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.

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the –completeness of the equational theories of action lattices of subsets of (...) a finite monoid and action lattices of binary relations on a finite universe. We also discuss possible applications in linguistics. (shrink)

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)

Dynamic topological logic is a multi-modal logic that was introduced for reasoning about dynamic topological systems, i.e. structures of the form $\langle{\mathfrak{X}, f}\rangle $, where $\mathfrak{X}$ is a topological space and $f$ is a continuous function on it. The problem of finding a complete and natural axiomatization for this logic in the original tri-modal language has been open for more than one decade. In this paper, we give a natural axiomatization of $\textsf{DTL}$ and prove its strong (...) completeness with respect to the class of all dynamic topological systems. Our proof system is infinitary in the sense that it contains an infinitary derivation rule with countably many premises and one conclusion. It should be remarked that $\textsf{DTL}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete. Moreover, we provide an infinitary axiomatic system for the logic ${\textsf{DTL}}_{\mathcal{A}}$, i.e. the $\textsf{DTL}$ of Alexandrov spaces, and show that it is strongly complete with respect to the class of all dynamical systems based on Alexandrov spaces. (shrink)

We describe an infinitary logic for metric structures which is analogous to Lω1,ω. We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients (...) of Banach spaces. (shrink)

Inspired by Pohlers' local predicativity approach to Pure Proof Theory and Howard's ordinal analysis of bar recursion of type zero we present a short, technically smooth and constructive strong normalization proof for Gödel's system T of primitive recursive functionals of finite types by constructing an ε 0 -recursive function [] 0 : T → ω so that a reduces to b implies [a] $_0 > [b]_0$ . The construction of [] 0 is based on a careful analysis of the Howard-Schütte (...) treatment of Gödel's T and utilizes the collapsing function ψ: ε 0 → ω which has been developed by the author for a local predicativity style proof-theoretic analysis of PA. The construction of [] 0 is also crucially based on ideas developed in the 1995 paper "A proof of strongly uniform termination for Gödel's T by the method of local predicativity" by the author. The results on complexity bounds for the fragments of T which are obtained in this paper strengthen considerably the results of the 1995 paper. Indeed, for given n let T n be the subsystem of T in which the recursors have type level less than or equal to n+2. (By definition, case distinction functionals for every type are also contained in T n .) As a corollary of the main theorem of this paper we obtain (reobtain?) optimal bounds for the T n -derivation lengths in terms of ω n+2 -descent recursive functions. The derivation lengths of type one functionals from T n (hence also their computational complexities) are classified optimally in terms of $ -descent recursive functions. In particular we obtain (reobtain?) that the derivation lengths function of a type one functional a ∈ T 0 is primitive recursive, thus any type one functional a in T 0 defines a primitive recursive function. Similarly we also obtain (reobtain?) a full classification of T 1 in terms of multiple recursion. As proof-theoretic corollaries we reobtain the classification of the IΣ n+1 -provably recursive functions. Taking advantage from our finitistic and constructive treatment of the terms of Gödel's T we reobtain additionally (without employing continuous cut elimination techniques) that PRA + PRWO ( $\varepsilon_0) \vdash \Pi^0_2$ - Refl(PA) and PRA + PRWO ( $\omega_{n+2}) \vdash \Pi^0_2$ - Refl(I Σ n+1 ), hence PRA + PRWO( $\epsilon_0) \vdash$ Con(PA) and PRA + PRWO( $\omega_{n+2}) \vdash$ Con(IΣ n+1 ). For programmatic reasons we outline in the introduction a vision of how to apply a certain type of infinitary methods to questions of finitary mathematics and recursion theory. We also indicate some connections between ordinals, term rewriting, recursion theory and computational complexity. (shrink)

