Foundations of Reasoning with Uncertainty via Real-valued Logics

• URL: http://arxiv.org/abs/2008.02429v3
• Date: Tue, 30 Aug 2022 21:42:24 GMT
• Title: Foundations of Reasoning with Uncertainty via Real-valued Logics
• Authors: Ronald Fagin, Ryan Riegel, Alexander Gray
• Abstract summary: We give a sound and strongly complete axiomatization that can be parametrized to cover essentially every real-valued logic. Our class of sentences are very rich, and each describes a set of possible real values for a collection of formulas of the real-valued logic.
• Score: 70.43924776071616
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Real-valued logics underlie an increasing number of neuro-symbolic approaches, though typically their logical inference capabilities are characterized only qualitatively. We provide foundations for establishing the correctness and power of such systems. We give a sound and strongly complete axiomatization that can be parametrized to cover essentially every real-valued logic, including all the common fuzzy logics. Our class of sentences are very rich, and each describes a set of possible real values for a collection of formulas of the real-valued logic, including which combinations of real values are possible. Strong completeness allows us to derive exactly what information can be inferred about the combinations of real values of a collection of formulas given information about the combinations of real values of several other collections of formulas. We then extend the axiomatization to deal with weighted subformulas. Finally, we give a decision procedure based on linear programming for deciding, for certain real-valued logics and under certain natural assumptions, whether a set of our sentences logically implies another of our sentences.

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