Approximate Categoricity in Continuous Logic

Vaught's Two-Cardinal Theorem and Quasi-Minimality in Continuous Logic

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 …

近似等价关系

Approximate equivalence relations

本文在可定义测度下研究近似等价关系，基于连续逻辑提出概率逻辑框架，利用Kim-Pillay空间完全分类二元相关性，推广范畴性定理与近似子群的李模型定理，并证明NIP理论中可定义测度自同构群的阿基米德有限维性。

Logic for approximate entailment in ordered universes of discourse

The Logic of Approximate Entailment (LAE) is a graded counterpart of classical propositional calculus, where conclusions that are only approximately correct can be drawn. This is achieved by equipping the underlying set of possible worlds with a similarity relation. When using this logic in applications, however, a disadvantage must be accepted; namely, in LAE it is not possible to combine conclusions in a conjunctive way. In order to overcome this drawback, we propose in this paper a modification of LAE where, at the semantic level, the underlying set of worlds is moreover endowed with an ord…

Logic of Approximate Entailment in quasimetric spaces

The logic LAE discussed in this paper is based on an approximate entailment relation. LAE generalises classical propositional logic to the effect that conclusions can be drawn with a quantified imprecision. To this end, properties are modelled by subsets of a distance space and statements are of the form that one property implies another property within a certain limit of tolerance. We adopt the conceptual framework defined by E. Ruspini; our work is towards a contribution to the investigation of suitable logical calculi. LAE is based on the assumption that the distance function is a quasimetr…

一阶 Łukasiewicz 逻辑超矢列演算的近似完备性

Approximate Completeness of Hypersequent Calculus for First-Order Łukasiewicz Logic

针对 Baaz 和 Metcalfe 的 GŁ∀ 近似完备性仅适用于前束公式的局限，本文将原证明推广至超矢列，证明其对任意一阶 Łukasiewicz 公式均近似完备。

仿射逻辑中的极值模型与直积分

Extremal models and direct integrals in affine logic

研究仿射逻辑中仿射理论的极值模型，提出度量结构可测场与直积分理论，证明单纯理论模型的极值分解定理，建立其Bauer-Poulsen二分结果，适配连续逻辑工具并给出多领域例子。

A proof of completeness for continuous first-order logic

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…

The Logic of Approximate Dependence

We extend the treatment of functional dependence, the basic concept of dependence logic, to include the possibility of dependence with a limited number of exceptions. We call this approximate dependence. The main result of the paper is a Completeness Theorem for approximate dependence atoms. We point out some problematic features of this which suggests that we should consider multi-teams, not just teams.

An approximate Herbrand's theorem and definable functions in metric structures

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in some absolutely ubiquitous structures from classical logic.

An approximate logic for measures

We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to give a short proof of the Szemerédi Regularity Lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.

The Real truth

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…

From rational Godel logic to continuous ultrametric logic

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. …

Epistemic BL-Algebras

Fuzzy Epistemic Logic is an important formalism for approximate reasoning. It extends the well known basic propositional logic BL, introduced by Hájek, by offering the ability to reason about possibility and necessity of fuzzy propositions. We consider an algebraic approach to study this logic, introducing Epistemic BL-algebras. These algebras turn to be a generalization of both, Pseudomonadic Algebras introduced by \cite{Bez2002} and serial, euclidean and transitive Bi-modal Gödel Algebras proposed by \cite{CaiRod2015}. We present the connection between this class of algebras and fuzzy possib…

Approximate Isomorphism of Metric Structures

We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov and by Ben Yaacov, Doucha, Nies, and Tsankov, which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two-sorted structures that witness approximate isom…

度量值域的近似AKE原理

An Approximate AKE Principle for Metric Valued Fields

本文基于连续逻辑研究度量值域，提出在投影线上的一种近似AKE原理，描述了等价性与余域和值群的关系，并证明了度量值差域在任意特征下无模型伴。