To be inserted into the Book as Section 5.8., because there is no Section 5.7 (of the
Content – “Theory of Concepts”), and the current 5.8. – “Philosophy of
mathematics” is going to become Section 5.7. !
5.8 Incomplete meanings
We have seen that TIL semantic philosophy is thoroughly anti-contextualistic, which may
seem to be unrealistic in particular when dealing with anaphoric sentences or sentences
containing indexicals. In this section we show that solving this problem is compatible
with TIL anti-contextualism.
The meaning of an expression E is a construction C expressed by the expression.
In case of an empirical expression C constructs an α-intension, i.e., the function of type
α denoted by E, where α is the value (extension, if any) of the intension referred to by
the expression E in a given state of affairs w, t. In particular in case of an empirical
sentence S the meaning of S is a construction Cp of a proposition P of type . The
construction Cp provides an instruction how to evaluate the truth conditions of the
sentence in any state of affairs w, t. In other words, Cp makes it in principle possible to
determine the value of P (if any) in any w, t. Which truth-value the denoted proposition
has in particular circumstances w, t is however not a matter of a priori logical
investigation, it is an a posteriori empirical matter. In other words, to discover the
expressed construction (instruction, procedure) is an a priori matter of logic. Performing
(executing) the procedure in a state of affairs w, t is an empirical matter.
But some empirical sentences have an incomplete meaning; they express an
incomplete instruction that does not make it possible to evaluate the truth conditions in
any w, t. The meaning of a sentence can be an open construction (containing as
constituents one or more free variables) that constructs a proposition only after a
valuation (i.e., the value of a pragmatic or anaphoric parameter) of the free variable(s) is
provided. The semantic conception can be illustrated by a schema (that a construction C
v-constructs an α-object is denoted by C v α):
Empirical sentence
expresses
a closed construction C
of a proposition P/, C  
an open construction C(x)
with a free variable x, C(x) v 
For instance, the sentence “He is a logician” expresses an open construction with a free
variable x (Logic(ian) / (), x  ):
wt [0Logicwt x],
which does not construct a proposition, it only v-constructs. It is not possible to evaluate
the truth conditions of the sentence until the value of the parameter x (the meaning of
‘he’) is provided. This value is provided by a context that can be (roughly speaking) of
two kinds: a pragmatic context (the situation of utterance1) or a linguistic context, which
is the case of anaphora.
On the other hand, the sentence “Pavel is a logician” has a complete meaning, it
expresses the closed construction2
wt [0Logikwt 0PM].
Evaluating truth conditionsthe “way to the truth value of a sentence” can be
performed only if we fully understand the sentence, which means we know its meaning.
We must not however confuse understanding a sentence, i.e., knowing the respective
construction conceived as an abstract procedureinstruction how to evaluate in a
particular w,twith an execution performed in space and time. Even the construction of
double execution is just an instruction to do so, not an execution proper. Well, if you read
for instance in an encyclopaedia the sentence “Snowflake, the Spanish gorilla, Twiga
Mweupe, the giraffe from Tanzania and Goolara, the koala from San Diego Zoo, USA,
are albinos” you are content with understanding the sentence and will not go to Tanzania,
San Diego and Spain to check empirically whether these animals have pigmentation
problems. But as an instruction the meaning of this sentence is complete. In any state of
affairs you can execute the instruction and find out whether it leads to True or not,
without knowing the situation of utterance or a language context in which it occurs.
The difference between the meaningabstract procedure and its concrete
execution is even more striking in mathematics. If we understand Goldbach hypothesis
By a pragmatic context we mean only a situation of utterance. Hence we don’t take into account, e.g.,
(interrogative, imperative, emotional) intentions of a speaker and other pragmatic aspects.
2
Now we do not take into account the broad problem of the semantics of proper names. We simply
conceive of the name ‘Pavel’ as a label of a concrete individual, namely Pavel Materna, that is constructed
by a simple concept 0PM.
1
(that every even number greater than two is a sum of two primes), then we know the
respective structured meaninginstruction how to arrive at a truth-value (True or False).
And yet we do not know the value, because we do not know how to execute the
instruction, we do not know what the following construction constructs3:
[0 x [[[0Even x]  [0> x 02]]  [0 yz [[0Prime y]  [0Prime z]  [x = [0+ y z]]]]]]
Types:  - the type of natural numbers,  / ( ( )), Even / ( ), Prime / ( ),
x, y, z  .
The following Figure 1 illustrates the whole TIL semantic conception.
Empirical sentence S
(expresses)
Closed construction C  
Open construction C(x) v 
(the value of x „v-supplied“ by context:)
linguistic
(x  C’)
pragmatic, situation
of utterance
v(x) = entity 
Proposition P / 
–––––––––––––––––––––––––––––––––––––––––––––––
Out of the scope of logic:
Empirical (a posteriori) evaluation of the truth conditions in w,t
(results at True, False, or no value)
Figure 1. Semantic conception of TIL
Which case do we deal with when analysing anaphoric sentences or sentences
with indexicals? The problem of anaphora will be dealt with in Section 5.8.2. Next
Section 5.8.1 deals with indexicals.
3
The respective hypotheses expresses a synthetic concept a priori. For details, see Duží-Materna (2004)
5.8.1 Indexicals
By now the way TIL explicates meaning has proved its power by being
successfully applied to solving some well-known semantic problems. Frege’s semantic
schema has been essentially modified by
a) shifting intensions (in the case of empirical expressions) so that they are denoted,
b) distinguishing between denotation and reference (in the case of empirical
expressions), and
c) letting constructions (i.e., abstract procedures) play the role of Fregean
‘sense’(universally).
Further, Parmenides principle has furnished us with a kind of optimism as for the
possibility to find for every meaningful expression its (relatively) best meaning analysis.
Up to now all examples of semantic analyses could have been done in the area of
pure semantics, i.e., we have been interested exclusively in the relations of abstract
expressions to extra-linguistic objects. We have never needed to study events called
utterances of expressions. Therefore no Quinean skepticism concerning meaning could
affect our analyses.
In all examples we have adduced neither meaning nor denotation were dependent
on the utterance of the given expression. On the other hand, Quine’s ‘semantics’
concerns just utterances: therefore it is rather a (behavioristic) pragmatics.
Yet there is an important class of expressions in (obviously) every natural
language where it at least seems that what is denoted is not independent of the utterance
of the given expression. The members of this class are expressions that contain some
indexicals (‘egocentric’ expressions). The latter are mostly pronouns, for example
personal pronouns (”I”, “you”, “they” etc.), demonstratives (“this”, “that” etc.),
possessive adjectives (“my”, “their” etc.), also some adverbs (“now 4”, “here” etc.).
Clearly, no unambiguous semantic analysis is possible if such expressions that contain
indexicals are simply written on the blackboard.
Compare such an inscription as
(1)
This hat is very expensive
Well, in those system that have an explicit temporalisation (like the TIL system) ’now’ does not have an
indexical character. Actually, ‘now’ denotes an identical function of type ().
4
with a situation where this sentence is uttered by my friend who just looks at a hat in a
store-window. No question of my consent or dissent with the sentence written on a
blackboard arises. On the other hand, I can agree or disagree with my friend in the second
case. It is not until a sentence of this kind is uttered (i.e., not until a specific event
happens) that I can understand what the sentence claims. Thus while the truth conditions
of our sentence are not defined we can define these conditions when knowing the
situation of an utterance of it. We can say that our sentence written on the blackboard
does not determine any definite proposition. Can such expressions be logically handled?
The blackboard case differs from the utterance case in that the latter makes it
possible to understand the proposition connected with a particular utterance of (1) (in
general, of any expression containing indexicals). It means that the utterance offers some
criterion that makes it possible to identify the respective proposition (in general,
intension). Can we explicate this criterion?
