## Abstract

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.

### Similar content being viewed by others

## Notes

“Wh” stands for the wh-question words.

Nomic explanations are explanation in terms of laws of nature.

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

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

Similar ideas also appeared in van Benthem (1991).

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.

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

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

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

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.

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

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

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.

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

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

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

## References

Artemov, S. (1995).

*Operational modal logic*. Technical report MSI 9529, Cornell University.Artemov, S. (2001). Explicit provability and constructive semantics.

*Bulletin of Symbolic Logic*,*7*(1), 1–36.Artemov, S. (2006). Justified common knowledge.

*Theoretical Computer Science*,*357*(1), 4–22.Artemov, S. (2008). The logic of justification.

*The Review of Symbolic Logic*,*1*(04), 477–513.Artemov, S., & Nogina, E. (2005). Introducing justification into epistemic logic.

*Journal of Logic and Computation*,*15*(6), 1059–1073.Bird, A. (1998).

*Philosophy of science*. Abingdon: Routledge.Bromberger, S. (1966). Questions.

*The Journal of Philosophy*,*63*(20), 597–606.Bucheli, S., Kuznets, R., & Studer, T. (2011). Justifications for common knowledge.

*Journal of Applied Non-classical Logics*,*21*(1), 35–60.Bucheli, S., Kuznets, R., & Studer, T. (2014). Realizing public announcements by justifications.

*Journal of Computer and System Sciences*,*80*(6), 1046–1066.Fagin, R., Halpern, J., Moses, Y., & Vardi, M. (1995).

*Reasoning about knowledge*. Cambridge: MIT Press.Fan, J., Wang, Y., & van Ditmarsch, H. (2015). Contingency and knowing whether.

*The Review of Symbolic Logic*,*8*, 75–107.Fitting, M. (2005). The logic of proofs, semantically.

*Annals of Pure and Applied Logic*,*132*(1), 1–25.Fitting, M. (2008). A quantified logic of evidence.

*Annals of Pure and Applied Logic*,*152*(1), 67–83.Fitting, M. (2016). Modal logics, justification logics, and realization.

*Annals of Pure and Applied Logic*,*167*(8), 615–648.Gattinger, M., van Eijck, J., & Wang, Y. (2017). Knowing values and public inspection. In

*Proceedings of ICLA ’17*(forthcoming).Groenendijk, J., & Stokhof, M. (1982). Semantic analysis of wh-complements.

*Linguistics and Philosophy*,*5*(2), 175–233.Halpern, J. Y., & Pucella, R. (2011). Dealing with logical omniscience: Expressiveness and pragmatics.

*Artificial Intelligence*,*175*(1), 220–235.Hempel, C. (1965).

*Aspects of scientific explanation and other essays in the philosophy of science*. Mankato: The Free Press.Hempel, C. G., & Oppenheim, P. (1948). Studies in the logic of explanation.

*Philosophy of Science*,*15*(2), 135–175.Hintikka, J. (1962).

*Knowledge and belief: An introduction to the logic of the two notions*(Vol. 181). Ithaca: Cornell University Press.Hintikka, J. (1981). On the logic of an interrogative model of scientific inquiry.

*Synthese*,*47*(1), 69–83.Hintikka, J. (1983).

*New foundations for a theory of questions and answers*(pp. 159–190). Dordrecht: Springer.Hintikka, J., & Halonen, I. (1995). Semantics and pragmatics for why-questions.

*The Journal of Philosophy*,*92*(12), 636–657.Kitcher, P. (1981). Explanatory unification.

*Philosophy of Science*,*48*(4), 507–531.Koura, A. (1988). An approach to why-questions.

