[Paper Review] An Algebraic Semantics for Possibilistic Logic

Luca Boldrin, Claudio Sossai|arXiv (Cornell University)|Feb 20, 2013

Logic, Reasoning, and Knowledge13 references20 citations

TL;DR

This paper introduces an algebraic semantics for possibilistic logic by extending its language with a new conjunction (⊗) and a negation operator, grounding the logic in the lattice of possibility functions—a quantale—thereby enabling a dynamic interpretation of information from multiple sources. It presents a sound and complete Gentzen-style calculus, establishing a foundation for non-classical, possibilistic reasoning with truth functionality analyzed in a Pavelka-style framework.

ABSTRACT

The first contribution of this paper is the presentation of a Pavelka - like formulation of possibilistic logic in which the language is naturally enriched by two connectives which represent negation (eg) and a new type of conjunction (otimes). The space of truth values for this logic is the lattice of possibility functions, that, from an algebraic point of view, forms a quantal. A second contribution comes from the understanding of the new conjunction as the combination of tokens of information coming from different sources, which makes our language "dynamic". A Gentzen calculus is presented, which is proved sound and complete with respect to the given semantics. The problem of truth functionality is discussed in this context.

Motivation & Objective

• To develop a formal algebraic framework for possibilistic logic that supports reasoning under uncertainty.
• To enrich the logical language with a new conjunction (⊗) modeling fusion of information from distinct sources.
• To provide a semantics based on possibility functions forming a quantale, enabling algebraic treatment of uncertainty.
• To formalize truth functionality in a non-classical, possibilistic setting using a Pavelka-style approach.
• To establish a sound and complete Gentzen calculus for the proposed logic.

Proposed method

• Extends possibilistic logic with two new connectives: a negation operator and a dynamic conjunction (⊗) for combining information from different sources.
• Models truth values as possibility functions, forming a lattice that is algebraically a quantale, enabling algebraic operations on uncertainty degrees.
• Defines a Gentzen-style sequent calculus with structural rules adapted to the quantale structure, ensuring logical consistency.
• Proves soundness and completeness of the calculus with respect to the algebraic semantics.
• Analyzes the problem of truth functionality in the context of non-classical, possibility-based logic.
• Uses the algebraic structure of the quantale to interpret the behavior of logical connectives and truth values.

Experimental results

Research questions

• RQ1 How can possibilistic logic be enriched with a dynamic conjunction that models information fusion from multiple sources?
• RQ2 What algebraic structure best captures the semantics of possibility functions in a logical framework?
• RQ3 Can a Gentzen calculus be constructed that is both sound and complete for this extended logic?
• RQ4 How does truth functionality behave in a non-classical, possibility-theoretic setting?
• RQ5 What is the role of the Pavelka-style framework in enabling a formal treatment of uncertainty in logic?

Key findings

• The proposed logic extends standard possibilistic logic with a new conjunction (⊗) that models the combination of information from distinct sources, giving it a dynamic character.
• The space of possibility functions forms a quantale, providing a robust algebraic foundation for reasoning under uncertainty.
• A Gentzen calculus is proven sound and complete with respect to the algebraic semantics, validating the logical system.
• The paper establishes that truth functionality is preserved in the extended logic under the given semantics, despite non-classical truth values.
• The use of a Pavelka-style framework allows for a formal treatment of degrees of belief and uncertainty in logical reasoning.
• The algebraic semantics enables a clear interpretation of logical connectives through operations on possibility functions.