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74.419 Artificial Intelligence Modal Logic see reference last slide Syntax of Modal Logic (□ and ◊) Formulae in (propositional) Modal Logic ML: The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML. If and are formulae in ML, then , , , , □, ◊ * are also formulae in ML. * Note: The operator ◊ is often later introduced and defined through □ . Semantics of Modal Logic (□ and ◊) The semantics of a modal logic ML is defined through: a set of worlds W = {w1, w2, ..., wn}, an accessibility relation RWW, and an interpretation function : {0,1} Semantics of Modal Logic ( and ) The interpretation in ML of a formula P, Q, ... of the propositional language of ML corresponds to its truth value in the "current world": w (P)=1 iff I(P) is true in w. w (PQ)=1 iff I(PQ) is true in w. ... Semantics of Modal Logic (□ and ◊) We extend the semantics with an interpretation of the operators □ and ◊, specified relative to a "current world" w. For all wW: w (□)=1 iff w': (w,w')R w' ()=1 ; 0 otherwise. w (◊)=1 iff w': (w,w')R w' ()=1 ; 0 otherwise. Note: Often, the operator ◊ is defined in terms of □: ◊ □ Semantics of Modal Logic (□ and ◊) We can also prove the equivalence of □ and ◊ for our definitions above: w (□)=1 iff (w (□)=1) (or w (□)=0) iff w': (w,w')R w' ()=1 iff w': (w,w')R w' ()=0 iff w': (w,w')R w' ()=1 iff w (◊)=1 This means: □ ◊ Exercise: Proof ◊ □ ! Semantics of Modal Logic (□ and ◊) Other logical operators are interpreted as usual, e.g. w (□)=1 iff w (□)=0 Semantics of ML - Complex Formulas The interpretation of a complex formula of ML is based on the interpretation of the atomic propositional symbols, and then composed using the interpretation function defined above, e.g. w (□)=1 iff (w': (w,w')R w' ()=1) iff w': (w,w')R w' ()=0 Let's say (PQ). w': (w,w')R w' (PQ)=0 w': (w,w')R (w' (P)=0 w' (Q)=0) "P or Q" is not necessarily true in world w, if there is a world w', accessible from w, in which P is false or Q is false. Semantics of Modal Logic - Grounding The interpretation in ML of a formula P, Q, ... of the propositional language of ML corresponds to its truth value in the "current world": w (P)=1 iff I(P) is true in w. w (PQ)=1 iff I(PQ) is true in w. ... Semantics of Modal Logic A formula is satisfied in a world w of a Model M=<W,R,>, if it is true in this world wW under the given interpretation , i.e. w ()=1. M, w |= A formula is true in a Model M=<W,R,>, if it is satisfied in all worlds wW of M. M |= A formula is valid, if it is true in all Models. |= A formula is satisfiable, if it is satisfied in at least one world wW of one Model M=<W,R,>. (or: If its negation is not valid.) Semantics of Modal Logic A formula is satisfied in a world w of a Model M=<W,R,>, if it is true in this world under the given interpretation , i.e. w ()=1. M, w |= A formula is true in a Model M=<W,R,>, if it is satisfied in all worlds wW of M. M |= A formula is valid, if it is true in all Models. |= A formula is satisfiable, if it is satisfied in at least one world wW of one Model M=<W,R,>. (or: If its negation is not valid.) A formula is a consequence of a set of formulas in M=<W,r,>, if in all worlds wW, in which is satisfied, is also satisfied. |= Semantics of Modal Logic: Terminology Sometimes the term "frame" is used to refer to worlds and their connection through the accessibility relation: A frame <W, R> is a pair consisting of a non-empty set W (of worlds) and a binary relation R on W. A model <F, > consists of a frame F, and a valuation that assigns truth values to each atomic sentence at each world in W. Textbooks on (Modal) Logic Richard A. Frost, Introduction to Knowledge-Base Systems, Collins, 1986 (out of print) Comments: one of my favourite books; contains (almost) everything you need w.r.t. foundations of classical and non-classical logic; very compact, comprehensive and relatively easy to understand. Allan Ramsay, Formal Methods in Artificial Intelligence, Cambridge University Press, 1988 Comments: easy to read and to understand; deals also with other formal methods in AI than logic; unfortunately out of print; a copy is on course reserve in the Science Library. Textbooks on (Modal) Logic Graham Priest, An Introduction to Non-Classical Logic, Cambridge University Press, 2001 Comments: the most poplar book (at least among philosophy students) on non-classical, in particular, (propositional) modal logic. Kenneth Konyndyk, Introductory Modal Logic, University of Notre-Dame Press, 1986 (with later reprints) Comments: relatively easy and nice to read; contains propositional as well as first-order (quantified) modal logic, and nothing else. Textbooks on (Modal) Logic J.C. Beall & Bas C. van Fraassen, Possibilities and Paradox, University of Notre-Dame Press, 1986 (with later re-prints) Comments: contains a lot of those weird things, you knew existed but you've never encountered in reality (during your university education). G.E. Hughes & M.J. Creswell, A New Introduction to Modal Logic, Routledge, 1996 Comments: Location: Elizabeth Dafoe Library, 2nd Floor, Call Number / Volume: BC 199 M6 H85 1996