In Tichý (1988), p. 194 we find an important characteristics of explications:
“It is to Frege that we owe the insight that the mathematical notion of
function is a universal medium of explication not just in mathematics but in
general.”
In our case the definiteness of a proposition connected with the given utterance
can be explained as follows: True, one and the same expression E containing indexicals
denotes one time one object (e.g., proposition), another time another object, but this is
due to the fact that there is a function F such that these various propositions are values of
F on various utterances of E. Thus the domain of F is the class of all utterances of E.5
This version (see however the footnote) explains the variability of the values of
expressions containing indexicals: the explanation (or: explication) shows that indexicals
point to the area of pragmatics. (Therefore Montague in his Pragmatics (@1974, pp. 95118) has among his indices not only possible worlds and times but also contextdependent ones.)
In Materna (1998, pp. 118-119) definitions of basic semantic properties of
‘pragmatically anchored’ expressions can be found, viz.
5
Another possibility is thinkable: F could be empirical in that it would have to be applied first to worldstimes and thereafter to the class that would be the value of the property‚ being an utterance of E.
i)
their meaning,
ii)
their pragmatic meaning in a situation S,
iii)
their pragmatic denotation in a situation S,
iv)
their WT-reference in a situation S.
Ad i): Pragmatically anchored expressions (PAEs) contain indexicals. Constructions that
are analyses of such expressions cannot represent indexicals in another way than
by free variables. The meaning of such a PAE is therefore the respective open
construction. (So the meanings of PAEs are not concepts.)
Ad ii): The pragmatic meaning of a PAE E in a situation S is the concept that arises from
the meaning of E via substituting respective objects given by the situation (of
utterance of E) for all free variables.
Ad iii): At most one intension / extension is identified by the pragmatic meaning of a
PAE E in a situation S. This intension / extension is the pragmatic denotation of E
in S.
Ad iv): If the pragmatic denotation of a PAE E in S is an extension then this is also the
WT-reference of E in S. Otherwise, i.e., if it is an intension, then its WT-value in
S is the WT-reference of E in S.
The most elaborate theory of indexicals is apparently David Kaplan@, in particular in his
(1978). His conception shares some points with the functional approach of TIL; compare,
e.g., his distinguishing between character and content: the former is just the function
from utterances (“contexts”) to contents, the latter might be our intensions (in particular
propositions). Inspiring could be Kaplan’s dthat (see Kaplan 1978a), where the idea that
utterances always take place in some world-time (see our preceding footnote!) is
exploited and the individual referred to in phrases of the form “Dthat [“…”] is F” is fixed
as follows: “The relevant individual is determined in the world in which the utterance
takes place”. TIL is a little reticent in this respect because the possible worlds used in
analyzing a language Λ are not characterized by such proto-properties which would be
defined by testing the use of expressions of the same Λ.
5.8.2 Anaphora and meaning (Semantic or pragmatic problem?)
5.8.2.1 Semantic pre-processing of the anaphoric reference
As stated above, the sentence “He is a logician“ has an incomplete meaning and
expresses thus an open construction wt [0Logicwt x]. If the sentence is uttered in the
situation when the speaker is pointing at Pavel Materna, then the meaning of the sentence
in this situation is the construction wt [0Logicwt 0PM], which though being v-equivalent
to the construction wt [0Logicwt x] is not however equivalent to this open construction;
in another situation we can obtain a different construction, because the variable x can vconstruct another individual. Hence the pragmatic meaning of the sentence in the given
situation of utterance is the closed construction, but the actual meaning is the open
construction6.
Now a question arises: If the sentence “He is a logician” occurs in a linguistic
context, does it also have an incomplete meaning? Since we advocate the anticontextualistic approach, the answer is yes, it has the same meaning in any context, which
means, it expresses always and in every context one and the same open construction.
However, when the sentence occurs in a linguistic context then we know to which
individual the anaphoric pronoun ‘he’ refers, so that the whole sentence must have a
complete meaning. For instance, in the following sentence (D) the pronoun ‘he’ refers to
Pavel
(D)
„If Pavel is rational, then he is a logician “,
and to understand the sentence completely we do not need any situation of utterance. The
sentence is a complete instruction for evaluating the truth conditions in any state of
affairs w, t. Hence its meaning has to be a closed construction constructing a proposition
without the mediation of a pragmatic or empirical factor.
But the meaning of (D) is not the following construction D2:
(Types: (Being) Ratio(nal) / ()), Logic(ian) / (), PM / )
(D2)
wt [wt [0Ratiowt 0PM]wt  wt [0Logicwt 0PM]wt],
(or, i-reduced wt [[0Ratiowt 0PM]  [0Logicwt 0PM]]),
because then the sentence (D) would be synonymous to the sentence
6
For details see Materna (1998, pp. 115-121)
If Pavel is rational then Pavel is a logician,
which is obviously not so.
A common objection against such a solution, namely that the first occurrence of
the name ‘Pavel’ can denote a different individual than the second one is however not
justified: the construction 0PM is simply a primitive concept of the given individual PM
regardless the way the individual is named. It constructs always, in every context, one
and the same individual.
But, there is a more serious objection: If the construction D2 were the meaning of
the sentence (D), then the meaning of the embedded clause “he is a logician” in this
context would have to be the construction wt [0Logicwt 0PM], and not the construction
wt [0Logicwt x], otherwise we would have to give up the Compositionality principle.
Thus it seems that we either have to give up the Compositionality principle, or the
principle of anti-contextualism. But our goal is to propose a solution that is fully in
accordance with TIL semantic principles (Compositionality and anti-contextualism).
A moment reflection on the way we understand the sentence (D) reveals the
solution.
Since the whole sentence (D) has a complete meaning, it is a complete instruction
for evaluating truth conditions in any state of affairs w, t; which means that as soon as we
understand (D), we know that a semantic pre-processing of the anaphoric reference is
specified. The pre-processing has to neither be specified by a pragmatic factor, nor
performed in the empirical level of its a posteriori evaluation. The instruction to preprocess the anaphoric reference has thus to be specified in the semantic level, it has to be
a constituent of the meaning of the whole sentence. We understand the sentence because
we know that the free variable (denoted by the pronoun name ‘he’) is to be substituted for
by the meaning of the expression to which ‘he’ refers. In our case the referred meaning is
the meaning of ‘Pavel’, viz. the construction 0PM. Hence anaphoric pronoun is not a
linguistic (elliptic) abbreviation, but a semantic abbreviation. The sentence expresses a
“two-phase instruction”: a) First pre-process the anaphoric reference by means of the
meaning of the antecedent expression, and then b) execute the adjusted meaning, namely
the pre-processed construction.
To specify the semantic pre-processing (phase a) of the two-phase construction)
we use the substitution function Subn that has been mentioned above (@Section 5.5.). Let
Subn / (*n *n *n *n) be the function that operates on constructions as follows: when being
applied to constructions C1, C2, C3, it returns as output the construction C that is the result
of correctly substituting C1 for C2 in C3. In the case of the sentence (D) we have n = 1,
hence we use the function Sub / (*1*1*1*1). The meaning of (D) is the construction (D1):
(D1)
wt [[0Ratiowt 0PM]  2[0SUB 00PM 0x 0[wt [0Logicwt x]]]wt].
Since (D1) seems to be rather complicated, we first make a type checking (using prefix
notation) and then we show that (D1) is an adequate analysis meeting our goals.
wt [ 0 [0Ratiowt 0PM]
2
[0SUB
00
PM
0
x
0
[wt [0Logicwt x]]]wt ]

(*1*1*1*1)

()
()

*1
*1 *1
*1 ( )




Now we first show what is constructed by the substitution (S)subconstruction of (D1)7:
(S)
[0SUB 00PM 0x 0[w’t’ [0Logicw’t’ x]]]  *1.
