[Paper Review] Partial Probability and Kleene Logic

Maurizio Negri|arXiv (Cornell University)|Oct 23, 2013

Advanced Algebra and Logic2 references3 citations

TL;DR

This paper unifies set-theoretic and linguistic approaches to probability using valued lattices and DMF-algebras, introducing partial probability within Kleene’s three-valued logic. It establishes a translation between event-based and formula-based probability, proves a weak form of Bayes’ Theorem for partial events, and provides an algebraic foundation for reasoning under uncertainty with incomplete information.

ABSTRACT

There are two main approach to probability, one of set-theoretic character where probability is the measure of a set, and another one of linguistic character where probability is the degree of confidence in a proposition. In this work we give an unified algebraic treatment of these approaches through the concept of valued lattice, obtaining as a by-product a translation between them. Then we introduce the concept of partial valuation for DMF-algebras (De Morgan algebras with a single fixed point for negation), giving an algebraic setting for probability of partial events. We introduce the concept of partial probability for propositions, substituting classical logic with Kleene's logic. In this case too we give a translation between set-theoretic and linguistic probability. Finally, we introduce the concept of conditional partial probability and prove a weak form of Bayes's Theorem.

Motivation & Objective

• To unify classical and linguistic probability approaches through algebraic structures based on valued lattices.
• To extend probability theory to partial events using De Morgan algebras with a single fixed point for negation (DMF-algebras).
• To develop a formal framework for partial probability in Kleene’s three-valued logic, replacing classical logic.
• To establish a bidirectional translation between event-based and formula-based probability in both classical and partial settings.
• To formalize conditional partial probability and prove a weak form of Bayes’ Theorem for partial events.

Proposed method

• Uses valued lattices as a general algebraic framework for probability, with valuations satisfying the modular equation $ v(a \vee b) = v(a) + v(b) - v(a \wedge b) $.
• Defines Boolean valuations as additive functions on Boolean algebras satisfying $ v(1) = 1 $ and $ v(a \wedge b) = 0 \Rightarrow v(a \vee b) = v(a) + v(b) $.
• Introduces partial valuations on DMF-algebras, assigning pairs $ (x,y) \in [0,1]^2 $ to elements, with operations respecting Kleene logic connectives.
• Constructs a translation from partial events in $ \mathcal{G}_A $ to formulas in a sentential language via morphisms from free DMF-algebras to event algebras.
• Defines partial probability functions $ \pi: F \to [0,1]^2 $ satisfying axioms based on Kleene logic, including $ \pi(\neg \alpha) = \sigma(\pi(\alpha)) $, where $ \sigma(x,y) = (y,x) $.
• Applies the representation theorem to construct a partial probability function $ \pi(\alpha) = \overline{v}(|\alpha|) $, where $ \overline{v} $ is a partial valuation derived from a measure $ \mu $ on partial events.

Experimental results

Research questions

• RQ1 How can set-theoretic and linguistic approaches to probability be unified algebraically using valued lattices?
• RQ2 What is the appropriate logical framework for reasoning about partial events, and how does it differ from classical logic?
• RQ3 Can a probability measure on partial events be translated into a probability function on formulas in a many-valued logic?
• RQ4 What is the algebraic structure of partial probability, and how does it extend classical probability theory?
• RQ5 Does a weak form of Bayes’ Theorem hold for partial events under the proposed framework?

Key findings

• A partial probability function $ \pi $ on formulas in Kleene logic is defined via a valuation $ \overline{v} $ on the Lindenbaum algebra of a sentential language, satisfying axioms that generalize classical probability.
• The translation from partial events $ (X,Y) \in \mathcal{G}_A $ to formulas $ \alpha \in \overline{\tau}(X,Y) $ ensures $ \mu(X,Y) = \pi(\alpha) $, establishing a one-to-many correspondence.
• The paper proves that $ \pi(\alpha \vee \beta) = \pi(\alpha) + \pi(\beta) - \pi(\alpha \wedge \beta) $, preserving the modular property of probability in the partial setting.
• It establishes that $ \pi(\neg \alpha) = \sigma(\pi(\alpha)) $, where $ \sigma(x,y) = (y,x) $, reflecting the duality of Kleene’s logic.
• A weak form of Bayes’ Theorem is derived for conditional partial probability, extending classical Bayesian reasoning to incomplete or intermediate truth values.
• The construction of $ \overline{v} = \mu \circ \eta $, where $ \eta $ is a morphism from a free DMF-algebra to the event algebra, ensures that the partial probability function is well-defined and consistent with the measure $ \mu $.