## Abstract

Epistemic logic has become a major field of philosophical logic ever since the groundbreaking work by Hintikka [58]. Despite its various successful applications in theoretical computer science, AI, and game theory, the technical development of the field has been mainly focusing on the propositional part, i.e., the propositional modal logics of “knowing that”. However, knowledge is expressed in everyday life by using various other locutions such as “knowing whether”, “knowing what”, “knowing how” and so on (knowing-wh hereafter). Such knowledge expressions are better captured in quantified epistemic logic, as was already discussed by Hintikka [58] and his sequel works at length. This paper aims to draw the attention back again to such a fascinating but largely neglected topic. We first survey what Hintikka and others did in the literature of quantified epistemic logic, and then advocate a new quantifier-free approach to study the epistemic logics of knowing-wh, which we believe can balance expressivity and complexity, and capture the essential reasoning patterns about knowing-wh. We survey our recent line of work on the epistemic logics of ‘knowing whether”, “knowing what” and “knowing how” to demonstrate the use of this new approach.

A mindmap of this paper is here: http://www.phil.pku.edu.cn/personal/wangyj/mindmaps/.

This is a preview of subscription content, access via your institution.

## Notes

- 1.
Hintikka was never happy with the term “possible worlds”, since in his models there may be no “worlds” but only situations or states, which are partial descriptions of the worlds. However, in this paper we will still use the term “possible worlds” for convenience.

- 2.
*How*is in general also considered as a wh-question word, besides*what, when, where, who, whom, which, whose*, and*why*. - 3.
The “knows X” search term can exclude the phrases such as “you know what” and count only the statements, while “know X” may appear in questions as well.

- 4.
Later on he singled out “knowing why” in his framework of interrogative models [65].

- 5.
It also makes sense to understand knowing-wh constructions by first understanding the semantics of questions, see [53] and references therein. Knowing-wh is then knowing a/the answer of the corresponding wh-question.

- 6.
[58] argued that the quantification into the modal context is necessary and not misleading, in contrast to Quine who was against such quantification due to the lack of substitution of identity in modal context.

- 7.
How the domain varies may affect the corresponding quantified modal axioms, see [17] for a overview on this issue in first-order modal logic.

- 8.
The above \(\textit{K}\forall x (\exists y\slash \textit{K})(M(x)\rightarrow T(x,y))\) is an example that cannot be expressed in standard first-order epistemic logic.

- 9.
- 10.
Modeling it globally can be done in propositional modal logic with new axioms like \(K_jp\rightarrow K_ip\), cf. e.g., [80].

- 11.
The absence of equality symbols also make the substitution of equal constants apparently irrelevant.

- 12.
- 13.
On the other hand, a slightly different axiom holds intuitively: knowing whether \(\varphi \leftrightarrow \psi \) and knowing whether \(\varphi \) does entail knowing whether \(\psi \).

- 14.
Fitch proved that you cannot know all the truths, e.g., \(p\wedge \lnot K_i p\) is not knowable by

*i*, which is demonstrated by the inconsistent Moore sentence: \(K_i(p\wedge \lnot K_i p)\) in the basic epistemic logic [45]. - 15.
- 16.
See [5] for more general versions of the knowing whether operator.

- 17.
Similar semantics has been applied to neighborhood structures [41].

- 18.
Note that if there is at most one successor of

*s*then every \(\varDelta \varphi \) formula holds. - 19.
The transitivity is hard, you need to enrich the two bisimulations a bit in connection with the middle model when proving it, see [40].

- 20.
See [43] for comparisons with other equivalent systems in the literature.

- 21.
Here we can also see the parallel of deduction and interrogation that [64] discussed.

- 22.
For some equivalent proof systems in the literature, see the survey and comparisons in [43].

- 23.
As we mentioned earlier, knowing the value can be seen as knowing the answer to a concealed question, see [4] and references therein for some recent discussions.

- 24.
Two people S and P are told respectively the sum and product of two natural numbers which are known to be below 100. The following conversation happens: P says: “I do not know the numbers.” S says: “I knew you didn’t.” P says: “I now know the numbers.” S says: “I now also know it.”.

- 25.
On the other hand, replacing \(\textit{Kv}\) with the knowing whether operator results in a consistent formula.