It constructs a construction of order 1, namely that one obtained by the substitution of
0
PM for the variable x into the construction w’t’ [0Logicw’t’ x]. The result is the
construction (S’) w’t’ [0Logicw’t’ 0PM], which evidently constructs the proposition P /
. But an argument of the logical connective implication () can be neither a
construction of proposition, nor a proposition, but a truth value. Since (S) constructs the
construction (S’), and (S’) constructs the proposition P, the execution steps have to be: a)
execute (S) to obtain the propositional construction, b) then execute the result to obtain
the proposition Phence double execution of (S)  , c) intensional descent of the
thus constructed proposition P/ with respect to the external w, t, namely:
[[2[0SUB 00PM 0x 0[w*t* [0Logikw*t* x]]]w] t] v .
7
To make the things clear, we α-renamed w,t variables.
This construction v-constructs the truth-value (True in those w, t, in which Pavel is a
logicianas it should be).
Hence the meaning of a sentence containing a clause with an anaphoric reference
is the procedure which is (in this case8) a two-phase instruction (which is specified by the
double execution). The instruction “says”:
First, execute the substitution of the meaning of antecedent for the anaphoric
variable, and then
second, the result (construction of a proposition) is to be executed again.
(You obtain a proposition that can be a posteriori evaluated in any empirical
context w, t.)
If = / () is the identity of truth values, then for any valuation v of variables w,
t it holds:
2
[0SUB 00PM 0x 0[w’t’ [0Logicw’t’ x]]]wt =
w’t’ [0Logicw’t’ 0PM]wt = [0Logicwt 0PM].
Hence the construction (D1) is equivalent to the construction (D2), but the meaning of
the sentence (D) is not the construction (D2), but (D1).
Note that the execution of the construction (S) corresponds to the pre-processing
of the sentence (D). This pre-processing is exactly specified by means of TIL, and it is a
constituent of the meaning (D1) of the sentence (D). Hence the semantic pre-processing
is specified. Moreover, the construction (D1) is closed (i.e., closed already “before” the
pre-processing the result of which “after” the pre-processing is (D2)). The variable x is
bound in (D1) by trivialisation, it is -bound.
At this point you might raise an objection. We said that the way we analyse an
expression is fully in accordance with the ‘Principle of Subject Matter’ coined by Tichý
Parmenides Principle9: An adequate analysis of an expression E contains only
constructions of those objects that receive mention by E. You may ask: “Which (sub-)
expression of (D) expresses the instruction to perform the substitution (S)?“ Then we
have to answer: “Anaphoric pronoun is a semantic abbreviation, and its full unpacked
8
In what follows we show that some execution steps (double execution or the intensional descent) can be
missing. It is so in those cases, when the expression with the anaphoric reference is used in a (hyper-)
intensional context.
9
For details see Duží – Materna (2005), Materna – Duží (2005).
meaning in a linguistic context is just the instruction to execute the semantic
substitution.” When unpacking the abbreviation, the sentence can be read as follows: “If
Pavel is rational then he (where by he is meant Pavel) is a logician”.
Let us now analyze by the above described method some more examples of
(A)a hyper-intensional attitude with anaphoric reference, (B)an intensional
(notional) attitude with anaphoric reference, and (C)an extensional attitude with
anaphoric reference.
(A)
5 + 7 = 12 and Charles knows it.
The embedded clause ‘Charles knows it’ does not express Charles relation(-in-intension)
to the truth value True, but to the procedure of calculating the result of 5 + 7 = 12. The
pronoun ‘it’ anaphorically refers to the meaning ‘5 + 7 = 12’the procedure, and not to
its output. Hence knowing is here a relation-in-intension of an individual to the
construction expressed by ‘5 + 7 = 12’. The meaning of (A) is thus a closed construction
(types: Know / (  *1), c / *2, c  *1):
’)
wt [[0=[0+ 0] 012] 
[0Sub 00[0=[0+ 0] 012] 0c 0[wt [0Know 0Charles
2
wt
c]]]wt].
The meaning of the sentence “Charles knows it” is an open construction
wt [0Knowwt 0Charles c].
The variable c is free here either for a pragmatic valuation (by the situation of utterance)
or for a substitution by means of the meaning of the antecedent that is referred to in the
linguistic context. The objectwhat is known by Charlescan be completed by a
situation of utterance or by a linguistic context. If the sentence occurs in another
linguistic context, then Sub2 substitutes a different construction of order 1 for the variable
c, namely the construction to which ‘it’ anaphorically refers.
(B)
Charles sought the mayor of Dunedin but (he) did not find him.
The function Sub creates a new construction from constructions, and thus it can
easily be iterated. Suppose now that Charles’ search concerned the office of the mayor of
Dunedin, he was finding out who holds the office (the de dicto reading of (B), see Section
5.5.@).
Types: Seek / (  ), Find / (  ), Charles / , Mayor (of something) / (),
D(unedin) / , x  , u  :
(B’)
wt [[0Seekwt 0Charles wt [0Mayorwt 0D]] 
2[0Sub 00Charles 0x [0Sub 0[wt [0Mayor 0D]] 0u 0[wt
wt
[0Findwt x u]]]]wt].
Again, the meaning of (B) is the closed construction (B’), and the meaning of the
embedded clause “He did not find him” is the open construction wt [0Findwt x u] with
two free variables, x (heCharles) and u (himthe office).
When this clause (sentence) occurs in a different linguistic context, its meaning is
the same. For instance, the de dicto reading of the sentence
(E)
Whomever Charles seeks he is not finding him,
where Seek is again a relation to a -office is analysed as follows.
Types: Seek / (  ), Charles / , Find / (  ), x  , u  , z  :
(E1’) wt z [[0Seekwt 0Charles z] 
2[0Sub 00Charles 0x [0Sub 0z 0u 0[wt
[0Findwt x u]]]]wt].
The construction (E1’) is again equivalent to the construction “after” the substitution
(E2’’) wt z [[0Seekwt 0Charles z]  [wt [0Findwt 0Charles z]]wt], which is iequivalent to
wt z [[0Seekwt 0Charles z]  [0Findwt 0Charles z]].
The meaning of (E) is however the construction (E1’) in which the semantic anaphoric
pre-processing is specified, and in which the meaning of the embedded clause is not
distorted.
The example of an extensional attitude is easy to analyse:
(C)
Charles met the Mayor of Dunedin and (he) carefully listened to him.
Types: Meet / (), (carefully) Listen / ()), x  , y  .
The meaning of the embedded clause is again an open construction:
[wt [0Listenwt x y]].
The substitution of the meaning of the first antecedent (0Charles) for the anaphoric
variable x is not a problem (the individual Charles is a priori given). But, for the variable
y we are to substitute the construction of that (unspecified) individual (if any) who in a
given external empirical context w, t of the sentence (C) plays the role of the mayor of
Dunedin. Since the embedded clause “He carefully listened to him” has its own empirical
context, in order to prevent the collision of variables we have to rename the variables
w, t:
[w’t’ [0Listenw’t’ x y]].
The resulting analysis of (C) is:
(C1’) wt [[0Meetwt 0Charles [0Mayorwt 0D]] 
2[0Sub 00Charles 0x [0Sub 0[0Mayor 0D] 0y 0[w’t’ [0Listen
wt
w’t’
x y]] ]]wt ],
which is equivalent to (C2’):
(C2’) wt [[0Meetwt 0Charles [0Mayorwt 0D]]  [0Listenwt 0Charles [0Mayorwt 0D]]].
The meaning of (C), i.e., the adequate analysis of (C), is the construction (C1’). The
construction [w’t’ [0Listenw’t’ x y]] is in (C1’) used with the de re supposition. Though
we first rather carelessly drew the de re occurrence of the concept of the mayor of
Dunedin into the de dicto (within the scope of the substitution) context of the
construction [w’t’ [0Listenw’t’ x y]], the resulting construction is correct due to the
double execution followed by the intensional descent (which turns [w’t’[0Listenw’t’ xy]]
into the de re occurrence within the whole (C1’) construction).
