arXiv subjects
Publications and source records attributed to Jouko Väänänen.
We argue that second order logic is a weaker form of set theory, despite the fact that the former is formalized in a second order langauge and the latter in a first order language. Along the way, we review the history of interactions between second order logic and set theory, as well as some modern trends.
We define a new Ehrenfeucht-Fraïssé game for dependence logic. The previously known rendition of such a game was based on moves that are teams. Since teams can be massive, making team moves may be quite complicated. To remedy this, our new Ehrenfeucht-Fraïssé game for dependence logic has only moves that consist of single elements, as in the classical Ehrenfeucht-Fraïssé game of first order logic. A new feature of the game is that a player can declare that their move is made on the basis of certain previous moves only and thereby in a sense independent of other moves. We show that our game characterizes elementary equivalence in dependence logic.
Dependence logic extends first-order logic with dependence atoms asserting that the value of a variable is determined by the values of certain other variables. The semantics of dependence logic has a second-order character and involves sets of assignments, called teams, instead of individual assignments as in the classical Tarski semantics. Since the model-checking problem is known to be NP-complete even for quantifier-free dependence logic (DQF) formulas, researchers have pursued conditions on formulas that make this problem tractable. In 2010, Jarmo Kontinen introduced the notion of k-coherence for dependence logic formulas, where k is a positive integer. This notion asserts that if the formula is satisfied in a structure by all k-element subteams of a given team, then the given team itself satisfies the formula. It has been proved that k-coherent DQF-formulas have a tame model-checking problem, because such formulas admit a first-order rewriting. In this paper, we investigate the structural and algorithmic aspects of coherence. We show that if a DQF-formula is first-order ewritable, then it is k-coherent for some positive integer k. Thus, for DQF-formulas, coherence is equivalent to first-order rewritability. Furthermore, we show that an analogous result holds for universally quantified dependence logic formulas under a stronger notion of coherence. After this, we focus on the complexity of deciding if a given dependence logic formula is k-coherent. We establish that this decision problem is highly undecidable for arbitrary dependence logic formulas, while for DQF-formulas this problem is co-recursively enumerable. Furthermore, we pinpoint the computational complexity of the coherence problem for propositional dependence logic formulas by showing that this problem is complete for the second level of the exponential hierarchy.
If $\mathcal{L}$ is an abstract logic (a.k.a. model theoretic logic), we can define the inner model $C(\mathcal{L})$ by replacing first order logic with $\mathcal{L}$ in Gödel's definition of the inner model $L$ of constructible sets. Set theoretic properties of such inner models $C(\mathcal{L})$ have been investigated recently and a spectrum of new inner models is emerging between $L$ and $\mathrm{HOD}$. The topic of this paper is the effect on $C(\mathcal{L})$ of a slight modification of $\mathcal{L}$ i.e. how sensitive is $C(\mathcal{L})$ on the exact definition of $\mathcal{L}$? The $Δ$-extension $Δ(\mathcal{L})$ of a logic is generally considered a "mild" extension of $\mathcal{L}$. We give examples of logics $\mathcal{L}$ for which the inner model $C(\mathcal{L})$ is consistently strictly smaller than the inner model $C(Δ(\mathcal{L}))$, and in one case we show this follows from the existence of $0^{\sharp}$.
We develop dimension theoretic methods for propositional team based logics. Such quantitative methods were defined for team based first-order logic in a recent paper by Hella, Luosto and the third author and were used to obtain strong hierarchy results in the first-order logic context. We show that in propositional logic and in several important cases, a team theoretical atom can be expressed in terms of atoms of lower arity. We estimate the `price' of such a reduction of arity, i.e. how much more complicated the new expression is. Our estimates involve as parameters the arity of the atoms involved, as well as the number of times the atom occurs in a formula. We also consider new variants of atoms and propositional operations, inspired by our work. We believe that our quantitative analysis leads to a deeper understanding of the scope and limits of propositional team based logic.
Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal $μ$, we prove the consistency of the existence of a wide $μ^+$-Aronszajn tree, i.e. a tree of height and cardinality $μ^+$ with no branches of length $μ^+$, into which every wide $μ^+$-Aronszajn tree can be embedded.
We develop Descriptive Set Theory in Generalized Baire Spaces without assuming $κ^{<κ}=κ$. We point out that without this assumption the basic topological concepts of these spaces have to be slightly modified in order to obtain a meaningful theory. This modification has no effect if $κ^{<κ}=κ$. After developing the basic theory we apply it to the question whether the orbits of models of a fixed cardinality $κ$ in the space $κ^κ$ are $κ$-Borel in our generalized sense. It turns out that this question depends, as is the case when $κ^{<κ}=κ$, on stability theoretic properties (structure vs. non-structure) of the first order theory of the model.
