Epistemic modal logic

The epistemic logic (from the Greek επιστήμη - science, knowledge), also knowledge logic deals with faith and knowledge in individuals and groups. Target for investigation by means of epistemic logic is often a dynamic and flexible model of opinions and knowledge states. This branch of philosophical logic is a subset of modal logic and falls in the range of beliefs and opinions ( beliefs ) often together with the doxastic logic.

Term

The epistemic logic is a classical logic widening philosophical logic, the elementary statements or predicate logic to

• An operator for the knowledge (knowledge operator "W" ) Advanced ( = epistemic logic in the narrow sense ( logic of knowledge) ) or more operators from the doxastic logic, eg for

• Faith ( Convinced - be ( strong faith ); For - likely - hold ( weak faith ) ) or
• For behavior - possible ( = epistemic logic in the broad sense ) ( logic of belief and knowledge).

The epistemic logic in its modern form, investigates the connections of the epistemic modalities to more complex calculi. The epistemic logic is thus the systematic relationships between the forms of knowledge on, for example, for possible hold - the - prerequisite knowledge for another, or self-reflection of knowledge, and reconstructs the basic concepts of the theory of knowledge in logic. She is concerned to show interested in when a statement is valid as proven, it is when they believed, claims known. She also deals with the concepts lies and errors and probability. The transitions to the logic of probabilities are fluid.

The epistemic logic can not be extensional (see Extension), but at most intensional interpretation. An intensional semantics is in the semantics of possible worlds. The basic idea is that someone believes that P if P is true in every world, which he considers possible. For more detailed syntactic and semantic characterizations of the different systems of epistemic and doxastic logic cf. modal logic.

Examples

Examples of valid and invalid statements from the epistemic logic (in the narrow sense)

• Validity: If a knows that P, then P is true.
• Validity: If a knows that P, and also know that Q, then a white that P and Q.
• Invalid: I do not know that P < => I know that not P.

Examples of valid and invalid statements from the doxastic logic

• Validity: If a is convinced that P and believes that Q, then a is also convinced that P and Q.
• Valid: a P holds for possible, if he is not convinced that P is not the case.
• Invalid: If a hold is likely that P and also probably holds that Q, then holds a is likely that P and Q.

Examples of in only in some systems valid statements

• If a knows that P, then a white and that he knows that P. ( So-called positive introspection axiom. )
• If a does not know that P, then a white, that he does not know that P. ( So-called negative introspection axiom. )

Application in the artificial intelligence

There are a number of approaches to formalize an epistemic logic, thus making computationally applicable. Background is the endeavor to implement inferential, based on faith and knowledge. A common approach is to start from the expression of statements or predicate logic and introduce two new operators ( modal operators ) for belief and knowledge. The special feature of these operators is that they presuppose the existence of a subject a, whose faith or knowledge, they allow to express:

Means as much as: the subject believes in the validity of a P.

Means as much as: the subject of a knows that P is valid.

To give another simple example in ( most systems) of the epistemic logic valid statements, whether here the mastery of the modus ponens by the subject called a:

(if a know P and also knows that PQ implies that a white and that Q).

There, different subjects, of course, believe different things or know that may even contradict. Such logical worlds are used for example in artificial intelligence for the realization of multi-agent systems.