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<p>Validity is usually taken to be a bivalent property: every inference is either valid or invalid, and never both. We argue for the controversial thesis that, if one endorses a many-valued semantics for the object language, then one likely has good reasons to also endorse a many-valued notion of validity. We present several logical systems (based on Belnap’s algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <mn mathvariant="bold">4</mn> </math></EquationSource> </InlineEquation>) whose notion of validity is non-bivalent: there are inferences that are both valid and invalid, and/or inferences that are neither valid nor invalid. We show that, under natural assumptions, the validity of metainferences is also non-bivalent in many of these systems. Lastly, we claim that our framework has fruitful applications to philosophical issues, such as the link between logical consequence and the conditional, and the link between logic and metalogic.</p>

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Non-Bivalent Validity

  • 期刊论文

  • 发表日期：

    2026年2月12日

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  • Eduardo Barrio,
  • Camillo Fiore,
  • Federico Pailos

摘要

Validity is usually taken to be a bivalent property: every inference is either valid or invalid, and never both. We argue for the controversial thesis that, if one endorses a many-valued semantics for the object language, then one likely has good reasons to also endorse a many-valued notion of validity. We present several logical systems (based on Belnap’s algebra \(\textbf{4}\) 4 ) whose notion of validity is non-bivalent: there are inferences that are both valid and invalid, and/or inferences that are neither valid nor invalid. We show that, under natural assumptions, the validity of metainferences is also non-bivalent in many of these systems. Lastly, we claim that our framework has fruitful applications to philosophical issues, such as the link between logical consequence and the conditional, and the link between logic and metalogic.