The anaphoric reference can occur not only in an embedded clause, it can also be
a part of a sentence. Let us analyse some other examples:
(F)
“John loves his mother “
(G)
“Everybody loves his mother “
First of all, the expression „his mother“ does not have a self-contained meaning10; the
elliptic pronoun “his” does not play the role of an anaphoric reference here. Only the
expression “loves his mother” has a self-contained meaning, it denotes a property LM of
individuals of type (). Other types:
John / , Love / ( ), Mother (of somebody) / (), Man / (); variables x, y  .
The property LM is now constructed by
wt y [0Lovewt y wt [0Motherwt y]wt], or by βi-reduced
(LM’) wt y [0Lovewt y [0Motherwt y]].
The sentence (F) predicates the property LM of John:
(F1)
wt [[wt y [0Lovewt y [0Motherwt y]]]wt 0John], or βi-equivalent
(F2)
wt [y [0Lovewt y [0Motherwt y]] 0John].
We keep analyzing in turns of the “Principle of subject matter” (Parmenides principle) described in
Materna – Duží (2005). @
10
The construction (F2) is an adequate (the best one with respect to a given ontology, i.e.,
the set of primitive concepts11) analysis of (F). Since 0John can never be improper, (F2)
can further be safely β-reduced into (F2’):
(F2’) wt [0Lovewt 0John [0Motherwt 0John]].
However, according to the Parmenides Principle adequate analyses of the sentence (F)
are the constructions (F1), (F2).
The sentence (G) now claims that every man has the property LM:
(G1)
wt x [[0Manwt x]  [[wt y [0Lovewt y [0Motherwt y]]]wt x]].
Another example shows that (F1) is the most adequate analyses of (F):
(F’)
“John loves his mother and so does Peter“
Hence: “John and Peter have a common property “12
The expression ‘so does’ plays the role of an anaphoric reference to the property of
“loving his own mother”, i.e., the property LM. The analysis of the embedded clause “so
does Peter” is thus an open construction with a free variable p  ():
wt [pwt 0Peter].
By substituting a construction of the property LM for the variable p we obtain the
meaning of (F’):
(F’1) wt [[[wt y [0Lovewt y [0Motherwt y]]]wt 0John] 
[ Sub 0[wt y [0Lovewt y [0Motherwt y]]] 0p 0[wt [pwt 0Peter]]]wt].
2 0
The construction (F’1) is equivalent to the construction that receives after the semantic
pre-processing (i.e., after the substitution and double execution are executed):
(F’2) wt [[[wt y [0Lovewt y [0Motherwt y]]]wt 0John] 
[wt [[wt y [0Lovewt y [0Motherwt y]]]wt 0Peter]]wt] =βi
(F’3) wt [[[wt y [0Lovewt y [0Motherwt y]]]wt 0John] 
[[wt y [0Lovewt y [0Motherwt y]]]wt 0Peter]]
The consequence that Peter and John have a common property, namely
11
A sel of simple concepts determines a conceptual system we work with. For details see Materna (2004)
@
12
We advocate here only the alternative of “strict reading” of the sentence (F’), namely: “John loves his
mother and so does Peter (i.e., Peter also loves his own mother)” and exclude the “sloppy reading“ “John
loves his mother and so does Peter (i.e., Peter loves John’s mother)”, see Neale (2004, p.63).
wt [p [[pwt 0Jan]  [pwt 0Petr]]],
is now trivially derivable by existential generalisation from (F’3). LM constructed by
[wt y [0Lovewt y [0Motherwt y]]] is the common property shared by John and Petr.
In so far analysed examples the analysis containing as a constituent the (double)
execution of the construction of the function Sub (i.e., the meaning of the whole sentence
with anaphoric reference) has been equivalent to the construction obtained after the
(double) execution of the substitution (and, as the case may be, the execution of the
intensional descent). The meaning of the antecedent to which the anaphoric pronoun
refers has been a) mentioned in a hyper-intensional context, or b) used with the de dicto
supposition in an intensional context, or c) used with the de re supposition in an
extensional context. The above mentioned equivalence is due to the fact that the
respective substitution is in a certain way homogeneous: we inserted a) a hyper-intension
(i.e., construction) into the hyper-intensional context, or b) an intension into the
intensional context, or c) an extension into the extensional context.
However, all the sentences containing an anaphoric reference are not so simple.
Problems may appear when there is a need to substitute a lower-order entity into a
higher-order context, namely an extension into an intensional or hyper-intensional
context, or an intension into the hyper-intensional context, because the higher-order
context is dominant.
It is in particular the case of de re attitudes (notional or
propositional ones) that we are going to analyse in the next section.
5.8.2.2 Analysis of de re attitudes with anaphoric reference.
5.8.2.2.1. De re notional (intensional) attitudes
In Section 5.5 we discussed notional attitudes viewed as intensional objects, namely
relations-in-intension of an agent to an intension: (  α)  objects. We analysed three
kinds of such attitudes, viz. sentences on a) wishing, b) seeking, and c) finding and
enunciated the ambiguity of such sentences. There are almost always at least two
alternative readings, viz. de dicto and de re reading. The de dicto reading is easy to
analyse, the agent is simply related to the α-intension the construction of which occurs
with the de dicto supposition. There is no existential presupposition, the agent’s intention,
and/or (unsuccessful) search may be a relation-in-intension even to a non-existent object,
i.e., to an intension that is actually not occupied or has an empty extension in a given
state of affairs w, t.
But we could also see that the de re reading of these sentences meets a problem:
the respective analysis has to respect the existential presupposition. Moreover, a more
fine-grained analysis reveals that the agent is again related to an intension, this time of
another type. In case of search the type of an object of the search is , which is usually
the location of a given individual to whom the agent is extensionally related, or, in case
of wishing the type is (), , respectively. Hence the construction of this intension
(-office, property, proposition) has to occur with the de dicto supposition. The problem
now consists in discovering a correct way of inserting the extension (usually individual)
to the latter de dicto intensional context of the construction of the intension.
First we encountered the problem in Section 5.5 when analysing the de re reading
of the sentence
(4)
Charles would like to talk to the Mayor of Dunedin
a coarse-grained analysis of which is (types WLTr / () (would like to talk), Charles /
, M(ayor of) / (), D(unedin) / ):
(4r)
wt [0WLTrwt 0Charles [wt [0Mwt 0D]wt].
An attempt to refine the analysis has been solved in Section 5.5 by means of
reformulating the sentence (4) into its passive form:
(4passive)
The Mayor of Dunedin is the man, to whom Charles would like to talk to.
We denoted by WChT / () the property of individuals (‘being wanted by Charles to
talk to’), and ascribed this property to the (actual, current) Mayor of Dunedin obtaining
thus (types Talk / (), WL1 / (  ())the relation-in-intension of an
individual to the property):
(4r’)
wt [[wt y [0WL1wt 0Charles [wt x [0Talkwt x y]]]]wt [wt [0Mwt 0D]]wt],
which is an adequate analysis of the de re passive reading of the sentence (4). The
variable y in (4 r’) is the meaning of the anaphoric pronoun ‘whom’ of (4passive). Then
the i-reduction leading to
(4r’)
wt [y [0WL1wt 0Charles [wt x [0Talkwt x y]]] [0Mwt 0D]]
can be performed.
This solution is correct. Anyway, we promised to analyse directly the active form of the
de re re adding of (4):
(4active)
Charles would like the (actual) Mayor of Dunedin that he (Charles) talked
to him.
Bjorn, is it a good English? Or, should it better be Charles wishes of the Mayor of
Dunedin that he (Charles) talked to him ? How would you formulate the de re active
reading of (4) in proper English?
