In logic, a normal modal logic 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): \( A\to B, A \vdash B; \)
Necessitation rule: \( \vdash A \) implies \( \vdash\Box A. \)
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 extensions of K. However a number of deontic 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 complete with respect to the frame classes given in the table, but they may correspond to a larger class of frames.
| Name | Axioms | Frame condition |
|---|---|---|
| K | — | all frames |
| T | T | reflexive |
| K4 | 4 | transitive |
| S4 | T, 4 | preorder |
| S5 | T, 5 or D, B, 4 | equivalence relation |
| S4.3 | T, 4, H | total preorder |
| S4.1 | T, 4, M | preorder, ∀\( \forall w\,\exists u\,(w\,R\,u\land \forall v\,(u\,R\,v\Rightarrow u=v)) \) |
| S4.2 | T, 4, G | directed preorder |
| GL, K4W | GL or 4, GL | finite strict partial order |
| Grz, S4Grz | Grz or T, 4, Grz | finite partial order |
| D | D | serial |
| D45 | D, 4, 5 | transitive, serial, and Euclidean |
References
Alexander Chagrov and Michael Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
