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
Slide 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.
Slide 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
Slide 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
Slide 6
6 Propositional logic (2) Syntax (well-formed formulas, wffs):
Any atomic proposition is a wff. If is a wff, then is a wff. If and
are wffs, then ( & ), ( v ), ( ), and ( ) are wffs. 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
Slide 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.
Slide 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
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)]
Slide 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)
Slide 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)
Slide 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)
Slide 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.
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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) =
Slide 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
Slide 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
Slide 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)
Slide 19
19 Semantics of operators If = , then [[]] M,w,g =1 iff w W,
[[]] M,w,g =1. If = , then [[]] M,w,g =1 iff there exists at least
one w W such that [[]] M,w,g =1.
Slide 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.
Slide 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
Slide 22
22 S5 Not adequate for all types of modality However, it is
commonly used for database work
Slide 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 iff I()(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 iff M,w v 4. M,w v & iff M,w v and M,w v 5.
M,w v ( x) iff M,w v [x/u] for all u U 6. M,w v iff M,w v for all w
W 7. M,w v [x.(x)](t) if M,w v [x/u] where u = g(t,w)
Slide 24
24 Human necessity is a human necessity iff it is true in all
worlds closest to the ideal If W is the modal base, wW there exists
wW such that: w w, and wW, if w w then is true in w is a human
possibility iff is not a human necessity
Slide 25
25 Backgrounds (Kratzer) Realistic: for each w, set of ps that
are true Totally realistic: set of ps that uniquely define w
Epistemic: ps that are established knowledge in w Stereotypical: ps
in the normal course of w Deontic: ps that are commanded in w
Teleological: ps that are related to aims in w Buletic: ps that are
wished/desirable in w Empty: the empty set of ps in any w
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
Slide 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)
