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Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy, theoretical computer science, artificial intelligence, economics and linguistics. While philosophers since Aristotle have discussed modal logic, and Medieval philosophers such as Avicenna, Ockham, and Duns Scotus developed many of their observations, it was C. I. Lewis who created the first symbolic and systematic approach to the topic, in 1912. It continued to mature as a field, reaching its modern form in 1963 with the work of Kripke.

Historical development

Many papers were written in the 1950s that spoke of a logic of knowledge in passing, but it was Finnish philosopher von Wright's paper An Essay in Modal Logic from 1951 that is seen as a founding document. It was not until 1962 that another Finn, Hintikka, would write Knowledge and Belief, the first book-length work to suggest using modalities to capture the semantics of knowledge rather than the alethic statements typically discussed in modal logic. This work laid much of the groundwork for the subject, but a great deal of research has taken place since that time. For example, epistemic logic has been combined recently with some ideas from dynamic logic to create dynamic epistemic logic, which can be used to specify and reason about information change and exchange of information in multi-agent systems. The seminal works in this field are by Plaza, Van Benthem, and Baltag, Moss, and Solecki.

Standard possible worlds model

Most attempts at modeling knowledge have been based on the possible worlds model. In order to do this, we must divide the set of possible worlds between those that are compatible with an agent's knowledge, and those that are not. This generally conforms with common usage. If I know that it is either Friday or Saturday, then I know for sure that it is not Thursday. There is no possible world compatible with my knowledge where it is Thursday, since in all these worlds it is either Friday or Saturday. While we will primarily be discussing the logic-based approach to accomplishing this task, it is worthwhile to mention here the other primary method in use, the event-based approach. In this particular usage, events are sets of possible worlds, and knowledge is an operator on events. Though the strategies are closely related, there are two important distinctions to be made between them:

• The underlying mathematical model of the logic-based approach are Kripke semantics, while the event-based approach employs the related Aumann structures.


• In the event-based approach logical formulas are done away with completely, while the logic-based approach uses the system of modal logic.

Typically, the logic-based approach has been used in fields such as philosophy, logic and AI, while the event-based approach is more often used in fields such as game theory and mathematical economics. In the logic-based approach, a syntax and semantics have been built using the language of modal logic, which we will now describe.

Syntax

The basic modal operator of epistemic logic, usually written K, can be read as "it is known that," "it is epistemically necessary that," or "it is inconsistent with what is known that not." If there is more than one agent whose knowledge is to be represented, subscripts can be attached to the operator (K1displaystyle mathit K_1, K2displaystyle mathit K_2, etc.) to indicate which agent one is talking about. So Kaφdisplaystyle mathit K_avarphi can be read as "Agent adisplaystyle a knows that φdisplaystyle varphi ." Thus, epistemic logic can be an example of multimodal logic applied for knowledge representation.^[1] The dual of K, which would be in the same relationship to K as ◊displaystyle Diamond is to ◻displaystyle Box , has no specific symbol, but can be represented by ¬Ka¬φdisplaystyle neg K_aneg varphi , which can be read as "adisplaystyle a does not know that not φdisplaystyle varphi " or "It is consistent with adisplaystyle a's knowledge that φdisplaystyle varphi is possible". The statement "adisplaystyle a does not know whether or not φdisplaystyle varphi " can be expressed as ¬Kaφ∧¬Ka¬φdisplaystyle neg K_avarphi land neg K_aneg varphi .

In order to accommodate notions of common knowledge and distributed knowledge, three other modal operators can be added to the language. These are EGdisplaystyle mathit E_mathit G, which reads "every agent in group G knows;" CGdisplaystyle mathit C_mathit G, which reads "it is common knowledge to every agent in G;" and DGdisplaystyle mathit D_mathit G, which reads "it is distributed knowledge to every agent in G." If φdisplaystyle varphi is a formula of our language, then so are EGφdisplaystyle mathit E_Gvarphi , CGφdisplaystyle mathit C_Gvarphi , and DGφdisplaystyle mathit D_Gvarphi . Just as the subscript after Kdisplaystyle mathit K can be omitted when there is only one agent, the subscript after the modal operators Edisplaystyle mathit E, Cdisplaystyle mathit C, and Ddisplaystyle mathit D can be omitted when the group is the set of all agents.

Semantics

As we mentioned above, the logic-based approach is built upon the possible worlds model, the semantics of which are often given definite form in Kripke structures, also known as Kripke models. A Kripke structure M for n agents over Φdisplaystyle Phi is a (n+2)-tuple (S,π,K1,...,Kn)displaystyle (S,pi ,mathcal K_1,...,mathcal K_n), where S is a nonempty set of states or possible worlds, πdisplaystyle pi is an interpretation, which associates with each state in S a truth assignment to the primitive propositions in Φdisplaystyle Phi , and K1,...,Kndisplaystyle mathcal K_1,...,mathcal K_n are binary relations on S for n numbers of agents. It is important here not to confuse Kidisplaystyle K_i, our modal operator, and Kidisplaystyle mathcal K_i, our accessibility relation.