We define a new inner model C2(omega) based on the fragment of second order logic in which second order variables range over countable subsets of the domain. We compare C2(omega) to the previously studied inner model C(aa). We argue that C2(omega) appears to be a much bigger inner model than C(aa), although this cannot be literally true in ZFC alone. However, we conjecture that it follows from large cardinal assumptions. For example, assuming large cardinals, C2(omega) contains, for every n, an inner model with n Woodin cardinals, while C(aa) contains, under the same assumption, no inner model with a Woodin cardinal. As to large cardinals in C2(omega), we show that, assuming a Woodin limit of Woodin cardinals, the cardinal omega_1 of V is Mahlo in C2(omega). A stronger result is proved for the combination C2(omega, aa) of C(aa) and C2(omega). We also show that the question whether HOD1, a variant of HOD, is the same as HOD cannot be decided on the basis of ZFC even if we add the assumption that there are supercompact cardinals.
This is a survey of the model theory of second order logic.
We bring an abstract model theory perspective to interpolation. We ask, what is the role of interpolation in the study of extensions of first order logic, such as infinitary logics, generalized quantifiers and higher order logics? The abstract model theory approach reveals the basic connections between various interpolation properties in isolation, on their own, as well as with respect to other model theoretic properties, such as compactness.
We propose a systematic study of the so-called flattening operator in team semantics. This operator was first introduced by Hodges in 1997, and has not been studied in more detail since. We begin a systematic study of the expressive power this operator adds to the most well-known team-based logics, such as dependence logic, anonymity logic, inclusion logic and exclusion logic.
We study dependence and independence concepts found in quantum physics, especially those related to hidden variables and non-locality, through the lens of team semantics and probabilistic team semantics, adapting a relational framework introduced by the first author in a prior paper. This leads to new developments also in independence logic and probabilistic independence logic.
We show, assuming PD, that every complete finitely axiomatized second order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g. supercompact) cardinals does not imply the categoricity of all finite complete second order theories. More exactly, we show that a non-categorical complete finitely axiomatized second order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second order theories with a model of a certain singular cardinality kappa of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming V=L, that every complete finitely axiomatized second order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second order theory with a countable model which is non-categorical.
We introduce a new inner model $C(aa)$ arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively $MM^{++}$, the regular uncountable cardinals of $V$ are measurable in the inner model $C(aa)$, the theory of $C(aa)$ is (set) forcing absolute, and $C(aa)$ satisfies CH. We introduce an auxiliary concept that we call club determinacy, which simplifies the construction of $C(aa)$ greatly but may have also independent interest. Based on club determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model $C(aa)$.
We introduce a new kind of infinitary logic that we call Boolean expansion of ${\mathcal L}_{κκ}$. This logic involves a new kind of variable, that we call generalised Boolean variable. These variables range over the powerset of a cardinal number in a way reminiscent of random variables. From this Boolean expansion, we extract a traditional infinitary logic, called Cartagena logic. We prove several model-theoretic properties of Cartagena logic, and give multiple examples of its expressive power. The main result is that Cartagena logic is a good syntactically defined approximation to Shelah's infinitary ${\mathcal L}^1_κ$. The latter is not known to have a generative syntax, while Cartagena logic does have a very clear one.
Lindström's Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negationless logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.
We introduce three measures of complexity for families of sets. Each of the three measures, that we call dimensions, is defined in terms of the minimal number of convex subfamilies that are needed for covering the given family: for upper dimension, the subfamilies are required to contain a unique maximal set, for dual upper dimension a unique minimal set, and for cylindrical dimension both a unique maximal and a unique minimal set. In addition to considering dimensions of particular families of sets we study the behaviour of dimensions under operators that map families of sets to new families of sets. We identify natural sufficient criteria for such operators to preserve the growth class of the dimensions. We apply the theory of our dimensions for proving new hierarchy results for logics with team semantics. First, we show that the standard logical operators preserve the growth classes of the families arising from the semantics of formulas in such logics. Second, we show that the upper dimension of k+1-ary dependence, inclusion, independence, anonymity, and exclusion atoms is in a strictly higher growth class than that of any k-ary atoms, whence the k+1-ary atoms are not definable in terms of any atoms of smaller arity.
Truth, consistency and elementary equivalence can all be characterised in terms of games, namely the so-called evaluation game, the model-existence game, and the Ehrenfeucht-Fraisse game. We point out the great affinity of these games to each other and call this phenomenon the strategic balance in logic. In particular, we give explicit translations of strategies from one game to another.