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					Modal logic(s)




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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

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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
            pq
            ¬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
       ((pq) & (pr)) 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:
    pq                  pq
       p                 qr
    --------             --------
       q                 pr
   Modus Tollens:      Disjunctive syllogism:
    pq                  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
            □(AB)  (□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)
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posted:12/8/2011
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