- 26.
[87] gave the following two extra introspection axioms on top of \(\mathbb {S}5\) to capture this announcement-free fragment without a proof: \(\textit{Kv}_ic\rightarrow \textit{K}_i\textit{Kv}_i c\) and \(\lnot \textit{Kv}_i c\rightarrow \textit{K}_i\lnot \textit{Kv}_i c\). Our later language will supersede this simple language.

- 27.
For example, I know that I have hands given that I am not a brain in a vat.

- 28.
Uniform substitution does not work for these new schemas.

- 29.
The decidability of \(\mathbf{ELKv }^r\) over epistemic models was shown by [102].

- 30.
- 31.
Instead of the ternary relation, it seems also natural to introduce an anti-equivalence relation \(R^c\) such that \(sR^ct\) intuitively means that

*s*and*t*do not agree on the value of*c*. However, this approach faces troubles due to the limited expressive power of the modal language, see [52] for a detailed discussion. - 32.
Clearly the corresponding models also satisfy more properties, such as \(sR_{i}^{c}uv\) only if \(v\not =u\). However, (1)-(3) are enough to keep the logic intact, see [52] for details.

- 33.
Note that here the maximal consistent sets are enough to build the canonical model due to the change of models, compared to canonical model for \(\mathbf{ELKv }^r\).

- 34.
Due to \(\texttt {SYM}\), we only need \(\texttt {DISTKv}^b\) and \(\texttt {NECKv}^b\) w.r.t. the first argument.

- 35.
A similar operator with propositional arguments was proposed by [51] in the setting of knowing whether, which can express that given the truth values of \(\varphi _1,\dots ,\varphi _n\) the agent

*i*knows whether \(\varphi \). - 36.
See [95] for a more detailed survey.

- 37.
See the collection of papers on the topic at philpaper edited by by John Bengson: http://philpapers.org/browse/knowledge-how.

- 38.
Such examples motivated intellectualists to propose an account other than treating knowledge-how simply as ability. A notable approach proposed in [90] breaks down “knowing how to

*F*” into: “There is a way such that I know it is a way to do F, and I entertain it in a*practical mode of presentation*.” Note that it essentially has the familiar shape \(\exists x\textit{K}\varphi (x)\), which also inspired the semi-formal treatment in [77]. - 39.
- 40.
We can derive \({\textit{U}}p\rightarrow {\textit{U}}{\textit{U}}p\) and \(\lnot {\textit{U}}p\rightarrow U\lnot {\textit{U}}p\) [95].

- 41.
For each maximal consistent set we build a canonical model [95].

- 42.
Recall that \(\lnot \textit{Kv}_i c\) is true if there are two states which disagree on

*c*. - 43.
In Hintikka’s terms, maybe it can be called the 1.5th generation of epistemic logics, since it is not as general as Hintikka’s idea of the second generation epistemic logics.

## References

Ågotnes T, Goranko V, Jamroga W, Wooldridge M (2015) Knowledge and ability. In: van Ditmarsch H, Halpern J, van der Hoek W, Kooi B (eds) Handbook of Epistemic Logic, College Publications, Chap 11, pp 543–589

Aloni M (2001) Quantification under conceptual covers. PhD thesis, University of Amsterdam

Aloni M (2016) Knowing-who in quantified epistemic logic. In: Jaakko Hintikka on knowledge and game theoretical semantics, Springer

Aloni M, Roelofsen F (2011) Interpreting concealed questions. Linguist Philos 34(5):443–478

Aloni M, Égré P, de Jager T (2013) Knowing whether A or B. Synthese 190(14):2595–2621

Aumann R (1989) Notes on interactive epistemology. In: Cowles foundation for research in economics working paper

Baltag A (2016) To know is to know the value of a variable. In: Advances in modal logic, vol 11, pp 135–155

Belardinelli F, van der Hoek W (2015) Epistemic quantified boolean logic: Expressiveness and completeness results. In: Proceedings of IJCAI ’15, AAAI Press, pp 2748–2754

Belardinelli F, van der Hoek W (2016) A semantical analysis of second-order propositional modal logic. In: Proceedings of AAAI’16, pp 886–892

Belardinelli F, Lomuscio A (2009) Quantified epistemic logics for reasoning about knowledge in multi-agent systems. Artif Intell 173(9–10):982–1013

