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Epistemic modal logic is a subfield ofmodal logic that is concerned with reasoning aboutknowledge. Whileepistemology has a long philosophical tradition dating back toAncient Greece, epistemic logic is a much more recent development with applications in many fields, includingphilosophy,theoretical computer science,artificial intelligence,economics, andlinguistics. While philosophers sinceAristotle have discussed modal logic, andMedieval philosophers such asAvicenna,Ockham, andDuns Scotus developed many of their observations, it wasC. 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 ofSaul Kripke.

Many papers were written in the 1950s that spoke of a logic of knowledge in passing, but the Finnish philosopherG. H. von Wright's 1951 paper titledAn Essay in Modal Logic is seen as a founding document. It was not until 1962 that another Finn,Jaakko Hintikka, would writeKnowledge and Belief, the first book-length work to suggest using modalities to capture the semantics of knowledge rather than thealethic 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 fromdynamic logic to createdynamic epistemic logic, which can be used to specify and reason about information change and exchange of information inmulti-agent systems. The seminal works in this field are by Plaza,Van Benthem, and Baltag, Moss, and Solecki.

Most attempts at modeling knowledge have been based on thepossible 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, theevent-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 areKripke semantics, while the event-based approach employs the relatedAumann structures based onset theory.
• 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 asgame theory andmathematical economics. In the logic-based approach, a syntax and semantics have been built using the language of modal logic, which we will now describe.

The basicmodal operator of epistemic logic, usually writtenK, 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 (${\displaystyle {\mathit{K}}_1}$,${\displaystyle {\mathit{K}}_2}$, etc.) to indicate which agent one is talking about. So${\displaystyle {\mathit{K}}_a\phi }$ can be read as "Agent${\displaystyle a}$ knows that${\displaystyle \phi }$." Thus, epistemic logic can be an example ofmultimodal logic applied forknowledge representation.^[1] The dual ofK, which would be in the same relationship toK as${\displaystyle ◊}$ is to${\displaystyle ◻}$, has no specific symbol, but can be represented by${\displaystyle \mathrm{\neg }{K}_a\mathrm{\neg }\phi }$, which can be read as "${\displaystyle a}$ does not know that not${\displaystyle \phi }$" or "It is consistent with${\displaystyle a}$'s knowledge that${\displaystyle \phi }$ is possible". The statement "${\displaystyle a}$ does not know whether or not${\displaystyle \phi }$" can be expressed as${\displaystyle \mathrm{\neg }{K}_a\phi \wedge \mathrm{\neg }{K}_a\mathrm{\neg }\phi }$.

In order to accommodate notions ofcommon knowledge (e.g. in theMuddy Children Puzzle) anddistributed knowledge, three other modal operators can be added to the language. These are${\displaystyle {\mathit{E}}_{\mathit{G}}}$, which reads "every agent in group G knows" (mutual knowledge);${\displaystyle {\mathit{C}}_{\mathit{G}}}$, which reads "it is common knowledge to every agent in G"; and${\displaystyle {\mathit{D}}_{\mathit{G}}}$, which reads "it is distributed knowledge to the whole group G." If${\displaystyle \phi }$ is a formula of our language, then so are${\displaystyle {\mathit{E}}_G\phi }$,${\displaystyle {\mathit{C}}_G\phi }$, and${\displaystyle {\mathit{D}}_G\phi }$. Just as the subscript after${\displaystyle \mathit{K}}$ can be omitted when there is only one agent, the subscript after the modal operators${\displaystyle \mathit{E}}$,${\displaystyle \mathit{C}}$, and${\displaystyle \mathit{D}}$ can be omitted when the group is the set of all agents.

As 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${\displaystyle \mathcal{M}=⟨S,\pi ,{\mathcal{K}}_1,\dots ,{\mathcal{K}}_n⟩}$ forn agents over${\displaystyle \mathrm{\Phi }}$, the set of all primitive propositions, is an${\displaystyle (n+2)}$-tuple, where${\displaystyle S}$ is a nonempty set ofstates orpossible worlds,${\displaystyle \pi }$ is aninterpretation, which associates with each state${\displaystyle s\in S}$ a truth assignment to the primitive propositions in${\displaystyle \mathrm{\Phi }}$, and${\displaystyle {\mathcal{K}}_1,...,{\mathcal{K}}_n}$ arebinary relations on${\displaystyle S}$ forn numbers of agents. It is important here not to confuse${\displaystyle K_i}$, our modal operator, and${\displaystyle {\mathcal{K}}_i}$, our accessibility relation.

The truth assignment tells us whether or not a proposition${\displaystyle p}$ is true or false in a certain state. So${\displaystyle \pi (s)(p)}$ tells us whether${\displaystyle p}$ is true in state${\displaystyle s}$ in model${\displaystyle \mathcal{M}}$. 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${\displaystyle \phi }$ is true at a certain world, one writes${\displaystyle (\mathcal{M},s)\models \phi }$, normally read as "${\displaystyle \phi }$ is true at${\displaystyle (\mathcal{M},s)}$," or "${\displaystyle (\mathcal{M},s)}$ satisfies${\displaystyle \phi }$".

