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<p>The object language of Kripke’s 1975 semantic theory of truth, based on the Strong Kleene valuation scheme, cannot contain a predicate that expresses the notion "ungroundedness" that Kripke provides an analysis of. This is unfortunate; it means that, in the object language of Kripke’s theory, there is no obvious way to express Kripke’s diagnostic insight about what causes semantic pathology. This paper shows how to introduce a “groundedness” predicate, G, to a Kripkean theory of truth that can fill this expressive gap. In the fixed-point construction that gives an interpretation for G, G’s anti-extension tracks networks of sentences that, due to predications of truth, result in non-terminating graphs of semantic dependence. In the fixed-point models that provide a class of intended interpretations for G: (i) every sentence with a classical semantic value is in the extension of G, (ii) every sentence in the anti-extension of G receives the value <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\frac{1}{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation>, and (iii) the anti-extension of G includes <i>all</i> the sentences that receive <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\frac{1}{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation> in the corresponding model of Kripke’s original theory. A language augmented with predicates for truth and groundedness possesses sufficient expressive resources to articulate Kripke’s diagnostic insight as it applies to itself.</p>

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A Groundedness Predicate for Kripke’s Theory of Truth

  • 期刊论文

  • 发表日期：

    2026年6月17日

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  • Torsten Odland

摘要

The object language of Kripke’s 1975 semantic theory of truth, based on the Strong Kleene valuation scheme, cannot contain a predicate that expresses the notion "ungroundedness" that Kripke provides an analysis of. This is unfortunate; it means that, in the object language of Kripke’s theory, there is no obvious way to express Kripke’s diagnostic insight about what causes semantic pathology. This paper shows how to introduce a “groundedness” predicate, G, to a Kripkean theory of truth that can fill this expressive gap. In the fixed-point construction that gives an interpretation for G, G’s anti-extension tracks networks of sentences that, due to predications of truth, result in non-terminating graphs of semantic dependence. In the fixed-point models that provide a class of intended interpretations for G: (i) every sentence with a classical semantic value is in the extension of G, (ii) every sentence in the anti-extension of G receives the value \(\frac{1}{2}\) 1 2 , and (iii) the anti-extension of G includes all the sentences that receive \(\frac{1}{2}\) 1 2 in the corresponding model of Kripke’s original theory. A language augmented with predicates for truth and groundedness possesses sufficient expressive resources to articulate Kripke’s diagnostic insight as it applies to itself.