1 Modal logic(s). 2 Encoding modality linguistically Auxiliary (modal) verbs can, should, may, must,...
Preview:
Citation preview
- Slide 1
- 1 Modal logic(s)
- Slide 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
- 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
- Slide 9
- 9 Formal logic and inferences DeMorgans Laws ( v ) ( & ) (
& ) ( v ) Conditional Laws ( ) ( v ) ( ) ( ) ( ) ( & )
Biconditional Laws ( ) ( ) & ( ) ( ) ( & ) v ( & )
- Slide 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)]
- 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.
- Slide 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) =
- 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
- Slide 26
- 26 Related notions Conditionals Counterfactuals Generics Tense
Intensionality Doxastics (belief models)
- Slide 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
- 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)