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A logic of knowing why


When we say “I know why he was late”, we know not only the fact that he was late, but also an explanation of this fact. We propose a logical framework of “knowing why” inspired by the existing formal studies on why-questions, scientific explanation, and justification logic. We introduce the \({{\mathcal {K}}{}\textit{y}}_i\) operator into the language of epistemic logic to express “agent i knows why \(\varphi \)” and propose a Kripke-style semantics of such expressions in terms of knowing an explanation of \(\varphi \). We obtain two sound and complete axiomatizations w.r.t. two different model classes depending on different assumptions about introspection. Finally we connect our logic with justification logic technically by providing an alternative semantics and an in-depth comparison on various design choices.

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  1. “Wh” stands for the wh-question words.

  2. Nomic explanations are explanation in terms of laws of nature.

  3. There are also various dimensions of each paradigm, e.g, probabilistic versus non-probabilistic, singular events or laws to be explained.

  4. See Schurz (1995) and Weber et al. (2013) for critical surveys.

  5. LP was invented to give an arithmetic semantics to intuitionistic logic under the Brouwer-Heyting-Kolmogorov provability interpretation.

  6. Similar ideas also appeared in van Benthem (1991).

  7. According to our semantics to be introduced later, it is also allowed to know why for different reasons (for different people), which might help to model mutual misunderstanding.

  8. E.g., there is no proof for the consistency of PA within PA, given PA is consistent.

  9. Though as Johan van Benthem pointed out via personal communication, in many cases there is often a best or shortest explanation for the fact.

  10. Note that here \(\rightarrow \) is a material implication which should not be treated as an arbitrary conditional.

  11. Martin Stokhof suggested an example where one explains to the other why he is the best candidate for a job, but in fact he is not, and his explanation may base on false premises.

  12. The situation is like in Padmanabha et al. (2018) where the bundled modality \(\exists \Box \) cannot distinguish constant domain and increasing domain models.

  13. Note that this is not one of the four introspection axioms of \({{\mathcal {K}}{}\textit{y}}_i\) mentioned earlier.

  14. The “S5 version” of justification logic \(\mathbf {JT45}\) also adds another condition about negative introspection: \(\overline{{\mathcal {E}}(t, \varphi )} \subseteq {\mathcal {E}}(?t, \lnot (t{}:{}\varphi ))\), and requires strong evidence, where ? is a new operation for justification terms in the language, cf. Artemov (2008). To simplify the discussion, we focus on \(\mathbf {LP}\) here.

  15. In the multi-agent setting, \(!_i\) was introduced to capture the proof check done by each agent (Yavorskaya 2006).

  16. The alternative semantics does not work if we just have only one evidence function.

  17. We may also discuss whether \({{\mathcal {K}}{}\textit{y}}_i\varphi \rightarrow {{\mathcal {K}}{}\textit{y}}_i{\mathcal {K}}_i\varphi \) is reasonable.

  18. There are also modal logic approaches to handle scientific explanations cf. e.g, Šešelja and Straßer (2013) and Sedlár and Halas (2015).


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We thank Albert Anglberger, Johan van Benthem, Huimin Dong, Mel Fitting, Dominik Klein, Fenrong Liu, Olivier Roy, Martin Stokhof, Che-Ping Su, Frank Veltman, Lu Wang, Wei Wang, Junhua Yu, Liying Zhang, and Yuncheng Zhou for discussions and useful suggestions on earlier versions of this paper. The insightful comments from the anonymous reviewers of this journal also helped in improving the paper. Yanjing Wang acknowledges the support from the National Program for Special Support of Eminent Professionals and NSSF major Project 12&ZD119. Thomas Studer is supported by the Swiss National Science Foundation Grant 200021_165549.

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Xu, C., Wang, Y. & Studer, T. A logic of knowing why. Synthese 198, 1259–1285 (2021).

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  • Knowing why
  • Why-questions
  • Scientific explanation
  • Epistemic logic
  • Justification logic
  • Axiomatization