanyi Type spaces, show its completeness, and prove both a de-nesting property and a unique extension theorem. We then prove that multi-agent interactive epistemology has greater complexity than its single-agent counterpart by showing that if the probability indices of the belief language are restricted to a finite set of rationals and there are finitely many propositional letters, then the canonical space for probabilistic beliefs with one age…
On the Definability of Strong Negation in Bilateral Logics
本文研究双边逻辑中强否定的可定义性，通过四类结果明确了不同构造性联结词及组合对强否定的定义能力，为N4相关片段提供了单联结词基础并刻画了特定联结词组合的定义条件。
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…
Dynamic Probability Logics: Axiomatization & Definability
本文引入动态概率逻辑DPL及其无穷扩张，给出希尔伯特式公理化并证明强完全性，进而研究保测、遍历、混合等性质的可定义性，并证明带初始分布的马尔可夫过程的平稳性、不变性、不可约性和常返性在无穷概率逻辑中可定义。
Definability of some $k$-ary Relations Over Second Order kinds of Logics
本文研究了计算机科学中重要的 $k$-hashing 关系在逻辑中的表达性，证明了该关系在 Monadic Second Order Logic (MSO) 中不可定义，但可在添加等基数关系的 MSO 扩展中表达。
The Svenonius theorem describes the (first-order) definability in a structure in terms of permutations preserving the relations of elementary extensions of the structure. In the present paper we prove a version of this theorem using permutations of sequences over the original structure (these are permutations of sequences of tuples of the structure elements as well). We say that such a permutation $φ$ almost preserves a relation if for every sequence of its arguments the value of the relation on an $n$-th element of the sequence and on its image under $φ$ coincide for almost all numbers $n.$ W…
We show that the investigation of universal models in Topos Theory can shed light on problems of definability in Logic as well as on the investigation of De Morgan's law and the law of excluded middle on Grothendieck toposes.
Drawing on the previous work on interpolation failure, we show that Beth's definability theorem does not hold for intuitionistic predicate logic of constant domains without identity.
Hanf Locality and Invariant Elementary Definability
本文引入了不变初等可定义性概念，扩展了一阶序不变可定义性，提出结构的表示概念，并在此框架下建立了两个局部性结果。
Constructive Interpolation and Concept-Based Beth Definability for Description Logics via Sequents
本文提出基于矢列系统的构造性方法，首次证明高表达力描述逻辑RIQ具有基于概念的Beth可定义性，并可从证明中计算插值式以提取隐式概念的显式定义。
Semiconic Idempotent Logic II: Beth Definability and Deductive Interpolation
本文研究半锥幂等逻辑 sCI 的扩张，通过分析其代数语义半锥幂等剩余格的共合与满同态性质，证明刚性合取类的强共合性，进而为多类 sCI 扩张建立投射 Beth 可定义性与演绎插值性质。
Found in Translation: at the limits of the Hudetz program
本文分析了一对广义理论在可相互定义意义上的等价性，针对非一阶逻辑表述的理论，推广了基于翻译的相对可解释性理论，提出在数学语言中而非相对特定结构的可定义性说明。
The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic $\vDash_Δ$ associated with an infinitary variety $Δ$ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of $\vDash_Δ$, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic $\vdash_Δ$ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion of consequence associated with $\vdash_Δ$ c…
Definability from Factorised Symmetry in Ultrapowers
本文分离Madarász定理的仿射几何成分，提出半线性忠实性假设，扩展有限关系坐标几何的可定义性刻画，其框架可应用于纯域外的多种几何场景。
Completeness and Well-Definability of a Provability Degree Measure in Sufficiently Powerful Formal Systems, and Finite-Time Effective Knowers
该研究证明足够强大形式系统中特定可证性程度测度的完备性与良定性，构造出最优有限时间定理认知者φ'，并揭示其与无界归纳计算的关联及对计算过程最终行为的意义。
Definability over $\mathrm BΣ^0_2$-models
本文研究$\mathsf{RCA}_0+Σ^0_2$-bounding且$Σ^0_2(A)$-归纳失效的模型，刻画了Ramsey、凝聚集和树定理的无解可定义实例及极小Turing度集合的跃变性质。
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.
The question of whether there is a logic that captures polynomial time was formulated by Yuri Gurevich in 1988. It is still wide open and regarded as one of the main open problems in finite model theory and database theory. Partial results have been obtained for specific classes of structures. In particular, it is known that fixed-point logic with counting captures polynomial time on all classes of graphs with excluded minors. The introductory part of this paper is a short survey of the state-of-the-art in the quest for a logic capturing polynomial time. The main part of the paper is concerned…
We prove preservation theorems for $\mathcal{L}_{ω_1, G}$, the countable fragment of Vaught's closed game logic. These are direct generalizations of the theorems of Łoś-Tarski (resp. Lyndon) on sentences of $\mathcal{L}_{ω_1, ω}$ preserved by substructures (resp. homomorphic images). The solution, in $ZFC$, only uses general features and can be extended to several variants of other strong first-order logic that do not satisfy the interpolation theorem; instead, the results on infinitary definability are used. This solves an open problem dating back to 1977. Another consequence of our approach …
In this article we formally define and investigate the computational complexity of the Definability Problem for open first-order formulas (i.e., quantifier free first-order formulas) with equality. Given a logic $\mathbf{\mathcal{L}}$, the $\mathbf{\mathcal{L}}$-Definability Problem for finite structures takes as input a finite structure $\mathbf{A}$ and a target relation $T$ over the domain of $\mathbf{A}$, and determines whether there is a formula of $\mathbf{\mathcal{L}}$ whose interpretation in $\mathbf{A}$ coincides with $T$. We show that the complexity of this problem for open first-orde…
A note on the definability of genus for Zariski geometries
该研究在Zariski几何中提出亏格概念，推广了Riemann--Hurwitz与Hurwitz定理，并证明该亏格概念不具备一阶可定义性。
Logic has its origins in basic questions about the nature of the real world and how we describe it. This article seeks to bring out the physical and epistemological relevance of some of the more recent technical work in logic and computability theory. "If you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until I have felt the mathematics lead me by the hand. It has led me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of…
Finite model theory for pseudovarieties and universal algebra: preservation, definability and complexity
本文研究了有限模型理论与伪变种及普遍代数之间的关系，揭示了有限代数在公理化和可定义性方面的复杂性。
We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Crit{è}res de compacit{é}" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces.
