1. TIL: notional attitudes of seeking and finding
Marie Duží

2. Logic of attitudes „propositional“ attitudes „notional“ attitudes
Tom Att1 (believes, knows, thinks), that P
Att1/(): relation of an individual to a proposition
Att1*/(n): relation of an individual to a hyper-proposition
„notional“ attitudes
Tom Att2 (seeks, looks for, finds, solves, wants to be, …) N. a a large group of sentences talking about attitudes like „hunt for“, „solve a problem“, „remember something“, „discover“, „be afraid of“, „be interested in“, „design“, …
Att2/(): relation of an individual to an intension
Att2*/(n): relation of an individual to an hyper-intension
Both kinds of attitudes can come in two variants: de dicto and de re
De re: Tom Att1 about something that P

3. Seeking (looking for) and finding
Such sentences are not used to describe an activity of obtaining something that we have got; about which we know what it is or where it is.
Seeking is an activity of identifying something we do not know.
A detective can be seeking a murderer provided the detective does not know who is the murderer. And of he knows that then he can try to identify the locality where the murderer is.
The seeker is primarily related to a condition and makes an attempt to find out what satisfies the condition, if anything.
We can be seeking a unicorn, Pegasus, and other non-existing entities.
For instance, when Schliemann was seeking Troy, he was for sure pretty much convinced that Troy did exist. Yet, it did not have to be so. And even if he happened to set foot on the Hissarlik hill, he would not be interested in this place unless he would find out that this is just the locality of Troy.
Hence the type of an entity to which the seeker is related is a construction or an intension. The seeker makes an attempt to find out which object (if any) is the value of the intension, or which object (if any) is constructed by a given construction.

4. Seeking and finding (de dicto)
a) De dicto seeking/finding is a relation to an -intension: Seek/()
The seeker wants to find out who/what is the actual value of the -intension.
Finding = success in the previous seeking: Find/()
Tom is seeking a unicorn.
wt [0Seekpwt 0Tom 0Unicorn], Seekp/(())
Police is seeking the murderer of JFK
wt [0Seekrwt 0Police wt [0Murderer-ofwt 0JFK]], Seekr/()
Police found the murderer of JFK
wt [0Findrwt 0Police wt [0Murderer-ofwt 0JFK]], Findr/()
The occurrences of 0Unicorn, wt [0Murder-ofwt 0JFK] are in the de dicto supposition.

5. Seeking and finding
Is the existence of the sought object a presupposition of finding?
Police found the murderer of JFK |= the murderer of JFK exists
Police Did not find the murderer of JFK |= ???
The failure in seeking can be cause by two resaons:
The murderer does not exist
Police did not work well, …
Hence, the existence of the sought object is not a presupposition of finding; it is merely entailed by finding
Hence the type of Find (after previous search) is the same as the type of previous seeking, (), ie. Relation to an -intension rather than to its value
Hence, existence of the sought object is only a necessary but not sufficient condition of finding.

6. Seeking and finding (de re)
Sentences containing the “seeking” constituent are ambiguous. There is another type of seeking; the seeker does not have attempt to identify the object occupying an office or having a property, he may know it, and yet look for the object.
Václav Havel is looking for Dagmar Havlova.
This is a relation of two individuals, VH and DH:
Look-for/(); VH, DH/,
The literal analysis comes down to this construction:
wt [0Look-forwt 0VH 0DH].
Well, but what then does Václav want to find out? Certainly not who Dagmar is, he knows the identity of his wife. Obviously, he does not know where she is. If we use a rather technical jargon, Ovšem zřejmě neví, kde se Dagmar právě nachází. Použijeme-li poněkud technický žargon, we would say that Vaclav attempts at localizing Dagmar.

7. Looking for, finding, de re
Hence we can explicate such a seeking as a relation to another role; this time not the role of individuals but of localities where a given individual can be.
Let us denote the attribute associating individuals with their positions as Loc(ality, position).
Furthermore, let us use a schematic type  for the type of a position (be it GPS coordinates or anything else). Thus we have
Loc/()
Now we can define Look-for as the relation between individuals x, y such that x is seeking the location of y; hence
Look-for/() is defined as follows:
0Look-for = wt xy [0Seekwt x wt [0Locwt y]].
wt [0Look-forwt 0VH 0DH] = wt [0Seekwt 0VH wt [0Locwt 0DH]].

