Transcript Slide 1

Transparent intensional logic,
-rule and Compositionality
Marie Duží
VSB-Technical University Ostrava
http://www.cs.vsb.cz/duzi
Attitude Logic(s)
 A reliable test on Compositionality
 Attitudes:
 Notional
 Propositional
 We are dealing with a fine difference between the
meanings of sentences like
(P1) Charles believes that the Pope is in danger
(P2) Charles believes of the Pope that he is in
danger
 Some authors even claim that (P1) is ambiguous, that
it can be also read as (P2).
Rules of substition
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Attitude logics and belief sentences
In our opinion it is not so. We can, for instance,
reasonably say (it may be true) that
Charles believes of the Pope that he is not the
Pope,
whereas the sentence
Charles believes that the Pope is not the Pope
cannot be true, unless our Charles is completely
irrational. The sentences like (P1) and (P2) have
different meanings, and their difference consists in
using ‘the Pope’ in the de dicto supposition (P1) vs.
the de re supposition (P2).
The two sentences are neither equivalent, nor is any of
them entailed by the other.
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Belief sentences in doxastic logics
 In the usual notation of doxastic logics the distinction
is characterised as the contrast between
BCharles D[p]
(de dicto)
(x) (x = p  BCharles D[x]
(de re)
 But there are worrisome questions (Hintikka, Sandu
1989):
Where does the existential quantifier come from in the de
re case? There is no trace of it in the original
sentence.
How can the two similar sentences be as different in their
logical form as they are?
 Hintikka, Sandu propose in their (1996) a remedy by
means of the Independence Friendly (IF) first-order
logic:
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Belief sentences in doxastic logics
 “Independence Friendly (IF) first-order logic deals
with a frequent and important feature of natural
language semantics. Without the notion of
independence, we cannot fully understand the logic of
such concepts as belief, knowledge, questions and
answers, or the de dicto vs. de re contrast.”
Hintikka, Sandu (1989): Informational Independence as a
Semantical Phenomenon. In J.E. Fenstad et el (eds.),
Logic, Methodology and Philosophy of Science,
Elsevier, Amsterodam 1989, pp. 571-589.
Hintikka, Sandu (1996): A revolution in Logic? Nordic
Journal of Philosophical Logic, Vol.1, No.2, pp. 169183.
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Belief sentences and IF semantics

Hinttika, Sandu solve the de dicto case as above, and propose the
de re solution with the independence indicator ‘/’:
BCharles D[p / BCharles]