Belardinelli F, Lomuscio A (2011) First-order linear-time epistemic logic with group knowledge: An axiomatisation of the monodic fragment. Fundamenta Informaticae 106(2–4):175–190

Belardinelli F, Lomuscio A (2012) Interactions between knowledge and time in a first-order logic for multi-agent systems: completeness results. J Artific Intell Res 45:1–45

Blackburn P, de Rijke M, Venema Y (2002) Modal logic. Cambridge University Press

Boer SE, Lycan WG (2003) Knowing who. The MIT Press

Bojańczyk M (2013) Modelling infinite structures with atoms. In: Libkin L, Kohlenbach U, de Queiroz R (eds) Proceedings of WoLLIC’13, Springer, pp 13–28

Bojańczyk M, David C, Muscholl A, Schwentick T, Segoufin L (2011) Two-variable logic on data words. ACM Trans Computat Logic 12(4):27:1–27:26

Braüner T, Ghilardi S (2007) First-order modal logic. In: Blackburn P, van Benthem J, Wolter F (eds) Handbook of modal logic, pp 549–620

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

Chaum D (1988) The dining cryptographers problem: unconditional sender and recipient untraceability. J Cryptol 1(1):65–75

Ciardelli I (2014) Modalities in the realm of questions: axiomatizing inquisitive epistemic logic. In: Advances in modal logic, vol 10, pp 94–113

Ciardelli I (2016) Questions in logic. PhD thesis, University of Amsterdam

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

Ciardelli I, Groenendijk J, Roelofsen F (2013) Inquisitive semantics: a new notion of meaning. Lang Linguist Comp 7(9):459–476

Cohen M, Dam M (2007) A complete axiomatization of knowledge and cryptography. In: Proceedings of LICS ’07, IEEE Computer Society, pp 77–88

Corsi G (2002) A unified completeness theorem for quantified modal logics. J Symbol Logic 67(4):1483–1510

Corsi G, Orlandelli E (2013) Free quantified epistemic logics. Studia Logica 101(6):1159–1183

Corsi G, Tassi G (2014) A new approach to epistemic logic. In: Logic, reasoning, and rationality, Springer, pp 25–41

Cresswell MJ (1988) Necessity and contingency. Studia Logica 47(2):145–149

Demri S (1997) A completeness proof for a logic with an alternative necessity operator. Studia Logica 58(1):99–112

Ding Y (2015) Axiomatization and complexity of modal logic with knowing-what operator on model class K, http://www.voidprove.com/research.html, unpublished manuscript

van Ditmarsch H (2007) Comments to ’logics of public communications’. Synthese 158(2):181–187

van Ditmarsch H, Fan J (2016) Propositional quantification in logics of contingency. J Appl Non-Classical Logics 26(1):81–102

van Ditmarsch H, van der Hoek W, Kooi B (2007) Dynamic epistemic logic. Springer

van Ditmarsch H, van der Hoek W, Illiev P (2012a) Everything is knowable how to get to know whether a proposition is true. Theoria 78(2)

van Ditmarsch H, van der Hoek W, Kooi B (2012b) Local properties in modal logic. Artific Intell 187–188:133–155

van Ditmarsch H, Fan J, van der Hoek W, Iliev P (2014) Some exponential lower bounds on formula-size in modal logic. In: Advances in modal logic, vol 10, pp 139–157

van Ditmarsch H, Halpern J, van der Hoek W, Kooi B (eds) (2015) Handbook of epistemic logic. College Publications

Egré P (2008) Question-embedding and factivity. Grazer Philos Studien 77(1):85–125

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

Fan J (2015) Logical studies for non-contingency operator. PhD thesis, Peking University, (in Chinese)

Fan J, van Ditmarsch H (2015) Neighborhood contingency logic. In: Proceedings of ICLA’15, pp 88–99

Fan J, Wang Y, van Ditmarsch H (2014) Almost neccessary. In: Advances in modal logic, vol 10, pp 178–196

Fan J, Wang Y, van Ditmarsch H (2015) Contingency and knowing whether. Rev Symbol Logic 8:75–107

Fine K (1970) Propositional quantifiers in modal logic. Theoria 36(3):336–346

Fitch F (1963) A logical analysis of some value concepts. J Symbol Logic 28(2):135–142

