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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 the Finnish philosopher G. H. von Wright's 1951 paper titled An Essay in Modal Logic 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 based on set 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 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 (${\displaystyle {\mathit {K}}_{1}}$, ${\displaystyle {\mathit {K}}_{2}}$, etc.) to indicate which agent one is talking about. So ${\displaystyle {\mathit {K}}_{a}\varphi }$ can be read as "Agent ${\displaystyle a}$ knows that ${\displaystyle \varphi }$." 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 ${\displaystyle \neg K_{a}\neg \varphi }$, which can be read as "${\displaystyle a}$ does not know that not ${\displaystyle \varphi }$" or "It is consistent with ${\displaystyle a}$'s knowledge that ${\displaystyle \varphi }$ is possible". The statement "${\displaystyle a}$ does not know whether or not ${\displaystyle \varphi }$" can be expressed as ${\displaystyle \neg K_{a}\varphi \land \neg K_{a}\neg \varphi }$.

In order to accommodate notions of common knowledge (e.g. in the Muddy Children Puzzle) and distributed 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 ${\displaystyle \varphi }$ is a formula of our language, then so are ${\displaystyle {\mathit {E}}_{G}\varphi }$, ${\displaystyle {\mathit {C}}_{G}\varphi }$, and ${\displaystyle {\mathit {D}}_{G}\varphi }$. 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.

Semantics

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 ${\displaystyle {\mathcal {M}}=\langle S,\pi ,K_{1},\dots ,K_{n}\rangle }$ for n agents over ${\displaystyle \Phi }$, the set of all primitive propositions, is an ${\displaystyle (n+2)}$-tuple, where ${\displaystyle S}$ is a nonempty set of states or possible worlds, ${\displaystyle \pi }$ is an interpretation, which associates with each state ${\displaystyle s\in S}$ a truth assignment to the primitive propositions in ${\displaystyle \Phi }$, and ${\displaystyle {\mathcal {K}}_{1},...,{\mathcal {K}}_{n}}$ are binary relations on ${\displaystyle S}$ for n 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 ${\displaystyle p}$Zdroj:https://en.wikipedia.org?pojem=Epistemic_modal_logic
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