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```74.419 Artificial
Intelligence
Modal Logic Systems
http://plato.stanford.edu/entries/logic-modal/#3
http://en.wikipedia.org/wiki/Semantics_of_modal_logic#Semantics_of_modal_logic
System K (Normal Modal Logics)
Distribution Axiom:
(AB)  ( A B )
Further:
(AB)  AB
AB  (AB)
Definition of "possible" : P = P
Non-Normal Modal Logics
There are also Modal Logics, to which the above
axioms do not apply. These are called "nonnormal".
The main characteristic of non-normal modal
logic is, that nothing is necessary, and everything
is possible, i.e.

is always false.

is always true.
Other Systems of Modal Logics
Other systems can be defined by adding axioms, e.g.
AA
Such axioms impose constraints on the structure of the
accessibility relation R and thus constrain the set of
models, which fulfill these axioms and are considered in
these logics.
The axiom above, for example, requests transitivity of
R. It is often used in Epistemic Logic, expressing: if
someone knows something, he knows that he knows it
(positive introspection).
Systems, Axioms and Frame Conditions
from Stanford Plato: http://plato.stanford.edu/entries/logic-modal/#3
Name Axiom
(D)
 A  A
Condition on Frames
u: wRu
R is...
Serial
(M)
 AA
wRw
Reflexive
(4)
 A   A
(wRv & vRu)  wRu
Transitive
(B)
A   A
wRv  vRw
Symmetric
(5)
 A   A
(wRv & wRu)  vRu
Euclidean
(CD)
 A  A
(wRv & wRu)  v=u
Unique
(□M)
( AA)
wRv  vRv
Shift Reflexive
(C4)
  A  A
wRv  u: (wRu & uRv)
Dense
(C)
  A    A (wRv & wRx)  u: (vRu & xRu) Convergent
Notation: &  
and
wRv  (w,v)R
Common Modal Axiom Schemata
from Wikipedia
name axiom
frame condition
T
reflexive
4
transitive
D
serial:
B
symmetric
5
Euclidean:
GL
R transitive, R-1 well-founded
Grz
R reflexive and transitive, R-1−Id well-founded
3
1
(a complicated second-order property)
2
http://en.wikipedia.org/wiki/Semantics_of_modal_logic#Semantics_of_modal_logic
Relationships Between Modal Logics
```
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