# Semantics

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```					Modal logic(s)

1
Encoding modality linguistically
   Auxiliary (modal) verbs
   can, should, may, must, could, ought to, ...
   possibly, perhaps, allegedly, ...
   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
   pq
   ¬p  ¬q
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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
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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.
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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 ( & )
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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)
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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) &

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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
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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

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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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