We are now going to use the method of semantic pre-processing described above. First,
the meaning of the embedded clause “he talked to him” is an open construction of a
proposition (variables x, y   are meanings of the anaphoric pronouns he, him):
(Emb)
wt [0Talkwt x y]
The analysis of the sentence (4active) has to respect the existential presupposition on the
Mayor of Dunedin, and Charles wants that the proposition v-constructed by (Emb) were
True for Charles and the (actual) Mayor of Dunedin. To this end we now use the WL2 /
(  ) objectthe relation of an individual to a proposition that the individual wants
to be True in w, t. Hence we need to insert the construction of Charles for x into (Emb),
which is not a problem: individual Charles is a priori given to us, its trivialisation cannot
be v-improper. But inserting the individual (if any) that plays the role MD of the Mayor
of Dunedin in w, t is a problem: We need to insert the construction of an
extensionnamely, the value (if any) of the individual office MD at w,t into the
intensional (de dicto) context of (Emb). To this end we use the functions Tr and Sub. Tr /
(*1 ) returns as its value the Trivialization (construction of order 1of type *1) of its
argumentan individual of type . Sub / (*n *n *n *n) operates on constructions in the
following manner: when applied to constructions A, B, C, it returns as its output the
construction D that is the result of correctly substituting A for B in C.
(4rAct)
wt [0WL2wt 0Charles
[ Sub 00Charles x [0Sub [0Tr [wt [0Mwt 0D]]wt] y 0[wt [0Talkwt x y]]]]].
2 0
We can easily check that this analysis is adequate: in any such state-of-affairs w, t in
which the Mayor of Dunedin fails to exist, the intensional descent [wt [0Mwt 0D]]wt fails
(i.e., is v-improper), as does the composition [0Tr [wt [0Mwt 0D]]wt], which is why Sub
receives no argument to operate on. Hence, the whole composition [0WL2wt 0Charles …]
is v-improper13, and the constructed proposition has no truth value. This is as it should be,
because otherwise, in those w, t where the Mayor does exist and the office is occupied
say by Mr. X, the result of the substitutions is the construction
[wt [0Talkwt 0Charles 0X]],
the execution of which constructs the proposition to which Charles is related. Hence
double execution is necessary: the first execution is the instruction to adjust (pre-process)
the construction wt [0Talkwt x y], the second execution constructs the respective
proposition that Charles wants to be True.
A similar method of the semantic pre-processing can also be applied to the de
dicto reading of the sentence (4) when conceiving ‘would like’ as denoting an object WL2
/ (  ). In Section 5.5 we proposed the analysis of the form
(4’’)
wt [0WL2wt 0Charles [wt [0Talkwt 0Charles [wt [0Mwt 0D]]wt]]].
This is an adequate analysis with respect to the de dicto supposition in which the concept
of the Mayor of Dunedin occurs; Charles has no idea who the mayor could be, he is
interested just in the office itself; (4’’) can be read as
(4d)
“Charles wishes that he (Charles) would talk to the Mayor of Dunedin.”
However, this proposal does not take into account the anaphoric pronoun ‘he’. Now we
have at our disposal the method of substitution by means of which the most adequate
analysis of (4d) is easy to discover (x  ):
(4’d)
wt [0WL2wt 0Charles 2[0Sub 00Charles x [wt [0Talkwt x [wt [0Mwt 0D]]wt]]]].
In Section 5.5 we actually analyzed Charles’ finding a location of the (actual
known) murderer of X in the above described way. (See Section 5.5, construction (7a’).)
Note that while the de dicto analysis (4’d) specifying the anaphoric pre-processing in the
adequate way is equivalent to the construction obtained after the pre-processing, namely
to (4’’), in the de re case it is not so. The most adequate analysis of the de re reading,
namely (4rAct), is not equivalent to the construction obtained after the respective
(double-) execution of the substitution, namely to [wt [0Talkwt 0Charles 0X]], as
explained above.
13
Partiality is being propagated upwards. For details see Duží (2003a).
Above we analyzed the de dicto reading of the sentence (B):
(B)
Charles sought the mayor of Dunedin but (he) did not find him.
Now we are going to analyze (perhaps a more natural) de re reading of this
sentence: namely (B) is uttered in such a situation when Charles knows who the Mayor
is, and he strived for locating this individual but did not succeed in his effort. A coarsegrained analysis relates Charles to the (unspecified) individual who plays the role of the
mayor. In the preliminary analysis we first do not take into account the anaphoric
character of the embedded clause:
Types: Look-For / (), Find / ()14, x  , y  :
(B’)
wt [[0Look-Forwt 0Charles [wt [0Mayorwt 0D]]wt] 
[0Findwt 0Charles [wt [0Mayorwt 0D]]wt]]
Now refining (B’) we have to take into account the fact that Charles sought the
place where the Mayor of Dunedin occurred. Charles’ (unsuccessful) search is thus
related not to the -office of the Mayor of Dunedin, but to the -office ‘Location of the
Mayor of Dunedin’. However, the concept of the former occurs in the de re supposition.
The fine-grained meaning of the complement clause ‘he did not find him’ is thus
an open construction (x  , y  , Loc(ation of somebody) / (), FindL / (  )):
wt [0FindLwt x wt [0Locwt y]],
where the variables x, y stand for the meaning of anaphoric pronouns ‘he’ and ‘him’,
respectively.
To find the fine-grained meaning of the first clause ‘Charles sought the mayor of
Dunedin’ on its de re reading we use the technique introduced above: we need to insert
the construction of an extensionnamely, the value (if any) of the individual office of
the Mayor of Dunedin at w,tinto the de dicto context of the location of y. Functions Tr
(*1 ) and Sub / (*n *n *n *n) serve to meet this goal.
Types: SeekL / (  ), y  .
Now, looking for the (unspecified) individual who holds the office of the Mayor of
Dunedin is explicated by the following construction. In any w, t the left-hand side and
right-hand side v-construct the same truth value:
In this section we use ‘Look for’ and ‘Find’ as denoting relation-in-intension of an individual to an
individual, whereas ‘SeekL’, ‘FindL’ as denoting relation-in-intension of an individual to a -office.
14
[0Look-Forwt 0Charles [wt [0Mayorwt 0D]]wt] =
[0SeekLwt 0Charles 2[0Sub [0Tr [wt [0Mayorwt 0D]]wt] 0y 0[wt [0Locwt y]]]].
(In any such state-of-affairs w,t in which the Mayor fails to exist, the intensional descent
[wt [0Mayorwt 0D]]wt fails (i.e., is v-improper), as does the composition [0Tr [wt
[0Murdwt 0X]]wt], which is why Sub receives no argument to operate on. Hence, the whole
composition is v-improper. In those w, t where the Mayor does exist and the office is
occupied, say by Y, the result of the substitution is the construction [wt [0Locwt 0Y]],
the execution of which constructs the respective -office to which Charles is related.)
Now it remains to apply the same technique to the fine-grained analysis of the
embedded clause so that to take into account the anaphoric character of the pronouns ‘he’
and ‘him‘. By combining the two clauses together (and performing the respective βireductions) we obtain the most adequate fine-grained analysis of the de re reading of the
sentence (B):
(Br)
wt 2[Sub [0Tr [0Mayorwt 0D]] 0y 0[0Sub 00Charles x
[[ SeekLwt 0Charles wt [0Locwt y]]  [0FindLwt x wt [0Locwt y]]]]].
0 0
Another alternative way to analyze the de re reading of the sentence (B) consists
in ascribing a property of being sought and not found by Charles to the holder of the
office of the mayor of Dunedin. We show that using this way we actually do not execute
the substitution. To make it clear, we can reformulate the sentence into its passive:
(Bpassive)
The mayor of Dunedin has been looked for but not found by Charles.
Now we construct the property of being looked for and not found by Charles.
Types: as above.
wt y [[0SeekLwt 0Charles wt [0Locwt y]]  [0FindLwt 0Charles wt [0Locwt y]]].
