Natural Language processing
Marie Duží
http://www.cs.vsb.cz/duzi/
Semantic scheme
Expression
encodes
denotes
procedure (construction)
v-constructs
denotation
Ontology: ramified hierarchy of types
Examples; exercise No. 2

1.
2.
3.
4.
5.
6.
All prime numbers greater than 2 are odd.
5 is a prime number
5 is greater than 2
––––––––––––––––––––––––––––––––––––
5 is an odd number
[0x [[[0Prime x]  [0> x 02]]  [0Odd x]]]
[0Prime 05]
[0> 05 02]
[[[0Prime 05]  [0> 05 02]]  [0Odd 05]] E, 5/x,1
[[0Prime 05]  [0> 05 02]]
I, 2,3
[0Odd 05]
modus ponens, 4,5
Examples; exercise No. 2








1.
2.
3.
4.
All dogs bark
Alík is a dog
–––––––––––––––
Alík barks
wt [[0All 0Dogwt] 0Barkwt]; All/((())())
wt [0Dogwt 0Alík]
How to prove that Alík barks? Let us define:
0All = mn x [[m x]  [n x]]; m, n  ()
[[0All m] n] = x [[m x]  [n x]]
[[0All 0Dogwt] 0Barkwt] = x [[0Dogwt x]  [0Barkwt x]] premise 1
[[0Dogwt 0Alík]  [0Barkwt 0Alík]]
E, x/Alík
[0Dogwt 0Alík]
premise 2
[0Barkwt 0Alík]
modus ponens, 2,3
Examples; exercise No. 2







Tom wants to be the president of ČR.
Prezident ČR is a husband of Ivana.
––––––––––––––––––––––––––––––
Tom wants to be a husband of Ivana.
Tom, CR, Ivana/; Prezident_of, Husband_of/();
Want_to_be/(): relation-in-intension of an individual to an
office; Prezident_of_CR, Husband_of_Ivana/; =/(): identity of
individuals
wt [0Want_to_bewt 0Tom wt [0Prezident_ofwt 0CR]]
wt [0= wt [0Prezident_ofwt 0CR]wt wt [0Husband_ofwt 0Ivana]wt]
-----------------------------------------------------------------------------------------wt [0Want_to_bewt 0Tom wt [0Husband_ofwt 0Ivana]]
The argument is invalid, because Want_to_be is a relation to an office
rather than to an individual. Hence, we can substitute only one and the
same office. That (according to the second premise) two different
offices (roles) happen to be occupied by the same individual is
irrelevant.
Example






Tom want to become the Pope.
The Pope is Francisco.
–––––––––––––––––––––––––––
Tom want to become Francisco.
wt [0Want_to_becomewt 0Tom 0Pope]
wt [0= 0Popewt 0Francisco]
-----------------------------------------------------------wt [0Want_to_becomewt 0Tom 0Francisco]
Want_to_become/(); Pope/; =/(): identity of individuals
The argument is invalid for the same reasons as above. Only one and the same
office (role) is substitutable. The second premise establishes a contingent fact
that the papal office happens to be occupied by Francisco, which is irrelevant for
the substitution.
No individual can miraculously change its identity. Individuals are bare; they are
given merely by their identity.
De dicto vs. de re


wt [0Want_to_bewt 0Tom 0Pope]
de dicto
wt [0= 0Popewt 0Francisco]
de re
De dicto / de re


Tom wants to be the Pope.
The Pope is the Bishop of Rome.
(read de dicto, as the identity of an office)




––––––––––––––––––––––––––––––––––––
Tom wants to be the Bishop of Rome.
wt [0Want_to_bewt 0Tom 0Pope]
wt [0=u 0Pope wt [0Bishop_ofwt 0Rome]]
------------------------------------------------------------------------wt [0Want_to_bewt 0Tom wt [0Bishop_ofwt 0Rome]]
Want_to_be/(); =u/(): identity of an office
The argument is valid. One and the same office can be substituted,
although the office is conceptualized (constructed) by two different
ways.
De dicto vs. de re (concerns the meaning of empirical expressions)


Let C  Intension/ is a constituent of D.
 Note. A constituent of a construction D is such a subconstruction C
of D that occurs in the execution mode. It means that if one wants
to execute the whole D they must execute also C. The occurrence
of C is not hyperintensional, i.e. within the scope of Trivialization.
The occurrence of C in D is in de dicto supposition, i.e. intensional,
if the whole function (Intension) is an object of predication,
i.e. the whole Intension is an argument of another function constructed within D.

The occurrence of C in D is in de re supposition, i.e. extensional, if
the value of the function (Intension) is an object of predication,
i.e. the value of the Intension in a given world w and time t is an argument of
another function constructed within D.


