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A logic with probabilistic Jaccard similarity.

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Title: A logic with probabilistic Jaccard similarity.
Authors: Dabić, Maja1 (AUTHOR), Stojanović, Nenad1 (AUTHOR), Ikodinović, NebojŠa2 (AUTHOR)
Source: Journal of Logic & Computation. Mar2026, Vol. 36 Issue 2, p1-24. 24p.
Subjects: Propositional calculus, Decidability (Mathematical logic), Artificial intelligence, Axioms
Abstract: We introduce an extension of classical probabilistic propositional logic |$\mathsf{LPP}_{1}$|⁠ , understood as an extension of classical propositional calculus with real-valued probability functions and iterated probability operators, by incorporating similarity operators based on the Jaccard index. The binary operators |$J_{\geqslant s}(\alpha ,\beta)$| and |$J_{\leqslant s}(\alpha ,\beta)$| allow us to formally reason about the degree of similarity between propositions, defined through the ratio of the probability of their conjunction and the probability of their disjunction. This addition enriches the expressive power of probabilistic logic and provides a natural way to capture relationships between formulas that go beyond absolute probability. We present the syntax and semantics of the resulting system |$\mathsf{LP}_{J}$|⁠ , establish a sound and complete axiomatization, and prove decidability by reducing satisfiability problems to finite systems of linear inequalities over real closed fields. The logic thus provides a mathematically robust framework that combines probability and similarity, with potential applications in artificial intelligence, knowledge representation and decision-making, especially in contexts where clustering and comparison of structured knowledge are essential. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 192182674
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  Data: A logic with probabilistic Jaccard similarity.
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  Data: <searchLink fieldCode="AR" term="%22Dabić%2C+Maja%22">Dabić, Maja</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Stojanović%2C+Nenad%22">Stojanović, Nenad</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Ikodinović%2C+NebojŠa%22">Ikodinović, NebojŠa</searchLink><relatesTo>2</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Logic+%26+Computation%22">Journal of Logic & Computation</searchLink>. Mar2026, Vol. 36 Issue 2, p1-24. 24p.
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  Data: <searchLink fieldCode="DE" term="%22Propositional+calculus%22">Propositional calculus</searchLink><br /><searchLink fieldCode="DE" term="%22Decidability+%28Mathematical+logic%29%22">Decidability (Mathematical logic)</searchLink><br /><searchLink fieldCode="DE" term="%22Artificial+intelligence%22">Artificial intelligence</searchLink><br /><searchLink fieldCode="DE" term="%22Axioms%22">Axioms</searchLink>
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  Data: We introduce an extension of classical probabilistic propositional logic |$\mathsf{LPP}_{1}$|⁠ , understood as an extension of classical propositional calculus with real-valued probability functions and iterated probability operators, by incorporating similarity operators based on the Jaccard index. The binary operators |$J_{\geqslant s}(\alpha ,\beta)$| and |$J_{\leqslant s}(\alpha ,\beta)$| allow us to formally reason about the degree of similarity between propositions, defined through the ratio of the probability of their conjunction and the probability of their disjunction. This addition enriches the expressive power of probabilistic logic and provides a natural way to capture relationships between formulas that go beyond absolute probability. We present the syntax and semantics of the resulting system |$\mathsf{LP}_{J}$|⁠ , establish a sound and complete axiomatization, and prove decidability by reducing satisfiability problems to finite systems of linear inequalities over real closed fields. The logic thus provides a mathematically robust framework that combines probability and similarity, with potential applications in artificial intelligence, knowledge representation and decision-making, especially in contexts where clustering and comparison of structured knowledge are essential. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
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  Data: <i>Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1093/logcom/exag004
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      – Code: eng
        Text: English
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      – SubjectFull: Propositional calculus
        Type: general
      – SubjectFull: Decidability (Mathematical logic)
        Type: general
      – SubjectFull: Artificial intelligence
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      – SubjectFull: Axioms
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      – TitleFull: A logic with probabilistic Jaccard similarity.
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              M: 03
              Text: Mar2026
              Type: published
              Y: 2026
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