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Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 5))

Abstract

Dependence logic extends the language of first order logic by means of dependence atoms and aims to establish a basic theory of dependence and independence underlying such seemingly unrelated subjects as causality, random variables, bound variables in logic, database theory, the theory of social choice, and even quantum physics. In this work we summarize the setting of dependence logic and recall the main results of this rapidly developing area of research.

J. Väänänen’s research partially supported by grant 40734 of the Academy of Finland and the EUROCORES LogICCC LINT programme.

P. Galliani’s research partially supported by the EUROCORES LogICCC LINT programme, by the Väisälä Foundation and by by Grant 264917 of the Academy of Finland.

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Notes

  1. 1.

    Before [19] it was an open question whether a compositional semantics can be given to independence friendly logic.

  2. 2.

    Or attributes, something that has a value.

  3. 3.

    The basic ideas can be applied to almost any logic, especially to modal logic.

  4. 4.

    See e.g. [26].

  5. 5.

    This was originally proved for random variables in [14] and then adapted for databases in [22].

  6. 6.

    In that paper, conditional independence logic is simply called “independence logic”. After all, the two logics are equivalent.

  7. 7.

    In order to guarantee that such a fixed point exist, \(R\) is required to appear only positively in \(\psi \).

  8. 8.

    That is, over finite structures with a built-in successor operator and two constants for the least and greatest elements.

  9. 9.

    The name “linear implication” is due to the similarity between the satisfaction conditions of this connective and the ones of the implication of linear logic. Another similarity is the following Galois connection: \(\theta \models \phi \multimap \psi \iff \theta \vee \phi \models \psi \) [1].

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Galliani, P., Väänänen, J. (2014). On Dependence Logic. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_4

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