1. Lecture 6 Three kinds of context
Marie Duží

2. Exercise 3 - examples The temperature in Amsterdam is rising.
The temperature in Amsterdam = The temperature in Prague.
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The temperature in Prague is rising.
Types. Terms ‘temperature in Amsterdam’ and ‘temperature in Prague’ denote magnitudes of type . Temperature_in/(), Prague, Amsterdam/, =/(), Rising/(): property of a function (here magnitude), that it is rising at a given argument, i.e., its derivative has a positive value, wt [0Temperature_inwt 0Amsterdam], wt [0Temperature_inwt 0Prague]  
wt [0Risingwt wt [0Temperature_inwt 0Amsterdam]]
wt [0= wt [0Temperature_inwt 0Amsterdam]wt wt [0Temperature_inwt 0Prague]wt]
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wt [0Risingwt wt [0Temperature_inwt 0Prague]]
The argument is invalid, because the second premise specifies a contingent identity of the values of the two (distinct) magnitude; however, Rising is a property of the whole magnitude. We could sbstitude only if the second premise specified the identity of magnitudes, which is not so.
The Closures wt [0Temperature_inwt 0Amsterdam] and wt [0Temperature_inwt 0Prague] occur in the first premise and in the conclusion with de dicto supposition, while in the second premise with de re supposition.

3. Exercise 3 The Mayor of Ostrava visited Brno.
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The Mayor of Ostrava exists.
wt [0Visitwt wt [0Mayor_ofwt 0Ostrava]wt 0Brno]
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wt [0Existwt wt [0Mayor_ofwt 0Ostrava]]
Types. Visit/(); Exist/()
Exist = wt u [0x [x = uwt]]; u v , x v , =/(): identity of individuals
Proof.
1. [0Visitwt wt [0Mayor_ofwt 0Ostrava]wt 0Brno] assumption
2. [0Impwt 0[wt [0Mayor_ofwt 0Ostrava]wt]] Def. Composition
3. [0Empty x [x = wt [0Mayor_ofwt 0Ostrava]wt]
4. [0x [x=wt [0Mayor_ofwt 0Ostrava]wt]]
5. [0Existwt wt [0Mayor_ofwt 0Ostrava]] Def. Exist
Imp/(): the property of a construction that it is v-improper in a given w, t
Empty/(()): the class of empty classes of individuals

4. Two principles de re a) Existential presupposition
The Mayor of Ostrava did/did not visit Brno.
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The Mayor of Ostrava exists.
b) Substitution of co-referential terms
The Mayor of Ostrava is Mr. X
The Mayor of Ostrava visited Brno.
X visited Brno

5. Exercise 3
For all numbers x holds that dividing x by 0 is improper.
Improper/(1): the class of constructions v-improper for every valuation v.
[0Improper 0[0: x 00]] – constructs T regardless of valuation of x, because [0: x 00] is v-improper for every valuation.
The variable x is o-bound, which is stronger than -binding, it occurs hyperintenally (not in execution mode)
Does it make sense to quantify over x occurring hyperintensionally?
[0x [0Improper 0[0: x 00]]] – constructs T, OK, but for instance,
[x [0Improper 0[0: x 00]] 05] constructs
[0Improper 0[0: x 00]] rather than [0Improper 0[0: 05 00]]
Hence quantifying into hyperintensional context is of no effect

6. Examples
There is a number y such that for any number x dividing x by y is v-improper.
[0Improper 0[0: x 00]]
 ???
[0y [0Improper 0[0: x y]]]
But the Composition [0: x y] is not v-improper for any valuation v; for instance, it is v(5/x, 1/y)-proper!
The existential quantifier has no effect here, because the variable y occurs hyperintensionally in 0[0: x y], it is not free for quantification
If we wanted to prove the above argument, it would not go:
[0Improper 0[0: x 00]] premise - T
[y [0Improper 0[0: x y]] 00] constructs F !!!

7. hyperintensional context
The whole construction C is an object of predication (argument); hence its product, i.e. the function f constructed by C (if any) is irrelevant
The construction C does not occur in the execution mode; it is just displayed (by Trivialization)
All the subconstructions of C (including variables) are also displayed, occur hyperintensionally as well
How then to operate in a hyperintensional context?
How to quantify into a hyperintensional context?
Solution, way in: Substitution method !!!

