Pills of Modal Logic (walking around possible worlds)

1. Pills of Modal Logic(walking around possible worlds) MIDI TALKS marco volpe 24 maggio 2007

2. Plan 1 / 16

3. 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… 2 / 16

4. 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 believesX when X is true in all the worlds he can imagine as possible (accessible, reachable, …). 3 / 16

5. 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…). 4 / 16

6. w1 C A w D w3 A w2 D C B D D Alice’s 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. 5 / 16

7. Alice’s adventures in possible worlds We can place conditions on the arrow relationship between worlds. 6 / 16

8. w1 C A w D w3 A w2 D C B D D Alice’s 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”. 7 / 16

9. Alice’s adventures in reflexive worlds In any world w, Alice believes X implies X. 8 / 16

10. w1 C A w D w3 A w2 D C B D D Alice’s 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. 9 / 16

11. Alice’s adventures in transitive worlds In any world w, Alice believes X implies Alice believes “Alice believes X”. (introspection = Doxastic Logic) 10 / 16

12. w1 C A w D w3 A w2 D C B D D Alice’s 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”. 11 / 16

13. Alice’s adventures in refl.+trans. worlds In any world w, Alice knows X is equivalent to Alice knows “Alice knows X”. (Epistemic Logic) 12 / 16

14. Frame <W,R> : whereW is a non-empty set of worlds R is a binary relation onW w1 C A w D w3 A w2 D C B D D w1 w w3 w2 Formally 13 / 16

15. Frame <W,R> : whereW is a non-empty set of worlds R is a binary relation onW Formally • Model <W,R,v> : whereW is a non-empty set of worlds R is a binary relation onW vis a function: W x P → {0,1} 13 / 16

16. Semantics M, w╞ p iff v(w, p) = 1 M, w╞  M, w╞ φ1 → φ2 iff M, w╞ φ1 or M, w╞ φ2 M, w╞ □φ iff wR w’implies M, w’╞ φ Formally • φ is valid in a modelM if it is true at every world of the model • φ is valid in a collection of framesF if it is valid in all models based on frames in F 14 / 16

17. Boxes and diamonds  P  ~ □~P 15 / 16

18. 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 16 / 16

19. What I am going to do

20. What (I believe) I am going to do • Labelled Deduction Systems φw:φ,w1R w2 M, w╞ φM╞ w:φ

21. temporal Labelled Temporal Logic modal

22. Grazie! there’s so many different worlds so many different suns and we have just one world but we live in different ones m. knopfler

23. 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?.