8. Seeking and finding; ambiguities
The sentence „Police seeks the murderer of JFK“ is ambiguous between de dicto and de re reading.
de dicto reading (seeking who the murderer is):
wt [0Seekrwt 0Police wt [0Murderer-ofwt 0JFK]], Seekr/()
If Lee Oswald really was the murderer of Kennedy (which many doubt), then another scenario was possible. Oswald did not have to (and should not) be shoot when being transported to police station, but he could have had escapen the police. In such a case the police in Dallas would announce “Lee Oswald, the murderer of JFK is wanted for questioning”, and looked for the location where this individuum is. We have
De re: wt [0Look-forwt 0Policie wt [0Murderer-ofwt 0JFK]wt]
Note that the Closure wt [0Murderer-ofwt 0JFK] does occur with de re supposition now.
Both principles de re hold.
If the only murderer does not exist (either Oswald was not the murderer or there were more of them) the so constructed proposition has a truth-value gap.
If Oswald was the murderer then the police is looking for Oswald.
wt [0Look-forwt 0Policie wt [0Murderer-ofwt 0JFK]wt] =
wt [xy [0Seekwt x wt [0Locwt y]] 0Policie wt [0Murderer-ofwt 0JFK]wt]] = … be careful!
Now we must apply -reduction by name, becuase we would draw the extensional (de re) occurrence of wt [0Murderer-ofwt 0JFK] into the intensional -generic context of wt [0Locwt y].

9. Seeking and finding; de re
We must apply the substitution method, -reduction by value
wt [0Look-forwt 0Police wt [0Murderer-ofwt 0JFK]wt] =
wt [xy [0Seekwt x wt [0Locwt y]] 0Police wt [0Vrahwt 0JFK]wt]] =
wt [0Seekwt 0Policie 2[0Sub [0Tr wt [0Murderer-ofwt 0JFK]wt] 0y 0[wt [0Locwt y]]]]
Now both the principles de re hold.
a) Existential presupposition:
If wt [0Murderer-ofwt 0JFK]wt is v-improper-nevlastní (the only murder does not exist), then the Composition
[0Tr wt [0Murderer-ofwt 0JFK]wt] is v-improper, hence also
[0Sub [0Tr wt [0Murderer-ofwt 0JFK]wt] 0y 0[wt [0Locwt y]]] and its Double Execution are v-improper, and
[0Seekwt 0Police 2[0Sub [0Tr wt [0Murderer-ofwt 0JFK]wt] 0y 0[wt [0Locwt y]]]] as well as its negation are v-improper.
b) Substitution of v-congruent kconstructions: In such a w,t in which the only murderer is Oswald, i.e. wt [0Murderer-ofwt 0JFK]wt = 0Oswald the following equivalences are valid:
[0Seekwt 0Police 2[0Sub [0Tr wt [0Murderer-ofwt 0JFK]wt] 0y 0[wt [0Locwt y]]]] =
[0Seekwt 0Police 2[0Sub [0Tr 0Oswald] 0y 0[wt [0Locwt y]]]] =
[0Seekwt 0Police 20[wt [0Locwt 0Oswald]]] =
[0Seekwt 0Police [wt [0Locwt 0Oswald]]]

10. Seeking and Finding of non-mathematical objects; summary
De dicto: seeking who, what is the value (if any) of an -intension: ()
usually (),(())
De re: seeking where is a given individual, seeking the location of an individual: ()
Finding as a success in foregoing seeking is of the same type as that seeking
Existence of the sought object is a necessary but not sufficient condition of finding  it is not a presupposition; rather, it is merely entailed

11. More examples on seeking and finding
de dicto reading of the sentence
Police is seeking but still not finding the murderer of JFK
Comes down to this construction
wt [[0Seekrwt 0Police wt [0Murderer-ofwt 0JFK]]  [[0Findrwt 0Police wt [0Murderer-ofwt 0JFK]]]
Types: Seekr, Findr/().

12. More on Finding
If the police finally succeeded and found the murderer, then we can derive not only that the murderer exists but also that the police knows who the murderer is or where he/she is
For the sake of simplicity we will now deal only with the first case, i.e. the success in seeking who the murderer is.
If the murderer happens to be Lee Oswald, then police identified Oswald as the murderer.
Hence existence of the murderer and its identification are merely entailed consequences of finding rather than its presuppositions.

13. More on Finding Ident / ():
Assume a hypothetical situation that Oswald is not only the murderer but also a champion in hammer throw. Does it mean that police also knows that Oswald is the hammer throw champion? Of course not. Police identified Oswald as the murderer rather than the champion.
To specify requisites of Finding we must determine the type of the entity Ident(ify something as something).
It cannot be a simple relation (-in-intension) of two individuals ().
If it were so, then on the assumption that the murderer of JFK is also the champion in hammer throw we could validly infer that police identified the hammer throw champion, which does not make sense.
Ident / ():
The relation (-in-intension) between an individual (who did identify) and another individual (whom he identify) and an individual role (as what).

14. Requisites of finding
Identification of the sought entity:
[0Req1 0Ident 0Findr] = wt [x r [[0Truewt wt [0Findrwt x r]]  [0Truewt wt [0Identwt x rwt r]]]]
existence of the sought entity:
[0Req2 0Exist 0Findr] =
wt [x r [[0Findrwt x r]  [0Existwt r]]]
Req1/(()()); Req2/(()()); Findr/(); Exist/(); Identr/(); x  ; r  .

