1. Natural language processing Lecture 7
Logic of Attitudes
Natural language processing
Lecture 7

2. 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

3. 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, not only that, it is a presupposition of knowing
Doxastic attitudes may be beliefs in false …

4. 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 “
d) The embedded clause P is empirical and does not contain mathematical terms  propositional / hyper-propositional
“Tom believes that Prague is larger than London“

5. 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

6. 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 [0Believe*wt 0Tom 0[[0All 0Prime] 0Odd]]
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.
Moreover, it is of no sense to believe a truth-value without any procedure specifying it
Similarly, the paradox of logical/mathematical omniscience would arise

7. 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*wt 0Tom 0[0= [ ] 02]]
0[0= [ ] 02]  0[0Undecidable 0Arithmetic]
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

8. 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/()
wt [0Requisite 0Mammal 0Whale]  

9. 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*wt 0Tom 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*wt 0Tom 0[0Requisite 0Unmarried 0Bachelor]]
0[0Requisite 0Unmarried 0Bachelor]  0[0Requisite 0Mammal 0Whale]
The paradox is blocked; /(nn): the non-identity of constructions

10. properties of propositions True, False, Undef/()
[0Truewt P] iff Pwt v-constructs T, otherwise F
[0Falsewt P] iff Pwt v-constructs T, otherwise F
[0Undefwt P] = [0Truewt P]  [0Falsewt P]
P,Q  
Requisites.
[0Req F G] =
wt x [[0Truewt wt [Gwt x]]  [0Truewt wt [Fwt 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.
[0Requisite 0Mammal 0Whale] =
wt x [[0Truewt wt [0Whalewt x]]  [0Truewt wt [0Mammalwt x]]

11. 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 “
(dec) = (hexa)
“Tom does not have to think that the number of Prague citizens is (hexa)“
Note that (dec), (hexa) denote one and the same number constructed in two different ways:
(dec) =
100000(hexa) =

12. Hyper-propositional attitudes
“Tom believes that the number of Prague citizens is “
Believe*/(n); Tom, Prague/; Number_of/(()); Citizen_of/(());
wt [0Believe*wt 0Tom [wt [0Number_of [0Citizen_ofwt 0Prague]] = ]]
Type-checking …. Yourself
“Tom believes that the number of people living in Prague is greater than one million“ ???
In a hyperintensional context only procedurally isomorphic construction is substitutable

13. 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  too permissive, any equivalent way of presenting the proposition must be known as well
Explicit Know*/(n): the relation-in-intension of an individual to a hyper-proposition  too restrictive, only the same way of presenting the proposition is known as well

14. Implicit knowledge
wt [0Knowwt 0Tom wt [0Largerwt 0London 0Prague]]
wt [0Knowwt 0Tom wt [0Smallerwt 0Prague 0London]]
Additional types. Larger, Smaller/()
Proof. In all worlds w and times t the following steps are truth-preserving:
[0Knowwt 0Tom wt [0Largerwt 0London 0Prague]] assumption
wt xy [[0Largerwt x y] =o [0Smallerwt y x]] axiom
[[0Largerwt 0London 0Prague] =o [0Smallerwt 0Prague 0London]] 2) Elimination of , 0London/x, 0Prague/y
wt [[0Largerwt 0London 0Prague] =o [0Smallerwt 0Prague 0London]] 3) Introduction of 
[wt [0Largerwt 0London 0Prague] =o wt [0Smallerwt 0Prague 0London]] 4) Introduction of 
[0Knowwt 0Tom wt [0Smallerwt 0Prague 0London]] 1),5) substitution of id.

15. Computational, inferable knowledge
Knowexp(a)wt  Knowinf(a)wt  Knowimp(a)wt
idiot a rational a omniscient 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 but resource bounded 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

16. Computational, inferable knowledge
“Tom believes that the number of Prague citizens is “ “Tom believes that the number of people living in Prague is greater than one million“ Inferable on the assumption that the set of rules Tom masters includes some elementary mathematics and the rule [0Citizen-ofwt x y] |-- [0Live-inwt x y]

17. 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 the Earth is flat? Futile question, because the Earth is not flat! (Unless you are in Terry Pratchett’s Discworld )
()[0Knowwt a P] ()[0Know*wt a C]
[0Truewt P] [0Truewt 2C]
Types. P  ; 2C  ; C  n.

18. “The Mayor of Ostrava doesn’t know that Tom knows that he (the mayor) is not going to the Alps”.
MO/: the individual office occupied by at most one individual
[0Know*wt 0MOwt 0[wt [0Know*wt 0Tom [0Sub [0Tr 0MOwt] 0he 0[wt [0Gowt he 0Alps]]]]]] |
[0Truewt 20[wt [0Know*wt 0Tom [0Sub [0Tr 0MOwt] 0he 0[wt [0Gowt he 0Alps]]]]]] |
[0Truewt [wt [0Know*wt 0Tom [0Sub [0Tr 0MOwt] 0he 0[wt [0Gowt he 0Alps]]]]]] |
[0Know*wt 0Tom [0Sub [0Tr 0MOwt] 0he 0[wt [0Gowt he 0Alps]]]] |
[0Truewt 2[0Sub [0Tr 0MOwt] 0he 0[wt [0Gowt he 0Alps]]]] |
2[0Sub [0Tr 0MOwt] 0he 0[wt [0Gowt he 0Alps]]]wt
Tübingen 2019

