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 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 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 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 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 = 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 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 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 = 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 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)