logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

, a normal

modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...

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

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

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

modus ponens In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference ...

''):

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'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: + ) 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 logics, 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. {{Logic-stub Modal logic