As information plays an ever more central role across disciplines, the lack of a precise and reusable definition of state impedes comparison, measurement, and verification. Building on Objective Information Theory (OIT), this paper proposes a logic-based framework that defines the state of an object or system at a time point (or interval) as the semantic valuation of a set of well-formed formulas over a given domain and interpretation. Within first-order and higher-order logic—extended to infinitary logic when needed—we show how finite and broad classes of infinite structures can be characterized, drawing on core results from model theory. We then instantiate the framework in economics, sociology, computer science, and natural language, demonstrating that logic provides a unifying language for representing, reasoning about, and relating states across domains. Finally, we refine OIT by supplying a universal state representation that supports cross-domain exchange, measurement, and verification.

Expressive power of infinitary logic and absolute co-Hopfianity

In (Barbero and Sandu 2020 Journal of Philosophical Logic, 50, 471-521), we showed that languages encompassing interventionist counterfactuals and causal notions based on them (as e.g. in Pearl’s and Woodward’s manipulationist approaches to causation) as well as information-theoretic notions (such as learning and dependence) can be interpreted in a semantic framework which combines the traditions of structural equation modeling and of team semantics. We now present a further extension of this framework (causal multiteams) which allows us to talk about probabilistic causal statements. We analyze the expressivity resources of two causal-probabilistic languages, one finitary and one infinitary. We show that many causal-probabilistic notions from the literature on causal inference can already be expressed in the finitary language, and we prove a normal form theorem that throws a new light on Pearl’s “ladder of causation”. In addition, we provide an exact semantic characterization of the infinitary language, which shows that this language captures precisely those causal-probabilistic statements that do not commit us to any specific interpretation of probability; and we prove that no usual, countable language is apt for this task.

This paper investigates big Ramsey degrees of unrestricted relational structures in (possibly) infinite languages. Despite significant progress in the study of big Ramsey degrees, the big Ramsey degrees of many classes of structures with finite small Ramsey degrees are still not well understood. We show that if there are only finitely many relations of every arity greater than one, then unrestricted relational structures have finite big Ramsey degrees, and give some evidence that this is tight. This is the first time finiteness of big Ramsey degrees has been established for a random structure in an infinite language. Our results represent an important step towards a better understanding of big Ramsey degrees for structures with relations of arity greater than two.

Abstract 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.

Abstract The present paper deals with a purely syntactic analysis of infinitary logic with infinite sequents. In particular, we discuss sequent calculi for classical and intuitionistic infinitary logic with good structural properties based on sequents possibly containing infinitely many formulas. A cut admissibility proof is proposed which employs a new strategy and a new inductive parameter. We conclude the paper by discussing related issues and possible themes for future research.

Boolean valued semantics for infinitary logics

Abstract The paper investigates from a proof-theoretic perspective various non-contractive logical systems, which circumvent logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (Grišin and Cantini). Systems with multiplicative quantifiers were proposed in the 2010s (Zardini), but they turned out to be inconsistent with the naive rules for truth or comprehension. We start by presenting a first-order system for disquotational truth with additive quantifiers and compare it with Grišin set theory. We then analyze the reasons behind the inconsistency phenomenon affecting multiplicative quantifiers. After interpreting the exponentials in affine logic as vacuous quantifiers, we show how such a logic can be simulated within a truth-free fragment of a system with multiplicative quantifiers. Finally, we establish that the logic for these multiplicative quantifiers (but without disquotational truth) is consistent, by proving that an infinitary version of the cut rule can be eliminated. This paves the way to a syntactic approach to the proof theory of infinitary logic with infinite sequents.

The Gödel-McKinsey-Tarski embedding allows to view intuitionistic logic through the lenses of modal logic. In this work, an extension of the modal embedding to infinitary intuitionistic logic is introduced. First, a neighborhood semantics for a family of axiomatically presented infinitary modal logics is given and soundness and completeness are proved via the method of canonical models. The semantics is then exploited to obtain a labelled sequent calculus with good structural properties. Next, soundness and faithfulness of the embedding are established by transfinite induction on the height of derivations: the proof is obtained directly without resorting to non-constructive principles. Finally, the modal embedding is employed in order to relate classical, intuitionistic and modal derivability in infinitary logic extended with axioms.

