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

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

- 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.
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.
- 12.
- 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.
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.
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.
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.
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.
The distinction between \(\mathcal {K}h\) and \(\mathcal {K}h^+\) is similar to the distinction between STIT and deliberative STIT.

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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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## Additional information

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