Moreover, this occurrence of C is not in D is not a subconstruction
of another construction occurring de dicto in D.
The higher intensional de dicto context is dominant over a lower
extensional de re context.
Two principles de re
1.
2.
Existential presupposition
Substitution of co-referential expressions
(with v-congruent meaning constructions)
Francisco is the Pope:
wt [0= 0Francisco 0Popewt]
-------------------------------------Hence, the Pope exists:
wt [0Existwt 0Pope] (the papal office is occupied)
Exist/(): the property of an office of being occupied
Two principles de re
2.
Substitution of co-referential expressions
(with v-congruent meaning constructions)
The Pope is Francisco
The Pope is wise
--------------------------------Francisco is wise
wt [0= 0Popewt 0Francisco]
wt [0Wisewt 0Popewt]
------------------------------------wt [0Wisewt 0Francisco]
Proof. In any w, t (elimination of ) the following steps are truth-preserving
1. [0= 0Popewt 0Francisco]
assumption
0
0
2. [ Wisewt Popewt]
assumption
3. [0Wisewt 0Francisco]
Leibniz: substitution of identicals
4. wt [0Wisewt 0Francisco]
introduction of 
Existence


Is not a property of bare individuals
 Aristotle in Analytica Posteriora, II, 7, 92b13 says “being is not a
genus”
 Kant in Critique of Pure Reason: “Being is … not a real predicate”
 Russell (Principia Mathematica, 2nd ed., p. 175): “… there is no
reason to suppose that a meaning of existence could be found
which would be applicable to immediately given subjects”.
Yet, non-trivial existence is predicated:

The Pope exists, the King of France does not exist, hobbits do not exist, …
Existence is a property, but not of individuals; rather, it is a
property of functional objects of a higher kind; it is a property of
functions/intensions of having a value at a given argument
 In our case it is a property of an individual office; namely
Exist/(): the property of being occupied in a given w and t

Existence
wt [0= 0Popewt 0Francisco]
-------------------------------------wt [0Existwt 0Pope]
How to prove it? Let us define, refine, calculemus …
Exist = wt u [0x [x = uwt]]; u v , x v , =/(): identity
[0Existwt 0Pope] = [u [0x [x = uwt]] 0Pope] = [0x [x = 0Popewt]]





[0= 0Popewt 0Francisco]
[x [0= 0Popewt x] 0 Francisco]
[0Empty x [0= 0Popewt x]]
[0x [0= 0Popewt x]]
[0Existwt 0Pope]
premise
-abstraction
def. of Composition
def. of 
def. of Exist
Substitution

De dicto context is intensional:




The whole constructed function (intension) f is an object of
predication
Substitution of a construction D for C (occurring de dicto) is
valid only if D v-constructs the same function f.
Hence C=D, the constructions are equivalent, i.e. vcongruent for every valuation v
De re kontext is extensional:



The value of the constructed function (intension) f is an
object of predication
Substitution of a construction D for C (occurring de re) is
valid only if D v-constructs the same value (even of a
different intension)
Hence C =v D, the constructions are v-congruent for a
given valuation v
Presupposition vs. (mere) entailement
(i) P is a presupposition of S:
(S |= P) and (non-S |= P)
Corollary: If P is not true, then neither S nor non-S is
true; S has no truth-value.
(ii) P is merely entailed by S, but P is not a
presupposition of S: (S |= P),
but neither (non-S |= P) nor (non-S |= non-P)
Hence if S is not true we cannot deduce anything
about the truth of P
Entailment: in any state-of-affairs w, t in which
premises are true the conclusion is true as well.
Existential presupposition de re




The pope is wise |= The pope exists
The pope is not wise |= The pope exists
Hence, if the pope does not exist, then the
two sentences have no truth-value; there is
no individual to ascribe wisdom to
Both sentences have the presupposition that
the Pope exists; i.e., that the papal office is
occupied
De dicto vs. de re




How to determine that a constituent occurs de re?
If both the principles de re do not hold then the occurrence is not de
re (it is either de dicto or a hyperintensional occurrence)
Auxiliary rule de re: C v  occurs de re, of C occurs in the
Composition Cwt with respect to w, t in which we evaluate, and this
occurence is not an occurence in another higher context; i.e., C does
not occur within the scope of -generic context or within the scope of
a Trivialization
The pope exists: wt [0Existwt 0Pope]



The object of predication is the whole office (that it is occupied), hence
0Pope occurs with de dicto supposition
Yet the above construction is equivalent to
wt [0x [x = 0Popewt]]
Here 0Pope is composed with w and t in which we evaluate, de re ???
NO, because 0Pope occurs in -generic context (x). The whole set (ie.
The whole function) is predicated to be non-empty, hence de dicto
Ambiguities

