1. 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

2. 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

3. Non-Normal Modal Logics There are also Modal Logics, to which the above axioms do not apply. These are called "non- normal". 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.

4. 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).

5. Systems, Axioms and Frame Conditions Name Axiom Condition on Frames R is... (D)  A   A  u: wRu 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 from Stanford Plato: http://plato.stanford.edu/entries/logic-modal/#3http://plato.stanford.edu/entries/logic-modal/#3 Notation: &   and wRv  (w,v)  R

6. Common Modal Axiom Schemata from Wikipedia nameaxiomframe condition T reflexive 4 transitive D serial: B symmetric 5 Euclidean: GL R transitive, R -1 well-foundedtransitivewell-founded Grz R reflexive and transitive, R -1 −Id well-foundedreflexivetransitivewell-founded 3 1 (a complicated second-order property)second-order 2 http://en.wikipedia.org/wiki/Semantics_of_modal_logic#Semantics_of_modal_logic

7. Relationships Between Modal Logics