Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 1 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) UTD Slide- 2 University of Texas at Dallas Modal Logic

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 3 University of Texas at Dallas

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 4 University of Texas at Dallas Modal Logic: Syntax

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 5 University of Texas at Dallas Semantics

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 6 University of Texas at Dallas Semantics  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 W. A formula  is valid,written ╞  if for every frame F = (W,R), for every interpretation M = (F,V) and for every w W, M,w ╞ .

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 7 University of Texas at Dallas Survey (Dekker)  Modal logic: extension to the concept that “X is true” “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)

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 8 University of Texas at Dallas Belief Modality  John believes A: In all worlds reachable from John’s current world, A is true  Two basic axioms:  John believes in all tautologies T (B1)  Forall w, John believes in X and in X → Y, then he believes Y. (B2)  B1 & B2 state that John is rational  Kripke semantics: Accessibility relation w1 → w2  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

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 9 University of Texas at Dallas Belief Modality

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 10 University of Texas at Dallas Doxastic Logic  Dekker Example on Page 5-6 (Peter)  Transitivity: Rule of introspection  Peter believes A → Peter believes Peter believes A (B4)  □ A → □ □ A (System S4)  Doxastic Logic = Logic of belief

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 11 University of Texas at Dallas Epistemic Logic

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 12 University of Texas at Dallas Deontic Logic

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 13 University of Texas at Dallas Temporal Logic

Applied Logic, Programming-Languages and Systems (ALPS) UTD Slide- 14 University of Texas at Dallas Temporal Logic