The truth assignment tells us whether or not a proposition p is true or false in a certain state. So π(s)(p)displaystyle pi (s)(p) tells us whether p is true in state s in model Mdisplaystyle mathcal M. Truth depends not only on the structure, but on the current world as well. Just because something is true in one world does not mean it is true in another. To state that a formula φdisplaystyle varphi is true at a certain world, one writes (M,s)⊨φdisplaystyle (M,s)models varphi , normally read as "φdisplaystyle varphi is true at (M,s)," or "(M,s) satisfies φdisplaystyle varphi ".

It is useful to think of our binary relation Kidisplaystyle mathcal K_i as a possibility relation, because it is meant to capture what worlds or states agent i considers to be possible. In idealized accounts of knowledge (e.g., describing the epistemic status of perfect reasoners with infinite memory capacity), it makes sense for Kidisplaystyle mathcal K_i to be an equivalence relation, since this is the strongest form and is the most appropriate for the greatest number of applications. An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The accessibility relation does not have to have these qualities; there are certainly other choices possible, such as those used when modeling belief rather than knowledge.

The properties of knowledge

Assuming that Kidisplaystyle mathcal K_i is an equivalence relation, and that the agents are perfect reasoners, a few properties of knowledge can be derived. The properties listed here are often known as the "S5 Properties," for reasons described in the Axiom Systems section below.

The distribution axiom

This axiom is traditionally known as K. In epistemic terms, it states that if an agent knows φdisplaystyle varphi and knows that φ⟹ψdisplaystyle varphi implies psi , then the agent must also know ψdisplaystyle ,psi . So,

(Kiφ∧Ki(φ⟹ψ))⟹Kiψdisplaystyle (K_ivarphi land K_i(varphi implies psi ))implies K_ipsi

This axiom is valid on any frame in relational semantics.

The knowledge generalization rule

Another property we can derive is that if ϕdisplaystyle phi is valid, then Kiϕdisplaystyle K_iphi . This does not mean that if ϕdisplaystyle phi is true, then agent i knows ϕdisplaystyle phi . What it means is that if ϕdisplaystyle phi is true in every world that an agent considers to be a possible world, then the agent must know ϕdisplaystyle phi at every possible world. This principle is traditionally called N.

if M⊨φ then M⊨Kiφ.displaystyle textif Mmodels varphi text then Mmodels K_ivarphi .,

This rule always preserves truth in relational semantics.

The knowledge or truth axiom

This axiom is also known as T. It says that if an agent knows facts, the facts must be true. This has often been taken as the major distinguishing feature between knowledge and belief. We can believe a statement to be true when it is false, but it would be impossible to know a false statement.

Kiφ⟹φdisplaystyle K_ivarphi implies varphi

This axiom is valid on any reflexive frame.

The positive introspection axiom

This property and the next state that an agent has introspection about its own knowledge, and are traditionally known as 4 and 5, respectively. The Positive Introspection Axiom, also known as the KK Axiom, says specifically that agents know that they know what they know. This axiom may seem less obvious than the ones listed previously, and Timothy Williamson has argued against its inclusion forcefully in his book, Knowledge and Its Limits.

Kiφ⟹KiKiφdisplaystyle K_ivarphi implies K_iK_ivarphi

This axiom is valid on any transitive frame.

The negative introspection axiom

The Negative Introspection Axiom says that agents know that they do not know what they do not know.

¬Kiφ⟹Ki¬Kiφdisplaystyle neg K_ivarphi implies K_ineg K_ivarphi

This axiom is valid on any Euclidean frame.

Axiom systems

Different modal logics can be derived from taking different subsets of these axioms, and these logics are normally named after the important axioms being employed. However, this is not always the case. KT45, the modal logic that results from the combining of K, T, 4, 5, and the Knowledge Generalization Rule, is primarily known as S5. This is why the properties of knowledge described above are often called the S5 Properties.

Epistemic logic also deals with belief, not just knowledge. The basic modal operator is usually written B instead of K. In this case though, the knowledge axiom no longer seems right—agents only sometimes believe the truth—so it is usually replaced with the Consistency Axiom, traditionally called D:

¬Bi⊥displaystyle neg B_ibot

which states that the agent does not believe a contradiction, or that which is false. When D replaces T in S5, the resulting system is known as KD45. This results in different properties for Kidisplaystyle mathcal K_i as well. For example, in a system where an agent "believes" something to be true, but it is not actually true, the accessibility relation would be non-reflexive. The logic of belief is called doxastic logic.