19. 0MO occurs extensionally, de re Two principles de re are valid
“The Mayor of Ostrava doesn’t know that Tom knows that he (the mayor) is not going to the Alps”.
0MO occurs extensionally, de re
Two principles de re are valid
Substitution of v-congruent constructions
Existential presupposition
Let [0Peter = 0MOwt].
Then the evaluation of the substitution comes down to this:
2[0Sub [0Tr 0MOwt] 0he 0[wt [0Gowt he 0Alps]]]wt (anaphora resolution)
“… that he (the Mayor of Ostrava) doesn’t go …”
= 20[wt [0Gowt 0Peter 0Alps]]wt
= [0Gowt 0Peter 0Alps]
Tübingen 2019

20. Propositional attitudes de dicto vs. de re
a believes that the Pope is wise (the Pope de dicto)
a believes of the Pope that the he is wise (the Pope de re)
De dicto: none of the two principles de re is valid;
the Pope does not have to exist;
if the Pope is Francisco, it doesn’t entail that a believes that Francisco is wise
De re: both the principles de re are valid
The sentence presupposes that the Pope exists
if the Pope is Francisco, then a believes that Francisco is wise
Both the attitudes are logically independent, i.e. neither is entailed by the other
That de dicto doesn’t entail de re is obvious;
De re ⊩ de dicto ??? No;
assume that Karol Wojtyla is the Pope; a may be in good friends with Karol from the very childhood, because he knows that Karol is wise; yet, he may have no idea that Wojtyla was elected for the Pope. The ‘reporter’ uses the papal office as a pointer to Wojtyla to make the report more sensational; hence the reporter must know that Wojtyla is the Pope rather than a

21. Propositional attitudes de dicto
a believes that the Pope is wise
wt [0Believewt a wt [0Wisewt 0Popewt]]
Types. Believe/(): intensional attitude to a proposition; a  ; Wise/(); Pope/.
Type checking. [0Wisewt 0Popewt]  ; wt [0Wisewt 0Popewt]  ; [0Believewt a wt [0Wisewt 0Popewt]]  ; wt [0Believewt a wt [0Wisewt 0Popewt]]  .
What can be derived? That
there is an office (rather than an individual) such that a believes that its holder is wise.
wt f [0Believewt a wt [0Wisewt fwt]]; f  .

22. Propositional attitudes de re
a believes of the Pope that the he is wise
There are two options.
To analyse the sentence in passive “The Pope is believed by a to be wise”
Literal analysis by applying the substitution method
Ad (1) Let BaW/() be the property of individuals of being believed by a to be wise.
Then a coarse-grained analysis of the sentence is
wt [0BaWwt 0Popewt]]; 0Pope occurs with de re supposition
Refinement of the analysis of the property BaW comes down to this:
0BaW = wt x [0Believewt a wt [0Wisewt x]]; x  .
Substituting the refined analysis for 0BaW we obtain
wt [wt x [0Believewt a wt [0Wisewt x]]wt 0Popewt];
or after applying the restricted -reduction by name (substituting w, t for w, t)
wt [x [0Believewt a wt [0Wisewt x]] 0Popewt]
0Pope occurs with de re supposition

23. Propositional attitudes de re
Ad (2) Literal analysis of “a believes of the Pope that the he is wise”
he   is an anaphoric reference to the holder of the papal office.
For resolving anaphoric references we apply the substitution method that exploits the functions Sub/(nnnn) and Tr/(n). Recall:
Sub/(nnnn) operates on constructions by substituting the product of the first for the product of the second into the product of the third; as a result, an adjusted construction arises
Tr/(n) returns as its value the Trivialization of the input -object
“he is wise”  wt [0Wisewt he]]; he  .
“of the Pope that he is wise”  [0Sub [0Tr 0Popewt] 0he 0[wt [0Wisewt he]]]  n
0Pope occurs with de re supposition
“a believes of the Pope that he is wise”  wt [Believewt a 2[0Sub [0Tr 0Popewt] 0he 0[wt [0Wisewt he]]]]
Question. How would you adjust the analysis if we had Believe*/(n) ?

24. Propositional attitudes de re
wt [x [0Believewt a wt [0Wisewt x]] 0Popewt]
wt [Believewt a 2[0Sub [0Tr 0Popewt] 0he 0[wt [0Wisewt he]]]]
Consider (1). What about going on with -reduction by name? As a result, we’d obtain wt [0Believewt a wt [0Wisewt 0Popewt]].
But this is the analysis of the de dicto attitude! How come?
First, there is a collision of variables; we must rename:
w0t0 [0Believew0t0 a w1t1 [0Wisew1t1 0Popew0t0]].
Does 0Pope occur de re here? No, still de dicto though being applied to the world w0 and time t0 of evaluation, because it occurs in the w1t1-generic intensional context.
De dicto and de re attitudes are logically independent so that (1) cannot be equivalent to the de dicto attitude. And it is not. The problem is due to -reduction by name, which is not an equivalent transformation.
We must apply -reduction by value:
wt [Believewt a 2[0Sub [0Tr 0Popewt] 0x 0[wt [0Wisewt x]]]]

25. Propositional attitudes de re
wt [x [0Believewt a wt [0Wisewt x]] 0Popewt]
wt [Believewt a 2[0Sub [0Tr 0Popewt] 0he 0[wt [0Wisewt he]]]]
These two constructions are provably equivalent.
What can be derived? That
there is an individual y such that a believes that y is wise.
wt y [0Believewt a wt [0Wisewt y]]; y  .
Not only that. The two principles de re are valid, as it should be.
Proof … home exercise