This is certainly a more plausible analysis, closer to the syntactic
form of the original sentence, and the independence indicator
indicates the essence of the matter:
There are two independent questions:
 ”Who is the pope” and
 ”What does Charles think of that person”.
Of course, Charles has to have a relation of an ”epistemic intimity”
to a certain individual, but he does not have to connect this
person with the office of the Pope (only the ascriber must do
so).
(Chisholm,R.(1976): Knowledge and Belief: ‘De dicto’ and ‘de re’.
Philosophical Studies 29 (1976), 1-20. )
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Belief sentences and
Intensional logics
BCharles D(p)
(de dicto)
x BCharles D(x)(p)
(de re)
But:
x BCharles D(x)(p)  BCharles D(p) !
(applying the rule of -reduction).
What then is the difference between de dicto
and de re?
Why is it “forbidden” here to perform the
fundamental rule of -calculi?
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Solomon Feferman (1995):
Logic of Definedness
Introduces the axioms (λp) for Partial Lambda Calculus
as follows (t↓ means - the term t is defined):
i.
λx.t ↓
ii. (λx.t(x))y  t(y).
The axiom (ii) corresponds to the trivial β-reduction,
but the limitation on instantiation in PLC restricts
its application to:
 s↓  (λx.t(x))s  t(s).
(but why this restriction?, proof?)
Our system (TIL) introduces a generally valid
β-reduction for the Partial Higher-Order Hyperintensional Lambda Calculus.
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Transparent Intensional Logic
Formally:
 The language of TIL constructions can be
viewed as a hyper-intensional -calculus
operating over partial functions.
 “hyper-intensional”: -terms are not
interpreted as set-theoretical mappings
(”modern functions”) but as algorithmically
structured procedures (which produce as an
output the (partial) mapping).
 Procedures, known as TIL constructions,
are objects sui generis: they can be not
only used but also mentioned within a
theory.
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TIL & beta-rule
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Suppositio (substitution)
 A lot of misunderstanding and many
paradoxes arise from confusing different
ways in which a meaningful expression can
be used.
 We are going to show that these different
ways consist in using and mentioning
entities (by means of an expression)
 In which way can an entity be used or
mentioned?
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TIL & beta-rule
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Using / Mentioning Entities
Expression
used
mentioned
to express its meaning:
procedure (‘TIL construction’)
de dicto / de re
mentioned
used
to produce a function:
mentioned
used to point at …
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Use / Mention
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TIL constructions
 Abstract procedures, structured from the algorithmic
point of view.
 structured meanings: Instructions specifying how to arrive
at less-structured entities.
 Being abstract, they are reachable only via a verbal
definition.
 The ‘language of constructions’: a modified version of the
typed -calculus, where Montague-like -terms denote, not
the functions constructed, but the constructions themselves.
 Henk Barendregt (1997): -terms denote functions, yet “...
in this interpretation the notion of a function is taken to be
(hyper-)intensional, i.e., as an algorithm.”
 Operate on input objects (of any type, even constructions)
and yield as output objects of any type: they realize
functions (mappings)
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Kinds of constructions
1.
Atomic: do not contain as a used constituent any
other construction but themselves (supply objects …)
 Variables x, y, p, c, … v-constructing
 Trivialisation of X: 0X
2. Compound.
 Composition [X X1…Xn]: the instruction to apply a
(partial) function f (constructed by X) to an
argument A (constructed by X1,…,Xn) to obtain the
value (if any) of f at A.
 -Closure [x1…xn X]: the instruction to abstract
over variables in order to obtain a function.
 Double execution 2X: the instruction to use a
higher-order construction X twice over as a
constituent.
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Constructions
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TIL Ramified Hierarchy of Types
The formal ontology of TIL is bidimensional.
One dimension is made up of
constructions.
The other dimension encompasses nonconstructions, i.e., partial functions
mapping (the Cartesian product of)
types to types.
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TIL Ramified Hierarchy of Types
1st-order: non-constructionsBase: , , , ,
partial functions ((())), (()), …, (01…n)
2nd-order: Base: *1 constructions of 1st-order
entities, partial functions involving such
constructions: (01…n), i = *1
3rd-order: Base: *2, constructions of 2nd-order
entities, partial functions involving such
constructions: (01…n), i = *2, or *1
And so on, ad infinitum
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-intensions; examples
 Functions of type ()
 Usually both modal and temporal
parameters: (())
 Abbreviation: 





Propositions / 
(individual) offices / 
Magnitudes / 
Empirical functions (attributes)/()
Attitudes / (n)
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Definition Used* vs. Mentioned*
Let C be a construction and D a sub-construction of C. Then an
occurrence of D is used* as a constituent of C iff:
 If D is identical to C (i.e., 0C = 0D) then the occurrence of D is
used* as a constituent of C.
 If C is identical to [X1 X2…Xm] and D is identical to one of the
constructions X1, X2,…,Xm, then the occurrence of D is used* as a
constituent of C.
 If C is identical to [x1…xmX] and D is identical to X, then the
occurrence of D is used* as a constituent of C.
 If C is identical to 1X or 2X and D is identical to X, then the
occurrence of D is used* as a constituent of C.
 If C is identical to 2X and X v-constructs a construction Y and D is
identical to Y, then the occurrence of D is used* as a constituent
of C.
 If an occurrence of D is used* as a constituent of an occurrence of
C’ and this occurrence of C’ is used* as a constituent of C, then
the occurrence of D is used * as a constituent of C.
If an occurrence of a sub-construction D of C is not used* as a
constituent of C, then the occurrence of D is mentioned* in C.
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Use / Mention
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Definition Used* vs. Mentioned*
Let C be a construction and D a subconstruction of C.
Then an occurrence of D is mentioned*
in C iff it is not necessary to execute
D in order to execute C;
Otherwise D is used* as a constituent
of C.
 Makes a fine individuation possible;
finer than just an equivalence.
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Use / Mention
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Two kinds of using a construction:
de dicto vs. de re supposition.
Roughly: C = [… D … ], D  ()
1. D occurs in C with de dicto supposition iff D is
not composed with a construction A  ;
 the respective function / () is just mentioned
2. D occurs in C with de re supposition iff D is
composed with a construction A  , and D
does not occur as a constituent of a de dicto
occurrence D’ (de dicto context is dominant);
 the respective function / () is used as a pointer
to its actual, current value / 
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Contextssuppositio substitution
The President of USA knows that John Kerry
wanted to become the President of USA.
The President of USA is (=) the husband of
Laura Bush.

