A logic with probabilistic Jaccard similarity

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.