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The Logic ILP for Intuitionistic Reasoning About Probability

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Abstract

We offer an alternative approach to the existing methods for intuitionistic formalization of reasoning about probability. In terms of Kripke models, each possible world is equipped with a structure of the form \(\langle H, \mu \rangle \) that needs not be a probability space. More precisely, though H needs not be a Boolean algebra, the corresponding monotone function (we call it measure) \(\mu : H \longrightarrow [0,1]_{\mathbb {Q}}\) satisfies the following condition: if \(\alpha \), \(\beta \), \(\alpha \wedge \beta \), \(\alpha \vee \beta \in H\), then \(\mu (\alpha \vee \beta ) = \mu (\alpha ) + \mu (\beta ) - \mu (\alpha \wedge \beta )\). Since the range of \(\mu \) is the set \([0,1]_{\mathbb {Q}}\) of rational numbers from the real unit interval, our logic is not compact. In order to obtain a strong complete axiomatization, we introduce an infinitary inference rule with a countable set of premises. The main technical results are the proofs of strong completeness and decidability.

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Authors and Affiliations

  1. Mathematical Institute of Serbian Academy of Sciences and Arts, Kneza Mihaila 36, Belgrade, 11000, Serbia

    Angelina Ilić-Stepić & Zoran Ognjanović

  2. Faculty of Transport and Traffic Engineering, University of Belgrade, Vojvode Stepe 305, Belgrade, 11000, Serbia

    Aleksandar Perović

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  1. Angelina Ilić-Stepić
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  2. Zoran Ognjanović
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  3. Aleksandar Perović
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Correspondence to Angelina Ilić-Stepić.

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Ilić-Stepić, A., Ognjanović, Z. & Perović, A. The Logic ILP for Intuitionistic Reasoning About Probability. Stud Logica 112, 987–1017 (2024). https://doi.org/10.1007/s11225-023-10084-z

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