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 $\mathbf {\Sigma }^0_2$, 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 Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable. (shrink)

A general approach to the synthesis of an optimal order of executing jobs in engineering systems with indeterminate times of job processing is presented. As a mathematical model of the system, a two-stage pipeline is taken whose first and second stages are, respectively, the input of data and its processing, and the corresponding mathematical apparatus is continuous logic and logic determinants.

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 $$1 \le p \le r \le 2$$. 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.

Linear continuous logic is the fragment of continuous logic obtained by restricting connectives to addition and scalar multiplications. Most results in the full continuous logic have a counterpart in this fragment. In particular a linear form of the compactness theorem holds. We prove this variant and use it to deduce some basic preservation theorems.

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 discrete first-order real rosy theories.

In the context of continuous logic, this paper axiomatizes both the class \ of lattice-ordered groups isomorphic to C for X compact and the subclass \ of structures existentially closed in \; shows that the theory of \ is \-categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \ and \; shows that \\in \mathcal {C}\) has a prime-model extension in \ just in case X is Boolean; and proves that in a sense relevant to continuous (...) logic, positive formulas admit in \ elimination of quantifiers to positive formulas. (shrink)

We show how to extend the Continuous Propositional Logic by means of an infinitary rule in order to achieve a Strong Completeness Theorem. Eventually we investigate how to recover a weak version of the Deduction Theorem.

The linear compactness theorem is a variant of the compactness theorem holding for linear formulas. We show that the linear fragment of continuous logic is maximal with respect to the linear compactness theorem and the linear elementary chain property. We also characterize linear formulas as those preserved by the ultramean construction.

In recent years, model theory has widened its scope to include metric structures by considering real-valued models whose underlying set is a complete metric space. We show that it is possible to carry out this work by giving presentation theorems that translate the two main frameworks into discrete settings. We also translate various notions of classification theory.

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.

The ultraproduct construction is generalized to p‐ultramean constructions () by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments of continuous logic and are very close to the constructions in real analysis. A powermean variant of the Keisler‐Shelah isomorphism theorem is proved for. It is then proved that ‐sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.

We prove Robinson consistency theorem as well as Craig, Lyndon and Herbrand interpolation theorems in linear continuous logic.

In this paper, we introduce a foundation for computable model theory of rational Pavelka logic and continuous logic, and prove effective versions of some related 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; we show that provability degree of a formula with respect to a linear theory (...) is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory is decidable. (shrink)

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 predicate, but also between the natural notion of quantification in the continuous semantics and the existing notion of quantification in continuous logic.Along the way, we formulate what it means for a given category to behave like the category of metric spaces, and afterwards show that any such category supports the aforementioned continuous semantics. As an application, we show that categories of presheaves of metric spaces are examples of such, and in fact even possess continuous subobject classifiers. (shrink)

We study expressive power of continuous logic in classes of metric groups defined by properties of their actions. We concentrate on unbounded continuous actions on metric spaces. For example, we consider the properties non‐OB, non‐FH and non‐FR.

We introduce a propositional many-valued modal logic which is an extension of the Continuous Propositional Logic to a modal system. Otherwise said, we extend the minimal modal logic to a Continuous Logic system. After introducing semantics, axioms and deduction rules, we establish some preliminary results. Then we prove the equivalence between consistency and satisfiability. As straightforward consequences, we get compactness, an approximated completeness theorem, in the vein of Continuous Logic, and a Pavelka-style (...) completeness theorem. (shrink)

If we imagine a chess-board with alternate blue and red squares, then this is something in which the individual red and blue areas allow themselves to be distinguished from each other in juxtaposition, and something similar holds also if we imagine each of the squares divided into four smaller squares also alternating between these two colours. If, however, we were to continue with such divisions until we had exceeded the boundary of noticeability for the individual small squares which result, then (...) it would no longer be possible to apprehend the individual red and blue areas in their respective positions. But would we then see nothing at all? Not in the least; rather we would see the whole chessboard as violet, i.e. apprehend it as something that participates simultaneously in red and blue. (shrink)

