A logic of goal-directed knowing how

Abstract

In this paper, we propose a decidable single-agent modal logic for reasoning about goal-directed “knowing how”, based on ideas from linguistics, philosophy, modal logic, and automated planning in AI. We first define a modal language to express “I know how to guarantee \(\varphi \) given \(\psi \)” with a semantics based not on standard epistemic models but on labeled transition systems that represent the agent’s knowledge of his own abilities. The semantics is inspired by conformant planning in AI. A sound and complete proof system is given to capture valid reasoning patterns, which highlights the compositional nature of “knowing how”. The logical language is further extended to handle knowing how to achieve a goal while maintaining other conditions.

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Notes

  1. 1.

    There is a cross-lingual fact: such knowing-wh sentences become ungrammatical if the verb “know” is replaced by “believe”, e.g., I believe how to swim. This may shed some shadow on philosophers’ usual conception of knowledge in terms of strengthened belief. Linguistically, this phenomenon occurs to many other verbs which can be roughly categorized using factivity, cf., e.g, (Egrè 2008).

  2. 2.

    Nevertheless Hintikka addressed some of those issues about first-order modal logic insightfully in the context of epistemic logic, see, e.g., a wonderful survey paper by Hintikka (1989). Many of those issues are also elegantly addressed in intensional first-order modal logic, cf. e.g., (Fitting and Mendelsohn 1998). There is a wonderful survey on quantified epistemic logic by Gochet and Gribomont (2006).

  3. 3.

    See (Wang forthcoming) for a survey with discussions of related work on quantified epistemic logic.

  4. 4.

    For example, knowing whether\(p\rightarrow q\) and knowing whether p together does not entail knowing whether q. Likewise, knowing how to p and knowing how to q does not entail knowing how to \(p\wedge q\). Moreover, you may not know why a tautology is a tautology which contradicts necessitation.

  5. 5.

    Fantl (2008) presents a survey of the debate. A comprehensive collection of the related papers (\(200^+\)) can be found at http://philpapers.org/browse/knowledge-how, edited by John Bengson.

  6. 6.

    Here knowing how to do an activity (like swimming) is not a typical example for our treatment, although we hope our formalism captures some common features shared also by them. As discussed in Gochet (2013), “knowing how” plus activities, though more philosophically interesting, is less demanding in logical structure than others.

  7. 7.

    The philosophical implication of a similar treatment is discussed by Lau and Wang (2016), where an intellectualistic account is advocated, which may help to reconcile intellectualism and anti-intellectualistic ability account.

  8. 8.

    Such conditions are rarely discussed in the philosophical literature of “knowing how” with few exceptions such as Noë (2005).

  9. 9.

    By using the condition, one can be said to know better how to swim than another if he can do it in a more hostile environment (thus weakening the condition) see Lau and Wang (2016).

  10. 10.

    Such conditionals are clearly not simple (material) implications and they are closely related to conditional probability and conditional belief [cf. e.g., Tillio et al. (2014)].

  11. 11.

    Taken from Wang and Li (2012), Yu et al. (2016).

  12. 12.

    The agent may have more abilities de facto than what he may realize. It is important to make sure the agent can knowingly guarantee the goal in terms of the ability he is aware of, cf. (McCarthy and Hayes 1969; Broersen 2008; Ågotnes et al. 2015).

  13. 13.

    Brown (1988) introduced a modality for can\(\varphi \) with the following \(\exists \forall \) schema over neighbourhood models: there is a relevant cluster of possible worlds (as the outcomes of an action) where \(\varphi \) is true in all of them.

  14. 14.

    This also distinguishes this work from our earlier philosophical discussion in Lau and Wang (2016), where intellectualism was defended by giving an \(\exists x \mathcal {K}\varphi (x)\)-like truth condition informally.

  15. 15.

    Note that \(\mathcal {U} \) is a very powerful modality in its expressiveness when combined with the standard \(\Box \) modality, cf. (Goranko and Passy 1992).

  16. 16.

    This is an analog of a requirement of “knowing how” by Moore (1985, p. 58): you need to make sure by doing the first step you will know how to continue.

  17. 17.

    This connects with the philosophical concept of knowledge as a strengthened notion based on justified true belief where the existence of a good justification suffices, cf. also justification logic proposed by Artemov (2008).

  18. 18.

    The distinction between \(\mathcal {K}h\) and \(\mathcal {K}h^+\) is similar to the distinction between STIT and deliberative STIT.

References

  1. Ågotnes, T., Goranko, V., Jamroga, W., & Wooldridge, M. (2015). Knowledge and ability, chapter 11. In H. van Ditmarsch, J. Halpern, W. van der Hoek, & B. Kooi (Eds.), Handbook of epistemic logic. London: College Publications.

    Google Scholar 

  2. Alur, R., Henzinger, T., & Kupferman, O. (2002). Alternating-time temporal logic. Journal of the ACM, 49, 672–713.

    Article  Google Scholar 

  3. Areces, C., Fervari, R., & Hoffmann, G. (2015). Relation-changing modal operators. Logic Journal of the IGPL, 23(4), 601–627.

