Epistemic Logic

The epistemic logic (from Greek ἐπιστήμη, science, knowledge '), also knowledge logic , deals with belief and knowledge in individuals and groups . The aim of investigations using epistemic logic is often a dynamic or flexible model of states of opinion and knowledge. This branch of philosophical logic is a sub-area of modal logic and often coincides with doxastic logic in the area of ​​beliefs and opinions (convictions) .

term

Epistemic logic is a philosophical logic that extends classical logic and that transforms elementary propositional or predicate logic

• an operator for knowledge (knowledge operator "W") expanded (= epistemic logic in the narrower sense (logic of knowledge) )

or further operators from doxastic logic, e.g. B. for

• Believe (being convinced (strong belief); believing -likely (weak belief) ) or
• Considering possible (= epistemic logic in the broader sense ) (logic of belief and knowledge) .

Epistemic logic in its modern form examines the connections between epistemic modalities and more complex calculi . The epistemic logic thus shows the systematic connections between the forms of knowledge, for example the presumed knowledge as further possible or the self-reflection of knowledge, and reconstructs the basic concepts of epistemology in logic. It is interested in showing when a statement is considered proven, when it is believed, asserted, and known. It also deals with the terms lie and error and probability . The transitions to the logic of probabilities are fluid.

The epistemic logic cannot be interpreted extensionally , but at best intensional . An intensional semantics is the semantics of possible worlds before. The basic idea is that someone is convinced that, if P is the case in whatever world he thinks possible. For more precise syntactic and semantic characterizations of the different systems of epistemic or doxastic logic ; see. Modal logic .

Examples

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

• Valid: If a knows that P, then P is true.
• Valid: If a knows that P, and also knows that Q, then a knows that P and Q.
• Invalid: I don't know that P I know that not P. ${\displaystyle \Leftrightarrow }$

Examples of valid and invalid statements from doxastic logic

• Valid: If a is convinced that P and is convinced that Q, then a is also convinced that P and Q.
• Valid: a thinks P is possible if he is not convinced that P is not the case.
• Invalid: If a believes that P is likely and also believes that Q is likely, then a considers that P and Q are likely.

Examples of statements that are only valid in some systems

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

Application in artificial intelligence

There are a number of approaches to formalize an epistemic logic and thus make it computationally applicable. The background is the endeavor to implement conclusions that are based on belief and knowledge. A common approach is to start from the expressive possibilities of propositional or predicate logic and to introduce two new operators ( modal operators ) for belief and knowledge. The peculiarity of these operators is that they presuppose the presence of a subject a, whose belief or knowledge they allow to express:

${\displaystyle G_{a}(P)}$ means something like: The subject a believes that P is true.

${\displaystyle W_{a}(P)}$ means something like: The subject a knows that P is true.

To give another simple example of statements that are valid in (most systems) of epistemic logic, the mastery of the modus ponens by the subject a should be mentioned here :

${\displaystyle W_{a}(P)\wedge W_{a}(P\rightarrow Q)\rightarrow W_{a}(Q)}$

(if a knows that P and also knows that P implies Q, then a also knows that Q).

Different subjects can of course believe or know different things, which can even contradict each other. Such logical worlds are used, for example, in artificial intelligence to implement multi- agent systems .

literature

• Georg Henrik von Wright : An Essay in Modal Logic. North-Holland Publishing Company, Amsterdam 1951.
• Jaakko Hintikka : Knowledge and Belief. Ithaka 1962. Reissued 2005, ISBN 978-1904987086 .
• Jaakko Hintikka: The Logic of Epistemology and the Epistemology of Logic. Springer Netherland, Berlin 1989.
• Wolfgang Lenzen : Belief, Knowledge and Probability. Vienna / New York, Springer 1980.
• Hans van Ditmarsch, Wiebe van der Hoek, Barteld Kooi: Dynamic Epistemic Logic. Springer 2007, ISBN 978-1402058387 .
• Ronald Fagin, Joseph Halpern, Yoram Moses, Moshe Y. Vardi: Reasoning about Knowledge. MIT Press 1995, ISBN 978-0262562003 .

Web links

• Entry in Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .

swell

1. ↑ ^{a b} Cf. Regenbogen / Meyer, Dictionary of Philosophical Terms (2005), doxastic logic
2. ↑ See previous version . Not mentioned in the sources
3. ^ Rainbow / Meyer, Dictionary of Philosophical Terms (2005), Epistemic Logic
4. ↑ Regenbogen / Meyer, Dictionary of Philosophical Terms (2005), Epistemic Logic: "undisputed law"
5. ↑ cf. Rainbow / Meyer: Dictionary of Philosophical Terms (2005), Epistemic Logic