Epistemic modal logic

Epistemic modal logical system is a subfield of modal logic that is concerned with reasoning approximately knowledge. While epistemology has the long philosophical tradition dating back to Ancient Greece, epistemic system of logic is a much more recent coding with a formal request to be considered for a position or to be enables to do or cause something. in numerous fields, including philosophy, theoretical computer science, artificial intelligence, economics together with linguistics. While philosophers since Aristotle clear discussed modal logic, together with Medieval philosophers such(a) as Avicenna, Ockham, and Duns Scotus developed numerous of their observations, it was C. I. Lewis who created the number one symbolic and systematic approach to the topic, in 1912. It continued to mature as a field, reaching its advanced form in 1963 with the take of Kripke.

Standard possible worlds model

Most attempts at modeling knowledge have been based on the possible worlds model. In outline to do this, we must divide the line of possible worlds between those that are compatible with an agent's knowledge, and those that are not. This loosely conforms with common usage. whether I know that it is for either Friday or Saturday, then I know forthat it is for not Thursday. There is no possible world compatible with my cognition where it is Thursday, since in all these worlds it is either Friday or Saturday. While we will primarily be study the logic-based approach to accomplishing this task, it is worthwhile to acknowledgment 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 gave between them:

Typically, the logic-based approach has been used in fields such(a) 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 Linguistic communication of modal logic, which we will now describe.

The basic multimodal logic applied for knowledge representation. The dual of K, which would be in the same relationship to K as is to , has no specific symbol, but can be represented by , which can be read as " does non know that non " or "It is consistent with 's knowledge that is possible". The calculation " does not know whether or not " can be expressed as .

In lines to accommodate notions of mutual knowledge; , which reads "it is common knowledge to every agent in G"; and , which reads "it is distributed knowledge to the whole combine G." If is a formula of our language, then so are , , and . Just as the subscript after can be omitted when there is only one agent, the subscript after the modal operators , , and can be omitted when the business is the shape of all agents.

As we noted above, the logic-based approach is built upon the possible worlds model, the semantics of which are often assumption definite form in Kripke structures, also so-called as Kripke models. A Kripke structure M for n agents over is an n + 2-tuple , where S is a nonempty set of states or possible worlds, is an interpretation, which associates with regarded and identified separately. state in S a truth assignment to the primitive propositions in the set of all primitive propositions, and are binary relations on S for n numbers of agents. It is important here not to confuse , our modal operator, and , our accessibility relation.

The truth assignment tells us whether or not a proposition p is true or false in astate. So tells us whether p is true in state s in framework . Truth depends not only on the structure, but on the current world as well. Just because something is true in one world does not intend it is true in another. To state that a formula is true at aworld, one writes , normally read as " is true at M,s," or "M,s satisfies ".

It is useful to think of our binary explanation as a possibility relation, because it is meant to capture what worlds or states agent i considers to be possible; In other words, if and only if , and such(a) 's are called epistemic alternatives for agent i. In idealized accounts of knowledge e.g., describing the epistemic status of perfect reasoners with infinite memory capacity, it enables sense for to be an equivalence relation, since this is the strongest form and is the nearly appropriate for the greatest number of applications. An equivalence report 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 picture rather than knowledge.