8. hyperintensional context
Substitution method
[0Improper 0[0: x 00]]
 !!!
[0y [0Improper [0Sub [0Tr y] 0z 0[0: x z]]]]
The variable y is now free in the Composition [0Improper … ], we can -bind it, and also the -quantifier functions here as it should
Proof.
[0Improper 0[0: x 00]] premise
[y [0Improper [0Sub [0Tr y] 0z 0[0: x z]]] 00] -abstraction
(constructs T, because [0Sub [0Tr y] 0z 0[0: x z]] v(0/y)-constructs the construction [0: x 00])
[0Empty y [0Improper [0Sub [0Tr y] 0z 0[0: x z]]]]

9. Exercise 3 a) [0Sub [0Tr x] 0y 0[0Sin y]]
b) 2[0Sub [0Tr x] 0y 0[0Sin y]]
Sub/(nnnn); Tr/(n); Sin/(); x, y v .
Valuation v(/x), i.e. valuation that assigns the number  to the variable x
ad a) [0Sub [0Tr x] 0y 0[0Sin y]] =* 0[0Sin 0]
ad b) 2[0Sub [0Tr x] 0y 0[0Sin y]] = 20[0Sin 0] = [0Sin 0] = 00
=*/(nn): identity of constructions
=/(): identity of numbers

10. De dicto vs. de re wt [0Want_to_bewt 0Tom 0Pope] de dicto
wt [0= 0Popewt 0Francisco]
de re
De dicto is a special case of an intensional occurence (of empirical notions)
De re is a special case of an extensional occurence (of empirical notions) in the sub-construction that is right after the first wt, i.e. world w and time t in which we are to evaluate

11. Three kinds of context Let C be a subconstruction of D. Then
The occurrence of C in D is hyperintensional, if the whole construction C is an argument of a function constructed within D. We say that such an occurrence of C is displayed in D.
Then all the subconstructions of this occurrence of C have also a hyperintensional occurrence.
Exmaple. Tom calculates Sin().
wt [0Calculatewt 0Tom 0[0Sin 0]]
hyperint
Calculate/(1)
Trivialization of a construction C (0C) raises the context up to the hyperintensional level
Dually, a Double Execution brings the context down to an intensional or extensional level: 20C = C

12. Three kinds of context
If an occurence of C in D is not hyperintensional, then the construction C occurs in the execution mode; we say that C is a constituent of D.
A constituent C of D can occur intensionally or extensionally.
Intensional occurrence of C v f: the whole function f v-constructed by C (f can be even a nullary function, i.e., an atomic object like a number or an individual) is an object of predication/argument of another function constructed within D.
Extensional occurrence of C v f/(): -value of the function f v-constructed by C (where f is at least unary) is an object of predication/argument of another function constructed within D.

13. wt [0Calculatewt 0Tom 0[0+ 03 05]]
Three kinds of context
wt [0Calculatewt 0Tom 0[ ]]
Constituents of [0Počítáwt 0Tom 0[ ]], i.e. those subconstructions that must be executed whenever the whole Composition is to be executed are these:
[0Calculatewt 0Tom 0[ ]]
[[0Calculate w]t]
[0Calculate w],
0Calculate, w, t,
0Tom,
0[ ].
The subconstruction [ ] is not a constituent, it is just an object of which we predicate that Tom wants to find out what this construction produces.

14. Three kinds of context Example. Sine is a periodic function.
Types. Periodic/(()): the set of periodic unary functions, Sine/().
[0Periodic 0Sine]
intensional
Sin()=0: [0= [0Sine 0] 00]
extensional

15. Intensional vs. extensional context
-Closure makes up a generic intensional context; it raises the context up to the intensional level.
Composition brings the context down to the extensional level.
Higher context is dominant over a lower one.
[0: x 00]
constituent 0: occurs here extensionally; the Composition is v-improper for any valuation v
x [0: x 00]
constituent 0: occurs here intensionally; Closure is not v-improper for any valuation; it constructs a degenerate function

16. Intensional vs. extensional context
[x y [0: x y] [0Cotg 0]]
The occurrence of 0Cotg is here extensional. The Composition is improper, it fails to construct anything, because [0Cotg 0] is improper; the function Cotg is not defined at the argument .
-reduction by name is not strictly equivalent !!!
[y [0: [0Cotg 0] y]]
The occurrence of 0Cotg is here intensional due to -generic context (y); the Composition produces a degenerate function; it is not improper
-reduction by value is strictly equivalent !!!
2[0Sub [0Tr [0Cotg 0]] 0x 0[y [0: x y]]
The occurrence of 0Cotg is here extensional. The Composition is improper, it fails to construct anything, because [0Cotg 0] is improper; partiality is „propagated up“.

17. Substitution Intensional context: Extensional context:
The whole function (intension) f is an object of predication
For an intensional occurence of C that v-constructs a function f we can substitute a construction D such that D v-constructs the same function f.
Hence C=D, i.e. C and D are equivalent, i.e. v-congruent for any valuation v
Extensional context:
The value (if any) of the v-constructed function (intension) f at a given argument is an object of predication
For an extensional occurrence of C we can substitute a construction D such that D v-constructs the same value (even of a different function) at a given argument a.
Hence C =v D, i.e. C and D are v-congruent for a certain valuation v(a/x)

18. Substitution Hyperintensional context:
The whole construction C is an object of predication
For a hyperintensional occurrence of C we can substitute the same construction C.
Only the same construction C? But then we don’t substitute, it turns out to be a too strong demand
We can substitute a construction D that is procedurally isomorphic with D
Hence 0C =i 0D, where =i/(nn) is the relation of procedural isomorphism
How to define this relation?

