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<p>We further investigate the metalogical properties of the ordered fragment. First, we provide a simplified proof of the satisfiability invariance under A. Herzig’s translation of the ordered fragment into modal logic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{KD}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">KD</mi> </math></EquationSource> </InlineEquation>. Second, based on the notion of bisimulation developed by B. Bednarczyk and R. Jaakkola, we show that each ordered formula is equivalent to a disjunction of ‘ordered types’. Third, we show that the fragment enjoys uniform interpolation, and that uniform interpolants can be effectively constructed from ‘ordered types’. Finally, we establish the Łoś-Tarski Preservation Theorem for the fragment, and therefore conclude that the ordered fragment is nice.</p>

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The Niceness of the Ordered Fragment of First-Order Logic

  • 期刊论文

  • 发表日期：

    2026年4月24日

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  • Hongkai Yin

摘要

We further investigate the metalogical properties of the ordered fragment. First, we provide a simplified proof of the satisfiability invariance under A. Herzig’s translation of the ordered fragment into modal logic \(\textbf{KD}\) KD . Second, based on the notion of bisimulation developed by B. Bednarczyk and R. Jaakkola, we show that each ordered formula is equivalent to a disjunction of ‘ordered types’. Third, we show that the fragment enjoys uniform interpolation, and that uniform interpolants can be effectively constructed from ‘ordered types’. Finally, we establish the Łoś-Tarski Preservation Theorem for the fragment, and therefore conclude that the ordered fragment is nice.