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New foundations of reasoning via real-valued first-order logics

Guillermo Badia, Ronald Fagin, Carles Noguera

TL;DR

The paper introduces real-valued first-order and modal extensions of MD-sentences, providing a uniform, parameterized axiomatization and completeness results for fixed domains and all domains, along with a 0-1 law for finitely-valued variants. It shows that MD-sentences enable genuine multi-valued reasoning beyond preservation of truth at 1, while acknowledging infinitary aspects hidden in the information sets S. For countable domains without equality, finitary consequence is complete for Gödel, Łukasiewicz, and Product logics, but the full framework is not recursively enumerable in general (especially with equality or infinite domains). The work offers a unified methodology to axiomatize a broad class of real-valued logics and clarifies the trade-offs between expressivity and decidability in finite versus infinite 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.

New foundations of reasoning via real-valued first-order logics

TL;DR

The paper introduces real-valued first-order and modal extensions of MD-sentences, providing a uniform, parameterized axiomatization and completeness results for fixed domains and all domains, along with a 0-1 law for finitely-valued variants. It shows that MD-sentences enable genuine multi-valued reasoning beyond preservation of truth at 1, while acknowledging infinitary aspects hidden in the information sets S. For countable domains without equality, finitary consequence is complete for Gödel, Łukasiewicz, and Product logics, but the full framework is not recursively enumerable in general (especially with equality or infinite domains). The work offers a unified methodology to axiomatize a broad class of real-valued logics and clarifies the trade-offs between expressivity and decidability in finite versus infinite 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.
Paper Structure (15 sections, 17 theorems, 26 equations)

This paper contains 15 sections, 17 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. The propositional case: an overview
  3. Axioms
  4. Inference rules
  5. The logic of a fixed domain
  6. First-order case (the logic of a fixed domain)
  7. Axioms
  8. Inference rules
  9. Modal logic (the logic of a fixed frame)
  10. Axioms
  11. Inference rules
  12. Axiomatizations of prominent first-order (and modal) real-valued logics
  13. A 0-1 law for MD-logics
  14. The logic of all domains
  15. Conclusion

Key Result

Lemma 4

Let $\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle$ be the premise of Rule (7) and assume that $G=\{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k})\}$ is closed under subformulas in the usual sense. Then, the conc

Theorems & Definitions (42)

  • Example 1
  • Example 2
  • Definition 3
  • Lemma 4
  • proof
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • Theorem 8: Completeness of the logic of a fixed domain
  • ...and 32 more