19. Identity of procedures
How hyper is hyper?
Identity of procedures
Functions in extension (sets, mappings, PWS-intensions) are extensionally individuated:
x [f(x) = g(x)]  f = g
Procedures are (hyper)intensionally individuated; Church’s ‘functions in intension’
Procedures are equivalent  produce the same object
Procedures are identical  consist of the same constituents
But: each procedure can be refined, ad infinitum, when are the constituents identical ???
Does it matter?
Hyperintensional contexts – only synonymous terms can be substituted
Synonymous terms  have the same meaning, are assigned the same meaning construction, procedure (?)

20. How hyper is hyper? Carnap’s intensional isomorphism
Church’s synonymous isomorphism
Any two terms or expressions whose respective meanings are procedurally isomorphic are deemed semantically indistinguishable, hence synonymous. Thus procedurally isomorphic constructions can be mutually substituted in any context, including hyperintensional ones.
Church’s Alternatives;
(A2) logical equivalence
(A1) includes - and -conversion; (A1’) + -conversion
(A0) includes -conversion and meaning postulates for atomic constants such as ‘bachelor’, ‘fortnight’, ‘prime’.
Procedural isomorphism;
(A½) Quid-relation (Materna); - and -conversion (Duží, Jespersen, Materna 2010)
(A¾); -, -conversion and restricted -conversion (Duží, Jespersen 2013)
(A1’’); modification of Church’s (A1), -conversion ‘by value’ (Duží, Jespersen 2014)
(A0’) ???

21. Procedural isomorfismus
Strict criterion:
We exclude -conversion and unrestricted -conversion ‘by name’
In the logic of partial functions such as TIL these conversions are not equivalent transformations
Different constituents
Loss of analytic information

22. Problems with - (/)-reduction
non-equivalence arises when drawing an extensional occurrence of a constituent into (hyper/) intensional context
Example:
[x [y [0+ x y]] [0Cotg 0]]
is an improper construction; it does not construct anything, because there is no value of the cotangent function at 
but its -reduced Composition
[y [0+ [0Cotg 0] y]]
constructs a degenerated function
The improper construction [0Cotg 0] has been drawn into the intensional context of the Closure [y [0+ x y]].
De re attitudes:
Tilman believes of the Pope that he is wise
wt [he [0Believewt 0Tilman w*t* [0Wisew*t* he]] 0Popewt] 
wt [0Believewt 0Tilman w*t* [0Wisew*t* 0Popewt]]

23. Procedural isomorphism
-conversion
The set of positive numbers:
x [0> x 00], y [0> y 00], z [0> z 00],… which if them? Any of them
restricted -conversion by name:
The Mayor of Ostrava is rich:
wt [0Richwt w’t’ [0Mayor_ofw’t’ 0Ostrava]wt] =r
wt [0Richwt [0Mayor_ofwt 0Ostrava]]
We don’t use -bound variables in a natural language
-conversion by value

24. Procedural isomorphism
Obviously it is wise to define several criteria of procedural isomorphism (hence of synonymy) depending on the discourse under scrutiny
Natural language  we don’t use -bound variables
-conversion, -conversion by value, restricted -conversion by name + meaning postulates for syntactically simple terms (Church’s A0)
In the language of mathematics and in a programming language variables can play a significant role
The most restrictive criterion: only meaning postulates for syntactically simple terms (Church’s A0)

25. Existential quantification
Extensional context – only in this case partiality plays a crucial role: the application of a function f() to the argument a   can fail if f is not defined at a (x  ):
[0Proper [f a]] | [Empty x [f x]] | [0 x [f x]]
Intensional context – we can derive that there is a function f (an object of predication) but not that there is a value of f
Tom is seeking a unicorn | unicorns exist
Not to turn logic to magic 

26. Existential generalization
Tom is seeking a unicorn
Tom is seeking something (there is something such that Tom is seeking it)
wt [0Seekwt 0Tom 0Unicorn]
wt [0p [0Seekwt 0Tom p]]
Seek/(()) : the relation-in-intension of an individual to a property the instances of which the individual wants to find, Unicorn/(), p v ()

27. Existential generalization
Tom calculates tg(/2)
Tom calculates something
(Tom calculates something but not a non-existing number)
wt [0Calculatewt 0Tom 0[0Tg [0/ 0 02]]
wt [0c [0Calculatewt 0Tom c]]
Calculate/(1) : the relation-in-intension of an individual to a construction, c v 1