Abstract We can measure the complexity of a logical formula by counting the number of alternations between existential and universal quantifiers. Suppose that an elementary first-order formula $\varphi $ (in $\mathcal {L}_{\omega ,\omega }$ ) is equivalent to a formula of the infinitary language $\mathcal {L}_{\infty ,\omega }$ with n alternations of quantifiers. We prove that $\varphi $ is equivalent to a finitary formula with n alternations of quantifiers. Thus using infinitary logic does not allow us to express a finitary formula in a simpler way.

A backdoor in a finite-domain CSP instance is a set of variables where each possible instantiation moves the instance into a polynomial-time solvable class. Backdoors have found many applications in artificial intelligence and elsewhere, and the algorithmic problem of finding such backdoors has consequently been intensively studied. Sioutis and Janhunen have proposed a generalised backdoor concept suitable for infinite-domain CSP instances over binary constraints. We generalise their concept into a large class of CSPs that allow for higher-arity constraints. We show that this kind of infinite-domain backdoors have many of the positive computational properties that finite-domain backdoors have: the associated computational problems are fixed parameter tractable whenever the underlying constraint language is finite. On the other hand, we show that infinite languages make the problems considerably harder: the general backdoor detection problem is W[2]-hard and fixed-parameter tractability is ruled out under standard complexity-theoretic assumptions. We demonstrate that backdoors may have suboptimal behaviour on binary constraints—this is detrimental from an AI perspective where binary constraints are predominant in, for instance, spatiotemporal applications. In response to this, we introduce sidedoors as an alternative to backdoors. The fundamental computational problems for sidedoors remain fixed-parameter tractable for finite constraint language (possibly also containing non-binary relations). Moreover, the sidedoor approach has appealing computational properties that sometimes leads to faster algorithms than the backdoor approach.

Abstract We analyze $\textrm {C}^{\ast }$-algebras, particularly AF-algebras, and their $K_{0}$-groups in the context of the infinitary logic $\mathcal {L}_{\omega _{1} \omega }$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_{0}$-groups as ordered unital groups, we prove that $K_{0}(A) \equiv _{\omega \cdot \alpha } K_{0}(B)$ implies $A \equiv _{\alpha } B$, where $M \equiv _{\beta } N$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $\beta $. This implication is proved using techniques from Elliott’s classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the metric setting. We use moreover this result to build a family $\{ A_{\alpha } \}_{\alpha < \omega _{1}}$ of pairwise non-isomorphic separable simple unital AF-algebras which satisfy $A_{\alpha } \equiv _{\alpha } A_{\beta }$ for every $\alpha < \beta $. In particular, we obtain a set of separable simple unital AF-algebras of arbitrarily high Scott rank. Next, we give a partial converse to the aforementioned implication, showing that $A \otimes \mathcal {K} \equiv _{\omega + 2 \cdot \alpha +2} B \otimes \mathcal {K}$ implies $K_{0}(A) \equiv _{\alpha } K_{0}(B)$, for every unital $\textrm {C}^{\ast }$-algebras $A$ and $B$.

We prove the following characterizations of nonstandard models of ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that have an expansion to a model of GB (Gödel-Bernays class theory) plus Δ11-CA (the scheme of Δ11-Comprehension). In what follows, M(α):=(V(α),∈)M, LM is the set of formulae of the infinitary logic L∞,ω that appear in the well-founded part of M, and Σ11-AC is the scheme of Σ11-Choice.Theorem A.The following are equivalent for a nonstandard modelMof ZFC of any cardinality:(a)M(α)≺LMMfor an unbounded collection ofα∈OrdM.(b)(M,X)⊨GB+Δ11-CA, whereXis the family ofLM-definable subsets ofM.(c)There isXsuch that(M,X)⊨GB+Δ11-CA.Theorem B.The following are equivalent for a countable nonstandard model of ZFC:(a)M(α)≺LMMfor an unbounded collection ofα∈OrdM.(b)There isXsuch that(M,X)⊨GB+Δ11-CA+Σ11-AC.

