Itaï Ben Yaacov, Alexander Usvyatsov
We develop continuous first order logic, a variant of the logic described in \cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson's logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.
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We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.
In this short note we compare the expressive power of real-valued continuous logic (or just continuous logic, in recent literature) with that of compact-valued continuous logic, proposed by Chang and Keisler. We conclude that the two logics have the same expressive power, and moreover, that this remains true if we replace the plethora of potential quantifiers of compact-valued logic with a single ''primordial'' one.
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.
We combine continuous and integral logics and found a logical framework for metric measure spaces equipped with a family of continuous relations and operations. We prove the ultraproduct theorem and deduce compactness and other usual results. We also give applications of the compactness theorem in metric measure theory.
We analyse logic actions of Polish groups which arise in continuous logic. We extend the generalised model theory of H.Becker to the case of Polish G-spaces when G is an arbitrary Polish group.
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…
In this paper, we present a generalized effective completeness theorem for continuous logic. The primary result is that any continuous theory is satisfied in a structure which admits a presentation of the same Turing degree. It then follows that any decidable theory is satisfied by a computably presentable structure. This modifies and extends previous partial effective completeness theorems for continuous logic given by Calvert and Didehvar, Ghasemloo, and Pourmahdian.
Uniqueness of constructible models in continuous logic
本文研究连续一阶逻辑中可构造模型的唯一性问题，证明任意完备连续一阶理论的可构造模型在同构意义下唯一。
We establish a characterization of extreme amenability of any Polish group in Fraïssé-theoretic terms in the setting of continuous logic, mirroring a theorem due to Kechris, Pestov and Todorcevic for closed subgroups of the permutation group of an infinite countable set.
Büchi and Owen studied algebraic structures called hoops. Hoops provide a natural algebraic semantics for a class of substructural logics that we think of as intuitionistic analogues of the widely studied Łukasiewicz logics. Ben Yaacov extended Łukasiewicz logic to get what is called continuous logic by adding a halving operator. In this paper, we define the notion of continuous hoop, or coop for short, and show that coops provide a natural algebraic semantics for continuous logic. We characterise the simple and subdirectly irreducible coops and investigate the decision problem for various the…
We give a presentation theorem for continuous first-order logic and Metric Abstract Elementary classes in terms of $L_{ω_1, ω}$ and Abstract Elementary Classes, respectively. This presentation is accomplished by analyzing dense subsets that are closed under functions. We extend this correspondence to types and saturation.
The ultraproduct construction is generalized to $p$-ultramean constructions ($1\leqslant p<\infty$) by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments $\mathscr L^p$ of continuous logic. A powermean variant of Keisler-Shelah isomorphism theorem is proved for $\mathscr L^p$. It is then proved that $\mathscr L^p$-sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.
Continuous Algebra: Algebraic Semantics for Continuous Propositional Logic
本文提出连续代数作为连续命题逻辑CPL的代数语义，并通过与2可除$\ell u$-群的对应关系，给出了CPL弱完备性定理的纯代数证明。
We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for classical real rosy theories.