Hence what ?
Did John Kerry want to become the husband of
Laura Bush?
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Contextssuppositio substitution
C1 wt [0= [wt [0Preswt 0USA]]wt [0Husbandwt 0Bush]]
extensional context: of using* de re
C2 wt [0Wwt 0K [wt [0Bwt 0K wt [0Pres 0USA]]] ]
intensional context: of using* de dicto
C3 wt [0Knowwt [wt [0Preswt 0USA]]wt
0[wt [0W 0K wt [0B 0K wt [0Pres 0USA]]]] ]
wt
wt
wt
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hyper-intensional context: of mentioning*21
Using / Mentioning Constructions
 Dividing six by three gives two and
dividing six by zero is improper.
Types: 0, 2, 3, 6 / , Div / (), Improper /
(1)the class of v-improper
constructions for all v
[[[0Div 06 03] = 02]  [0Improper 0[0Div 06 00]]]
used* constituents
Rules of substition
mentioned*
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Using / Mentioning Constructions
There is a number such that dividing any number
by it is improper.
Types: Div / (), Improper / (1), ,/(()), x, y  .
Exists x for all y [0Improper 0[0Div x y]].
But x, y occur in the hyper-intensional context of mention*;
they are not free for evaluation or substitution.
How to quantify? To this end we use functions Sub and Tr:
Sub / (1111)the mapping which takes a construction C1,
variable x, and a construction C2 to the resulting
construction C3, where C3 is the result of substituting C1
for x in C2.
Tr / (1)the mapping which takes a number and returns
its trivialisation

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Using / Mentioning Constructions
(*)
[0y [0x [0Improper
[0Sub [0Tr y] 0y’ [0Sub [0Tr x] 0x’ 0[0Div x’ y’ ]]]]]].
 Let a valuation v assign 0 to y and 6 to x. Then
the sub-construction
[0Sub [0Tr y] 0y’ [0Sub [0Tr x] 0x’ 0[0Div x’ y’ ]]]
 v-constructs the construction [0Div 06 00], which
belongs to the class Improper. This is true for any
valuation v’ that differs from v at most by
assigning another number to x.
 The construction (*) constructs True.
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De dicto / de re supposition



The temperature in Amsterdam equals the temperature in
Prague.
The temperature in Amsterdam is increasing.
-------------------------------------------------------- The temperature in Prague is increasing.
Types: Temp(erature in …)/(), Amster(dam), Prague/,
Increas(ing)/().

wt [wt [0Tempwt 0Amster]wt = (de re)
wt [0Tempwt 0Prague]wt]

wt [0Increaswt [wt [0Tempwt 0Amst]]
the magnitude is
mentioned.
(de dicto)
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Rules of Substitution
(logic of partial functions !)
 “Homogeneous” substitution: no problemLebniz’s law