15. Rules for finding [0Findrwt x r] (I)  [0Identwt x rwt r]
Since the first occurrence of the variable u in the consequent is extensional (de re), the principle of the substitution of v-congruent constructions is valid, and the office r must be occupied:
[0Identwt x rwt r], [rwt = r’wt] [0Identwt x rwt r]
(IIa)  (IIb) 
[0Identwt x r’wt r] [0Existwt r]
by tranzitivity we obtain the rule
(III) 
[0Existwt r]
Note that none of these rules is valid in case of the failure in seeking.

16. Rules for Finding
Similar rules are valid for Finding as a success in foregoing search of the location.
It is true that Schliemann found the location of the ancient Troy:
wt [0Findlwt 0Schliemann [wt [0Locwt 0Troywt]]]
Findl/(); Troy/.
Hence we can validly derive that Troy existed and that if Hissarlik is the location of Troy, i.e.
wt [= 0Hissarlik [wt [0Locwt 0Troywt]]wt]
then Schliemann identified Hissarlik as the location of Troy:
wt [0Identwt 0Schliemann 0Hissarlik [wt [0Locwt 0Troywt]]]

17. Accidental finding
There is another kind of finding, namely an accidental finding.
Imagine Tilman walks along a river in India, kicks a little stone and after his coming home finds out that it is a the most valuable precious stone, the largest cut diamond Koh-i-Noor.
„Tilman found the largest cut diamond“.
Yet this time Tilman is not related to the role of the largest cut diamond, because he was not looking for it. He just happened to stumble over it and only afterwards found out that it is the largest cut diamond. Hence,
Finda/()
wt [0Findawt 0Tilman w’t’ [0Largest [0Cut 0Diamond]w’t’]wt]
where wt [0Largest [0Cut 0Diamond]wt] is the construction of the role of the largest cut diamond that occurs here with de re supposition.
Additional types. Largest/(()): the function that selects from a set of individuals the largest one; Cut/(()()): property modifier; Diamond/().

18. Accidental finding
wt [0Findawt 0Tilman w’t’ [0Largest [0Cut 0Diamond]w’t’]wt]
Since w’t’ [0Largest [0Cut 0Diamond]w’t’] occurs with de re supposition, both principles de re are valid
For instance, if Koh-i-Noor is the largest cut diamond, then Tilman found Koh-i-Noor
And of course, existential presupposition (rather than mere entailment as it is the case of finding after a foregoing search);
Hence, the largest cut diamond must exist

19. Hyperintensional seeking
Tilman is seeking the last decimal of 
Tilman is seeking something
where something is restricted type-theoretically, so that the conclusion states that Tilman is seeking something of one particular type.
wt [0Seek*wt 0Tilman 0[0Last_Dec 0]]

wt [0*c [0Seek*wt 0Tilman c]]
Seek*/(n); Tilman/; Last_Dec/(): the function that associates a number with its last decimal digit; /; c/2 v 1.

20. Hyperintensional seeking
Tilman is seeking the last decimal of 
There is a number such that Tilman is seeking its last decimal
wt [0Seek*wt 0Tilman 0[0Last_Dec 0]]

wt [0x [0Seek*wt 0Tilman [0Sub [0Tr x] 0y 0[0Last_Dec y]]]]
Additional types: x, y/1 v ; /(()).

21. Hyperintensional seeking
If Tilman is seeking the millionth digit of the decimal expansion of , he may succeed in his effort and identify that number. Thus we have another argument:
Tilman found the millionth digit of the decimal expansion of 
There is a number such that Tilman identified it as the millionth digit of the decimal expansion of 
wt [0Findwt 0Tilman 0[0Mill_Dec 0]]

wt [0x [[x = [0Mill_Dec 0]]  [0Identwt 0Tilman x 0[0Mill_Dec y]]]]
Types: Find/(n); : the type of naturals; Mill_Dec/(): the function that associates a real number with its millionth decimal digit; Ident/(n).

22. Hyperintensional seeking
Suppose that Tilman is solving the equation (x2 + x – 2) = 0 and that his effort meets with success:
Tilman has solved the equation (x2 + x – 2) = 0
––––––––––––––––––––––––––––––––––––––––––
There is something that Tilman has solved
wt [0Solvedwt 0Tilman 0[x [x2 + x – 2] = 0]]
––––––––––––––––––––––––––––––––––––––––
wt [0c [0Solvedwt 0Tilman c]] c  n
If Tilman has solved the equation x2 + x – 2 = 0 then there is an ()-object (in this case the set {1, –2}) satisfying the equation, and Tilman has identified this set as the product of the construction [x [x2 + x – 2] = 0]. Thus we have (variable s v ()):
–––––––––––––––––––––––––––––––––––––––––––
wt [0s [[s=[x [x2 + x – 2] = 0]]  [0Identwt 0Tilman s 0[x [x2 + x – 2] = 0]]]]