Problems with the possible world model and modal model of knowledge

The notion of knowledge discussed does not take into account computational constraints on inference. If we take the possible worlds approach to knowledge, it follows that our epistemic agent a knows all the logical consequences of his or her or its beliefs. If Qdisplaystyle Q is a logical consequence of Pdisplaystyle P, then there is no possible world where Pdisplaystyle P is true but Qdisplaystyle Q is not. So if a knows that Pdisplaystyle P, it follows that all of the logical consequences of Pdisplaystyle P are true of all of the possible worlds compatible with a 's beliefs. Therefore, a knows Qdisplaystyle Q. It is not epistemically possible for a that not-Qdisplaystyle Q given his knowledge that Pdisplaystyle P. This consideration was a part of what led Robert Stalnaker to develop two dimensionalism, which can arguably explain how we might not know all the logical consequences of our beliefs even if there are no worlds where the propositions we know come out true but their consequences false.^[2]

Even when we ignore possible world semantics and stick to axiomatic systems, this peculiar feature holds. With K and N (the Distribution Rule and the Knowledge Generalization Rule, respectively), which are axioms that are minimally true of all normal modal logics, we can prove that we know all the logical consequences of our beliefs. If Qdisplaystyle Q is a logical consequence of Pdisplaystyle P, then we can derive Ka(P→Q)displaystyle mathcal K_a(Prightarrow Q) with N and the conditional proof and then KaP→KaQdisplaystyle mathcal K_aPrightarrow mathcal K_aQ with K. When we translate this into epistemic terms, this says that if Qdisplaystyle Q is a logical consequence of Pdisplaystyle P, then a knows that it is, and if a knows Pdisplaystyle P, a knows Qdisplaystyle Q. That is to say, a knows all the logical consequences of every proposition. This is necessarily true of all classical modal logics. But then, for example, if a knows that prime numbers are divisible only by themselves and the number one, then a knows that 8683317618811886495518194401279999999 is prime (since this number is only divisible by itself and the number one). That is to say, under the modal interpretation of knowledge, when a knows the definition of a prime number, a knows that this number is prime. It should be clear at this point that a is not human. This shows that epistemic modal logic is an idealized account of knowledge, and explains objective, rather than subjective knowledge (if anything).^[3]

See also

• Common knowledge


• Epistemic closure


• Epistemology


• Logic in computer science


• Modal logic


• Philosophical Explanations


• Two-dimensionalism

Notes

References

• Anderson, A. and N. D. Belnap. Entailment: The Logic of Relevance and Necessity. Princeton: Princeton University Press, 1975. ASIN B001NNPJL8.


• Brown, Benjamin, Thoughts and Ways of Thinking: Source Theory and Its Applications. London: Ubiquity Press, 2017. [1].


• van Ditmarsch Hans, Halpern Joseph Y., van der Hoek Wiebe and Kooi Barteld (eds.), Handbook of Epistemic Logic, London: College Publications, 2015.


• 
  Fagin, Ronald; Halpern, Joseph; Moses, Yoram; Vardi, Moshe (2003). Reasoning about Knowledge. Cambridge: MIT Press. ISBN 978-0-262-56200-3.. A classic reference.


• Ronald Fagin, Joseph Halpern, Moshe Vardi. "A nonstandard approach to the logical omniscience problem." Artificial Intelligence, Volume 79, Number 2, 1995, p. 203-40.


• Hendricks, V.F. Mainstream and Formal Epistemology. New York: Cambridge University Press, 2007.


• 
  Hintikka, Jaakko (1962). Knowledge and Belief - An Introduction to the Logic of the Two Notions. Ithaca: Cornell University Press. ISBN 978-1-904987-08-6..


• Meyer, J-J C., 2001, "Epistemic Logic," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.


• Montague, R. "Universal Grammar". Theoretica, Volume 36, 1970, p. 373-398.


• 
  Rescher, Nicolas (2005). Epistemic Logic: A Survey Of the Logic Of Knowledge. University of Pittsburgh Press. ISBN 978-0-8229-4246-7..


• 
  Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7.. See Chapters 13 and 14; downloadable free online.

External links

• 
  "Dynamic Epistemic Logic". Internet Encyclopedia of Philosophy.
  


• 
  Hendricks, Vincent; Symons, John. "Epistemic Logic". In Zalta, Edward N. Stanford Encyclopedia of Philosophy.
  


• 
  Garson, James. "Modal logic". In Zalta, Edward N. Stanford Encyclopedia of Philosophy.
  


• 
  Vanderschraaf, Peter. "Common Knowledge". In Zalta, Edward N. Stanford Encyclopedia of Philosophy.
  


• 
  Epistemic modal logic at PhilPapers
  


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• "Epistemic modal logic"—Ho Ngoc Duc.