Used* de re  extensional context de re
Used* de dicto  intensional context de dicto
Mentioned* construction  hyperintensional context
 Used* constructions – constituents:
 De re (extensional) context: [Cx] = [C’y]
 co-incidental constructions substitutable
 De dicto (intensional context): C = C’
 equivalent constructions substitutable
 Mentioned* (hyper-intensional) context: 0C = 0C’
 Only identical constructions substitutable
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Rules of Substitution
(logic of partial functions !)
 Heterogeneous substitutions.
 Construction of a lower-order into a higher-
order context (which is dominant):
 We must not carelessly draw a construction
D occurring in a lower-order context into a
higher-order context.
 Why not? The substitution would not be correct
even if there is no collision of variables,
due to partiality
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De re rules
 The president of CR is (is not) an economist.
 de re
The president of CR exists.
 The president of CR is eligible.

The president of CR may not exist.
de dicto
 In the de re case there is an existential
presupposition, unlike the de dicto case.
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Charles believes of the president of CR that
he is an economist.
Types:
Ch/, B/(), Pr(esident of …)/(), CR/, Ec/()
Synthesis (h  , a free variablethe meaning of “he”):
He is an economist: wt [0Ecwt h] v  (anaphora)
The President of CR: wt [0Prwt 0CR]  
a) The President of CR is believed by Charles to be an
economist – the passive variant
wt [h [0Bwt 0Ch wt [0Ecwt h]] wt [0Prwt 0CR]wt ]
Now, can we perform -reduction ???
Yes, but only the trivial one:
wt [0Prwt 0CR]wt | [0Prwt 0CR]
Collision of variables? Let us rename them:
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Charles believes of the president of CR that
he is an economist.
-reduction “by name” :
wt [h [0Bwt 0Ch w’t’ [0Ecw’t’ h]] [0Prwt 0CR] ]
| ??? wt [0Bwt 0Ch w’t’ [0Ecw’t’ [0Prwt 0CR]]]
No collision of variables,
But. [h [0Bwt 0Ch w’t’ [0Ecw’t’ h]] [0Prwt 0CR] ]

[0Existwt wt [0Prwt 0CR]] =
[0x [x = [0Prwt 0CR]]
Unlike the latter.
Therefore, don’t perform -reduction (!?!)
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Charles believes of the president of CR that
he is an economist.
b) The direct analysis of the active form,
using Tr and Sub.
-reduction “by value”:
Now we have to substitute for h the construction
of the individual (if any) that actually plays the
role of the president:
b) wt [0Belivewt 0Charles
2[0Sub [0Tr wt [0Pr
0CR] ] 0h (extens.)
wt
wt
0[wt [0Ec
(intens.)
wt h]]]]
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2-phase -reduction:
how does it work?
wt [0Belwt 0Ch 2[0Sub [0Tr wt [0Prwt 0CR]wt] 0h
0[wt [0Ec
1.
wt
h]]]]
Let wt [0Prwt 0CR]wt be v-improper (the president does not
exist).

Then [0Tr wt [0Prwt 0CR]wt] is v-improper and

The function Sub does not have an argument to operate on:

[0Sub [0Tr wt [0Prwt 0CR]wt] 0h 0[wt [0Ecwt h]]]

v-improper. (And so is the Double execution.)
The so-constructed proposition does not have a truth-value, as
it should be (the existential presupposition)
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Substitution by value (-reduction)
wt [0Belwt 0Ch 2[0Sub [0Tr wt [0Prwt 0CR]wt] 0h
0[wt [0Ec
2.
wt
h]]]]
Let wt [0Prwt 0CR]wt be v-proper (the president exist). Then

the construction [0Prwt 0CR] v-constructs particular individual
Y (For instance V. Klaus.) Then

[0Tr wt [0Prwt 0CR]wt] v-constructs 0Y, and Sub inserts it
for the variable h.

the result is the construction: [wt [0Ecwt 0Y]] that is
executed (Double execution) in order to construct the
proposition that is believed by Charles.
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Substitution by value (-reduction)
Type checking:
2[Sub
[0Tr [0Prwt 0CR]] 0h
(*1 )
(*1*1*1*1)
*1

0[
wt [0Ecwt h]]]

