Natural language processing Lecture 7 Logic of attitudes 1 propositional attitudes Tom Att 1 believes knows that P a Att 1 relationinintension of an individual to a ID: 615823
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Slide1
Logic of Attitudes
Natural language processing
Lecture
7Slide2
Logic of attitudes
1)
‘propositional’ attitudes
Tom
Att
1
(believes, knows) that
P
a)
Att
1
/(
)
: relation-in-intension of an individual to a
proposition
b)
Att
1
*/(
n
)
: relation-in-intension of an individual to a ;
hyper-proposition
2)
‘notional’ attitudes
Tom
Att
2
(seek
s
, find
s
,
is
solving, wishing, wanting to, …)
P
a)
Att
2
/(
)
: relation-in-intension of an individual to an
intension
b)
Att
2
*/(
n
)
: relation-in-intension of an individual to a
hyper-intension
Moreover, both kinds of attitudes come in two variants;
de
dicto
and
de reSlide3
Propositional attitudes
1)
doxastic
(ancient Greek
δόξ
α; from verb δοκεῖν
dokein
, "to appear", "to seem", "to think" and "to accept")
“a believes that P”
2)
epistemic
(ancient Greek; ἐπ
ίστ
αμαι, meaning "to know, to understand, or to be acquainted with“)
“a knows that P”
Epistemic attitudes represent
factiva
;
what is known must be true
Doxastic attitudes may be false beliefsSlide4
Propositional attitudes
a
) The embedded clause
P
is
mathematical
or
logical
hyper-propositional
“
Tom believes that all prime numbers are odd”
b
) The embedded clause
P
is
analytically true/false and contains empirical terms
hyper-propositional
“
Tom does not believe that whales are mammals
“
c
) The embedded clause
P
is empirical and contains mathematical terms
hyper-propositional
“
Tom thinks that the number of Prague citizens is 1048576
“
d
) The embedded clause
P
is empirical and does not contain mathematical terms
propositional / hyper-propositional
“
Tom believes that Prague is larger than London
“ Slide5
a) Attitudes to mathematical propositions
“
Tom believes that all prime numbers are odd”
Believe*
must be a relation to a construction;
otherwise
the
paradox of an idiot
; Tom would believe every false mathematical sentence
“
Tom knows that some prime numbers are even”
Know*
must be a relation to a construction;
otherwise
the
paradox of logical/mathematical omniscience
; Tom would know every true mathematical sentenceSlide6
a) Attitudes to mathematical propositions
“
Tom believes that all prime numbers are odd”
Types
. Believe*
/(
n
)
;
Tom
/;
All
/(()()): restricted quantifier;
Prime, Odd
/()
Synthesis.
w
t
[
0
Believe*
wt
0
Tom
0
[[
0
All
0
Prime
]
0
Odd
]]
Type-checking … (yourself)
If the analysis were not
hyperintensional
, i.e., as an attitude to a
construction
, then Tom would believe every analytic False, e.g. that 1+1=3; the paradox of an idiot
Similarly, the
paradox of logical/mathematical omniscience
would arise Slide7
the
paradox of logical/mathematical omniscience
Tom knows that 1+1=2
1+1=2
iff
arithmetic is undecidable
-------------------------------------------------------
Tom knows that arithmetic is undecidable
Iff
/(
): the identity of truth-values
w
t
[
0
Know*
wt
0
Tom
0
[
0
= [
0
+
0
1
0
1]
0
2]]
0
[
0
= [
0
+
0
1
0
1]
0
2]
0
[
0
Undecidable
0
Arithmetic
]
The paradox is blocked;
/(
n
n
)
: the
non-identity
of constructions
All true (false) mathematical sentences denote the truth-value
T
(
F
); yet not in the same way. They
construct
a truth-value in different waysSlide8
the
paradox of logical/mathematical omniscience
Similarly, an attitude to an analytically true (false) sentence must be
hyperintensional
; otherwise – the paradox of logical omniscience (idiocy)
Analytically true sentence denotes
True
: the proposition that takes the truth-value
T
in all worlds
w
and times
t
Analytically false sentence denotes
False
: the proposition that takes the truth-value
F
in all worlds
w
and times
t
Example
.
Whales are mammals
denotes
True
;
Read in
de
dicto
way; the property being a mammal is a requisite of the property of being a whale
Requisite
/(
()
()
);
Whale
,
Mammal
/
()
[
0
Requisite
0
Mammal
0
Whale
]Slide9
the
paradox of logical/mathematical omniscience
b
) The embedded clause
P
is
analytically true/false and contains empirical terms
hyper-propositional
“
Tom does not believe that whales are mammals
“
w
t
[
0
Believe*
wt
0
Tom
0
[
0
Requisite
0
Mammal
0
Whale
]]
“
Tom knows that no bachelor is married
“
“
No bachelor is married
”
iff
“Whales are mammals”
Iff
/(
)
: the identity of propositions
“
Tom knows that whales are mammals
“ ??? No, not necessarily
w
t
[
0
Know*
wt
0
Tom
0
[
0
Requisite
0
Unmarried
0
Bachelor
]]
0
[
0
Requisite
0
Unmarried
0
Bachelor
]
0
[
0
Requisite
0
Mammal
0
Whale
]
The paradox is blocked;
/(
n
n
)
: the
non-identity
of constructionsSlide10
properties of
propositions
True
,
False
,
Undef
/(
)
[
0
True
wt
P
]
iff
P
wt
v-
constructs
T
,
otherwise
F
[
0
False
wt
P
]
iff
P
wt
v-
constructs
F
,
otherwise
T
[
0
Undef
wt
P
]
=
[
0
True
wt
P
]
[
0
False
wt
P
]
P,Q
Requisites.
