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Fredrik Engström

Publications and source records attributed to Fredrik Engström.

12 recordsLinked to original sources
arXiv2026

Intuitionistic Implication in Elementary Team Logics

The logic FOT is a team-based logic whose expressive power coincides with first-order logic at the level of both sentences and open formulas. In contrast to dependence and independence logics, which can define stronger second-order team properties, FOT is designed to capture exactly elementary team properties, modulo the empty team. In this paper we consider two modifications of FOT. First, we investigate essentially the inclusion atom free fragment of FOT. Our main result establishes quantifier elimination for the fragment in the empty signature. Second, we study an extension of FOT by the intuitionistic implication. The main conclusion is that adding this single connective increases the expressive strength so that every second-order sentence can be encoded by an open formula evaluated on the full team. Consequently, validity of formulas is equivalent to validity of full second-order logic.

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arXiv2026

The Propositional Logic of Team Properties

Since its introduction by Hodges and refinement by Väänänen, team semantic constructions have been used to generate expressively enriched logics preserving some desirable properties, such as compactness or decidability. By contrast, these logics fail to be substitutional, limiting any algebraic treatment and rendering schematic uniform proof systems impossible. This shortcoming can be attributed to the flatness principle, commonly adhered to when generating team semantics. Investigating the formation of team semantics from algebraic semantics, and disregarding the flatness principle, we present the Logic of Team Properties (LTP), a substitutional logic in which important propositional team logics are axiomatisable as fragments. Starting from classical propositional logic and Boolean algebras, we give a semantics for LTP by considering the algebras that are powersets of Boolean algebras B, that is, of the form P(B), equipped with internal pointwise and external set-theoretic connectives. Furthermore, we present a well-motivated sound and complete labelled natural deduction system for LTP.

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arXiv2024

Generalized quantifiers using team semantics

Dependence logic provides an elegant approach for introducing dependencies between variables into the object language of first-order logic. In [1] generalized quantifiers were introduced in this context. However, a satisfactory account was only achieved for monotone increasing generalized quantifiers. In this paper, we modify the fundamental semantical guideline of dependence logic to create a framework that adequately handles both monotone and non-monotone generalized quantifiers. We demonstrate that this new logic can interpret dependence logic and possesses the same expressive power as existential second-order logic (ESO) on the level of formulas. Additionally, we establish truth conditions for generalized quantifiers and prove that the extended logic remains conservative over first-order logic with generalized quantifiers and is able to express the branching of continuous generalized quantifiers.

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arXiv2014

Implicitly definable generalized quantifiers

We give a new elementary proof of the main theorem of [Fef12]: Quantifiers implicitly definable in pure second-order logic equipped with Henkin semantics implies are (explicitly) definable in first-order logic.

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arXiv2013

Invariance and definability, with and without equality

The dual character of invariance under transformations and definability by some operations has been used in classical work by for example Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this paper, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in $\La_{\infty\infty}$ so as to cover the cases (quantifiers, logics without equality) that are of interest in the logicality debates, getting McGee's theorem about quantifiers invariant under all permutations and definability in pure $\La_{\infty\infty}$ as a particular case. We also prove some optimality results along the way, regarding the kind of relations which are needed so that every subgroup of the full permutation group is characterizable as a group of automorphisms.

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arXiv2013

Dependence Logic with Generalized Quantifiers: Axiomatizations

We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the meaning that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as "there exists uncountable many." Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences.

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arXiv2012

Generalized quantifiers in Dependence Logic

We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindström, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle quantifier scope dependencies of generalized quantifiers in a satisfying way the dependence atom in Dependence logic is not well suited and that the multivalued dependence atom is a better choice. This atom is in fact definably equivalent to the \emph{independence atom} recently introduced by Väänänen and Grädel.

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arXiv2010

Non-permutation invariant Borel quantifiers

Every permutation invariant Borel subset of the space of countable structures is definable in $\La_{ω_1ω}$ by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed subgroup $G$ of the symmetric group $S_{\infty}$, there is a closed binary quantifier $Q$ such that the $G$-invariant subsets of the space of countable structures are exactly the $\La_{ω_1ω}(Q)$-definable sets.

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arXiv2008

A note on standard systems and ultrafilters

Let $(M,\scott X) \models \ACA$ be such that $P_\scott X$, the collection of all unbounded sets in $\scott X$, admits a definable complete ultrafilter and let $T$ be a theory extending first order arithmetic coded in $\scott X$ such that $M$ thinks $T$ is consistent. We prove that there is an end-extension $N \models T$ of $M$ such that the subsets of $M$ coded in $N$ are precisely those in $\scott X$. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory $T$ extending first order arithmetic is the standard system of a recursively saturated model of $T$.

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arXiv2004

Expansions, omitting types, and standard systems

Recursive saturation and resplendence are two important notions in models of arithmetic. Kaye, Kossak, and Kotlarski introduced the notion of arithmetic saturation and argued that recursive saturation might not be as rigid as first assumed. In this thesis we give further examples of variations of recursive saturation, all of which are connected with expandability properties similar to resplendence. However, the expandability properties are stronger than resplendence and implies, in one way or another, that the expansion not only satisfies a theory, but also omits a type. We conjecture that a special version of this expandability is in fact equivalent to arithmetic saturation. We prove that another of these properties is equivalent to β-saturation. We also introduce a variant on recursive saturation which makes sense in the context of a standard predicate, and which is equivalent to a certain amount of ordinary saturation. The theory of all models which omit a certain type p(x) is also investigated. We define a proof system, which proves a sentence if and only if it is true in all models omitting the type p(x). The complexity of such proof systems are discussed and some explicit examples of theories and types with high complexity, in a special sense, are given. We end the thesis by a small comment on Scott's problem. We prove that, under the assumption of Martin's axiom, every Scott set of cardinality <2^{\aleph_0} closed under arithmetic comprehension which has the countable chain condition is the standard system of some model of PA. However, we do not know if there exists any such uncountable Scott sets.

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arXiv2002

Satisfaction classes in nonstandard models of first-order arithmetic

A satisfaction class is a set of nonstandard sentences respecting Tarski's truth definition. We are mainly interested in full satisfaction classes, i.e., satisfaction classes which decides all nonstandard sentences. Kotlarski, Krajewski and Lachlan proved in 1981 that a countable model of PA admits a satisfaction class if and only if it is recursively saturated. A proof of this fact is presented in detail in such a way that it is adaptable to a language with function symbols. The idea that a satisfaction class can only see finitely deep in a formula is extended to terms. The definition gives rise to new notions of valuations of nonstandard terms; these are investigated. The notion of a free satisfaction class is introduced, it is a satisfaction class free of existential assumptions on nonstandard terms. It is well known that pathologies arise in some satisfaction classes. Ideas of how to remove those are presented in the last chapter. This is done mainly by adding inference rules to M-logic. The consistency of many of these extensions is left as an open question.

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