74.419 Artificial Intelligence Modal Logic Systems

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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_lo
  gicSemantics_of_modal_logic

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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

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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.

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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).
  

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Systems, Axioms and Frame Conditions
from Stanford Plato http//plato.stanford.edu/ent
ries/logic-modal/3

• 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) ? vu 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
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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_lo
gicSemantics_of_modal_logic
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Relationships Between Modal Logics