    Article  Google Scholar 

  4. Artemov, S. (2008). The logic of justification. The Review of Symbolic Logic, 1(04), 477–513.

    Article  Google Scholar 

  5. Belardinelli, F. (2014). Reasoning about knowledge and strategies: Epistemic strategy logic. In Proceedings of SR, 2014 (pp. 27–33)

  6. Broersen, J. (2008). A logical analysis of the interaction between ‘obligation-to-do’ and ‘knowingly doing’. In Proceedings of DEON, 08 (pp. 140–154)

  7. Broersen, J., & Herzig, A. (2015). Using STIT theory to talk about strategies. In J. van Benthem, S. Ghosh, & R. Verbrugge (Eds.), Models of strategic reasoning: Logics, games, and communities (pp. 137–173). New York: Springer.

    Google Scholar 

  8. Brown, M. A. (1988). On the logic of ability. Journal of Philosophical Logic, 17(1), 1–26.

    Article  Google Scholar 

  9. Carr, D. (1979). The logic of knowing how and ability. Mind, 88(351), 394–409.

    Article  Google Scholar 

  10. Ciardelli, I., & Roelofsen, F. (2015). Inquisitive dynamic epistemic logic. Synthese, 192(6), 1643–1687.

    Article  Google Scholar 

  11. Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2013). Inquisitive semantics: A new notion of meaning. Language and Linguistics Compass, 7(9), 459–476.

    Article  Google Scholar 

  12. Egré, P. (2008). Question-embedding and factivity. Grazer Philosophische Studien, 77(1), 85–125.

    Article  Google Scholar 

  13. Fagin, R., Halpern, J., Moses, Y., & Vardi, M. (1995). Reasoning about knowledge. Cambridge, MA: MIT Press.

    Google Scholar 

  14. Fan, J., Wang, Y., & van Ditmarsch, H. (2014). Almost necessary. Advances in Modal Logic, 10, 178–196.

    Google Scholar 

  15. Fan, J., Wang, Y., & van Ditmarsch, H. (2015). Contingency and knowing whether. The Review of Symbolic Logic, 8, 75–107.

    Article  Google Scholar 

  16. Fantl, J. (2008). Knowing-how and knowing-that. Philosophy Compass, 3(3), 451–470.

    Article  Google Scholar 

  17. Fitting, M., & Mendelsohn, R. L. (1998). First-order modal logic. Synthese library. New York: Springer.

    Google Scholar 

  18. Gochet, P. (2013). An open problem in the logic of knowing how. In J. Hintikka (Ed.), Open problems in epistemology. Helsinki: The Philosophical Society of Finland.

    Google Scholar 

  19. Gochet, P., & Gribomont, P. (2006). Epistemic logic. In D. M. Gabbay & J. Woods (Eds.), Handbook of the history of logic (Vol. 7). North Holland: Elsevier.

    Google Scholar 

  20. Goranko, V., & Passy, S. (1992). Using the universal modality: Gains and questions. Journal of Logic and Computation, 2(1), 5–30.

    Article  Google Scholar 

  21. Gu, T., & Wang, Y. (2016). “Knowing value” logic as a normal modal logic. Advances in Modal Logic, 11, 362–381.

    Google Scholar 

  22. Harrah, D. (2002). The logic of questions. In D. Gabbay (Ed.), Handbook of philosophical logic (Vol. 8). New York: Springer.

    Google Scholar 

  23. Hart, S., Heifetz, A., & Samet, D. (1996). Knowing whether, knowing that, and the cardinality of state spaces. Journal of Economic Theory, 70(1), 249–256.

    Article  Google Scholar 

  24. Herzig, A. (2015). Logics of knowledge and action: Critical analysis and challenges. Autonomous Agents and Multi-Agent Systems, 29(5), 719–753.

    Article  Google Scholar 

  25. Herzig, A., & Troquard, N. (2006). Knowing how to play: Uniform choices in logics of agency. In Proceedings of AAMAS, 06 (pp. 209–216)

  26. Herzig, A., Lorini, E., & Walther, D. (2013). Reasoning about actions meets strategic logics. In Proceedings of LORI, 2013 (pp. 162–175)

  27. Hintikka, J. (1962). Knowledge and belief: An introduction to the logic of the two notions. Ithaca, NY: Cornell University Press.

    Google Scholar 

  28. Hintikka, J. (1989). Reasoning about knowledge in philosophy: The paradigm of epistemic logic. In The logic of epistemology and the epistemology of logic, synthese library (Vol. 200, pp. 17–35). Dordrecht: Springer

  29. Hintikka, J. (2007). Socratic epistemology: Explorations of knowledge-seeking by questioning. Cambridge: Cambridge University Press.

    Google Scholar 

  30. van der Hoek, W., & Lomuscio, A. (2003). Ignore at your peril: Towards a logic for ignorance. In Proceedings of AAMAS, 03 (pp. 1148–1149).

  31. Jamroga, W., & van der Hoek, W. (2004). Agents that know how to play. Fundamenta Informaticae, 63(2–3), 185–219.

    Google Scholar 

  32. Kracht, M., & Wolter, F. (1997). Simulation and transfer results in modal logic: A survey. Studia Logica, 59(1), 149–177.