It is useful to think of our binary relation${\displaystyle {\mathcal{K}}_i}$ as apossibility relation, because it is meant to capture what worlds or states agenti considers to be possible; In other words,${\displaystyle w{\mathcal{K}}_{i}v}$ if and only if${\displaystyle \mathrm{\forall }\phi [(w\models K_i\phi )⟹(v\models \phi )]}$, and such${\displaystyle v}$'s are called epistemic alternatives for agenti. In idealized accounts of knowledge (e.g., describing the epistemic status of perfect reasoners with infinite memory capacity), it makes sense for${\displaystyle {\mathcal{K}}_i}$ to be anequivalence 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 isreflexive,symmetric, andtransitive. 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.

Assuming that${\displaystyle {\mathcal{K}}_i}$ 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.

This axiom is traditionally known asK. In epistemic terms, it states that if an agent knows${\displaystyle \phi }$ and knows that${\displaystyle \phi ⟹\psi }$, then the agent must also know${\displaystyle \psi }$. So,

${\displaystyle (K_i\phi \wedge K_i(\phi ⟹\psi ))⟹K_i\psi }$

This axiom is valid on any frame inrelational semantics. This axiom logically establishesmodus ponens as arule of inference for every epistemically possible world.

Another property we can derive is that if${\displaystyle \varphi }$ is valid (i.e. atautology), then${\displaystyle K_i\varphi }$. This does not mean that if${\displaystyle \varphi }$ is true, then agent i knows${\displaystyle \varphi }$. What it means is that if${\displaystyle \varphi }$ is true in every world that an agent considers to be a possible world, then the agent must know${\displaystyle \varphi }$ at every possible world. This principle is traditionally calledN (Necessitation rule).

${\displaystyle \text{if }\models \phi \text{ then }M\models K_i\phi .}$

This rule always preserves truth inrelational semantics.

This axiom is also known asT. 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 toknow a false statement.

${\displaystyle K_i\phi ⟹\phi }$

This axiom can also be expressed in itscontraposition as agents cannotknow a false statement:

${\displaystyle \phi ⟹\mathrm{\neg }{K}_i\mathrm{\neg }\phi }$

This axiom isvalid on anyreflexiveframe.

This property and the next state that an agent has introspection about its own knowledge, and are traditionally known as4 and5, respectively. The Positive Introspection Axiom, also known as the KK Axiom, says specifically that agentsknow that they know what they know. This axiom may seem less obvious than the ones listed previously, andTimothy Williamson has argued against its inclusion forcefully in his book,Knowledge and Its Limits.

${\displaystyle K_i\phi ⟹K_i{K}_i\phi }$

Equivalently, this modal axiom4 says that agentsdo not knowwhat they do not know that they know

${\displaystyle \mathrm{\neg }{K}_i{K}_i\phi ⟹\mathrm{\neg }{K}_i\phi }$

This axiom isvalid on anytransitive frame.

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

${\displaystyle \mathrm{\neg }{K}_i\phi ⟹K_i\mathrm{\neg }{K}_i\phi }$

Or, equivalently, this modal axiom5 says that agentsknowwhat they do not know that they do not know

${\displaystyle \mathrm{\neg }{K}_i\mathrm{\neg }{K}_i\phi ⟹K_i\phi }$

This axiom is valid on anyEuclidean frame.

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 ofK,T,4,5, and the Knowledge Generalization Rule, is primarily known asS5. This is why the properties of knowledge described above are often called the S5 Properties. However, it can be proven that modal axiomB is a theorem in S5 (viz.${\displaystyle S5⊢\mathbf{B}}$), which says thatwhat an agent does not know that they do not know is true:${\displaystyle \mathrm{\neg }{K}_i\mathrm{\neg }{K}_i\phi ⟹\phi }$. The modal axiomB is true on any symmetric frame, but is very counterintuitive in epistemic logic: How canthe ignorance on one's own ignorance imply truth? It is therefore debatable whether S4 describes epistemic logic better, rather than S5.

Epistemic logic also deals with belief, not just knowledge. The basic modal operator is usually writtenB instead ofK. 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 calledD:

${\displaystyle \mathrm{\neg }{B}_i\mathrm{\perp }}$

which states that the agent does not believe a contradiction, or that which is false. WhenD replacesT in S5, the resulting system is known as KD45. This results in different properties for${\displaystyle {\mathcal{K}}_i}$ 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 calleddoxastic logic.

When there are multiple agents in thedomain of discourse where each agenti corresponds to a separate epistemic modal operator${\displaystyle K_i}$, in addition to the axiom schemata for each individual agent listed above to describe the rationality of each agent, it is usually also assumed that the rationality of each agent iscommon knowledge.