Building on previous work by Mints, Buchholz and Schwichtenberg, a simplified version of continuous normalization for the untyped λ-calculus and Gödel’s is presented and analysed in the coalgebraic framework of non-wellfounded terms with so-called repetition constructors.The primitive recursive normalization function is uniformly continuous w.r.t. the natural metric on non-wellfounded terms. Furthermore, the number of necessary repetition constructors is locally related to the number of reduction steps needed to reach the normal form and its size.It is also shown how (...) continuous normal forms relate to derivations of strong normalizability in the typed λ-calculus and how this leads to new bounds for the sum of the height of the reduction tree and the size of the normal form.Finally, the methods are extended to an infinitary λ-calculus with ω-rule and permutative conversions and this is used to derive a strong form of normalization for an iterative version of Gödel’s system , leading to a value table semantics for number-theoretic functions. (shrink)

Let κ and λ be infinite cardinals such that κ ≤ λ (we have new information for the case when $\kappa ). Let T be a theory in L κ +, ω of cardinality at most κ, let φ(x̄, ȳ) ∈ L λ +, ω . Now define $\mu^\ast_\varphi (\lambda, T) = \operatorname{Min} \{\mu^\ast:$ If T satisfies $(\forall\mu \kappa)(\exists M_\chi \models T)(\exists \{a_i: i Our main concept in this paper is $\mu^\ast_\varphi (\lambda, \kappa) = \operatorname{Sup}\{\mu^\ast(\lambda, T): T$ is a theory (...) in L κ +, ω of cardinality κ at most, and φ (x, y) ∈ L λ +, ω }. This concept is interesting because of THEOREM 1. Let $T \subseteq L_{\kappa^+,\omega}$ of cardinality ≤ κ, and φ (x̄, ȳ) ∈ L λ +, ω . If $(\forall\mu then $(\forall_\chi > \kappa) I(\chi, T) = 2^\chi$ (where I(χ, T) stands for the number of isomorphism types of models of T of cardinality χ). Many years ago the second author proved that $\mu^\ast (\lambda, \kappa) \leq \beth_{(2^\lambda)^+}$ . Here we continue that work by proving. THEOREM 2. $\mu^\ast (\lambda, \aleph_0) = \beth_{\lambda^+}$ . THEOREM 3. For every κ ≤ λ we have $\mu^\ast (\lambda, \kappa) \leq \beth)_{(\lambda^\kappa)}^+$ . For some κ or λ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem. THEOREM 4. For every $\kappa \leq \lambda, T \subseteq L_{\kappa^+,\omega}$ , and any set of formulas $\Lambda \subseteq L_{\lambda^+,\omega}$ such that $\Lambda \subseteq L_{\kappa^+,\omega}$ , if T is (Λ,μ)-unstable for μ satisfying μ μ * (λ, κ) = μ then T is Λ-unstable (i.e. for every χ ≥ λ, T is (Λ, χ)-unstable). Moreover, T is L κ +, ω -unstable. In the second part of the paper, we show that always in the applications it is possible to replace the function I(χ, T) by the function IE(χ, T), and we give an application of the theorems to Boolean powers. (shrink)

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau )$ be the set of countable structures with universe $\omega $ in vocabulary $\tau $ topologized by the Scott topology. We show that an invariant set $X\subseteq Mod(\tau )$ is $\Pi ^0_\alpha $ in the Borel hierarchy of this topology if and only if it is definable by a $\Pi ^p_\alpha $ -formula, a positive $\Pi ^0_\alpha $ formula in the infinitary logic $L_{\omega (...) _1\omega }$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $\mathcal {K}$ be positively computably embeddable in $\mathcal {K}'$ by $\Phi $, then for every $\Pi ^p_\alpha $ formula $\xi $ in the vocabulary of $\mathcal {K}'$ there is a $\Pi ^p_\alpha $ formula $\xi ^{*}$ in the vocabulary of $\mathcal {K}$ such that for all $\mathcal {A}\in \mathcal {K}$, $\mathcal {A}\models \xi ^{*}$ if and only if $\Phi (\mathcal {A})\models \xi $. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings. (shrink)

Infinitary action logic can be naturally expanded by adding exponential and subexponential modalities from linear logic. In this article we shall develop infinitary action logic with a subexponential that allows multiplexing (instead of contraction). Both non-commutative and commutative versions of this logic will be considered, presented as infinitary sequent calculi. We shall prove cut admissibility for these calculi, and estimate the complexity of the corresponding derivability problems: in both cases it will turn out (...) to be between complete first-order arithmetic and the \(\omega ^\omega \) level of the hyperarithmetical hierarchy. Here the complexity upper bound is much lower than that for the system with a subexponential that allows contraction. The complexity lower bound in turn is much higher than that for infinitary action logic. (shrink)

This paper deals with the infinitary modal propositional logic Kω1, featuring countable disjunctions and conjunc- tions. It is known that the natural infinitary extension LK.

