logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

, a normal

modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...

is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautologies; * All instances of the

Kripke Kripke is a surname. Notable people with the surname include: * Dorothy K. Kripke (1912–2000), American author of Jewish educational books, and the mother of Saul Kripke * Eric Kripke (born 1974), American television writer, director, and produc ...

schema:

\Box(A\to B)\to(\Box A\to\Box B)

and it is closed under: * Detachment rule (''

modus ponens In propositional logic, (; MP), also known as (), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P'' implies ''Q.'' ''P'' is true. Therefore, ''Q'' must ...

''):

A\to B, A \in L

implies

B \in L

; * Necessitation rule:

A \in L

implies

\Box A \in L

. The smallest logic satisfying the above conditions is called K. Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g.

C. I. Lewis Clarence Irving Lewis (April 12, 1883 – February 3, 1964) was an American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted logician, he later branched into epis ...

's S4 and S5, are normal (and hence are extensions of K). However a number of

deontic In moral philosophy, deontological ethics or deontology (from Greek: and ) is the normative ethical theory that the morality of an action should be based on whether that action itself is right or wrong under a series of rules and principles, ...

and

epistemic logic 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 appl ...

s, for example, are non-normal, often because they give up the Kripke schema. Every normal modal logic is regular and hence classical.

Common normal modal logics

The following table lists several common normal modal systems. The notation refers to the table at Kripke semantics § Common modal axiom schemata. Frame conditions for some of the systems were simplified: the logics are ''sound and complete'' with respect to the frame classes given in the table, but they may ''correspond'' to a larger class of frames.

References

*Alexander Chagrov and Michael Zakharyaschev, ''Modal Logic'', vol. 35 of Oxford Logic Guides, Oxford University Press, 1997. Modal logic {{Logic-stub