University of Texas at Dallas
Modal Logic
Gopal Gupta
Department of Computer Science
The University of Texas at Dallas
Based on the paper
Possible Worlds, Belief, and Modal Logic: a Tutorial
By Anthony H. Dekker
Applied Logic, Programming-Languages and Systems (ALPS) Lab @ UTD Slide- 1
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Modal Logic
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Modal Logic: Syntax
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Semantics
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Semantics
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Satisfiability and validity
A formula is satisfiable if there exists a frame F = (W,R) and
an interpretation M = (F,V) such that M,w ╞  for some w ϵ W.
A formula is valid ,written ╞ if for every frame F = (W,R), for
every interpretation M = (F,V) and for every w ϵ W, M,w ╞ .
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Survey (Dekker)
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Modal logic: extension to the concept that “X is true”
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“X is believed to be true”
“X is known to be true”
“X ought to be true”
“X is eventually true”
“X is necessarily true”
These extensions make sense in “possible worlds” or “alternative
unverses”
Alternative universes are logically consistent (2+2 is always 4, but
Obama may not be the President of USA)
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Belief Modality
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John believes A: In all worlds reachable from John’s current world,
A is true
Two basic axioms:
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John believes in all tautologies T
(B1)
Forall w, John believes in X and in X → Y, then he believes Y. (B2)
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B1 & B2 state that John is rational
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Kripke semantics: Accessibility relation w1 → w2
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If John lives in world w1, then he would think of w2 as possible
In any world v: John believes X ↔ X is true in all wi’s with v → wi
Tautologies are true in all worlds, so John believes them
Likewise for B2
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Belief Modality
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Doxastic Logic
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Dekker Example on Page 5-6 (Peter)
Transitivity: Rule of introspection
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Peter believes A → Peter believes Peter believes A (B4)
□ A → □ □ A (System S4)
Doxastic Logic = Logic of belief
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Epistemic Logic
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Deontic Logic
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Temporal Logic
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Temporal Logic
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