*Synthese*,*74*(2), 191–206.Kuznets, R., & Studer, T. (2012). Justifications, ontology, and conservativity. In T. Bolander, T. Braüner, S. Ghilardi, & L. Moss (Eds.),

*Advances in modal logic*(Vol. 9, pp. 437–458). London: College Publications.Kuznets, R., & Studer, T. (2013). Update as evidence: Belief expansion. In S. Artemov & A. Nerode (Eds.),

*Logical foundations of computer science*, LFCS 2013, Springer, LNCS, Vol. 7734, pp. 266–279.Kuznets, R., & Studer, T. (2019).

*Justification logic*. London: College Publications.Padmanabha, A., Ramanujam, R., & Wang, Y. (2018). Bundled fragments of first-order modal logic: (Un)decidability. In

*Proceedings of FSTTCS 2018*(pp. 43:1–43:20).Pischke, N. (2017). Dynamic extensions for the logic of knowing why with public announcements of formulas. arXiv e-prints 1707.05617.

Plaza, J. (2007). Logics of public communications.

*Synthese*,*158*(2), 165–179.Renne, B. (2008).

*Dynamic epistemic logic with justification*. PhD thesis, New York, NY, USA, aAI3310607.Renne, B. (2012). Multi-agent justification logic: Communication and evidence elimination.

*Synthese*,*185*(1), 43–82.Salmon, W. (1984).

*Scientific explanation and the causal structure of the world*. Princeton: Princeton University Press.Schurz, G. (1995). Scientific explanation: A critical survey.

*Foundations of Science*,*1*(3), 429–465.Schurz, G. (1999). Explanation as unification.

*Synthese*,*120*(1), 95–114.Schurz, G. (2005). Explanations in science and the logic of why-questions: Discussion of the Halonen–Hintikka-approachand alternative proposal.

*Synthese*,*143*(1), 149–178.Sedlár, I., & Halas, J. (2015). Modal logics of abstract explanation frameworks. In

*Abstract in proceedings of CLMPS 15*.Šešelja, D., & Straßer, C. (2013). Abstract argumentation and explanation applied to scientific debates.

*Synthese*,*190*(12), 2195–2217.van Benthem, J. (1991). Reflections on epistemic logic.

*Logique and Analyse*,*34*(133–134), 5–14.van Ditmarsch, H., Halpern, J. Y., van der Hoek, W., & Kooi, B. (Eds.). (2015).

*Handbook of epistemic logic*. London: College Publications.van Fraassen, B. C. (1980).

*The scientific image*. Oxford: Oxford University Press.Wang, Y. (2015). A logic of knowing how. In

*Proceedings of LORI-V*(pp. 392–405).Wang, Y. (2017). A new modal framework for epistemic logic. In

*Proceedings of TARK ’17*(pp. 515–534).Wang, Y., & Fan, J. (2013). Knowing that, knowing what, and public communication: Public announcement logic with Kv operators. In

*Proceedings of IJCAI’13*(pp. 1139–1146).Wang, Y. (2018a). Beyond knowing that: A new generation of epistemic logics. In H. van Ditmarsch & G. Sandu (Eds.),

*Jaakko Hintikka on knowledge and game theoretical semantics, outstanding contributions to logic*(Vol. 12, pp. 499–533). Berlin: Springer.Wang, Y. (2018b). A logic of goal-directed knowing how.

*Synthese*,*195*(10), 4419–4439.Wang, Y., & Fan, J. (2014). Conditionally knowing what.

*Advances in Modal Logic*,*10*, 569–587.Weber, E., van Bouwel, J., & De Vreese, L. (2013).

*Scientific explanation*. Berlin: Springer.Yavorskaya, T. (2006). Multi-agent explicit knowledge. In D. Grigoriev, J. Harrison, & E. A. Hirsch (Eds.),

*Proceedings of CSR 2006*(pp. 369–380). Berlin: Springer.

## Acknowledgements

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.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

Xu, C., Wang, Y. & Studer, T. A logic of knowing why.
*Synthese* **198**, 1259–1285 (2021). https://doi.org/10.1007/s11229-019-02104-0

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11229-019-02104-0