    Article  Google Scholar 

  33. Lau, T., Wang, Y. (2016). Knowing your ability. The Philosophical Forum, pp. 415–424

  34. Lenzen, W. (1978). Recent work in epistemic logic. Acta Philosophica Fennica, 30(2), 1–219.

    Google Scholar 

  35. Li, Y., Wang, Y. (2017). Achieving while maintaining: A logic of knowing how with intermediate constraints. In: Proceedings of ICLA’17 (forthcoming)

  36. McCarthy, J. (1979). First-order theories of individual concepts and propositions. Machine Intelligence, 9, 129–147.

    Google Scholar 

  37. McCarthy, J., & Hayes, P. J. (1969). Some philosophical problems from the standpoint of artificial intelligence. In Machine intelligence (pp. 463–502). Edinburgh University Press

  38. Moore, R. C. (1985). A formal theory of knowledge and action. In J. R. Hobbs & R. C. Moore (Eds.), Formal theories of the commonsense world. Norwood: Ablex Publishing Corporation.

    Google Scholar 

  39. Morgenstern, L. (1986). A first order theory of planning, knowledge, and action. In Proceedings TARK 86 (pp. 99–114). San Francisco: Morgan Kaufmann Publishers Inc.

  40. Noë, A. (2005). Against intellectualism. Analysis, 65, 278–290.

    Article  Google Scholar 

  41. Plaza, J. A. (1989). Logics of public communications. In M. L. Emrich, M. S. Pfeifer, M. Hadzikadic, Z. W. Ras (Eds.), Proceedings of the 4th international symposium on methodologies for intelligent systems (pp. 201–216)

  42. Ryle, G. (1946). Knowing how and knowing that: The presidential address. Proceedings of the Aristotelian Society, 46, 1–16.

    Article  Google Scholar 

  43. Smith, D. E., & Weld, D. S. (1998). Conformant graphplan. In: AAAI 98 (pp. 889–896).

  44. Stanley, J. (2011). Know how. Oxford: Oxford University Press.

    Google Scholar 

  45. Stanley, J. (2001). Knowing how. The Journal of Philosophy, 98, 411–444.

    Article  Google Scholar 

  46. Tillio, A. D., Halpern, J. Y., & Samet, D. (2014). Conditional belief types. Games and Economic Behavior, 87, 253–268.

    Article  Google Scholar 

  47. van Benthem, J., van Eijck, J., & Kooi, B. (2006). Logics of communication and change. Information and Computation, 204(11), 1620–1662.

    Article  Google Scholar 

  48. van Ditmarsch, H., Halpern, J. Y., van der Hoek, W., & Kooi, B. (2015). Handbook of epistemic logic. London: College Publications.

    Google Scholar 

  49. Vendler, Z. (1967). Linguistics in philosophy. Ithaca: Cornell University Press.

    Google Scholar 

  50. Von Wright, G. H. (1951). An essay in modal logic. Amsterdam: North Holland.

    Google Scholar 

  51. Wang, R. J. (2011). Timed modal epistemic logic. PhD thesis, City University of New York

  52. Wang, Y. (2015a). A logic of knowing how. In Proceedings of LORI, 2015 (pp. 392–405)

  53. Wang, Y. (2015b). Representing imperfect information of procedures with hyper models. In Proceedings of ICLA, 2015 (pp. 218–231)

  54. Wang, Y. (forthcoming). Beyond knowing that: A new generation of epistemic logics. In H. van Ditmarsch & G. Sandu (Eds.), Jaakko Hintikka on knowledge and game theoretical semantics. New York: Springer.

  55. Wang, Y., & Fan, J. (2013). Knowing that, knowing what, and public communication: Public announcement logic with Kv operators. In Proceedings of IJCAI, 13 (pp. 1147–1154)

  56. Wang, Y., & Fan, J. (2014). Conditionally knowing what. Advances in Modal Logic, 10, 569–587.

    Google Scholar 

  57. Wang, Y., & Li, Y. (2012). Not all those who wander are lost: Dynamic epistemic reasoning in navigation. Advances in Modal Logic, 9, 559–580.

    Google Scholar 

  58. Yu, Q., Li, Y., & Wang, Y. (2016). A dynamic epistemic framework for conformant planning. In Proceedings of TARK’15, EPTCS (pp. 298–318)

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Acknowledgements

The author acknowledges the support from the National Program for Special Support of Eminent Professionals and NSSF key projects 12&ZD119. The author thanks Frank Veltman, Maria Aloni and the two anonymous reviewers of this journal for their helpful comments on the earlier versions of the paper. The author is grateful to John Maier who proofread the paper.

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Correspondence to Yanjing Wang.

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A preliminary version of this paper appeared in the proceedings of LORI-V (Wang 2015a).

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Wang, Y. A logic of goal-directed knowing how. Synthese 195, 4419–4439 (2018). https://doi.org/10.1007/s11229-016-1272-0

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Keywords

  • Knowing how
  • Epistemic logic
  • Conformant planning
  • Modal logic