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On the Transitivity of Logical Consequence without Assuming Monotonicity

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Abstract

We generalize Ripley’s results on the transitivity of consequence relation, without assuming a logic to be monotonic. Following Gabbay, we assume nonmonotonic consequence relation to be inclusive and cautious monotonic, and figure out the implications between different forms of transitivity of logical consequence. Weaker frameworks without inclusiveness or cautious monotonicity are also discussed. The paper may provide basis for the study of both non-transitive logics and nonmonotonic ones.

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Notes

  1. Note that for brevity we write \(\varphi \vdash \psi \) for \(\{\varphi \}\vdash \psi \) for all \(\varphi ,\psi \in {\mathcal {F}}\).

  2. Another common substitution of monotonicity in nonmonotonic logic is rational monotonicity, i.e., if \(\Gamma \,|\hspace{-0.5em}\sim \varphi \) and \(\Gamma \,|\hspace{-0.5em}\sim \hspace{-1em}/~\lnot \psi \), then \(\Gamma , \psi \,|\hspace{-0.5em}\sim \varphi \). But rational monotonicity is a controversial property. Stalnaker [18] gave a counterexample to it. Moreover, rational monotonicity is not relevant to any form of transitivity, so we will not consider it here.

  3. We pronounce kc ‘kinda complete’, following Ripley.

  4. Similar results in Set-Set framework can be obtained correspondingly through Ripley’s counterparts.

  5. For Set-Fmla properties, this is obvious. For Set-Set properties, though there are multiple conclusions, these conclusions state different possibilities. That is, \(\Gamma \,|\hspace{-0.5em}\sim \Delta \) means \(\bigwedge \Gamma \,|\hspace{-0.5em}\sim \bigvee \Delta \), to which cautious monotonicity could not be applied. For instance, we cannot get \(\varphi , \Gamma \,|\hspace{-0.5em}\sim \Delta \) from \(\Gamma \,|\hspace{-0.5em}\sim \Delta , \varphi \), since \(\Gamma \,|\hspace{-0.5em}\sim \Delta , \varphi \) does not mean \(\Gamma \,|\hspace{-0.5em}\sim \Delta \) and \(\Gamma \,|\hspace{-0.5em}\sim \varphi \).

  6. The consequence relation of default logic is defined as follow: \(W\vdash _D \varphi \) iff \(\varphi \) is included in all extensions of the default theory (W, D). For details, see [1]. It is easy to come up with a counterexample for each linking property.

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  1. Institute of Logic and Cognition, Sun Yat-sen University, No. 135, Xingang Xi Road, Guangzhou, 510275, China

    Lin Chen & Xuefeng Wen

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  1. Lin Chen
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  2. Xuefeng Wen
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This research was supported by the 2021 Humanities and Social Science General Program sponsored by the Ministry of Education of China (Grant No. 21YJA72040001).

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Chen, L., Wen, X. On the Transitivity of Logical Consequence without Assuming Monotonicity. Log. Univers. (2024). https://doi.org/10.1007/s11787-024-00345-3

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