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Modal logic(s) 1 Encoding modality linguistically Auxiliary (modal) verbs can, should, may, must, could, ought to, ... Adverbs possibly, perhaps, allegedly, ... Adjectives useful, possible, inflammable, edible, ... Many languages are much richer 2 Modal-based ambiguity in NL John can sing. Fred would take Mary to the movies. The dog just ran away. Dave will discard the newspaper. Jack may come to the party. 3 Propositional logic (review) Used to represent properties of propositions Formal properties, allows for wide range of applications, usable crosslinguistically Has three parts: vocabulary, syntax, semantics 4 Propositional logic (1) Vocabulary: Atoms representing whole propositions: p, q, r, s, … Logic connectives: &, V, , , Parentheses and brackets: (, ), [, ] Examples John is hungry.: p John eats Cheerios.: q pq ¬p ¬q 5 Propositional logic (2) Syntax (well-formed formulas, wff’s): Any atomic proposition is a wff. If is a wff, then is a wff. If and are wff’s, then ( & ), ( v ), ( ), and ( ) are wff’s. Nothing else is a wff. Examples & pq is not a wff ((pq) & (pr)) is a wff (p v q) s is a wff ((((p & q) v r) s) t) is a wff 6 Propositional logic (3) Semantics: V() = 1 iff V() = 0. V( & ) = 1 iff V() = 1 and V() = 1. V( v ) = 1 iff V() = 1 or V() = 1. V( ) = 1 iff V() = 0 or V() = 1. V( ) = 1 iff V() = V(). The valuation function V is all- important for semantic computations. 7 Logical inferences Modus Ponens: Hypothetical syllogism: pq pq p qr -------- -------- q pr Modus Tollens: Disjunctive syllogism: pq pvq q p --------- -------- p q 8 Formal logic and inferences DeMorgan’s Laws ( v ) ( & ) ( & ) ( v ) Conditional Laws ( ) ( v ) ( ) ( ) ( ) ( & ) Biconditional Laws ( ) ( ) & ( ) ( ) ( & ) v ( & ) 9 Lexical items and predication …sneezed x.(sneeze(x)) …saw… y.x.(see(x,y)) … laughed and is not a woman x.(laugh(x) & ¬woman(x)) … respects himself x.respect(x,x) …respects and is respected by… y.x.[respect(x,y) & respect(y,x)] 10 The function of lambdas Lambdas fill open predicates’ variables with content John sneezed. John, x.(sneeze(x)) x.(sneeze(x)) (John) x.(sneeze(x)) (John) sneeze(John) 11 The basic op: -conversion In an expression (x.W)(z), replace all occurrences of the variable x in the expression W with z. (x.hungry(x))(John) hungry(John) (x.[¬married(x) & male(x) & adult(x)])(John) ¬married(John) & male(John) & adult(John) 12 Contingency and truth true statements false statements non-contingent contingent non-contingent possibly true statements not possibly true (= not necessarily false) (= necessarily false) not possibly false possibly false statements (= necessarily true) (= not necessarily true) 13 Two necessary ingredients Background: premises from which conclusions are drawn Relation: “force” of the conclusion John may be the murderer. John must be the murderer. 14 Model-theoretic valuation M = <U,V> where U is domain of individuals V is a valuation function For example, U = {mary, bill, pc23} V (likes) = {<mary, bill>, <bill, pc23>} V (hungry) = {mary, bill} V (is broken) = {pc23} V (is French) = Ø 15 Model-theoretic valuation [[Mary is hungry]]M = [[is hungry]]([[Mary]]) = [V(hungry)](mary) is true iff mary ∈V(hungry) =1 [[my computer likes Mary]]M = 1 iff <[[my computer]],[[Mary]]> ∈ [[likes]] iff <pc23,mary> ∈ V(likes) =0 So far, have only used constants BUT variables are also possible function g assigns to any variable an element from U 16 Possible worlds Variants, miniscule or drastic, from the actual context (world) W is the set of all possible worlds w’, w’’, w’’’, ... Ordering can be induced on the set of all possible worlds The ordering is reflexive and transitive Modal logic: evaluates truth value of p w/rt each of the possible worlds in W 17 Modal logic Build up a useful system from propositional logic Add two operators: ◊: It is possible that ... □: It is necessary that ... K Logic: propositional logic plus: If A is a theorem, then so is □A □(AB) (□A □B) 18 Semantics of operators If ψ = □φ, then [[ψ]]M,w,g=1 iff ∀w∈W, [[φ]]M,w,g=1. If ψ = ⃟φ, then [[ψ]]M,w,g=1 iff there exists at least one w∈W such that [[φ]]M,w,g=1. 19 Notes on K Obvious equivalencies: ◊A = ¬□¬A Operators behave very much like quantifiers in predicate calculus K is too weak, so add to it: M: □A A The result is called the T logic. 20 Notes on T Still too weak, so: (4) □A □□A (5) ◊A □◊A Logic S4: adding (4) to T Logic S5: adding (5) to T 21 S5 Not adequate for all types of modality However, it is commonly used for database work 22 O say what is (modal) truth? Let M = <U, W, I> be a model with mapping I, and V be a valuation in the model; then: 1. M,w ⊨v φ iff I(φ)(w) = true 2. If R(t1,...,tk) is atomic, M,w ⊨v R(t1...tk) iff <V(t1,w),...V(tk,w)> ∈ V(R)(w) 3. M,w ⊨v ¬ φ iff M,w ¬⊨v φ 4. M,w ⊨v φ & ψ iff M,w ⊨v φ and M,w ⊨v ψ 5. M,w ⊨v φ (∀x)φ iff M,w ⊨v φ[x/u] for all u ∈ U 6. M,w ⊨v □ φ iff M,w ⊨v φ for all w ∈ W 7. M,w ⊨v [λx.φ(x)](t) if M,w ⊨v φ[x/u] where u = g(t,w) 23 Human necessity φ is a human necessity iff it is true in all worlds closest to the ideal If W is the modal base, ∀ w∈W there exists wʹ∈W such that: w ≤ wʹ, and ∀ wʹʹ∈W , if wʹ ≤ wʹʹ then φ is true in wʹʹ φ is a human possibility iff ¬φ is not a human necessity 24 Backgrounds (Kratzer) Realistic: for each w, set of p’s that are true Totally realistic: set of p’s that uniquely define w Epistemic: p’s that are established knowledge in w Stereotypical: p’s in the normal course of w Deontic: p’s that are commanded in w Teleological: p’s that are related to aims in w Buletic: p’s that are wished/desirable in w Empty: the empty set of p’s in any w 25 Related notions Conditionals Counterfactuals Generics Tense Intensionality Doxastics (belief models) 26 The Fitting paper Applies modal logic to databases model-theoretic, S5, formulas tableau methods for proofs, derived rules Operator that associates, combines semantic items compositionally Predicates, entities Variables 27 The Fitting paper db records: possible worlds access: ordering on possible worlds two types of axioms: constraint axioms instance axioms Queries: modal logic expressions Proofs and derivations: tableau methods (several rules) 28

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posted: | 12/8/2011 |

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