If we take the possible worlds approach to knowledge, it follows that our epistemic agenta knows all thelogical consequences of their beliefs (known as logical omniscience^[2]). If${\displaystyle Q}$ is a logical consequence of${\displaystyle P}$, then there is no possible world where${\displaystyle P}$ is true but${\displaystyle Q}$ is not. So ifa knows that${\displaystyle P}$ is true, it follows that all of the logical consequences of${\displaystyle P}$ are true of all of the possible worlds compatible witha's beliefs. Therefore,a knows${\displaystyle Q}$. It is not epistemically possible fora that not-${\displaystyle Q}$ given his knowledge that${\displaystyle P}$. This consideration was a part of what ledRobert Stalnaker to developtwo-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.^[3]

Even when we ignore possible world semantics and stick to axiomatic systems, this peculiar feature holds. WithK andN (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${\displaystyle Q}$ is a logical consequence of${\displaystyle P}$ (i.e. we have thetautology${\displaystyle \models (P\to Q)}$), then we can derive${\displaystyle K_a(P\to Q)}$ withN, and using aconditional proof with the axiomK, we can then derive${\displaystyle K_{a}P\to K_{a}Q}$ withK. When we translate this into epistemic terms, this says that if${\displaystyle Q}$ is a logical consequence of${\displaystyle P}$, thena knows that it is, and ifa knows${\displaystyle P}$,a knows${\displaystyle Q}$. 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, ifa knows that prime numbers are divisible only by themselves and the number one, thena 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, whena knows the definition of a prime number,a knows that this number is prime. This generalizes to any provable theorem in any axiomatic theory (i.e. ifa knows all the axioms in a theory, thena knows all the provable theorems in that theory). It should be clear at this point thata is not human (otherwise there would not be any unsolved conjectures in mathematics, likeP versus NP problem orGoldbach's conjecture). This shows that epistemic modal logic is an idealized account of knowledge, and explains objective, rather than subjective knowledge (if anything).^[4]

Inphilosophical logic, themasked-man fallacy (also known as theintensional fallacy or epistemic fallacy) is committed when one makes an illicit use ofLeibniz's law in an argument. The fallacy is "epistemic" because it posits an immediate identity between a subject's knowledge of an object with the object itself, failing to recognize that Leibniz's Law is not capable of accounting forintensional contexts.

The name of the fallacy comes from the example:

• Premise 1: I know who Bob is.
• Premise 2: I do not know who the masked man is
• Conclusion: Therefore, Bob is not the masked man.

Thepremises may be true and the conclusion false if Bob is the masked man and the speaker does not know that. Thus the argument is a fallacious one.

In symbolic form, the above arguments are

• Premise 1: I know who X is.
• Premise 2: I do not know who Y is.
• Conclusion: Therefore, X is not Y.

Note, however, that thissyllogism happens in the reasoning by the speaker "I"; Therefore, in the formalmodal logic form, it'll be

• Premise 1: The speakerbelieves he knows who X is.
• Premise 2: The speaker believes he does not know who Y is.
• Conclusion: Therefore, the speaker believes X is not Y.

Premise 1${\displaystyle {\mathcal{B}}_s\mathrm{\forall }t(t=X\to K_s(t=X))}$ is a very strong one, as it islogically equivalent to${\displaystyle {\mathcal{B}}_{\mathcal{s}}\mathrm{\forall }t(\mathrm{\neg }{K}_s(t=X)\to t\ne X)}$. It is very likely that this is afalse belief:${\displaystyle \mathrm{\forall }t(\mathrm{\neg }{K}_s(t=X)\to t\ne X)}$ is likely a false proposition, as the ignorance on the proposition${\displaystyle t=X}$ does not imply the negation of it is true.

Another example:

• Premise 1: Lois Lane thinks Superman can fly.
• Premise 2: Lois Lane thinks Clark Kent cannot fly.
• Conclusion: Therefore, Superman and Clark Kent are not the same person.

Expressed in doxastic logic, the above syllogism is:

• Premise 1:${\displaystyle {\mathcal{B}}_{\text{Lois}}{\text{Fly}}_{\text{(Superman)}}}$
• Premise 2:${\displaystyle {\mathcal{B}}_{\text{Lois}}\mathrm{\neg }{\text{Fly}}_{\text{(Clark)}}}$
• Conclusion:${\displaystyle \text{Superman}\ne \text{Clark}}$

The above reasoning is invalid (not truth-preserving). The valid conclusion to be drawn is${\displaystyle {\mathcal{B}}_{\text{Lois}}(\text{Superman}\ne \text{Clark})}$.

• Epistemic closure
• Epistemology
• Dynamic epistemic logic
• Logic in computer science
• Philosophical Explanations

• 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."ArtificialIntelligence, 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.{{cite book}}:ISBN / Date incompatibility (help).
• 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.

• "Dynamic Epistemic Logic".Internet Encyclopedia of Philosophy.
• Hendricks, Vincent; Symons, John."Epistemic Logic". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
• Garson, James."Modal logic". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
• Vanderschraaf, Peter."Common Knowledge". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
• Epistemic modal logic atPhilPapers
• "Epistemic modal logic"—Ho Ngoc Duc.

• Modal logic
• Artificial intelligence
• Formal epistemology
• Epistemic logic

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