Fitting M, Mendelsohn RL (1998) First-Order Modal Logic. Springer

van Fraassen B (1980) The scientific image. Oxford University Press, Oxford

Gattinger M, van Eijck J, Wang Y (2016) Knowing value and public inspection, unpublished manuscript

Gochet P (2013) An open problem in the logic of knowing how. In: Hintikka J (ed) Open Problems in Epistemology. The Philosophical Society of Finland

Gochet P, Gribomont P (2006) Epistemic logic. In: Gabbay DM, Woods J (eds) Handbook of the history of logic, vol 7

Goranko V, Kuusisto A (2016) Logics for propositional determinacy and independence. https://arxiv.org/abs/1609.07398

Gu T, Wang Y (2016) Knowing value logic as a normal modal logic. In: Advances in modal logic, vol 11, pp 362–381

Harrah D (2002) The logic of questions. In: Gabbay D (ed) Handbook of philosophical logic, vol 8

Hart S, Heifetz A, Samet D (1996) "knowing whether", "knowing that", and the cardinality of state spaces. J Econom Theor 70(1):249–256

Heim I (1979) Concealed questions. In: Bäuerle R, Egli U, von Stechow A (eds) Semantics from different points of view, pp 51–60

Herzig A (2015) Logics of knowledge and action: critical analysis and challenges. Autonom Agents Multi-Agent Syst 29(5):719–753

Herzig A, Lorini E, Maffre F (2015) A poor man’s epistemic logic based on propositional assignment and higher-order observation. In: Proceedings of LORI-V, pp 156–168

Hintikka J (1962) Knowledge and belief: an introduction to the logic of the two notions. Cornell University Press, Ithaca N.Y

Hintikka J (1989a) On sense, reference, and the objects of knowledge. In: The logic of epistemology and the epistemology of logic: selected essays, Springer, pp 45–61

Hintikka J (1989b) Reasoning about knowledge in philosophy: The paradigm of epistemic logic. In: The logic of epistemology and the epistemology of logic: selected essays, Springer, pp 17–35

Hintikka J (1996) Knowledge acknowledged: knowledge of propositions vs. knowledge of objects. Philos Phenomenol Res 56(2):251–275

Hintikka J (1999) What is the logic of experimental inquiry? In: Inquiry as inquiry: a logic of scientific discovery, Springer, pp 143–160

Hintikka J (2003) A second generation epistemic logic and its general significance. In: Jørgensen KF, Pedersen SA (eds) Hendricks VF. Knowledge Contributors, Springer, pp 33–55

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

Hintikka J, Halonen I (1995) Semantics and pragmatics for why-questions. J Philos 92(12):636–657

Hintikka J, Sandu G (1989) Informational independence as a semantical phenomenon. In: Fenstad JE, Frolov IT, Hilpinen R (eds) Logic, methodology and philosophy of science 8, Elsevier, pp 571–589

Hintikka J, Symons J (2003) Systems of visual identification in neuroscience: lessons from epistemic logic. Philos Sci 70(1):89–104

Hodkinson IM (2002) Monodic packed fragment with equality is decidable. Studia Logica 72(2):185–197

Hodkinson IM, Wolter F, Zakharyaschev M (2000) Decidable fragment of first-order temporal logics. Ann Pure Appl Logic 106(1–3):85–134

Hodkinson IM, Wolter F, Zakharyaschev M (2002) Decidable and undecidable fragments of first-order branching temporal logics. In: Proceedings of LICS’02, pp 393–402

van der Hoek W, Lomuscio A (2004) A logic for ignorance. Electron Notes Theor Comput Sci 85(2):117–133

Holliday WH, Perry J (2014) Roles, rigidity, and quantification in epistemic logic. In: Smets S (ed) Baltag A. Johan van Benthem on Logic and Information Dynamics, Springer, pp 591–629

Humberstone L (1995) The logic of non-contingency. Notre Dame J Form Logic 36(2):214–229

Kaneko M, Nagashima T (1996) Game logic and its applications. Studia Logica 57(2/3):325–354

Khan MA, Banerjee M (2010) A logic for multiple-source approximation systems with distributed knowledge base. J Philos Logic 40(5):663–692

Kuhn S (1995) Minimal non-contingency logic. Notre Dame J Form Logic 36(2):230–234