Ascribing this property to the holder of the office of the Mayor leads to the alternative
analysis (Br’):
(Br’)
wt [y [[0SeekLwt 0Charles wt [0Locwt y]] 
[0FindLwt 0Charles wt [0Locwt y]]] [0Mayorwt 0D]].
Summarising, …@
5.8.2.2.2 Hyper-intensional de re ‘propositional attitudes’
Now we are going to examine the case of so-called ‘propositional attitudes’ that are
actually hyper-intensional. The agent is not related to a proposition but to its
construction.
In Section 5.4@ we distinguished two kinds of propositional attitudes, viz.
implicit and explicit ones. Implicit attitudes are ascribed to the agent from the outside;
once the agent knows (believes, doubts, etc.,) a proposition P, he/she/it knows (believes,
doubts, etc.) all the logical consequences of P, or (in a restricted Montague-Scoot @
version) all the propositions equivalent to P. Thus handling implicit attitudes in this way
leads inevitably to a variant of the paradox of logical / mathematical omniscience. The
tightest restriction that can be obtained by “implicit approach” is the restriction up to
equivalence of statements, because equivalent sentences cannot be distinguished by an
intensional approach.
When dealing with an information base of a real (computational) agent in a multiagent world, intensional approach does not suffice. An agent in a multi-agent world has
to be able to reason about his/her/its own knowledge (beliefs, hypotheses), as well as
about that of the other agents. We need a more fine-grained analysis so that to prevent the
paradox of omniscience, hence we adhere to the hyper-intensional approach.
The need for a fine-grained hyper-intensional analysis of the information content
of particular agents’ information base can be illustrated by the following example:
Example Imagine that we design a multi-agent system in which the three agents a, b and
c operate, and in which c knows
(i)
that a believes of the capital of the Czech Republic that the number of its
inhabitants who are in danger equals decimal 1048576(10);
(ii)
that b believes of the capital of the Czech Republic that the number of its
inhabitants who are in danger equals hex number 100000(16).
Though the possible-world propositions denoted by the complement clauses are identical,
c cannot deduce that a’s and b’s respective pieces of information are identical. In case of
emergency the security system could slide into a chaotic state, if c supposed that the
information base of a and b were identical. For, if a and/or b does not master the rules of
transition from the decimal to the hexadecimal number system, they may not be able to
properly communicate.15 What is more, if the capital of the Czech Republic ceases to
exist, while c supposes that both (i) and (ii) are true, the system is heading for
inconsistency. In such a situation c won’t know what either a or b believes of the capital
of the Czech Republic. Both the proposition that a (or b) believes of the capital that it is
thus and so, and the proposition that a (or b) does not believe of the capital that it is thus
and so will be without truth-value.
(i) and (ii) are examples of constructional de re attitudes, the analyses of which
have to respect the existential presupposition that the capital of the Czech Republic
should exist.
Types: CCR (the capital of the Czech Republic) / , Number (of elements of a set of
individuals) / (()), (being) Inh(abitant of something) / (), (being in)Danger /
(), a, b, c / , 1048576(10), 100000(16) / , Know / (  *1), Believe / (  *1).
The distinct meanings of the embedded clauses with an anaphoric reference ‘its’
are open constructions (with the free variable y standing for the meaning of the anaphoric
pronoun ‘its’) which v-construct one and the same proposition.
For the sake of simplicity, we will now use a common mathematical infix notation
without trivialisation in the two different constructions of one and the same magnitude
(namely, number of inhabitants of y who are in danger, x  , y  ):
(DEC)
wt [0Number x [[0Inhwt x y]  [0Dangerwt x]] =
[106 + 4.104 + 8.103 + 5.102 + 7.10 + 6]],
(HEX)
wt [0Number x [[0Inhwt x y]  [0Dangerwt x]] = 165].
However, further we will abbreviate the above different constructions of one and the
same number, viz. [106 + 4.104 + 8.103 + 5.102 + 7.10 + 6] and 165, respectively, using
the usual notation ‘1048576=(10)’ instead of ‘[106 + 4.104 + 8.103 + 5.102 + 7.10 + 6]’ and
‘100000(16)’ instead of ‘165’.
Now when analysing (i) and (ii), the construction of the individual office CCR has
to occur in the de re supposition and we need to insert the construction of an
extensionnamely, the value (if any) of the individual office CCR at w, tinto the
15
This transition is only superficially to do with shifting between notational systems. What is at stake is, at
heart, a shift from one calculation to another. Relations to calculations are relations to constructions, not
formulae belonging to some system of mathematical notation; though the attitude must be reported by
means of a particular such system.
hyperintensional (constructional) contexts of a’s and b’s pieces of information. To this
end we again use the functions Tr / (*1 ) and Sub / (*n *n *n *n) described above.
(i’)
wt [0Believewt 0a [0Sub [0Tr 0CCRwt] 0y
0
(ii’)
[wt [0Number x [[0Inhwt x y]  [0Dangerwt x]] = 1048576(10)]]]]
wt [0Believewt 0b [0Sub [0Tr 0CCRwt] 0y
0
[wt [0Number x [[0Inhwt x y]  [0Dangerwt x]] = 100000(16)]]]].
We can easily check that these analyses are adequate: in any such state-of-affairs w, t in
which the capital of the Czech Republic fails to exist, the intensional descent 0CCRwt fails
(i.e, is v-improper), as does the composition [0Tr 0CCRwt], which is why Sub receives no
argument to operate on. Hence, the whole composition [0Believewt 0a ….] is v-improper,
and the constructed proposition has no truth value. This is as it should be, because
otherwise, in those w,t where the capital does exist and the office is occupied (say) by
Prague, the results of the substitution are the constructions
[wt [0Number x [[0Inhwt x 0Prague]  [0Dangerwt x]] = 1048576(10)]]
[wt [0Number x [[0Inhwt x 0Prague]  [0Dangerwt x]] = 100000(16)]],
to which a, b, respectively, are related.
The exterior agent c attempting to draw valid inferences about the interior agents
a and b’s information bases must take into account a’s and b’s inferential capabilities16.
5.8.2.3 Anaphora and quantification (“Donkey sentences”)
The following example is a variant of the well-known problem of a „donkey sentence“:
(K)
“If somebody has got a new car then he often washes it“.
The analysis of the embedded clause “he often washes it“ with anaphoric
pronouns ‘he’ and ‘it’ is again an open construction with two free variables x1who
(washes), x2 what (is washed), x1  , x2  , Wash / (  ):
wt [0Washwt x1 x2].
If we also want to analyze the frequency of washing, i.e., the meaning of ‘often’, then we
use a function Freq(uently)17 / ((())). The function Freq associates each time-point T
In Duží, Jespersen, Müller (2006)@ we introduced the notion of ‘Inferable knowledge’ which is
computed as a closure of a knowledge base relatively to particular set of inference rules.
16
with a set of those time-intervals (of type (())) that are frequent in T (for instance
once a week). The analysis of “he often washes it“ is then
wt [0Freqt t’[0Washwt’ x1 x2]].
However, since rendering the frequency of washing does not influence the way of solving
the problem of anaphora in a “donkey sentence“, we will use for the sake of simplicity the
first construction.
A problem to solve is however rather a logical form18 of a “donkey sentence“,
than the problem of the anaphoric reference. P. Geach (1962, p.126) proposes a structure
that can be rendered in the 1st order predicate logic as follows (NC – new car):
xy ((NC(y)  Has(x, y))  Wash(x, y)).
However, to this solution Russell19 objected that the expression “a new car” is an
indefinite description, which is not rendered by Geach’s analysis. Hence Russell
proposed an analysis that corresponds to the formula of the 1st order predicate logic:
x (y (NC(y)  Has(x, y))  Wash(x, y)).
But the last occurrence of the variable y (marked in bold) is free in this formulaout of
the scope of the existential quantifier that should bind the variable.
Neale in his (1990) proposed a solution that combines both the above proposals.
