Pills of Modal Logic walking around possible worlds

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Pills of Modal Logic (walking around possible
worlds)

• MIDI TALKS
• marco volpe
• 24 maggio 2007

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Plan
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Why Modal Logic?

• We want to qualify the truth of a judgement
• it is necessary that
• it is possible that
• it is obligatory that
• it is permitted that
• it is forbidden that
• it will always be the case that
• it will be the case that
• A believes that

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Informally

• We can think of classical logic as dealing with a
  single world, where sentences are either true or
  not.
• We can think of a modal logic as dealing with a
  family of possible worlds.
• One believes X when X is true in all the worlds
  he can imagine as possible (accessible,
  reachable, ).

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Two weeks ago

• Alice and Bob are married.
• They want to get a divorce.
• They are rich (who is the richest?).
• They have a car (who got it?).
• They do strange things (e.g. they flip coins
  over the telephone).

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Alices adventures in possible worlds
A Bob loves Alice B Alice loves Bob C Bob
loves Eve D 2 2 4

• In the world w, Alice believes
• Alice believes A,
• Alice believes B,
• Alice believes C,
• Alice believes D.

• In the world w, Alice believes B, C, D.
• In the world w1, Alice believes everything.
• In the world w2, Alice believes A, B, C, D.
• In the world w3, Alice believes everything.

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Alices adventures in possible worlds
We can place conditions on the arrow
relationship between worlds.
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Alices adventures in reflexive worlds
A Bob loves Alice B Alice loves Bob C Bob
loves Eve D 2 2 4

• In the world w, Alice believes B, D.
• In the world w1, Alice believes A, B, C, D.
• In the world w2, Alice believes A, D.
• In the world w3, Alice believes A, B, A, D.

• In the world w, Alice believes
• Alice believes D.

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Alices adventures in reflexive worlds
In any world w, Alice believes X implies X.
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Alices adventures in transitive worlds
A Bob loves Alice B Alice loves Bob C Bob
loves Eve D 2 2 4

• In the world w, Alice believes
• Alice believes A,
• Alice believes B,
• Alice believes C,
• Alice believes D.

• In the world w, Alice believes D.
• In the world w1, Alice believes everything.
• In the world w2, Alice believes A, B, C, D.
• In the world w3, Alice believes everything.

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Alices adventures in transitive worlds
In any world w, Alice believes X implies Alice
believes Alice believes X. (introspection
Doxastic Logic)
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Alices adventures in refl. trans. worlds
A Bob loves Alice B Alice loves Bob C Bob
loves Eve D 2 2 4

• In the world w, Alice believes D.
• In the world w1, Alice believes A, B, C, D.
• In the world w2, Alice believes A, D.
• In the world w3, Alice believes A, B, A, D.

• In the world w, Alice believes
• Alice believes D.

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Alices adventures in refl. trans. worlds
In any world w, Alice knows X is equivalent to
Alice knows Alice knows X. (Epistemic Logic)
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Formally

• Frame
• lt W, R gt where W is a non-empty set of worlds
• R is a binary relation
  on W

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Formally

• Frame
• lt W, R gt where W is a non-empty set of worlds
• R is a binary relation
  on W

• Model
• lt W, R, v gt where W is a non-empty set of
  worlds
• R is a binary relation
  on W
• v is a function W x P ? 0,1

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Formally

• Semantics
• M, w p iff v (w, p) 1
• M, w ?
• M, w f1 ? f2 iff M, w f1 or M, w
  f2
• M, w ?f iff w R w implies M, w
  f

• f is valid in a model M
  if it is true at every
  world of the model
• f is valid in a collection of frames F
  if it is valid in all models
  based on frames in F

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Boxes and diamonds

• ? P ? ? ? P

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Map

• Classical axiomatic system necessitation
  rule
• K all frames ? ( P ? Q ) ? ( ?P ? ?Q )
• T reflexive frames ?P ? P K
• K4 transitive frames ?P ? ??P K
• S4 refl. trans. frames ?P ? P ?P ? ??P
  K

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What I am going to do
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What (I believe) I am going to do

• Labelled Deduction Systems
• f ? w f , w1 R w2
• M, w f ? M w f

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Labelled Temporal Logic
modal
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• Grazie!

theres so many different worlds so many
different suns and we have just one world but we
live in different ones m. knopfler
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References

• B. Chellas, Modal Logic An Introduction,
  Cambridge Univ. Press, 1980.
• G. Hughes and M. Cresswell, An Introduction to
  Modal Logic, Methuen, 1968.
• A. Dekker, Possible Worlds, Belief and Modal
  Logic a Tutorial, 2004.
• L. Viganò, Labelled Non-Classical Logics, Kluwer,
  2001.
• Dire Straits, Brothers in arms, 199?.