[Paper Review] Independence and abstract multiplication

Dov Gabbay, Karl Schlechta|arXiv (Cornell University)|Jan 18, 2010

Logic, Reasoning, and Knowledge7 references3 citations

TL;DR

This paper introduces a formal framework for independence in nonmonotonic logics through the concept of abstract size multiplication, establishing a strong connection between independence and the compositional behavior of logical systems. It demonstrates that when a function f preserves composition under two operations ∘ and ∘′, and allows bidirectional recovery of components, it enables semantical interpolation and supports key properties like Rational Monotony and modular revision.

ABSTRACT

We investigate the notion of independence, which is at the basis of many, seemingly unrelated, properties of logic like Rational Monotony in non-monotonic logics, and interpolation theorems.

Motivation & Objective

• To formalize the notion of independence in logical systems, particularly in nonmonotonic logics, where components can be composed and recovered independently.
• To investigate the connection between independence and the multiplication of abstract size in logical structures, especially in preferential and nonmonotonic logics.
• To establish conditions under which semantical interpolation holds, particularly in monotonic and nonmonotonic settings, via the independence framework.
• To analyze the role of independence in theory revision and distance-based reasoning, especially in modular and hierarchical logical systems.
• To address the relevance and limitations of independence in inconsistent or non-modular contexts, such as when inconsistency obscures component recovery.

Proposed method

• Proposes a formal definition of independence via a triple ⟨f, ∘, ∘′⟩, where f(Σ₁∘Σ₂) = f(Σ₁)∘′f(Σ₂), and f(Σ₁), f(Σ₂) can be recovered from f(Σ₁∘Σ₂) without reapplying f.
• Applies the independence framework to ranked structures, classical logic with disjoint language fragments, and preferential logic, showing that independence holds under specific conditions.
• Introduces two scenarios for size multiplication: nested subsets and product spaces, each leading to different structural properties (e.g., rankedness vs. modularity).
• Uses model operators μ and M(φ) to demonstrate independence in preferential and classical logics, showing that minimal models and model sets behave multiplicatively under composition.
• Applies the framework to revision systems, particularly Parikh-style modular revision, by linking it to the independence condition and size multiplication.
• Employs logical rules such as (GH), (GH+), (s*s), and (b*b⇔b) to formalize the behavior of abstract size and independence in various logical systems.

Experimental results

Research questions

• RQ1 Under what conditions does a function f preserve composition and allow bidirectional recovery of components under operations ∘ and ∘′?
• RQ2 How is independence in logical systems related to the multiplication of abstract size in nonmonotonic and monotonic logics?
• RQ3 In what settings does semantical interpolation follow from the independence condition, particularly in preferential and classical logics?
• RQ4 Why does independence fail in the presence of inconsistency, and what remedies can be applied to restore component recovery?
• RQ5 How can the independence framework be extended to support modular revision and distance-based reasoning in logical systems?

Key findings

• Independence holds in ranked structures when f is the minimal model operator μ and ∘ = ∘′ = ∪, as μ(X∪Y) = μ(X)∪μ(Y) and components can be recovered via intersection.
• Independence is satisfied in classical logic for consistent formulas with disjoint language fragments, where f(φ∧ψ) = M(φ)∩M(ψ), and M(φ) is recoverable as the projection of the joint model set.
• Independence fails for inconsistent formulas such as M(a∧¬a∧b), as the inconsistency cannot be uniquely attributed to a single component, preventing recovery.
• Preferential logic satisfies independence when μ(X×Y) = μ(X)×μ(Y) and the consequence relation ∼∣ satisfies the ∼∣∘∼∣ rule, enabling nonmonotonic interpolation.
• The independence framework supports modular revision and distance-based revision, as seen in the compatibility of (GHD) and (GHD+) rules with the independence condition.
• The paper identifies that the direction A×B⊆X×Y ⇒ A⊆X, B⊆Y is more intuitively justified than the reverse, due to proportionality concerns in size multiplication.