On the one hand the existential character of an indefinite description is saved (Russell’s
demand), and on the other side the anaphoric variable is bound by the general quantifier
(Geach’s solution). Neale introduces the so-called restricted quantifiers, the semantics of
which is not clear and it is questionable whether it can be expressed by means of the 1st
order predicate logic:
[every x: man x and [a y: new-car y](x owns y)]([whe z: car z and x owns z]
(x often washes z))20.
The sentence (K) does not affirm that if the man owns more than one new car then he/she
does not often wash some of them. Hence we can reformulate the sentence into (K’):
See Tichý (1986, p.263), Duží (2004)
See Duží, Materna (2005)
19
See Russell (1995) @
20
Neale (1990, p. 236). Note: Neale takes into account that the sentence is true even if a man owns more
than one new car. To avoid singularity he thus claims that the description used in his analysis does not have
to be singular (definite) but plural: his abbreviation "whe F" stands for "the F or the Fs".
17
18
(K’)
Every man who owns some new cars (he often) washes all of them (any of the
new cars he/she owns).
However, the following sentence (K’’) claims something else:
(K’’) Every man who owns some new cars (he often) washes some of them (some of the
new cars he/she owns).
Types: Own / ( ), Wash / ( ), NC (being a new car) / (),
variables: x  , y  , z  .
The analysis of (K’) that in principle corresponds to Geach’s proposal is:
(K’3) wt xy [[[0NCwt y]  [0Ownwt x y]] 
2 0
[ Sub 0x 0x1 [0Sub 0y 0x2 0[wt [0Washwt x1 x2]]]]wt],
which is (βi-)equivalent to the construction after the executing substitution
(K’4) wt xy [[[0NCwt y]  [0Ownwt x y]]  [0Washwt x y]].
But there is Neale’s objection that can be applied against these analyses, namely that in
the original sentence (K) the anaphoric pronoun ‘it’ stands out of the scope of the
quantifier occurring in the antecedent.
To meet this demand, we introduce a different type of quantifiers. Besides the
common quantifiers ,  / (()) that “operate” on a set of individuals (returning True
iff this set is the whole universe () / non-empty (), respectively) we use quantifiers of
another type, namely21:
Some / ((()) ()), All / ((()) ()).
Some is a function that associates the argumenta set Swith the set of all those sets
which have a non-empty intersection with S.
All is a function that associates the argumenta set Swith the set of all those sets
which contain S as a subset.
Thus for instance the sentence “Some students are stupid” is analyzed using Some as
follows (Student / (), Stupid / ()):
wt [[0Some 0Studentwt] 0Stupidwt].
See also Materna – Duží (2005), Duží (2003a)@. These quantifiers obviously correspond to Neale’s
‘restricted quantifiers’.
Type checking:
wt [ [0Some
0
((()) ())
Studentwt]
0
Stupidwt]
()
(())
()


Similarly the sentence “All students are stupid” is analyzed as follows:
wt [[0All 0Studentwt] 0Stupidwt]
Let us analyze first the embedded clauses of (K’), (K’’), namely:
E1:
“he washes all of them”
E2:
“he washes some of them”.
The anaphoric reference ‘them’ refers here to the set of individuals, viz. new cars which a
man x owns. Therefore we use a variable p  () standing for the meaning of ‘them’.
The analyses of E1, E2 are:
wt [[0All p] x2 [wt [0Washwt x1 x2]]wt],
wt [[0Some p] x2 [wt [0Washwt x1 x2]]wt], or, βi-reduced
E1’
wt [[0All p] x2 [0Washwt x1 x2]]
E2’
wt [[0Some p] x2 [0Washwt x1 x2]].
Since the variable p refers to the set of new cars that the man x owns, we need to
substitute a construction of this set for p. Further we have to substitute the variable x
(anybody) for the variable x1 (who washes), and then the pre-processed construction has
to be executed (thus double execution: first execute Sub, then the adjusted construction).
Finally, thus v-constructed proposition has to be intensionally descended (applied to w, t)
to a truth-value, in order that the second argument for the implicative connective  is
obtained. To prevent collision of variables, we rename “internal variables” w, t.
(For the sake of clearness we now use the full notation for quantifiers , .)
Types: Man / (), N(ew)C(ar) / (), Own / (), Wash / (),
 / (()), / (()), All / ((())()), Some / ((())()), x, y, x1, x2  , p  ().
The analysis of (K’) gets into:
(K’1) wt [0x [[[0Manwt x]  [0y [[0NCwt y]  [0Ownwt x y]]]] 
[ Sub 0[y [[0NCwt y]  [0Ownwt x y]]] 0p [0Sub 0x 0x1
2 0
0
[w’t’ [[0All p] x2 [0Washw’t’ x1 x2]]]]]wt]].
(K’1) reads: It holds for every man that if the man owns some new cars then he
(the man x) washes all of them (i.e., of his new cars p).
This construction can be viewed as the most adequate analysis of (K’), because it meets
Russell’s demand on in definite description in the antecedent and the scope of the
existential quantifier  does no exceed the antecedent. Now (K’1) is equivalent to the
construction that would be obtained after the pre-processing (i.e., executions of the
respective substitutions):
(K’2) wt [0x [[[0Manwt x]  [0y [[0NCwt y]  [0Ownwt x y]]]] 
[[0All [y [[0NCwt y]  [0Ownwt x y]]]] x2 [0Washwt x x2]]]].
(K’2) reads: It holds for every man that if the man owns some new cars then the man
washes all the new cars the man owns.
The second possible reading of (K) is now analyzed in a similar way using Some instead
of All:
(K’’1) wt [0x [[[0Manwt x]  [0y [[0NCwt y]  [0Ownwt x y]]]] 
[ Sub 0[y [[0NCwt y]  [0Ownwt x y]]] 0p [0Sub 0x 0x1
2 0
0
[w’t’ [[0Some p] x2 [0Washw’t’ x1 x2]]]]]wt]].
(K’’1) reads: It holds for every man that if the man owns some new cars then he
(the man x) washes some of them (i.e., of his new cars p).
(K’’1) is also equivalent to the construction that would be obtained after the preprocessing (i.e., executions of the respective substitutions):
(K’’2) wt [0x [[[0Manwt x]  [0y [[0NCwt y]  [0Ownwt x y]]]] 
[[0Some [y [[0NCwt y]  [0Ownwt x y]]]] x2 [0Washwt x x2]]]].
Note: As stated above, it is not clear how to exactly understand the sentence (K), the
sentence is ambiguous; in our opinion both the readings (K’), (K’’) are plausible.
However, Neale obviously took into account only the first reading (K’).
5.8.2.4 Indefinite (incomplete) descriptions
The problem of indefinite descriptions has been a subject of much dispute among
philosophers and logicians just when connected with an anaphoric reference. S. Neale
characterizes the indefinite descriptions in the proceedings Descriptions and Beyond
(2004, p.32) as follows:
The label ‘incomplete description’ is misleading. But we need to begin somewhere,
so let us have some preliminary definitions. Let us say for the moment that a
description is proper if, and only if, its nominal—or its superficial matrix in some
standard system of representation—is true of exactly one thing, and improper
otherwise. And let us say that an improper description is empty if it is true of
nothing, and incomplete if it is true of more than one thing.
Now we have to make a comment: From the point of view of TIL the condition of a
description being ‘proper’, namely „its nominal—or its superficial matrix in some
standard system of representation—is true of exactly one thing” expresses analytical
unicity (singularity). It means that it characterizes a definite description. In every state of
affairs w, t there is just one entity denoted by the description. On the other hand, an
improper description can be contingently in some w, t unique, but it is not necessarily so
(not in all w, t). In other w, t an ‘improper description’ can refer to more than one object
(incomplete description), or even to none (empty description). We will use a term
‘indefinite description’ for the latterinstead of Neale’s term ‘improper description’.
