Propositional and Notional Attitudes in Logic and Natural Language Processing
Logic of Attitudes
Natural language processing
Lecture
7
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 re
Propositional attitudes
1)
doxastic
(ancient Greek δόξα; from verb δοκεῖν
dokein
, "to appear",
"to seem", "to think" and "to accept")
•
“a believes that P”
2)
epistemic
(ancient Greek; ἐπίσταμαι, meaning "to know, to
understand, or to be acquainted with“)
•
“a knows that P”
•
Epistemic attitudes represent factiva;
what is known must be true
what is known must be true
•
Doxastic attitudes may be false beliefs
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
“
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
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
paradox of logical/mathematical omniscience
; Tom
would know every true mathematical sentence
a) Attitudes to mathematical propositions
•
“
Tom believes that all prime numbers are odd”
1.
Types
. Believe*
/(
n
)
;
Tom
/
;
All
/(
(
)(
)): restricted
quantifier;
Prime, Odd
/(
)
2.
Synthesis.
w
t
[
0
Believe*
wt
0
Tom
0
[[
0
All
0
Prime
]
0
Odd
]]
3.
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
paradox of logical/mathematical omniscience
would arise
the
paradox of logical/mathematical omniscience
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 ways
the
paradox of logical/mathematical omniscience
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
]
the
paradox of logical/mathematical omniscience
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
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
n
n
)
)
: the
non-identity
non-identity
of constructions
p
r
o
p
e
r
t
i
e
s
o
f
p
r
o
p
o
s
i
t
i
o
n
s
T
r
u
e
,
F
a
l
s
e
,
U
n
d
e
f
/
(
)
•
[
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.
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
]]
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
0
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 …. yourself
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-proposition
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:
1.
[
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
3.
[[
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
6.
[
0
Know
wt
0
Tom
w
t
[
0
Smaller
wt
0
Prague
0
London
]]
5) substitution of id.
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
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
.
Computational, inferable knowledge
Know
Know
exp
exp
(a)
(a)
wt
wt
Know
Know
inf
inf
(a)
(a)
wt
wt
Know
Know
imp
imp
(a)
(a)
wt
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
Explore the intricate concepts of propositional and notional attitudes in the context of logic and natural language processing. Dive into the distinctions between belief, knowledge, seeking, finding, solving, wishing, and wanting within the realms of individual intensions and hyper-intensions. Understand the variations of de dicto and de re attitudes and the implications of mathematical, logical, and empirical hyper-propositional clauses. Grasp the essence of doxastic and epistemic attitudes, where knowing signifies truth and belief may involve false notions. Delve into attitudes towards mathematical propositions, deciphering the fine line between belief in construction and paradoxical scenarios of idiocy and omniscience.
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Logic of Attitudes Natural language processing Lecture 7
Logic of attitudes 1) propositional attitudes Tom Att1(believes, knows) that P a) Att1/( ) : relation-in-intension of an individual to a proposition b) Att1*/( n) : relation-in-intension of an individual to a ; hyper-proposition 2) notional attitudes Tom Att2(seeks, finds, is solving, wishing, wanting to, ) P a) Att2/( ) : relation-in-intension of an individual to an intension b) Att2*/( n) : relation-in-intension of an individual to a hyper-intension Moreover, both kinds of attitudes come in two variants; de dicto and de re
Propositional attitudes 1) doxastic (ancient Greek ; from verb dokein, "to appear", "to seem", "to think" and "to accept") a believes that P 2) epistemic (ancient Greek; , 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 beliefs
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
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 sentence
a) Attitudes to mathematical propositions Tom believes that all prime numbers are odd 1. Types. Believe*/( n) ; Tom/ ; All/( ( )( )): restricted quantifier; Prime, Odd/( ) 2. Synthesis. w t [0Believe*wt0Tom 0[[0All 0Prime] 0Odd]] 3. 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
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 [0Know*wt0Tom 0[0= [0+01 01] 02]] 0[0= [0+01 01] 02] 0[0Undecidable0Arithmetic] 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 ways
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/( ) [0Requisite 0Mammal 0Whale]
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 [0Believe*wt0Tom 0[0Requisite 0Mammal 0Whale]] 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 [0Know*wt0Tom 0[0Requisite 0Unmarried 0Bachelor]] 0[0Requisite 0Unmarried 0Bachelor] 0[0Requisite 0Mammal 0Whale] The paradox is blocked; /( n n): the non-identity of constructions
Undef/( /( properties of propositions True True, False False, Undef ) ) [0Truewt P] iff Pwtv-constructs T, otherwise F [0Falsewt P] iff Pwtv-constructs F, otherwise T [0Undefwt P] = [0Truewt P] [0Falsewt P] P,Q Requisites. [0ReqF G] = w t x [[0Truewt w t [Gwtx]] [0Truewt w t [Fwtx]] 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. [0Requisite 0Mammal 0Whale] = w t x [[0Truewt w t [0Whalewt x]] [0Truewt w t [0Mammalwt x]]
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)= 1.106+ 0.105+ 4.104+ 8.103+ 5.102+ 7.101+ 6.100 100000(hexa)= 1.165+ 0.164+ 0.163+ 0.162+ 0.161+ 0.160
Hyper-propositional attitudes Tom thinks that the number of Prague citizens is 1048576 Think*/( n) ; Tom, Prague/ ; Number_of/( ( )); Citizen_of/(( ) ) ; w t [0Think*wt0Tom 0[ w t [0Number_of [0Citizen_ofwt0Prague]] = 01048576]] Type-checking . yourself
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-proposition
Implicit knowledge w t [0Knowwt0Tom w t [0Largerwt0London 0Prague]] --------------------------------------------------------------------------- w t [0Knowwt0Tom w t [0Smallerwt0Prague 0London]] Additional types. Larger, Smaller/( ) Proof. In all worlds w and times t the following steps are truth-preserving: 1. [0Knowwt0Tom w t [0Largerwt0London 0Prague]] 2. w t xy [[0Largerwt x y] =o[0Smallerwt y x]] 3. [[0Largerwt 0London 0Prague] =o[0Smallerwt 0Prague 0London]] 2) Elimination of , 0London/x, 0Prague/y 4. w t [[0Largerwt 0London 0Prague] =o[0Smallerwt 0Prague 0London]] 3) Introduction of 5. w t [[0Largerwt 0London 0Prague] =o w t [0Smallerwt 0Prague 0London]] 4) Introduction of 6. [0Knowwt0Tom w t [0Smallerwt 0Prague 0London]] assumption axiom 5) substitution of id.
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 ) ( )[0Knowwta P] ---------------------- [0TruewtP] Types. P ; 2C ; C n. ( )[0Know*wta C] -------------------------- [0Truewt2C]
Computational, inferable knowledge Knowexp(a)wt idiot a How to compute inferable knowledge? K0(a)wt= Knowexp(a)wt K1(a)wt= [Inf(R) Knowexp(a)wt] K2(a)wt= [Inf(R) K1(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 Knowinf(a)wt rational a Knowimp(a)wt omniscient a
