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NEW FOUNDATIONS OF REASONING VIA REAL-VALUED FIRST-ORDER LOGICS

Part of: General logic

Published online by Cambridge University Press:  17 March 2025

GUILLERMO BADIA Open the ORCID record for GUILLERMO BADIA [Opens in a new window] ,
RONALD FAGIN  and
CARLES NOGUERA Open the ORCID record for CARLES NOGUERA [Opens in a new window]
GUILLERMO BADIA
Affiliation:
UNIVERSITY OF QUEENSLAND AUSTRALIA e-mail: g.badia@uq.edu.au
RONALD FAGIN
Affiliation:
IBM ALMADEN RESEARCH CENTER USA e-mail: fagin@us.ibm.com
CARLES NOGUERA*
Affiliation:
UNIVERSITY OF SIENA ITALY
*
e-mail: carles.noguera@unisi.it
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Abstract

Many-valued logics, in general, and real-valued logics, in particular, usually focus on a notion of consequence based on preservation of full truth, typically represented by the value $1$ in the semantics given in the real unit interval $[0,1]$. In a recent paper [Foundations of Reasoning with Uncertainty via Real-valued Logics, Proceedings of the National Academy of Sciences 121(21): e2309905121, 2024], 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 a rich class of sentences, multi-dimensional sentences, that talk about combinations of any possible truth values of real-valued formulas. They have proved a strong completeness result that allows one to derive exactly what information can be inferred about the combinations of truth values of a collection of formulas given information about the combinations of truth values of a finite number of other collections of formulas. In this paper, we extend that work to the first-order (as well as modal) logic of multi-dimensional sentences. We give a parameterized axiomatic system that covers any reasonable logic and prove a corresponding completeness theorem, 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 zero-one law for finitely-valued versions of these logics. Since several first-order real-valued logics are known not to have recursive axiomatizations but only infinitary ones, our system is by force akin to infinitary systems.

Keywords

mathematical fuzzy logicmany-valued logicsreal-valued logicsfirst-order fuzzy logicsmodal fuzzy logicscompleteness theoremszero-one laws

MSC classification

Primary: 03B50: Many-valued logic 03B52: Fuzzy logic; logic of vagueness

Information

Type
Article
Information
Bulletin of Symbolic Logic , Volume 31 , Issue 2 , June 2025 , pp. 319 - 349
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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