This paper continues the investigation, started in Lávička and Noguera : 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to (...) lattices of filters over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics. (shrink)

For each regular cardinal κ, we set up three systems of infinitary type logic, in which the length of the types and the length of the typed syntactical constructs are $\Sigma _{}$, the global system $\text{g}\Sigma _{}$ and the τ-system $\tau \Sigma _{}$. A full cut elimination theorem is proved for the local systems, and about the τ-systems we prove that they admit cut-free proofs for sequents in the τ-free language common to the local and global systems. These (...) two results follow from semantic completeness proofs. Thus every sequent provable in a global system has a cut-free proof in the corresponding τ-systems. It is, however, an open question whether the global systems in themselves admit cut elimination. (shrink)

Dynamic First Order Logic results from interpreting quantification over a variable v as change of valuation over the v position, conjunction as sequential composition, disjunction as non-deterministic choice, and negation as test for continuation. We present a tableau style calculus for DFOL with explicit binding, prove its soundness and completeness, and point out its relevance for programming with DFOL, for automated program analysis including loop invariant detection, and for semantics of natural language. We also extend this to an (...) class='Hi'>infinitary calculus for DFOL with iteration and connect up with other work in dynamic logic. (shrink)

We develop a series of small infinitary epistemic logics to study deductive inference involving intra-/interpersonal beliefs/knowledge such as common knowledge, common beliefs, and infinite regress of beliefs. Specifically, propositional epistemic logics GL are presented for ordinal α up to a given αo so that GL is finitary KDn with n agents and GL allows conjunctions of certain countably infinite formulae. GL is small in that the language is countable and can be constructive. The set of formulae Lα is increasing (...) up to α = ω but stops at ω We present Kripke-completeness for GL for each α ≤ ω, which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL has a sufficient expressive power to discuss intra-/interpersonal beliefs with infinite lengths. As applications, we discuss the explicit definability of Axioms T, 4, 5, and of common knowledge in GL Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL, α ≤ ω. (shrink)

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.

ABSTRACT A new type of calculi is proposed for a first order linear temporal logic. Instead of induction-type postulates the introduced calculi contain a similarity saturation principle, indicating some form of regularity in the derivations of the logic. In a finitary case we obtained the finite set of saturated sequents, showing that ?nothing new? can be obtained continuing the derivation process. Instead of the ?-type rule of inference, an infinitary saturated calculus has an infinite set of saturated (...) sequents, showing that only a ?similar? sequents can be obtained continuing the dérivation process. The saturation calculi have some resemblance with resolution-like calculi. (shrink)

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.

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 show how to encode, by classical structures, both the objects and the morphisms of the category of complete metric spaces and uniformly continuous maps. The result is a category of, what we call, cognate metric spaces and cognate maps. We show this category relativizes to all models of set theory. We extend this encoding to an encoding of complete metric structures by classical structures. This provide us with a general technique for translating results about infinitary logic (...) on classical structures to the setting of infinitary continuous logic on continuous structures. Our encoding will also allow us to talk about not only the relations between complete metric structures, but also the potential relations between complete metric structures, i.e. those which are satisfied in some larger model of set theory. For example we will show that given any two complete metric structures we can determine if they are potentially isomorphic by looking at any admissible set which contains them both. (shrink)

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)

The article deals with infinitary modal logic. We first discuss the difficulties related to the development of a satisfactory proof theory and then we show how to overcome these problems by introducing a labelled sequent calculus which is sound and complete with respect to Kripke semantics. We establish the structural properties of the system, namely admissibility of the structural rules and of the cut rule. Finally, we show how to embed common knowledge in the infinitary calculus and (...) we discuss first-order extensions of infinitary modal logic. (shrink)

In this article, a cut-free system TLMω1 for infinitary propositional modal logic is proposed which is complete with respect to the class of all Kripke frames.The system TLMω1 is a kind of Gentzen style sequent calculus, but a sequent of TLMω1 is defined as a finite tree of sequents in a standard sense. We prove the cut-elimination theorem for TLMω1 via its Kripke completeness.

