Pills of Modal Logic (walking around possible worlds)

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## 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**Plan**1 / 16**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**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**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**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**Alice’s adventures in possible worlds**We can place conditions on the arrow relationship between worlds. 6 / 16**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**Alice’s adventures in reflexive worlds**In any world w, Alice believes X implies X. 8 / 16**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**Alice’s adventures in transitive worlds**In any world w, Alice believes X implies Alice believes “Alice believes X”. (introspection = Doxastic Logic) 10 / 16**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**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**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**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**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**Boxes and diamonds** P ~ □~P 15 / 16**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**What (I believe) I am going to do**• Labelled Deduction Systems φw:φ,w1R w2 M, w╞ φM╞ w:φ**temporal**Labelled Temporal Logic modal**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**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?.