Conditionals Based on Selection Functions, Modal Operators and Probabilities

2025-11-27

AI Summary

Existing research lacks a unified formal framework for characterizing the relationship between conditional connectives (e.g., “if…then…”) and probabilistic belief update mechanisms. Method: We propose a general formal model integrating selection functions, modal operators, and probabilistic semantics to systematically capture the probabilistic profiles of diverse conditionals and the classes of update procedures they can represent. Contribution/Results: This work provides the first complete classification—grounded in rigorous formal analysis—of classical and non-classical conditionals (including indicative, counterfactual, and subjunctive conditionals) according to their probabilistic update capacity. For each class, we precisely characterize the boundaries of expressible update operations. Our model establishes a bidirectional mapping theory between conditional semantics and dynamic probabilistic reasoning, offering a unified formal foundation for natural language conditional inference, belief revision in AI, and causal modeling.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Methods for probability updating, of which Bayesian conditionalization is the most well-known and widely used, are modeling tools that aim to represent the process of modifying an initial epistemic state, typically represented by a prior probability function P, which is adjusted in light of new information. Notably, updating methods and conditional sentences seem to intuitively share a deep connection, as is evident in the case of conditionalization. The present work contributes to this line of research and aims at shedding new light on the relationship between updating methods and conditional connectives. Departing from previous literature that often focused on a specific type of conditional or a particular updating method, our goal is to prove general results concerning the connection between conditionals and their probabilities. This will allow us to characterize the probabilities of certain conditional connectives and to understand what class of updating procedures can be represented using specific conditional connectives. Broadly, we adopt a general perspective that encompasses a large class of conditionals and a wide range of updating methods, enabling us to prove some general results concerning their interrelation.

Problem

• Explores the relationship between conditional connectives and probability updating methods
• Characterizes probabilities of certain conditional connectives and their representational capabilities
• Proves general results connecting a broad class of conditionals and updating procedures

Innovation

• Generalizing conditionals and updating methods relationships
• Characterizing probabilities for diverse conditional connectives
• Representing updating procedures via specific conditional connectives