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Modal logic(s)

Modal logic(s). 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. Modal-based ambiguity in NL.

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Modal logic(s)

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  1. Modal logic(s)

  2. 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

  3. 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.

  4. 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

  5. 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 • p  q • ¬p  ¬q

  6. 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 • ((pq) & (pr)) is a wff • (p v q)  s is a wff • ((((p & q) v  r)  s)  t) is a wff

  7. 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.

  8. Modus Ponens:p  q p-------- q Modus Tollens:p  q q---------  p Hypothetical syllogism:p  qq  r--------p  r Disjunctive syllogism:p v q p-------- q Logical inferences

  9. Formal logic and inferences • DeMorgan’s Laws • ( v )  ( & ) • ( & )  ( v ) • Conditional Laws • ( )  ( v ) • ( )  (  ) • ( )   ( & ) • Biconditional Laws • ( )  (  ) & (  ) • ( )  ( & ) v ( & )

  10. 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)]

  11. 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)

  12. 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)

  13. true statements false statements non-contingent contingent non-contingent possibly true statements(= not necessarily false) not possibly true(= necessarily false) not possibly false(= necessarily true) possibly false statements(= not necessarily true) Contingency and truth

  14. 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.

  15. 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

  16. 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

  17. 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

  18. 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 • □(AB)  (□A  □B)

  19. 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.

  20. 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.

  21. 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

  22. S5 • Not adequate for all types of modality • However, it is commonly used for database work

  23. 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: • M,w ⊨v φ iff I(φ)(w) = true • If R(t1,...,tk) is atomic, M,w ⊨v R(t1...tk) iff <V(t1,w),...V(tk,w)> ∈ V(R)(w) • M,w ⊨v ¬φ iff M,w ¬⊨v φ • M,w ⊨v φ & ψ iff M,w ⊨v φ and M,w ⊨v ψ • M,w ⊨v φ (∀x)φ iff M,w ⊨v φ[x/u] for all u ∈ U • M,w ⊨v □φ iff M,w ⊨v φ for all w ∈ W • M,w ⊨v [λx.φ(x)](t) if M,w ⊨v φ[x/u] where u = g(t,w)

  24. 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

  25. 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

  26. Related notions • Conditionals • Counterfactuals • Generics • Tense • Intensionality • Doxastics (belief models)

  27. 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

  28. 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)

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