*1
*1 ( )
*1
1. step
2. step (if the 1st did not fail):
1[wt [0Ec
0
wt Y]]  
wt [0Belwt 0Ch 20[wt [0Ecwt 0Y]]  
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-reduction, another example
(*)
(*n)
(*v)
[y [0Deg z [0: z y]] 0x ]
( = square root)
(Deg/(())-a degenerated function)
-reduced “syntactically-by-name”:
[0Deg z [0: z 0x]]
 [[0Exist x] 0] ??? NO
-reduced “by value”:
2[0Sub [0Tr 0x] 0y 0[0Deg
z [0: z y]]]
for:
value of (*) of (*n)
of (*v)
x>0
False
False
False
x=0
True
True
True
x<0
Undefined True
Rules of substition
Undefined
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Valid rule of -reduction (2-phase)
Let C(y) be a construction with a free
variable y, y  , and let D  . Then
[[y C(y)] D]  2[0Sub [0Tr D] 0y 0C(y)]
is a valid rule (proof, see above).
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Rules of inference:
Types:
y  β, x  , D / (β), [Dx]  β, C(y)  α, y C(y)  (αβ),
[[y C(y)] [Dx]]  α.
Compositionality:
[0Improper 0[Dx]] | [0Improper 0[[y C(y)] [Dx]]]
[0Improper 0[Dx]] | [0Improper 02[0Sub [0Tr [Dx]] 0y 0C(y)]]
[0Proper 0[Dx]] | 2[0Sub [0Tr [Dx]] 0y 0C(y)] = [[y C(y)] [Dx]] =
C(y/[Dx])
Special case: Existential presupposition de re
Exist / (( (β)) )the property of a (β)-function of being defined at a
-argument, [Exist x]  ( (β))
[[0Exist x] D] | [0Improper 0[[y C(y)] [Dx]]]
[[0Exist x] D] | [0Improper 02[0Sub [0Tr [Dx]] 0y 0C(y)]]
But not: C(y/[Dx]) | [[0Exist x] D] …
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The two “de re principles”:
a) existential presupposition
Example:
[y [0Deg z [0: z y]] [0x]] | [[0Exist x] 0]
2[0Sub [0Tr [0x]] 0y 0[0Deg z [0: z y]]] | [[0Exist x] 0]
Indeed: The square root does not exist for x < 0; for x < 0
the left-hand side is (v-)improper. If the left-hand side
is true or false, then the square root exists and x  0.
However, the result of the “syntactical” β-reduction does
not meet these rules:
[0Deg z [0: z 0x]] and not (for x < 0) [[0Exist x] 0 ].
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The two “de re principles”:
b) inter-substitutivity of co-incidentals
[Dx] = [D’ ]

[[y C(y)] [Dx]] = [[y C(y)] [D’ ]] =
2[0Sub [0Tr [D’ ]] 0y 0C(y)]
Example:
The US President is the husband of Laura.
The US President is a Republican.
Hence: The husband of Laura is a Republican.
But not: John Kerry wanted to become the
husband of Laura.
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Substitutions in general
Types: c  n, 2c  , A  , y  
a) “by name” (homogeneous substitution):
2[0Sub 00A 0c 0C(c)] = C(c/0A)
2[0Sub 0A 0y 0C(y)] = C(y/A)
b) “by value” (generally valid, even for
heterogeneous substitution):
2[0Sub [0Tr A] 0y 0C(y)] = [y [C(y)] A]
 C(y/A)
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42
Conclusions
 The top-down, fine-grained approach of TIL makes it
possible to adequately model structured meanings,
and thus:




to formulate meaning-driven (non ad hoc) rules of
substitution taking into account the Use/Mention
distinction at all levels;
to adhere to Compositionality and anti-contextualism
(even in the cases of anaphora, de re attitudes with
anaphoric reference, hyper-intensional attitudes, …);
to take into account partiality;
to meet the two de re extensional principles (existential
presupposition, inter-substitutivity of co-referentials).
Rules of substition
43