An Invitation to Modal Logic:Lecture 4Philosophy 150Eric PacuitStanford Universityai.stanford.edu/~epacuitDecember 3, 2007Eric Pacuit: Invitation to Modal Logic, Philosophy 150 1PlanX Motivating ExamplesX Formalizing the muddy children puzzle, Introductionto Modal LogicX More about truth of modal formulas12/3: Focus on Epistemic Logic.Digression: A small experiment.12/5: Multiagent Epsitemic Logic, Dynamics in Logic12/7: Dynamics in Logic IIEric Pacuit: Invitation to Modal Logic, Philosophy 150 2Plan for TodayFirst half of the lecture: Epsitemic Logic.Second half of the lecture: a small experiment.Eric Pacuit: Invitation to Modal Logic, Philosophy 150 3SummaryLanguage: The modal language extends a propositional languagewith formulas of the form P and ♦P.IThe modal language can formalize natural language sentencesinvolving modalities (eg. knows, believes, necessary, etc.).IThe modal language expresses properties of relationalstructures.Semantics: A Kripke structure, or more generally relationalstructures, is a set W of states with a relation R on W .Truth:Iw |= P iff for all v , if wRv then v |= PIw |= ♦P iff there exists v such that wRv and v |= P .Eric Pacuit: Invitation to Modal Logic, Philosophy 150 4SummaryLanguage: The modal language extends a propositional languagewith formulas of the form P and ♦P.IThe modal language can formalize natural language sentencesinvolving modalities (eg. knows, believes, necessary, etc.).IThe modal language expresses properties of relationalstructures.Semantics: A Kripke structure, or more generally relationalstructures, is a set W of states with a relation R on W .Truth:Iw |= P iff for all v , if wRv then v |= PIw |= ♦P iff there exists v such that wRv and v |= P .Eric Pacuit: Invitation to Modal Logic, Philosophy 150 4SummaryLanguage: The modal language extends a propositional languagewith formulas of the form P and ♦P.IThe modal language can formalize natural language sentencesinvolving modalities (eg. knows, believes, necessary, etc.).IThe modal language expresses properties of relationalstructures.Semantics: A Kripke structure, or more generally relationalstructures, is a set W of states with a relation R on W .Truth:Iw |= P iff for all v , if wRv then v |= PIw |= ♦P iff there exists v such that wRv and v |= P .Eric Pacuit: Invitation to Modal Logic, Philosophy 150 4SummaryLanguage: The modal language extends a propositional languagewith formulas of the form P and ♦P.IThe modal language can formalize natural language sentencesinvolving modalities (eg. knows, believes, necessary, etc.).IThe modal language expresses properties of relationalstructures.Semantics: A Kripke structure, or more generally relationalstructures, is a set W of states with a relation R on W .Truth:Iw |= P iff for all v , if wRv then v |= PIw |= ♦P iff there exists v such that wRv and v |= P .Eric Pacuit: Invitation to Modal Logic, Philosophy 150 4Two issues to remember1. Modal formulas are interpreted locally.sKuNNo modal formula can distinguish between s and u .Can you think of a first-order formula that can distinguish theKripke structures?Eric Pacuit: Invitation to Modal Logic, Philosophy 150 5Two issues to remember1. Modal formulas are interpreted locally.2. Modal logic can express interesting properties of Kripkestructures.• P → P corresponds to the reflexivity property.• P → P corresponds to the transitivity property.Eric Pacuit: Invitation to Modal Logic, Philosophy 150 5Next lecture: Dynamics in logic.Questions?Email: [email protected]: ai.stanford.edu/~epacuitOffice: Gates 258Eric Pacuit: Invitation to Modal Logic, Philosophy 150