Tameness in least fixed-point logic and McColm's conjecture
本文研究有限结构族上最小不动点逻辑的四个驯良性性质，证明其仅依赖初等极限理论，并证实对模型论驯良的极限理论，McColm猜想成立。
Ordered henselian valued fields: definability and Borel sets
研究有序赫恩塞利赋值域的可定义性与博雷尔集性质，通过量化消除和维度缩减定理，揭示其与NIP域分类猜想的联系。
We extend first-order logic to include variadic function symbols, and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.
We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, `Talagrand's stability', and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove the `almost definability' and `Baire~1 definability' of coheirs assuming NIP. We show that a formula $ϕ(x,y)$ has the strict order property if and only if there is a convergent sequence of continuous functions on the space of $ϕ$-types such that its limit is not…
Analogues of Scott's isomorphism theorem, Karp's theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An "interpolation theorem" (of a particular sort introduced by Barwise and van Benthem) for the infinitary quantificational boolean logic $L_{\infty ω}$ holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definability property.
We investigate atomicity of free algebras and various forms of amalgamation for BL and MV algebras, and also Heyting algebras, though the latter algebras may not be linearly ordered, so strictly speaking their corresponding intuitionistic logic does not belong to many valued logic. Generalizing results of Comer proved in the classical first order case; working out a sheaf duality on the Zarski topology defined on the prime spectrum of such algebras, we infer several definability theorems, and obtain a representation theorem for theories as continuous sections of Sheaves. We also prove an omitt…
$Σ_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders
本文研究高基数上借助选择公理构造的集合能否由仅用κ及遗传基数小于κ的集合作参数的Σ₁公式定义，通过完美集定理推广Mathias关于极大几乎不相交族的非定义性结果，并确定奇异强极限基数上相关对象不存在Σ₁定义的精确一致性强度。
We show that for any square-free natural number $n$ and any global field $K$ with $(\text{char}(K), n)=1$ containing the $n$th roots of unity, the pairs $(x,y)\in K^*\times K^*$ such that $x$ is not a norm of $K(\sqrt[n]{y})/K$ form a diophantine set over $K$. We use the Hasse norm theorem, Kummer theory, and class field theory to prove this result. We also prove that for any $n\in \mathbb{N}$ and any global field $K$ with $(\text{char}(K), n)=1$, $K^*\setminus K^{*n}$ is diophantine over $K$. For a number field $K$, this is a result of Colliot-Thélène and Van Geel, proved using results on the…
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。
We introduce a general version of singular compactness theorem which makes it possible to show that being a $Σ$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair. We also prove the dual result for Gorenstein injective modules.
Sahlqvist formulas are a syntactically specified class of modal formulas proposed by Hendrik Sahlqvist in 1975. They are important because of their first-order definability and canonicity, and hence axiomatize complete modal logics. The first-order properties definable by Sahlqvist formulas were syntactically characterized by Marcus Kracht in 1993. The present paper extends Kracht's theorem to the class of `generalized Sahlqvist formulas' introduced by Goranko and Vakarelov and describes an appropriate generalization of Kracht formulas.
Infinitary Logics and Abstract Elementary Classes
本文证明了抽象初等类可以在特定无限逻辑中被公理化，并引入了canonical tree概念，探讨了其与新逻辑的关系。
In arXiv:1604.08705 we introduced the propositional modal logic $\textbf{TSC}$ (which stands for Turing Schmerl Calculus) which adequately describes the provable interrelations between different kinds of Turing progressions. In arXiv:1709.04715 we defined a model $\mathcal{J}$ which is proven to be a universal model for $\textbf{TSC}$ based on the intensively studied Ignatiev's universal model for the closed fragment of $\textbf{GLP}$ (Gödel Löb's polymodal provability logic). In the current paper we present a new universal frame $\mathcal{H}$, which is a slight modification of $\mathcal{J}$, …
Trace definability IV: higher arity notions
本文引入了k-迹可定义性概念，证明了k-迹可定义于NIP理论的理论是k-NIP，且2-迹可定义于稳定理论的理论是2-NFOP。所有已知的k-NIP理论都可以k-迹可定义于NIP理论中，并构造了通用理论D_k(T)。
In this article, we study logics of unknown truths and false beliefs under neighborhood semantics. We compare the relative expressivity of the two logics. It turns out that they are incomparable over various classes of neighborhood models, and the combination of the two logics are equally expressive as standard modal logic over any class of neighborhood models. We propose morphisms for each logic, which can help us explore the frame definability problem, show a general soundness and completeness result, and generalize some results in the literature. We axiomatize the two logics over various cl…
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