The primary purpose of this article is to show that a certain natural set of axioms yields a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an \emph{approximated} form of strong completeness, whereby $Σ\vDashφ$ (if and) only if $Σ\vdashφ\dotminus 2^{-n}$ for all $n<ω$. This approximated form of strong completeness asserts that if $Σ\vDashφ$, then proofs from $Σ$, being fin…
We exhibit Pontryagin duality as a special case of Stone duality in a continuous logic setting. More specifically, given an abelian topological group $A$, and $\mathcal F$ the family (group) of continuous homomorphisms from $A$ to the circle group $\mathbb T$, then, viewing $(A,+)$ equipped with the collection $\mathcal F$ as a continuous logic structure $M$, we show that the local type space $S_\mathcal F(M)$ is precisely the Pontryagin dual of the group $\mathcal F$ where the latter is considered as a discrete group. We conclude, using Pontryagin duality (between compact and discrete abelian…
A characterization of Continuous Logic by using quantale-valued logics
论文将 Continuous Logic 推广到 co-quantale 值距离，并在结构条件下证明 Tarski-Vaught 判据与 Łoś 定理，从而通过放松条件构造新逻辑。
This paper is devoted to systematic studies of some extensions of first-order Gödel logic. The first extension is the first-order rational Gödel logic which is an extension of first-order Gödel logic, enriched by countably many nullary logical connectives. By introducing some suitable semantics and proof theory, it is shown that the first-order rational Gödel logic has the completeness property, that is any (strongly) consistent theory is satisfiable. Furthermore, two notions of entailment and strong entailment are defined and their relations with the corresponding notion of proof is studied. …
Ehrenfeucht-Fraïssé Games for Continuous First-Order Logic
本文在度量模型论和连续一阶逻辑中定义了Ehrenfeucht-Fraïssé博弈，证明长度为$n$的获胜策略等价于量词秩$n$内的初等等价，并通过实例和无穷逻辑的联系展示了其应用价值。
We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than the unit ball approach (which has been the common approach so far to Banach space structures in continuous logic), as well as of applying in situations where the unit ball approach does not apply (i.e., when the unit ball is not a definable set). We also introduce the process of single point \emph{emboundment} (closely related to the topological single point compactification), allowing to bring unbou…
We show that a topometric space $X$ is topometrically isomorphic to a type space of some continuous first-order theory if and only if $X$ is compact and has an open metric (i.e., satisfies that $\{p : d(p,U) < \varepsilon\}$ is open for every open $U$ and $\varepsilon > 0$). Furthermore, we show that this can always be accomplished with a stable theory.
Preservation under Reduced Products in Continuous Logic
本文提出连续逻辑中一种在缩减积下保持的片段，扩展了经典理论在保持性和稳定性方面的结果，并通过保持性质表征了相关片段。
The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. Here we investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out. We also introduce, in the context of Ban…
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.
Type-free systems of logic are designed to consistently handle significant instances of self-reference. Some consistent type-free systems also have the feature of allowing the sort of general abstraction or comprehension principle that infamously leads to paradox in naive set theory. Because type-free systems possess these features, and avoid the hierarchy of types that is felt to be unnatural in some contexts, they have the potential to play an important role in the foundations of mathematics, the theory of classes (producing a richer notion of class than that currently used in set theory and…
Strong completeness for the predicate logic of the continuous t-norms
本文扩展Hájek基于BL-链的一阶BL逻辑公理系统，通过加入平方量词公理和一条无穷推理规则，证明其对连续t-模语义的强完全性。
We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbfΣ^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth--Kechris about discrete structures. As a different application, we also give a new proof of Kechr…
We define sofic, weakly sofic, linear sofic and hyperlinear metric groups and discuss some issues involving axiomatizability of these classes in continuous logic.
We suggest an ordering for the predicates in continuous logic so that the semantics of continuous logic can be formulated as a hyperdoctrine. We show that this hyperdoctrine can be embedded into the hyperdoctrine of subobjects of a suitable Grothendieck topos. For this embedding we use a simplification of the hyperdoctrine for continuous logic, whose category of equivalence relations is equivalent to the category of complete metric spaces and uniformly continuous maps.
Definably amenable groups in Continuous logic
本文为连续逻辑中的群引入了可定义顺从性与极端可定义顺从性概念，利用不动点性质进行刻画，证明了稳定群与超紧群具备该性质，并揭示了依赖理论中该性质与f-generic类型的等价关系及一阶群的随机化结果。
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]。
We study a real valued propositional logic with unbounded positive and negative truth values that we call R-valued logic. Such logic slightly extends continuous propositional logic which, in turn, builds on Lukasiewicz many-valued logic. After presenting the deduction machinery and the semantics of R-valued logic, we prove a completeness theorem for finite theories. Then we define unital and Archimedean theories, in accordance with the theory of Riesz spaces. In the unital setting, we prove the equivalence of consistency and satisfiability and an approximated completeness theorem similar to th…
Stability in affine logic
本文研究了 affine逻辑中稳定性理论的基础，证明了稳定性在直接积分下的保持性，并推导了极值模型稳定性与理论稳定性之间的关系。
We study expressive power of continuous logic in classes of (locally compact) groups. We also describe locally compact groups which are separably categorical structures.
