Intensional Logic TIL Marie Duží httpwwwcsvsbczduzi Natural Language Processing The most important applications Logical analysis of natural language Multiagent systems agents communication and reasoning ID: 555942
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Slide1
Natural Language Processing (Transparent Intensional Logic)TIL
Marie Duží
http://www.cs.vsb.cz/duzi/
Slide2
Natural Language Processing The most important applications of TIL logical
system
in NLP
Logical analysis of natural language
Multi-agent systems; agent’s communication and reasoning
T
he
TIL-Script
functional programming language
Materials to study:
http://www.cs.vsb.cz/duzi/TIL.html
Duží M., Jespersen B.,
Materna
P. (2010):
Procedural Semantics for
Hyperintensional
Logic.
SpringerSlide3
Logical semanticsLogic is
about reasoning,
about argumentation that is going from premises to a conclusion
the
analysis and appraisal of arguments
When you do logic, you try to clarify reasoning and separate good from bad reasoning, i.e.,
Separate valid arguments from invalid ones
Valid argument
(
example
)
If you overslept, you are late
You are not late
You didn’t oversleep
The conclusion is
logically entailed
by the premisesSlide4
Valid arguments
An argument is
valid
if it would be contradictory (impossible) to have the premises all true and conclusion false
.
When you do reasoning, you use
natural language
When you
analyse
arguments, you must
analyse
premises
in a fine
-grained way so that not to infer something that is not entailed by the premises.
Our goal: to build up an
inference machine
that neither over-infers (
paradoxes), nor under-infers ( lack of knowledge)
The more fine-grained the analysis of the
meaning
of premises is, the better inference machine can we build upSlide5
Logical semanticsPropositional logic
Is very limited; semantics reduced to assigning
T
(rue)
,
F
(
alse
)
to atomic propositions and composing these propositions by means of truth-value functions
(Boolean) algebra of truth-values
1
st
-order predicate logic (FOL)
Limited analysis of the structure of atomic propositions
–
up to assigning properties and relations to individuals
Apt for
mat
h
emati
cs
,
problems with natural languageSlide6
Coarse-grained analysis - problemsSome prime numbers are even
Some odd numbers are even
Some students are lazy
Formalization in FOL
:
x
[
P
(
x
)
Q
(
x
)]
Questions
:
How come that the sentences
(1), (2), (3)
have the same analysis
?
is analytically true sentence
is analytically false sentence
is empirical sentence, maybe true, maybe false
How come that the formula
x
[
P
(
x
)
Q
(
x
)]
has interpretations in which it is true and other interpretations in which it is false if it is the analysis of (1) or (2)
?
How does the translation of natural language sentences into the language of FOL contribute to understanding their meaning
?Slide7
Coarse-grained analysisNo bachelor is married
No bachelor is rich
FOL
:
x
[
P
(
x
)
Q
(
x
)]
or
x
[
P
(
x
)
Q
(
x
)]
Questions
:
How come that both the sentences have t
w
o analyses, and which of them is the ‘correct’ one?
After all, (1) is
analytically true
(true at all state-of-affairs), while (2) is
empirical statement
, which might be true at some state-of-affairs and false in othersSlide8
Coarse-grained analysis of premisesDoes it matter? If we always could
validly derive conclusions
that are entailed by the premises, the coarse-grained analysis would be OK
Does the analysis in FOL make it possible to validly derive consequences entailed by the premises
?
Unfortunately, it does not.
Coarse-grained analysis of premises yields
paradoxes
inferring something that is not entailed by the premises (over-inferring) and under-inferringSlide9
ParadoxesNecessarily, 8 5
The number of planets = 8
––––––––––––––––––––––––––––––– ???
Necessarily, the number of planets
5
Quine’s paradox
Solution: Modal logic, introducing an operator
□
It is ordered to deliver a letter
If the letter is delivered, then it is delivered or burnt out
––––––––––––––––––––––––––––––––––––––––––– ???
It is ordered to deliver the letter or burn out
Ross’s paradox
Deontic logic, introducing operator
O
Substitution of
identicals
is not allowed within the scope of operators
opaque,
intensional
contexts
It removes symptoms but does not cure the causeSlide10
ParadoxesJohn believes that Prague has 1.048.576 citizens 1.048.576 = 100 000
(16)
–––––––––––––––––––––––––––––––––
???
John believes that Prague has
100 000
(16)
citizens
Doxastic
/
Epist
emic logics
,
introduce operators
B
, K
John knows that
1
+1=2
1+1=2
Sin(
) =
0
–––––––––––––––––––––––
???
John knows that
Sin(
) =
0
Doxastic
/
Epist
emic logics
,
introduce operators
B
(
elieve
) and
K
(now)
The paradox of logical/mathematical omniscienceSlide11
ParadoxesJohn calculates 2 + 52 + 5 = 7
––––––––––––––
???
John calculates
7
Oidipus
seeks the murderer of his father
Oidipus
is the murderer of his father
–––––––––––––––––––––––––
???