The notion of a continuously variable quantity can be regarded as a generalization of that of a particular quantity, and the properties of such quantities are then akin to, and derived from, the properties of constants. For example, the continuous, real-valued functions on a topological space behave like the field of real numbers in many ways, but instead form a ring. Topos theory permits one to apply this same idea to logic, and to consider continuously variable sets . (...) In this expository paper, such applications are explained to the non-specialist. Some recent results are mentioned, including a new completeness theorem for higher-order logic. (shrink)

In classical modal semantics, a binary accessibility relation connects worlds. In this paper, we present a uniform and systematic treatment of modal semantics with a continuous accessibility relation alongside the continuous accessibility modal logics that they model. We develop several such logics for a variety of philosophical applications. Our main conclusions are as follows. Modal logics with a continuous accessibility relation are sound and complete in their natural classes of models. The class of Kripke frames where a (...) continuous accessibility relation has a magnitude characterizing its degree of accessibility is not modally definable, and this has unappreciated significance to completeness proofs for such logics, revealing a methodological advantage of using classical multimodal semantics over fuzzy modal semantics. There is a pseudometric space modal logic that is complete in the class of pseudometric spaces, a natural semantic setting for quantitative modal reasoning about similarity. There is a metric space modal logic that is complete in the class of metric spaces, a natural semantic setting for quantitative modal reasoning about neighborhoods and counterfactual stability. There is a real line continuous temporal logic that is canonical for real lines, a natural semantic setting for quantitative modal reasoning about time. (shrink)

The notion of a continuously variable quantity can be regarded as a generalization of that of a particular (constant) quantity, and the properties of such quantities are then akin to, and derived from, the..

The paper analyzes dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the latter, used as an analytical aid. We show maps (...) induced by action model transformations continuous with respect to the Stone topology and present results on the recurrent behavior of said maps. (shrink)

Axiomatizations are presented for fuzzy logics characterized by uninorms continuous on the half-open real unit interval [0,1), generalizing the continuous t-norm based approach of Hájek. Basic uninorm logic BUL is defined and completeness is established with respect to algebras with lattice reduct [0,1] whose monoid operations are uninorms continuous on [0,1). Several extensions of BUL are also introduced. In particular, Cross ratio logic CRL, is shown to be complete with respect to one special uninorm. A (...) Gentzen-style hypersequent calculus is provided for CRL and used to establish co-NP completeness results for these logics. (shrink)

The paper deals with fuzzy Horn logic which is a fragment of predicate fuzzy logic with evaluated syntax. Formulas of FHL are of the form of simple implications between identities. We show that one can have Pavelka-style completeness of FHL w.r.t. semantics over the unit interval [0, 1] with left-continuous t-norm and a residuated implication, provided that only certain fuzzy sets of formulas are considered. The model classes of fuzzy structures of FHL are characterized by closure properties. (...) We also give comments on related topics proposed by N. Weaver. (shrink)

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, as well as of applying in situations where the unit ball approach does not apply. We also introduce the process of single point emph{emboundment}, allowing to bring unbounded structures back into the setting of bounded continuous first order logic. Together (...) with results from cite{BenYaacov:Perturbations} regarding perturbations of bounded metric structures, we prove a Ryll-Nardzewski style characterisation of theories of Banach spaces which are separably categorical up to small perturbation of the norm. This last result is motivated by an unpublished result of Henson. (shrink)

We continue our work [5] on the logic of multisets (or on the multiset semantics of linear logic), by interpreting further the additive disjunction . To this purpose we employ a more general class of processes, called free, the axiomatization of which requires a new rule (not compatible with the full LL), the cancellation rule. Disjunctive multisets are modeled as finite sets of multisets. The -Horn fragment of linear logic, with the cut rule slightly restricted, is sound (...) with respect to this semantics. Another rule, which is a slight modification of cancellation, added to HF makes the system sound and complete. (shrink)