We study expressive power of continuous logic in classes of metric groups defined by properties of their actions. For example we consider properties non-OB, non-FH and non-FR. The paper substantially extends Section 2 of the paper A.Ivanov, "Locally compact groups and continuous logic", arxiv:1206.5473.
In continuous first-order logic, the union of definable sets is definable but generally the intersection is not. This means that in any continuous theory, the collection of $\varnothing$-definable sets in one variable forms a join-semilattice under inclusion that may fail to be a lattice. We investigate the question of which semilattices arise as the collection of definable sets in a continuous theory. We show that for any non-trivial finite semilattice $L$ (or, equivalently, any finite lattice $L$), there is a superstable theory $T$ whose semilattice of definable sets is $L$. We then extend t…
Over the last century, the principle of "induction on the continuum" has been studied by different authors in different formats. All of these different readings are equivalent to one of the three versions that we isolate in this paper. We also formalize those three forms (of "continuous induction") in first-order logic and prove that two of them are equivalent and sufficiently strong to completely axiomatize the first-order theory of the real closed (ordered) fields. We show that the third weaker form of continuous induction is equivalent with the Archimedean property. We study some equivalent…
We prove the following continuous analogue of Vaught's Two-Cardinal Theorem: if for some $κ>λ\geq \aleph_0$, a continuous theory $T$ has a model with density character $κ$ which has a definable subset of density character $λ$, then $T$ has a model with density character $\aleph_1$ which has a separable definable subset. We also show that if we assume that $T$ is $ω$-stable, then if $T$ has a model of density character $\aleph_1$ with a separable definable set, then for any uncountable $κ$ we can find a model of $T$ with density character $κ$ which has a separable definable subset. In order to …
We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and only if it has IP or SOP.
We generalize pp elimination for modules, or more generally abelian structures, to a continuous logic environment where the abelian structure is equipped with a homomorphism to a compact (Hausdorff) group. We conclude that the continuous logic theory of such a structure is stable.
In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of Łukasiewicz logic) and continuous logic, and prove effective versions of some theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; show that provability degree of a formula w.r.t. a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consist…
A Logic of Stability: Formalizing Similarity in Counterfactual Reasoning
本文针对反事实推理中可能世界的相似性问题，提出量化相似程度的超度量方法，开发并公理化多模态相似性逻辑\ud835\udc4b\ud835\udc5b\ud835\udc5f，证明其可靠性与完备性，建立超度量语义与反事实理论的联系。
We use the compactness theorem of continuous logic to give a new proof that $L^r([0,1]; \mathbb{R})$ isometrically embeds into $L^p([0,1]; \mathbb{R})$ whenever $1 \leq p \leq r \leq 2$. We will also give a proof for the complex case. This will involve a new characterization of complex $L^p$ spaces based on Banach lattices.
We explore approximate categoricity in the context of distortion systems, introduced in our previous paper, which are a mild generalization of perturbation systems, introduced by Ben Yaacov. We extend Ben Yaacov's Ryll-Nardzewski style characterization of separably approximately categorical theories from the context of perturbation systems to that of distortion systems. We also make progress towards an analog of Morley's theorem for inseparable approximate categoricity, showing that if there is some uncountable cardinal $κ$ such that every model of size $κ$ is 'approximately saturated,' in the…
Using the category of metric spaces as a template, we develop a metric analogue of the categorical semantics of classical/intuitionistic logic, and show that the natural notion of predicate in this "continuous semantics" is equivalent to the a priori separate notion of predicate in continuous logic, a logic which is independently well-studied by model theorists and which finds various applications. We show this equivalence by exhibiting the real interval [0,1] in the category of metric spaces as a "continuous subobject classifier" giving a correspondence not only between the two notions of pre…
Cut-free Deductive System for Continuous Intuitionistic Logic
本文提出了一种无割推理论系统，用于连续直觉逻辑，通过代数方法证明了其完备性和割可接受性。
Elements of affine model theory
本文介绍了仿射连续模型理论的基础，通过避免连续逻辑中的合取与析取连接词，利用超均值构造等仿射化工具，为具有非范畴理论的紧致结构提供了新的模型论研究框架。
The amalgamation property and Urysohn structures in continuous logic
本文研究了连续逻辑中 amalgamation 性质与 Urysohn 结构的关系，证明了相关类别的 Fraïssé 性质和 EPPA，以及自同构群的 universal 性质。
One of the nice properties of the first-order logic is the compactness of satisfiability. It state that a finitely satisfiable theory is satisfiable. However, different degrees of satisfiability in many-valued logics, poses various kind of the compactness in these logics. One of this issues is the compactness of $K$-satisfiability. Here, after an overview on the results around the compactness of satisfiability and compactness of $K$-satisfiability in many-valued logic based on continuous t-norms (basic logic), we extend the results around this topic. To this end, we consider a reverse semantic…
We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{ω_1 ω}$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property {\bf (T)} can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.