Lau T, Wang Y (2016) Knowing your ability. The Philosophical Forum 47(3–4):415–423

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

Liu F, Wang Y (2013) Reasoning about agent types and the hardest logic puzzle ever. Minds Mach 23(1):123–161

Lomuscio A, Ryan M (1999) A spectrum of modes of knowledge sharing between agents. In: Proceedings of ATAL’99, pp 13–26

McCarthy J (1979) First-Order theories of individual concepts and propositions. Mach Intell 9:129–147

Montgomery H, Routley R (1966) Contingency and non-contingency bases for normal modal logics. Logique et Anal 9:318–328

Moore RC (1977) Reasoning about knowledge and action. In: Proceedings of IJCAI’77, pp 223–227

Petrick RPA, Bacchus F (2004a) Extending the knowledge-based approach to planning with incomplete information and sensing. In: Zilberstein S, Koehler J, Koenig S (eds) Proceedings of ICAPS’04), AAAI Press, pp 2–11

Petrick RPA, Bacchus F (2004b) PKS: Knowledge-based planning with incomplete information and sensing. In: Proceedings of ICAPS’04

Pizzi C (2007) Necessity and relative contingency. Studia Logica 85(3):395–410

Plaza JA (1989) Logics of public communications. In: Emrich ML, Pfeifer MS, Hadzikadic M, Ras ZW (eds) Proceedings of the 4th international symposium on methodologies for intelligent systems, pp 201–216

Ryle G (1949) The concept of mind. Penguin

Stanley J (2011) Know how. Oxford University Press

Stanley J, Williamson T (2001) Knowing how. J Philos 98:411–444

Sturm H, Wolter F, Zakharyaschev M (2000) Monodic epistemic predicate logic. In: Proceedings of JELIA’00, pp 329–344

Väänänen J (2007) Dependence logic: a new approach to independence friendly logic. Cambridge University Press

Van Ditmarsch H, Herzig A, De Lima T (2011) From situation calculus to dynamic epistemic logic. J Logic Comput pp 179–204

Von Wright GH (1951) An essay in modal logic. North Holland, Amsterdam

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

Wang Y (2015b) Representing imperfect information of procedures with hyper models. In: Proceedings of ICLA’15, pp 218–231

Wang Y (2017) A new modal framework for epistemic logic. In: Proceedings of TARK’17: 493–512

Wang Y (2017) A logic of goal-directed knowing how. Synthese (forthcoming)

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, Fan J (2014) Conditionally knowing what. In: Proceedings of AiML Vol.10

Wolter F (2000) First order common knowledge logics. Studia Logica 65(2):249–271

Xiong S (2014) Decidability of \({\mathbf{ELKv}}^{\bf r}\). Bachelor’s thesis, Peking University (in Chinese)

Xu C (2016) A logic of knowing why. Master’s thesis, Peking University, (in Chinese)

Li Y, Yu Q, Wang Y (2017) More for free: a dynamic epistemic framework for conformant planning over transition systems. J Logic Comput (forthcoming)

Zolin E (1999) Completeness and definability in the logic of noncontingency. Notre Dame J Form Logic 40(4):533–547

Zolin E (2001) Infinitary expressibility of necessity in terms of contingency. In: Striegnitz K (ed) Proceedings of the sixth ESSLLI student session, pp 325–334

## Acknowledgements

The author acknowledges the support from the National Program for Special Support of Eminent Professionals and NSSF key projects 12&ZD119. The author is grateful to Hans van Ditmarsch for his very detailed comments on an early version of this paper. The author also thanks the anonymous reviewer who gave many constructive suggestions including the observation in footnote 14.

## Author information

### Authors and Affiliations

### Corresponding author

## Editor information

### Editors and Affiliations

## Rights and permissions

## Copyright information

© 2018 Springer International Publishing AG

## About this chapter

### Cite this chapter

Wang, Y. (2018). Beyond Knowing That: A New Generation of Epistemic Logics. In: van Ditmarsch, H., Sandu, G. (eds) Jaakko Hintikka on Knowledge and Game-Theoretical Semantics. Outstanding Contributions to Logic, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-62864-6_21

### Download citation

DOI: https://doi.org/10.1007/978-3-319-62864-6_21

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-319-62863-9

Online ISBN: 978-3-319-62864-6

eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)