Corlett (2019) raises two groups of challenges against the two-factor theory of delusions: One focuses on weighing “the evidence for … the two-factor theory”; the other aims to question “the logic of the two-factor theory” (p. 166). McKay (2019) has robustly defended the two-factor theory against the first group. But the second group, which Corlett believes is in many aspects independent of the first group and Darby (2019, p. 180) takes as “[t]he most important challenge to the two-factor theory (...) raised by Dr. Corlett”, has by large remained. Here I offer my two cents in response to the second group. More specifically, I argue that Corlett’s challenges to the logic of the two-factor theory, concerning modularity, double dissociation and cognitive penetration, seem to be based on some misunderstandings of the two-factor theory. (shrink)

It is shown that the propositional modal logic IRM (interpretability logic with Montagna's principle and with witness comparisons in the style of Guaspari's and Solovay's logicR) is sound and complete as the logic ofII 1-conservativity over each∑ 1-sound axiomatized theory containingI∑ 1. The exact statement of the result uses the notion of standard proof predicate. This paper is an immediate continuation of our paper [HM]. Knowledge of [HM] is presupposed. We define a modal logic, called IRM, (...) which includes both ILM andR (the logic of [GS]) and prove an arithmetical completeness theorem in the style of [GS], thus showing that IRM is the logic ofII 1-conservativity with witness comparisons. The reader is recommended to have Smoryński's book [Sm] at his/her disposal. (shrink)

This paper begins with a brief account of how I started out as a young logician studying modal logic with the hope that it would be useful when applied to evaluating real examples of arguments found in natural language texts. The exposition moves on to relate how my interests shifted to the study of argumentation in informal logic, and from there to computational systems combining defeasible argumentation schemes with argument mapping (diagramming). The story ends by leading to recent (...) collaborations with three young logicians and computer scientists who are continuing on the path of my journey. (shrink)

It is shown the complete equivalence between the theory of continuous (enumeration) fuzzy closure operators and the theory of (effective) fuzzy deduction systems in Hilbert style. Moreover, it is proven that any truth-functional semantics whose connectives are interpreted in [0,1] by continuous functions is axiomatizable by a fuzzy deduction system (but not by an effective fuzzy deduction system, in general).

The purpose of this paper is to investigate continuity properties arising in elementary (i.e., first-order) logic in the hope of illuminating the special status of this logic. The continuity properties turn out to be closely related to conditions which characterize elementary logic uniquely, and lead to various further questions.

A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic . The proofs of soundness and completeness are constructive and the computational content of their composition is, in particular, a β-normalisation-by-evaluation program for simply typed lambda calculus with sum types. Although the inspiration comes from Danvyʼs type-directed partial evaluator for the same lambda calculus, the use of delimited control operators is avoided. The role (...) of polymorphism is crucial – dropping it allows one to obtain a notion of model complete for classical predicate logic. (shrink)

Is logical empiricism incompatible with a critical social science? The longstanding assumption that it is incompatible has been prominent in recent debates about welfare economics. Sen’s development of a critical and descriptively rich welfare economics is taken by writers such as Putnam, Walsh and Sen to involve the excising of the influence of logical empiricism on neo-classical economics. However, this view stands in contrast to the descriptively rich contributions to political economy of members of the left Vienna Circle, such as (...) Otto Neurath. This paper considers the compatibility of the meta-theoretical commitments of Neurath and others in the logical empiricist tradition with this first-order critical political economy. (shrink)

In his early writings on logic Collingwood offered a powerful critique of contemporary theories, including subjective idealism and realism to which he continued to be opposed throughout his career. Simultaneously these same early writings present a sustained attack on dichotomous forms of thought, which are also carried through to his later writings. Throughout Collingwood maintains a critical respect for Hegel. Subjectivity and objectivity are not to be severed from each other, nor are identities to be excluded from one another. (...) Continuity is not to be understood apart from discontinuity. Collingwood's early critiques inspire a variety of later doctrines such as the scale of forms, the logic of question and answer, metaphysics as a science of absolute presuppositions, and his conception of the relation between mind and civilisation. The later theories are to be understood as continuous with the earlier doctrines, but not deducible from, nor reducible to them. (shrink)

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen :168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen, that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we (...) get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem. (shrink)