In this paper we consider the classes of all continuous $\mathcal {L}$ -(pre-)structures for a continuous first-order signature $\mathcal {L}$. We characterize the moduli of continuity for which the classes of finite, countable, or all continuous $\mathcal {L}$ -(pre-)structures have the amalgamation property. We also characterize when Urysohn continuous $\mathcal {L}$ -(pre)-structures exist, establish that certain classes of finite continuous $\mathcal {L}$ -structures are countable Fraïssé classes, prove the coherent EPPA for these classes of finite (...) continuous $\mathcal {L}$ -structures, and show that actions by automorphisms on finite $\mathcal {L}$ -structures also form a Fraïssé class. As consequences, we have that the automorphism group of the Urysohn continuous $\mathcal {L}$ -structure is a universal Polish group and that Hall’s universal locally finite group is contained in the automorphism group of the Urysohn continuous $\mathcal {L}$ -structure as a dense subgroup. (shrink)

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.

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.

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

We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism), which implies, in particular, that isometry of locally compact Polish metric spaces is Borel reducible to graph isomorphism. We show that potentially $\boldsymbol {\Pi }^{0}_{\alpha + 1}$ isomorphism relations are Borel reducible to equality on hereditarily countable sets of rank $\alpha $, $\alpha \geq 2$. We also study (...) approximations of the Hjorth-isomorphism game, and formulate a condition ruling out classifiability by countable structures. (shrink)

Provability, Computability and Reflection.

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...) an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is(I) strong enough to express interesting properties not expressible by the classical language, but(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.

We introduce a logic for a class of probabilistic Kripke structures that we call type structures, as they are inspired by Harsanyi type spaces. The latter structures are used in theoretical economics and game theory. A strong completeness theorem for an associated infinitary propositional logic with probabilistic operators was proved by Meier. By simplifying Meier’s proof, we prove that our logic is strongly complete with respect to the class of type structures. In order to do that, (...) we define a canonical model (in the sense of modal logics), which turns out to be a terminal object in a suitable category. Furthermore, we extend some standard model-theoretic constructions to type structures and we prove analogues of first-order results for those constructions. (shrink)

We show that the modal propositional logic G, originally introduced to describe the modality "it is provable that", is also sound for various interpretations using filters on ordinal numbers, for example the end-segment filters, the club filters, or the ineffable filters. We also prove that G is complete for the interpretation using end-segment filters. In the case of club filters, we show that G is complete if Jensen's principle □ κ holds for all $\kappa ; on the other hand, (...) it is consistent relative to a Mahlo cardinal that G be incomplete for the club filter interpretation. (shrink)

It is known that a theory in S5‐epistemic logic with several agents may have numerous models. This is because each such model specifies also what an agent knows about infinite intersections of events, while the expressive power of the logic is limited to finite conjunctions of formulas. We show that this asymmetry between syntax and semantics persists also when infinite conjunctions (up to some given cardinality) are permitted in the language. We develop a strengthened S5‐axiomatic system for such (...) infinitary logics, and prove a strong completeness theorem for them. Then we show that in every such logic there is always a theory with more than one model. (shrink)

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)

We prove a completeness theorem for Kmath image, the infinitary extension of the graded version K0 of the minimal normal logic K, allowing conjunctions and disjunctions of countable sets of formulas. This goal is achieved using both the usual tools of the normal logics with graded modalities and the machinery of the predicate infinitary logics in a version adapted to modal logic.

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

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 provide a strongly complete infinitary proof system for hybrid logic. This proof system can be extended with countably many sequents. Thus, although these logics may be non-compact, strong completeness proofs are provided for infinitary hybrid versions of non-compact logics like ancestral logic and Segerberg’s modal logic with the bounded chain condition. This extends the completeness result for hybrid logics by Gargov, Passy, and Tinchev.