“Pope is the head of Catholic church”
a)
de dicto reading; the office is defined as the head of Catholic church


existence of the Pope is neither entailed nor presupposed
It is analytic, necessary truth, i.e. true in all w, t, even in those where the Pope
does not exist
[0=r 0Pope wt [0Head_ofwt 0Church]]
=r/(): the identity of offices;
wt [0=r 0Pope wt [0Head_ofwt 0Church]
Note. All such constitutional and normative sentences are to be read de
dicto.
Example.
The US president is the head of United States.
The US president is the commander in chief of the Armed Services.
Only a natural-born citizen of the United States is eligible to serve as the US
president


These are intensional, de dicto statements
They specify the requisites of the office of US president; there is a necessary relation
between intensions
Ambiguities
de re reading; the individual who happens
b)
to hold the papal office, occupies the office
of the head of Catholic Church as well



The two principles de re hold; in particular, that
the Pope exists
The sentence is not analytic truth.
In those w, t, in which the Pope exists, the
proposition takes the value T, otherwise it has no
truth-value (existential presupposition de re):
=i /(): identity of individuals
wt [0=i 0Popewt wt [0Head_ofwt 0Church]wt]
Ambiguities
The Pope might not have been the head of Catholic Church
a)
b)
de dicto  analytically necessary False
wt [0=r 0Pope wt [0Head_ofwt 0Church]]
de re  empirical „almost necessary truth“, True
wt w* t* [0=i 0Papežwt wt [0Head_ofwt 0Church]w*t*]
In those w, t in which the Pope exists, the proposition takes T,
because no individual has a non-trivial empirical property or holds
an office necessarily
If in a given w, t the Pope does not exist, the proposition has no
truth-value (existential presupposition de re)
Propositional attitudes

Tom believes that the Pope is wise



Believe/(): the relation-in-intension to a proposition;
Wise/(); Pope/.

0Pope

wt [0Believewt 0Tom [w1t1 [0Wisew1t1 0Popew1t1]]



Tom believes that the proposition that the Pope is wise is true –
the Pope occurs in supposition de dicto
wt [0Believewt 0Tom [wt [0Wisewt 0Popewt]]
occurs de dicto, though it is Composed with w, t, why?
Occurs in -generic context (w1t1)
The office is not extensionalized with respect to those w,t in which
we evaluate
Tom believes of the Pope that he is wise

Hence Tom believes that the individual that actually holds the
papal office that he is wise – the Pope occurs de re
Attitudes de re
Tom believes of the Pope that he is wise







a)
b)
wt [0Believewt 0Tom [w1t1 [0Wisew1t1 0Popewt]] ???
Believe/(); Wise/(); Pope/.
still de dicto !!! Due to -generic context (w1t1).
If the Pope does not exist in a given w,t, Tom believes that
the (degenerate) proposition is true, which is possible
It is necessary to get 0Popewt out of the generic context
(w1t1)
Two ways:
The Pope has the property that Tom believes of
him to be wise
Application of the substitution method, literal
analysis
Attitudes de re
Let BTW is a property of individuals that Tom believes of them to be
wise
Then a coarse-grained analysis comes down to: wt [0BTWwt 0Popewt]
Let us refine the 0BTW:
0BTW = wt x [0Believe 0Tom [wt [0Wise x]]
wt
wt
a)
Apply to 0Popewt:
wt [x [0Believewt 0Tom [wt [0Wisewt x]] 0Popewt]
OK, but what about -reduction? We obtain:
wt [0Believewt 0Tom [wt [0Wisewt 0Popewt]]]
But this is the de dicto case! Where is the mistake?
First, there is a collision of variables; we must rename:
wt [0Believewt 0Tom [w1t1 [0Wisew1t1 0Popewt]]]
But this is still a de dicto occurrence of 0Pope ! Where is the mistake?
The problem is -reduction by name; in the logic of partial functions like TIL this
is not a valid rule
Attitudes de re
Solution: application of substitution method using functions Sub a Tr:
b)
wt [0Believewt 0Tom 2[0Sub [0Tr 0Popewt] 0he 0[wt [0Wisewt he]]]]
Additional types:
Sub/(nnnn): operates on constructions: [0Sub what for-what to]; as a result
we obtain an adjusted construction
Tr /(n ): v-constructs the Trivialization of an individual
[0Tr 0Popewt] v-constructs the Trivialization of that individual who happens to be the
Pope in a given w, t (variables w,t are free here !!!).
[0Sub [0Tr 0Popewt] 0he 0[wt [0Wisewt he]]] v-constructs a construction of a
proposition.
2[0Sub [0Tr 0Pope
wt]
0he 0[wt [0Wise
he]]] the second execution constructs the
proposition to which Tom is related
wt

If Francisco is the Pope then the result is [wt [0Wisewt 0Francisco]]

If the Pope does not exist then [0Tr 0Popewt] is v-improper; hence the
Substitution and Double Execution are v-improper as well; the soconstructed proposition has no truth-value; existential presupposition de re.