[
0
Req
F G
] =
w
t
x
[[
0
True
wt
w
t
[
G
wt
x
]]
[
0
True
wt
w
t
[
F
wt
x
]]
F, G
()
Gloss.
The property F is a requisite of the property G
iff
necessarily, for all x holds: if it is true that x is a G then it is true that is x an F
Example
.
If it is true that Tom stopped smoking then it is true that Tom previously smoked.
[
0
Requisite
0
Mammal
0
Whale
] =
w
t
x
[[
0
True
wt
w
t
[
0
Whale
wt
x
]]
[
0
True
wt
w
t
[
0
Mammal
wt
x
]]
Slide11
Hyper-propositional attitudes
c
) The embedded clause
P
is empirical and contains mathematical terms
hyper-propositional
“
Tom thinks that the number of Prague citizens is 1048576
“
1048576
(
dec
)
= 100000
(
hexa
)
“Tom does not have to think that the number of Prague citizens is 100000
(
hexa
)
“
Note that 1048576
(
dec
)
, 100000
(
hexa
)
denote one and the same number
constructed in two different ways
:
1048576
(
dec
)
= 1.10
6
+ 0.10
5
+ 4.10
4
+ 8.10
3
+ 5.10
2
+ 7.10
1
+ 6.10
0
100000
(
hexa
)
= 1.16
5
+ 0.16
4
+ 0.16
3
+ 0.16
2
+ 0.16
1
+ 0.16
0Slide12
Hyper-propositional attitudes
“
Tom thinks that the number of Prague citizens is 1048576
“
Think
*/(
n
)
;
Tom, Prague
/;
Number_of
/(());
Citizen_of
/(())
;
w
t
[
0
Think*
wt
0
Tom
0
[
w
t
[
0
Number_of
[
0
Citizen_of
wt
0
Prague
]] =
0
1048576
]]
Type-checking …. yourselfSlide13
Propositional attitudes
d
) The embedded clause
P
is empirical and does not contain mathematical terms
propositional / hyper-propositional
“
Tom knows that London is larger than Prague
“
iff
“
Tom knows that Prague is smaller than London
“
iff
“
Tom knows that (London is larger than Prague and whales are mammals)
“
Implicit
Know
/(
)
: the relation-in-intension of an individual to a proposition
Explicit
Know*
/(
n
)
: the relation-in-intension of an individual to a hyper-propositionSlide14
Implicit knowledge
w
t
[
0
Know
wt
0
Tom
w
t
[
0
Larger
wt
0
London
0
Prague
]]
---------------------------------------------------------------------------
w
t
[
0
Know
wt
0
Tom
w
t
[
0
Smaller
wt
0
Prague
0
London
]]
Additional types.
Larger
,
Smaller
/
(
)
Proof
. In all worlds
w
and times
t
the following steps are truth-preserving:
[
0
Know
wt
0
Tom
w
t
[
0
Larger
wt
0
London
0
Prague
]] assumption
w
t
xy
[[
0
Larger
wt
x y
]
=
o
[
0
Smaller
wt
y x
]] axiom
[[
0
Larger
wt
0
London
0
Prague
]
=
o
[
0
Smaller
wt
0
Prague
0
London
]]
2) Elimination of
,
0
London
/
x,
0
Prague
/
y
w
t
[[
0
Larger
wt
0
London
0
Prague
]
=
o
[
0
Smaller
wt
0
Prague
0
London
]]
3) Introduction of
w
t
[[
0
Larger
wt
0
London
0
Prague
]
=
o
w
t
[
0
Smaller
wt
0
Prague
0
London
]]
4) Introduction of
[
0
Know
wt
0
Tom
w
t
[
0
Smaller
wt
0
Prague
0
London
]] 5) substitution of id.Slide15
Knowing is
factivum
What is known must be true
Agent
a
knows that
P
P
is true
Agent
a
does not know that
P
P
is true
P
being true is a
presupposition
of knowing
Do you know that Earth is flat?
Futile question, because the Earth is not flat! (Unless you are in a Terry Pratchett’s Discworld )
(
)[
0
Know
wt
a P
] (
)[
0
Know*
wt
a C
]
---------------------- --------------------------
[
0
True
wt
P
] [
0
True
wt
2
C
]
Types.
P
;
2
C
;
C
n
.Slide16
Computational, inferable knowledge
Know
exp
(a)
wt
Know
inf
(a)
wt
Know
imp
(a)
wt
idiot a rational a omniscient a
How to compute inferable knowledge?
K
0
(a)
wt
=
Know
exp
(a)
wt
K
1
(a)
wt
=
[
Inf
(R)
Know
exp
(a)
wt
]
K
2
(a)
wt
=
[
Inf
(R)
K
1
(a)
wt
]
…
Non-descending sequence of known hyper-propositions
There is a fixed point – computational, inferable knowledge of a rational agent who masters the set of rules
R
Inf
(R)
/((
n
)(
n
))
is a function that associates a given set
S
of constructions (hyper-propositions) with the set
S
’
of those constructions that are derivable from
S
by means of the rules
R