Motivation Basic Modal Logic Logic Engineering 10—Modal Logic I CS 5209: Foundation in Logic and AI Martin Henz and Aquinas Hobor March 25, 2010 Generated on Thursday 25 th March, 2010, 16:24 CS 5209: Foundation in Logic and AI 10—Modal Logic I 1
Motivation Basic Modal Logic Logic Engineering Motivation 1 Basic Modal Logic 2 Logic Engineering 3 CS 5209: Foundation in Logic and AI 10—Modal Logic I 2
Motivation Basic Modal Logic Logic Engineering Motivation 1 Basic Modal Logic 2 Logic Engineering 3 CS 5209: Foundation in Logic and AI 10—Modal Logic I 3
Motivation Basic Modal Logic Logic Engineering Necessity You are crime investigator and consider different suspects. Maybe the cook did it with a knife? Maybe the maid did it with a pistol? But: “The victim Ms Smith made the call before she was killed.” is necessarily true. “Necessarily” means in all possible scenarios (worlds) under consideration. CS 5209: Foundation in Logic and AI 10—Modal Logic I 4
Motivation Basic Modal Logic Logic Engineering Notions of Truth Often, it is not enough to distinguish between “true” and “false”. We need to consider modalities if truth, such as: necessity (“in all possible scenarios”) morality/law (“in acceptable/legal scenarios”) time (“forever in the future”) Modal logic constructs a framework using which modalities can be formalized and reasoning methods can be established. CS 5209: Foundation in Logic and AI 10—Modal Logic I 5
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Motivation 1 Basic Modal Logic 2 Syntax Semantics Equivalences Logic Engineering 3 CS 5209: Foundation in Logic and AI 10—Modal Logic I 6
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Syntax of Basic Modal Logic ⊤ | ⊥ | p | ( ¬ φ ) | ( φ ∧ φ ) φ ::= | ( φ ∨ φ ) | ( φ → φ ) | ( φ ↔ φ ) | ( � φ ) | ( ♦ φ ) CS 5209: Foundation in Logic and AI 10—Modal Logic I 7
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Pronunciation and Examples Pronunciation If we want to keep the meaning open, we simply say “box” and “diamond”. If we want to appeal to our intuition, we may say “necessarily” and “possibly” (or “forever in the future” and “sometime in the future”) Examples ( p ∧ ♦ ( p → � ¬ r )) � (( ♦ q ∧ ¬ r ) → � p ) CS 5209: Foundation in Logic and AI 10—Modal Logic I 8
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Kripke Models Definition A model M of basic modal logic is specified by three things: A set W , whose elements are called worlds ; 1 A relation R on W , meaning R ⊆ W × W , called the 2 accessibility relation; A function L : W → P ( Atoms ) , called the labeling function. 3 CS 5209: Foundation in Logic and AI 10—Modal Logic I 9
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Who is Kripke? How do I know I am not dreaming? Kripke asked himself this question in 1952, at the age of 12. His father told him about the philosopher Descartes. Modal logic at 17 Kripke’s self-studies in philosophy and logic led him to prove a fundamental completeness theorem on modal logic at the age of 17. Bachelor in Mathematics from Harvard is his only non-honorary degree At Princeton Kripke taught philosophy from 1977 onwards. Contributions include modal logic, naming, belief, truth, the meaning of “I” CS 5209: Foundation in Logic and AI 10—Modal Logic I 10
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Example W { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } = R { ( x 1 , x 2 ) , ( x 1 , x 3 ) , ( x 2 , x 2 ) , ( x 2 , x 3 ) , ( x 3 , x 2 ) , ( x 4 , x 5 ) , ( x 5 , x 4 ) , ( x 5 , x 6 ) } = L { ( x 1 , { q } ) , ( x 2 , { p , q } ) , ( x 3 , { p } ) , ( x 4 , { q } ) , ( x 5 , {} ) , ( x 6 , { p } ) } = x 2 p , q x 1 p x 3 q p x 4 x 6 q x 5 CS 5209: Foundation in Logic and AI 10—Modal Logic I 11
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences When is a formula true in a possible world? Definition Let M = ( W , R , L ) , x ∈ W , and φ a formula in basic modal logic. We define x � φ via structural induction: x � ⊤ x � � ⊥ x � p iff p ∈ L ( x ) x � ¬ φ iff x � � φ x � φ ∧ ψ iff x � φ and x � ψ x � φ ∨ ψ iff x � φ or x � ψ ... CS 5209: Foundation in Logic and AI 10—Modal Logic I 12
