In
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 premises ...
, 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 other ...
is a set ''L'' of modal formulas such that ''L'' contains:
* All propositional
tautologies;
* All instances of the
Kripke schema:
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. ...
''):
implies
;
* Necessitation rule:
implies
.
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), usually cited as C. I. Lewis, was an American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted logic ...
'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 language, Greek: + ) is the normative ethics, normative ethical theory that the morality of an action should be based on whether that action itself is right or wrong under a s ...
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 applica ...
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.
{{Logic-stub
Modal logic