1. 1 Modal logic(s)

2. 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. 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. 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. 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. 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. 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. 8 Logical inferences Modus Ponens: p  q p -------- q Modus Tollens: p  q  q ---------  p Hypothetical syllogism: p  q q  r -------- p  r Disjunctive syllogism: p v q  p -------- q

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

10. 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. 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. 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. 13 Contingency and truth non-contingent contingent true statements false statements possibly true statements (= not necessarily false) not possibly true (= necessarily false) not possibly false (= necessarily true) possibly false statements (= not necessarily true)

14. 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. 15 Model-theoretic valuation M = where U is domain of individuals V is a valuation function For example, U = {mary, bill, pc23} V (likes) = {, } V (hungry) = {mary, bill} V (is broken) = {pc23} V (is French) = Ø

16. 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 ∈ [[likes]] iff ∈ 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. 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. 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. 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. 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. 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. 22 S5 Not adequate for all types of modality However, it is commonly used for database work

23. 23 O say what is (modal) truth? Let M = 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(t 1,...,t k ) is atomic, M,w ⊨ v R(t 1...t k ) iff ∈ 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)

24. 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. 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. 26 Related notions Conditionals Counterfactuals Generics Tense Intensionality Doxastics (belief models)

27. 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. 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)