[Paper Review] New foundations of reasoning via real-valued first-order logics

Guillermo Badía, Ronald Fagin+1 more|arXiv (Cornell University)|Jun 30, 2022

Advanced Algebra and Logic4 citations

TL;DR

This paper introduces a new framework for first-order and modal reasoning using real-valued multi-dimensional sentences (MD-sentences), extending prior propositional work to handle partial truth-values and complex truth-value combinations. It provides sound and complete axiomatic systems for fixed and varying domains, including a 0-1 law for finitely-valued versions, and establishes completeness over finite domains despite non-recursive validity in such settings.

ABSTRACT

Many-valued logics in general, and fuzzy logics in particular, usually focus on a notion of consequence based on preservation of full truth, typical represented by the value 1 in the semantics given the real unit interval [0,1]. In a recent paper (\emph{Foundations of Reasoning with Uncertainty via Real-valued Logics}, arXiv:2008.02429v2, 2021), Ronald Fagin, Ryan Riegel, and Alexander Gray have introduced a new paradigm that allows to deal with inferences in propositional real-valued logics based on multi-dimensional sentences that allow to prescribe any truth-values, not just 1, for the premises and conclusion of a given entailment. In this paper, we extend their work to the first-order (as well as modal) logic of multi-dimensional sentences. We give axiomatic systems and prove corresponding completeness theorems, first assuming that the structures are defined over a fixed domain, and later for the logics of varying domains. As a by-product, we also obtain a 0-1 law for finitely-valued versions of these logics.

Motivation & Objective

• To extend the propositional framework of multi-dimensional sentences in real-valued logics to first-order and modal settings.
• To address the limitation of standard many-valued logics that focus only on full truth (value 1) and ignore partial truth-value combinations.
• To provide a uniform, parameterized axiomatization for valid inferences in first-order real-valued logics, including those not recursively enumerable.
• To prove completeness theorems for both fixed and varying domains, and to establish a 0-1 law for finitely-valued variants.
• To demonstrate that the system remains finitistic and complete when restricted to finite domains, despite undecidability in the general case.

Proposed method

• Extends the syntax of multi-dimensional sentences (MD-sentences) to first-order and modal logics, where each MD-sentence is of the form ⟨σ₁,…,σₖ; S⟩ with S ⊆ [0,1]ᵏ representing allowed truth-value combinations.
• Defines semantics such that an MD-sentence is true in a model if the tuple of truth-values of its component formulas lies in the set S.
• Develops a finitary axiomatic system with rules that capture logical consequence between MD-sentences, including a generalized Rule (7)* for handling truth-value sets.
• Proves completeness for fixed domains using a strategy analogous to the propositional case, relying on maximal consistent sets and model construction.
• Adapts the system to varying domains by removing domain-size constraints and proving completeness over all domains.
• Applies Oberschelp’s result and reductions from classical first-order logic to show that the satisfiability of S′ = ∅ in Rule (7)* is undecidable, explaining non-recursiveness.

Experimental results

Research questions

• RQ1 Can a uniform, parameterized axiomatization be developed for first-order real-valued logics that captures inference over partial truth-values beyond full truth (1)?
• RQ2 How can multi-dimensional sentences be extended to first-order and modal logics while preserving soundness and completeness?
• RQ3 What is the logical behavior of these systems over finite domains, and can completeness be preserved despite undecidability?
• RQ4 Does a 0-1 law hold for finitely-valued versions of these first-order multi-dimensional logics?
• RQ5 Can the system be adapted to handle varying domains while maintaining completeness and finitary reasoning?

Key findings

• A complete and sound axiomatic system is established for first-order real-valued logics with multi-dimensional sentences over fixed domains, generalizing the propositional completeness result.
• The system is not recursively enumerable due to the undecidability of checking whether the truth-value set S′ is empty in Rule (7)*, which is shown via reduction from finite validity in classical first-order logic.
• A 0-1 law is proven for finitely-valued versions of the logic, indicating that asymptotically, almost all sentences are either almost surely true or almost surely false.
• Completeness is also established for the logic over arbitrary domains (varying domain semantics), extending the framework beyond fixed-size models.
• When restricted to finite domains, the system becomes finitistic and complete, with Γ ⊢ θ if and only if Γ ⊨ finite θ, despite the general non-recursiveness of the system.
• The framework subsumes and generalizes rational Pavelka logic and other fuzzy logics with truth-constants, offering greater expressivity by allowing arbitrary subsets S of [0,1]ᵏ as information sets.