Oidipus
seeks
Oidip
us
Attitude logics, …Slide12
ParadoxesThe US President is the husband of MelaniaHillary wanted to become the US president
––––––––––––––––––––––––––––––––––––––––
Hillary wanted to become the husband of
Melania
Tom
believes that the King of France is wise
–––––––––––––––––––––––––––––––––––––––
???
The King of France exists
Tom is seeking an abominable snowman (yeti)
–––––––––––––––––––––––––––––––––––––––
???
Abominable snowman exists
Logic – magic ???Slide13
Extensional vs. intensional (opaque) contextWhen is the context extensional?The context is extensional if the extensional rules like
s
ubstitution
of
identit
i
cal
s
a
nd
existen
tial
generaliza
tion
are valid
A
nd
when are these rules valid
?
In an extensional context
Hmmm
We stir clear of this circle by
Defining three kinds of context first
Defining universally valid rules Slide14
Transparent Intensional Logic (TIL)There is a
spreading
and
still growing tree of particular logics
It has
been growing bottom
up
Is
it
OK?
Shouldn’t here be just one universal logic?
Aren’t logical rules valid universally?
TIL – univers
al logical framework
„top
down
“
approach
Logical rules are valid universally, only they have to be properly appliedSlide15
Procedural semantics of TIL Expression
S
ense
(procedur
e
,
c
onstruc
tion
)
denot
ation
Ontolog
y of
TIL: r
amified
hierarchy of types
15Slide16
TIL: three kinds of context
Hyperintensional
;
construction
of the denoted function is an object of predication
Tom computes
Sin
(
)
Tom
believes that
the
Pope
is wise
but does not
believe that
the Bishop of Rome
is wise
Intensional
; the denoted
function itself
is an object of predication
Sin
e
is a periodic function
Tom wants to become the
Pope
Extensional
; value of the denoted function is an object of predicationSin() = 0The Pope is wise.Slide17
17
TIL
Ontolog
y (types of order 1)
(
non
-
procedural
objects)
Basic types
truth-values {T, F}
(
)
universe of discourse {individuals}
(
)
times or real numbers
(
)
possible worlds
(
)
Functional types
(
1
…
n
)
partial
functions
(
1
…
n
)
PWS Intensions
– entities of type
((
))
;
Slide18
Functional approach; sets and relations All the denoted objects are functions
, possibly in an extreme case 0-ary functions without arguments, i.e. atomic objects like individuals of type
or numbers of type
How then do we model
sets
and
relations
(-in-extension)?
By
characteristic functions
. Hence, a set of -elements is an object of type (), Binary relation between - and -objects is an object of type ()
Examples.
The set of prime numbers is an object of type (
); in symbols
Prime
/
(
)
The set of solutions of the equation
Sin(x) =
0, i.e. the set of multiples of is also an object of type
(
)
The relation > defined on numbers is an object of type
(
)Slide19
Possible worldsno sci-phi !No multiple universesUniverse of discourse: the collection of bare individuals –
abstract hangers (determined just by an ID) to hang particular traits and relations on
Possible world
: chronology of maximal consistent distributions of these basic traits among individuals
PWS-intensions
/ ((
)
); or
for shortSlide20
Examples of PWS-intensionspropositions of type (()
) or
for short;
denoted by sentences like “Tom is a student”
properties
of individuals of type ((
(
)
)
) or
(
)
for short;
denoted by nouns or adjectives like ‘(being a) student’, ‘round’, …
binary relations
-in-intension between individuals of type
(
)
;
denoted by verbs like ‘to kick’, ‘to like’, …individual offices (or roles) of type ; denoted by definite descriptions like ‘the Pope’, ‘the US president’, ‘Miss World 2019’, … Slide21
Examples of extensions (not functions with the domain )Logical objects like truth-functions and quantifiers are extensional
(conjunction),
(disjunction) and
(implication)
are of type (
), and
(Boolean negation)
of type (
).
Quantifiers
,
are type-theoretically polymorphic total functions of type (
(
)), for an arbitrary type
, defined as follows.
The
universal quantifier is a function that associates a class A of -elements with T if A contains all elements of the type , otherwise with F. The existential quantifier
is a function that associates a class A of
-elements with T if A is a non-empty class, otherwise with F
. Slide22
22
Constructions
Variables
x, y
, p, w, t, … v
-construct
Trivializa
tion
0
C
constructs
C
(of any type)
a
fixed pointer
to
C
and the
dereference
of the pointer.
In order to operate on
C
,
C
needs to be grabbed, or ‘called’, first. Trivialization is such a grabbing mechanism.