Generic Stability and Modes of Convergence
该研究在经典与连续逻辑中探究泛型稳定型/测度，分析其与随机化及型/测度收敛模式的联系，推进相关逻辑领域研究。
Continuous logic in a classical setting
本文在经典框架下推广了基于正公式的连续逻辑方法，为更一般的紧致豪斯多夫空间上的模型类引入了初等性、紧致性、饱和性等基本模型论工具。
We survey discrete and continuous model-theoretic notions which have important connections to general topology. We present a self-contained exposition of several interactions between continuous logic and $C_p$-theory which have applications to a classification problem involving Banach spaces not including $c_0$ or $l^p$, following recent results obtained by P. Casazza and J. Iovino for compact continuous logics. Using $C_p$-theoretic results involving Grothendieck spaces and double limit conditions, we extend their results to a broader family of logics, namely those with a first countable weak…
We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately $\aleph_0$-saturated up to arbitrarily small perturbations are isomorphic up to arbitrarily small perturbations (where the notion of perturbation is part of the data). As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories all of whose separable models are isomorphic up to arbitrarily small perturbations.
Star sorts, Lelek fans, and the reconstruction of non-$\aleph\_0$-categorical theories in continuous logic
本文针对可数语言下连续/经典逻辑的任意理论，构建了以开波兰拓扑群胚$\boldsymbol{G}^*(T)$为形式的双解释不变量，其基与Lelek扇形同胚，通过两步构造法将结果纳入统一框架。
Positive Logic: An Introduction for Model Theorists
本文为模型论学者提供正逻辑的自洽导论，讲解正模型论基础、可数范畴性、巨型模型、单纯与稳定理论等内容，介绍其在超虚元素、连续逻辑中的应用并附内容速览。
Affine logic with the integration operator
本文提出了仿射积分逻辑，扩展了仿射连续逻辑，并通过两种方法证明了紧致性定理，同时给出了证明系统和完备性定理，利用Keisler-Shelah同构定理的变体来表征非分叉扩展。
Continuous logic extends the multi-valued Lukasiewicz logic by adding a halving operator on propositions. This extension is designed to give a more satisfactory model theory for continuous structures. The semantics of these logics can be given using specialisations of algebraic structures known as hoops. As part of an investigation into the metatheory of propositional continuous logic, we were indebted to Prover9 for finding a proof of an important algebraic law.
Generically Stable Measures and Distal Regularity in Continuous Logic
该研究将经典逻辑的相关结论推广至连续逻辑框架，在NIP度量理论中发展生成稳定与光滑Keisler测度理论，给出其等价刻画，并结合弱正交性结果与切割引理，得到远端正则性与强Erdős-Hajnal性质的解析版本。
In aiming to apply to a broader class of examples the Avigad-Iovino "ultraproducts and metastability" approach to obtaining uniformity for convergence of sequences, we construct a framework using continuous logic that in particular is able to handle discontinuous functions in its domain of discourse. This setup weakens the usual continuity requirements for functions, but compensates for the loss of control by introducing a notion of "linear structure" that captures in a quite general way the situation of having geodesics between every pair of points, and has as a special case the vector space …
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