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences When is a formula true in a possible world? Definition (continued) Let M = ( W , R , L ) , x ∈ W , and φ a formula in basic modal logic. We define x � φ via structural induction: ... x � φ → ψ iff x � ψ , whenever x � φ x � φ ↔ ψ iff ( x � φ iff x � ψ ) x � � φ iff for each y ∈ W with R ( x , y ) , we have y � φ x � ♦ φ iff there is a y ∈ W such that R ( x , y ) and y � φ . CS 5209: Foundation in Logic and AI 10—Modal Logic I 13
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Example x 2 p , q x 1 p x 3 q p x 4 x 6 q x 5 x 1 � q x 1 � ♦ q , x 1 � � � q x 5 � � � p , x 5 � � � q , x 5 � � � p ∨ � q , x 5 � � ( p ∨ q ) x 6 � � φ holds for all φ , but x 6 � � ♦ φ CS 5209: Foundation in Logic and AI 10—Modal Logic I 14
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Formula Schemes Example We said x 6 � � φ holds for all φ , but x 6 � � ♦ φ Notation Greek letters denote formulas, and are not propositional atoms. Formula schemes Terms where Greek letters appear instead of propositional atoms are called formula schemes . CS 5209: Foundation in Logic and AI 10—Modal Logic I 15
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Entailment and Equivalence Definition A set of formulas Γ entails a formula ψ of basic modal logic if, in any world x of any model M = ( W , R , L ) , whe have x � ψ whenever x � φ for all φ ∈ Γ . We say Γ entails ψ and write Γ | = ψ . Equivalence We write φ ≡ ψ if φ | = ψ and ψ | = φ . CS 5209: Foundation in Logic and AI 10—Modal Logic I 16
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Some Equivalences De Morgan rules: ¬ � φ ≡ ♦ ¬ φ , ¬ ♦ φ ≡ � ¬ φ . Distributivity of � over ∧ : � ( φ ∧ ψ ) ≡ � φ ∧ � ψ Distributivity of ♦ over ∨ : ♦ ( φ ∨ ψ ) ≡ ♦ φ ∨ ♦ ψ � ⊤ ≡ ⊤ , ♦ ⊥ ≡ ⊥ CS 5209: Foundation in Logic and AI 10—Modal Logic I 17
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Validity Definition A formula φ is valid if it is true in every world of every model, i.e. iff | = φ holds. CS 5209: Foundation in Logic and AI 10—Modal Logic I 18
Motivation Syntax Basic Modal Logic Semantics Logic Engineering Equivalences Examples of Valid Formulas All valid formulas of propositional logic ¬ � φ ↔ ♦ ¬ φ � ( φ ∧ ψ ) ↔ � φ ∧ � ψ ♦ ( φ ∨ ψ ) ↔ ♦ φ ∨ ♦ ψ Formula K : � ( φ → ψ ) ∧ � φ → � ψ . CS 5209: Foundation in Logic and AI 10—Modal Logic I 19
Valid Formulas wrt Modalities Motivation Properties of R Basic Modal Logic Correspondence Theory Logic Engineering Preview: Some Modal Logics Motivation 1 Basic Modal Logic 2 Logic Engineering 3 Valid Formulas wrt Modalities Properties of R Correspondence Theory Preview: Some Modal Logics CS 5209: Foundation in Logic and AI 10—Modal Logic I 20
Valid Formulas wrt Modalities Motivation Properties of R Basic Modal Logic Correspondence Theory Logic Engineering Preview: Some Modal Logics A Range of Modalities In a particular context � φ could mean: It is necessarily true that φ It will always be true that φ It ought to be that φ Agent Q believes that φ Agent Q knows that φ After any execution of program P , φ holds. Since ♦ φ ≡ ¬ � ¬ φ , we can infer the meaning of ♦ in each context. CS 5209: Foundation in Logic and AI 10—Modal Logic I 21
Valid Formulas wrt Modalities Motivation Properties of R Basic Modal Logic Correspondence Theory Logic Engineering Preview: Some Modal Logics A Range of Modalities From the meaning of � φ , we can conclude the meaning of ♦ φ , since ♦ φ ≡ ¬ � ¬ φ : � φ ♦ φ It is necessarily true that φ It is possibly true that φ It will always be true that φ Sometime in the future φ It ought to be that φ It is permitted to be that φ Agent Q believes that φ φ is consistent with Q ’s beliefs Agent Q knows that φ For all Q knows, φ After any run of P , φ holds. After some run of P , φ holds CS 5209: Foundation in Logic and AI 10—Modal Logic I 22
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