Closure
[
x
1
…
x
n
X
]
(
1
…
n
)
1
n
Composition
[
F X
1
…
X
n
]
(
1
…
n
)
1
n
Execution
1
X
, Double Execution
2
XSlide23
23
TIL Ontology (
higher-order types)
Constructions of order 1
(
1
)
construct entities belonging to a type of order 1
/ belong to
1
:
type of order
2
Constructions of order 2
(
2
)
construct entities belonging to a type of order 2 or 1
/ belong to
2
:
type of order
3
Constructions of order n
(
n
)
construct entities belonging to a type of order
n
1
/ belong to
n
:
type of order
n + 1
Functional entities
:
(
1
…
n
)
/ belong to
n
(
n
:
the highest of the types to which
,
1
, …,
n
belong)
And so on,
ad infinitumSlide24
explicit intensionalization and temporalization constructions of possible-world intensions directly encoded in the logical syntax:
w
t
[…
w
….
t
…]
w
;
t
;
0
Happy
(
)
;
0
Pope
w t [0
Happywt
0Pope
wt]
In any possible world (
w
) at any time (
t
):
Take the property of being happy (
0
Happy
)
Take the papal office (
0
Pope
)
Extensoinalize
both of them (
0
Happy
wt
,
0
Pope
wt
)
Check whether the holder of the Papal office is happy at that
w
,
t
of evaluation (
[
0
Happy
wt
0
Pope
wt
]
) Slide25
Method of analysisAssing typ
es
to objects that are mentioned by the expression
E
,
i.e
.
to the objects denoted by some subexpression of
E.
Compose constructions
of objects
ad 1)
to construct the object denoted by
E
.
Semantically simple expressions (including idioms) are furnished with Trivialization of the denoted object as their meaning
Type checking
.
Slide26
Example: „The Mayor of Ostrava“
Typ
es
:
Mayor_of
/(((
)
)
) –
abbr
. (
)
:attribute;
Ostrava
/
,
Mayor_of
_Ostrav
a
/((
)
) – abbr.
Synthesis: wt [0Mayor_ofwt 0Ostrava]Type checking: w t [[[0Mayor_of w] t] 0Ostrava
] ((()
))
(()
)
(
)
(
)
((
)
)
abbreviated as
(
individu
al office
)Slide27
„The Mayor of Ostrava is rich“
Additional t
yp
e
:
Rich
/()
Synt
hesis
:
w
t
[
0
Rich
wt
w
t
[
0
Mayor_of
wt
0Ostrava]]wt]Typechecking
(shortened): w t [[[0Richwt wt [0Mayor_ofwt 0Ostrava]]wt] ()
()
(()
) abbr.
(
propo
s
i
tion
)Slide28
TIL vs. Montague’s ILIL is an extensional logic, since the axiom of extensionality is valid: x (Ax = Bx
)
A = B
.
This is a good thing. However, the price exacted for the simplification of the language (due to ghost variables) is too high;
the law of universal instantiation, lambda conversion and Leibniz’s Law do not generally hold, all of which is rather unattractive.
Worse, IL does
not
validate the Church-Rosser ‘diamond’
. It is a well-known fact that an ordinary typed
-calculus will have this property. Given a term
x
(
A
)
B
(the
redex
), we can simplify the term to the form [
B
/
x
]
A
, and the order in which we reduce particular redexes does not matter. The resulting term is uniquely determined up to -renaming variables.
TIL does not have this defect; it validates the Church-Rosser property though it works with n-ary partial functionsthe functions of TY2 are restricted to unary total functions (Schönfinkel)Slide29
TIL: logical coreconstructions +
type hierarchy
(simple and ramified)
The
ramified
type hierarchy organizes all higher-order objects:
constructions (types
n
)
,
as well as functions with domain or range in constructions.
The
simple
type hierarchy organizes first-order objects:
non-constructions
like extensions (individuals, numbers, sets, etc.), possible-world intensions (functions from possible worlds) and their arguments and values. Slide30
Hyperintensionality was
born
out
of
a
negative
need, to block invalid inferences
Carnap
(1947,
§§13
ff
)
;
there
are
contexts
that
are
neither
extensional
nor
intensional
(attitudes)Cresswell; any context in which substitution of necessary equivalent terms fails is hyperintensionalYet, which inferences are valid in hyperintensional contexts?How hyper are hyperintensions?
Which contexts are intensional / hyperintensional?TIL definition is positive: a context is hyperintensional if the very meaning procedure is an object of predication
30Slide31
Three kinds of contexthyperintensional context
: a meaning construction occurs
displayed
so that the very
construction
is an object of predication
though a construction at least one order higher need to be executed in order to produce the displayed construction
intensional
context
: a meaning construction occurs
executed
in order to produce a function
f
so that
the whole
function
f is an object of predication
moreover, the executed construction does not occur within another displayed construction
extensional context
: the meaning construction is
executed
in order to produce a particular value of the so-constructed function
f
at its argument
so that
the
value
of the function f is an object of predication
moreover, the executed construction does not occur within another
intensional or hyperintensional context. Slide32
HyperintensionalityExtensional logic of hyperintensionsTransparency: no context is opaqueThe same (extensional) logical rules are valid in all kinds of context;
Leibniz’s substitution of identicals, existential quantification even into hyperintensional contexts, …
Only the types of objects these rules are applied at differ according to a context
Anti-contextualism: constructions are assigned to expressions as their context-invariant meanings