A new framework to measure distances (similarity) between formal languages and between grammars based on distances between words is introduced. It is based on approximating languages by their finite subsets and using monotone sequences of such finite approximations to define an infinite language in the limit. Distances between finite languages are defined and extended to distances between monotone sequences of finite languages leading to distances between infinite languages. The framework captures several distances studied in the literature. Context-free grammars with energy are introduced to enable finite approximations emphasizing “syntactically important” parts of words. Grammars with energy are also used to extend distances between monotone sequences of finite languages to distances between context-free grammars. A basic toolkit for monotone sequences of finite languages and distances between languages resp. grammars is provided. As part of this toolkit a non-symmetric version of distances is defined, providing additional characterisation of distances in general. Additional properties of distances between grammars are derived by restricting the“energy use” of grammars with energy. Some methods of estimating the distances are presented to be used in cases where the distance is not computable or difficult to compute.

Abstract We study propositional and first-order Gödel logics over infinitary languages, which are motivated semantically by corresponding interpretations into the unit interval $[0,1]$. We provide infinitary Hilbert-style calculi for the particular (propositional and first-order) cases with con-/disjunctions of countable length and prove corresponding completeness theorems by extending the usual Lindenbaum–Tarski construction to the infinitary case for a respective algebraic semantics via complete linear Heyting algebras. We provide infinitary hypersequent calculi and prove corresponding cut-elimination theorems in the Schütte–Tait style. Initial observations are made regarding truth-value sets other than $[0,1]$.

de Finetti's coherence and exchangeability in infinitary logic

Abstract In this paper I discuss Ernst Zermelo’s ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences. The presentation of Zermelo’s ideas is accompanied with some remarks concerning the development of infinitary logic. I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1

In Schwichtenberg (Studies in logic and the foundations of mathematics, vol 90, Elsevier, pp 867–895, 1977), Schwichtenberg fine-tuned Tait’s technique (Tait in The syntax and semantics of infinitary languages, Springer, pp 204–236, 1968) so as to provide a simplified version of Gentzen’s original cut-elimination procedure for first-order classical logic (Gallier in Logic for computer science: foundations of automatic theorem proving, Courier Dover Publications, London, 2015). In this note we show that, limited to the case of classical propositional logic, the Tait–Schwichtenberg algorithm allows for a further simplification. The procedure offered here is implemented on Kleene’s sequent system G4 (Kleene in Mathematical logic, Wiley, New York, 1967; Smullyan in First-order logic, Courier corporation, London, 1995). The specific formulation of the logical rules for G4 allows us to provide bounds on the height of cut-free proofs just in terms of the logical complexity of their end-sequent.

Abstract Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs where possible, followed by a survey of recent advances.

The logic \({\mathbb{L}}_\theta ^1\) was introduced in [She12]; it is the maximal logic below \({{\mathbb{L}}_{\theta, \theta}}\) in which a well ordering is not definable. We investigate it for θ a compact cardinal. We prove that it satisfies several parallels of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are \({\mathbb{L}}_\theta ^1\)-equivalent iff for some ω-sequence of θ-complete ultrafilters, the iterated ultrapowers by it of those two models are isomorphic.Also for strong limit λ>θ of cofinality \({\aleph _0}\), every complete \({\mathbb{L}}_\theta ^1\)-theory has a so-called special model of cardinality λ, a parallel of saturated. For first order theory T and singular strong limit cardinal λ, T has a so-called special model of cardinality λ. Using “special” in our context is justified by: it is unique (fixing T and λ), all reducts of a special model are special too, so we have another proof of interpolation in this case.