Abstract
Causal multiteam semantics is a framework where probabilistic notions and causal inference can be studied in a unified setting. We study a logic (\(\mathcal {PCO}\)) that features marginal probabilities, observations and interventionist counterfactuals, and allows expressing conditional probability statements, do expressions and other mixtures of causal and probabilistic reasoning. Our main contribution is a strongly complete infinitary axiomatisation for \(\mathcal {PCO}\).
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Notes
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- 3.
Throughout the paper, the semantic relation in terms of which \(T^\alpha \) is defined will be the semantic relation for language \(\mathcal{C}\mathcal{O}\), which will soon be defined.
- 4.
Whereas \(\Pr (\alpha )>0\) could be replaced with \((\lnot \alpha )^C\), and \((X=x)^C\) could be also expessed as \(\bigvee _{x' \ne x} X = x'\), the use of probability atoms in \((X\ne x)^C\) seems essential.
- 5.
Save for some inessential differences, this is is the content of Theorem 3.4 from [9].
- 6.
It seems to us that an axiom analogous to P3b should be added also to the system in [32].
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Acknowledgments
Fausto Barbero was partially supported by the DFG grant VI 1045-1/1 and by the Academy of Finland grants 316460 and 349803. Jonni Virtema was partially supported by the DFG grant VI 1045-1/1 and by the Academy of Finland grant 338259.
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Barbero, F., Virtema, J. (2023). Strongly Complete Axiomatization for a Logic with Probabilistic Interventionist Counterfactuals. In: Gaggl, S., Martinez, M.V., Ortiz, M. (eds) Logics in Artificial Intelligence. JELIA 2023. Lecture Notes in Computer Science(), vol 14281. Springer, Cham. https://doi.org/10.1007/978-3-031-43619-2_44
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