Further Neale in (2004, p.36) characterizes in which sense a description can be
incomplete:
So what sorts of things have we really been attributing incompleteness to for the
past sixty years? The remarks by Quine and Sellars quoted above suggest we have
been talking all along about incomplete uses or utterances of descriptions. Recall
that they brought the suggestive word ‘elliptical’ into the debate in the course of
sketching their own answers to the question the Russellian must answer. They talk
of elliptical ‘uses’ (Quine) or elliptical ‘utterances’ (Sellars) of descriptions, and
not of descriptions per se being elliptical. According to Sellars, an utterance of ‘the
table’ will typically be elliptical for an utterance the speaker could have made of a
richer description such as ‘the table over here’ or ‘the table beside me’. The
connection between ellipsis and incompleteness in Sellars’s thinking manifests
itself when he says (i) that ‘in ellipsis the context completes the utterance and
enables it to say something which it otherwise would not, different contexts
enabling it to say different things,’ (ii) that some ‘utterances … are not complete
and are only made complete by the context in which they are uttered,’ and (iii) that
‘statements which are non-elliptical … do not depend on their contexts for their
completion’. Drawing upon these early discussions, we might talk of incomplete
‘utterances’ of descriptions.
Here Neale talks about the difference that has been introduced at the beginning of this
section above, namely the difference between an incomplete / complete meaning
(dependent / independent of the situation of utterance) and a pragmatic meaning in a
given situation of utterance.
From all that has been said till now it is obvious that the sentence “The mountain
is high“ has an incomplete meaning. It expresses an open construction with a free
variable. Well, it does not express a complete instruction for evaluating truth conditions
in any empirical context w, t. If used, we do not understand it unless somebody /
something provides an additional piece of information on which mountain (one from
many possible mountains) we are to deal with.
Let us now consider the way in which the definite article ‘the’ is used in English.
Roughly speaking, there are in principle (at least) two ways of using “the F”:
a) The expression ‘F’ denotes an office (role) F, i.e., an intension of type α (where
α  (β) for any type β). Hence this F is analyticallynecessarily unique. The
extension of F is necessarily, at every w,t, at most one object of type α. The
expressions like ‘the Pope’, ‘the President of USA’, ‘the highest mountain on the
Earth’ can serve as an example. In Slavonic languages which do not use a definite
article (like in Czech) this way of using ‘the’ does not correspond to any
expression, because the necessary uniqueness is determined by the meaning of the
expression ‘F’. This is the case of definite description.
b) The expression ‘F’ denotes a property F, i.e., an intension of type (α) which
can contingently at some w, t have a unique extension (“population of the
property”), but at other w, t the extension of F is of more than one element or even
empty. In this case the expression ‘the F’ does not have a complete meaning. A
sentence in which ‘the F’ is used does not denote a proposition; it is not possible
to evaluate its truth conditions in any w, t, unless an additional piece of
information is provided that makes a unique selection of an α-object (the element
of a many-valued population) possible. In Slavonic languages (like Czech) this
way of using the definite article ‘the’ corresponds to using a definite pronoun
(like ‘ten’, ‘ta’, ‘to’, etc.) that stands for the reference to a context. This is the
case of indefinite description.
Sentences containing definite descriptions have been analyzed in Sections 5.8.3.2 and
5.8.3.3. If a definite description of type α (α  (β) for any type β) is used de re, then it
serves as a “pointer” to (an unspecified) entity of type α (mostly an individual of type ).
In the de re case two principles hold: First, the principle of existential presupposition
(with the exception of sentences claiming (non-)existence): if the respective intension
does not have a value at w, t, then the whole sentence does not have any truth value at
this w, t. Second, the principle of inter-substitutivity salva veritate of co-referential
expressions.22.
When analyzing the sentence “The mountain is high” we obviously deal with the
case ad b), i.e., the expression ‘the mountain’ serves as an indefinite description. Its
meaning is thus an open construction (High / (), Mountain / (), x  , p  (),
I / (())the so-called singularizer: it returns the only member of a singleton, otherwise
it is not defined at an empty set or a many-element set):
(HM) wt [0Highwt [0I x [[0Mountainwt x]  [pwt x]]]].
A valuation of the subsidiary parameter p should provide an additional property so that
the set v-constructed by the construction [0I x [[0Mountainwt x]  [pwt x]]] be a singleton.
The construction (HM) does not construct a proposition, it only v-constructs for
some valuations of p. To complete the meaning (HM) we have to substitute a property for
p. The property can be provided either by a linguistic context (the case of anaphoric
reference), or by a pragmatic context of the situation of utterance. For instance, in the
situation when there is just one mountain in the sky-line (HM) is v-equivalent to the
construction
22
For details, see Duží (2004)
wt [0Highwt [0I x [[0Mountainwt x]  [0Sky-linewt x]]]],
(HMP)
which is the pragmatic meaning of the sentence in the described situation. It constructs a
proposition P that takes the value True at those w, t at which there is just one mountain in
the sky-line and the mountain is high, False if the only mountain in the sky-line is not
high, and in those w, t at which there are more mountains in the sky-line or none the
proposition P has no truth value. Of course, (HMP) is not equivalent to (HM), it is only vequivalent, because for some other valuations of p (HM) may construct a different
proposition.
If the sentence occurs in a linguistic context, the article ‘the’ has an anaphoric
character; it refers to the meaning of an antecedent expression that denotes a property.
For instance, using the substitution method described above, the meaning of the
following sentence
(N)
“There is a mountain at the sky-line and the mountain is high”
is obtained (Types: Sky-line / (), Mountain / (), High / (), p  (), x  ,
y  , z  ):
(N1)
wt [x [[0Mountainwt x]  [0Sky-linewt x]] 
[ Sub 0[wt z [0Sky-linewt z]] 0p
2 0
0[w’t’ [0High
w’t’
[0I y [[0Mountainw’t’ y]  [pw’t’ y]]]]]]wt].
Now (N1) constructs the same proposition P as described above (the proposition
constructed by the pragmatic meaning (HMP) in the situation when there is just one
mountain in the sky-line).
On the other hand, the sentence
(N’)
“There is a mountain in the sky-line which is high”
is not a case of using indefinite description. It is simply the case of anaphoric referring to
a quantified variable:
(N’1) wt x [[0Mounainwt x]  [0Sky-linewt x] 
[ SUB 0x 0y 0[w’t’ [0Highw’t’ y]]]wt].
2 0
An indefinite description can be combined together with a pragmatic (indexical)
variable, as it is the case, for instance, of the sentence (L):
(L)
“The boy believes that he is immortal “.
The sentence (L) does not have a complete meaning. To be able to evaluate its truth
conditions, i.e., in order that the sentence denoted a proposition, the indefinite description
‘the boy’ has to be completed for uniqueness by a situation of utterance; in other words,
the situation of utterance provides a valuation of a free variable p denoted by the article
‘the’.
Types: Boy / (), Believe / (  ), Immortal / (), x  , y  , p  ().
(L1)
wt [0Believewt [0I x [[0Boywt x]  [pwt x]]]
2 0Sub [0Tr [0I
[
x [[0Boywt x]  [pwt x]]]] 0y 0[wt [0Immortalwt y]]]]
We analyzed (L) as an intensional attitude to a proposition the construction of which
occurs thus in the de dicto supposition. After the execution of the substitution the second
execution step is another execution (thus double execution) of the adjusted construction
that v-constructs a proposition P that he is immortal. The proposition P is not further
intensionally descended because just to this P the boy is related. Since we need to insert
an individual (v-constructed by the meaning of ‘the boy’[0I x [[0Boywt x]  [pwt x]]]) to
a higher intensional context, we must not draw this construction into the de dicto context
but have to use the function Tr. Given w, t, and a situation of utterance, the construction
[0I x [[0Chlwt x]  [pwt x]]] may v-construct, say, the individual Charles; the result of the
substitution and a pragmatic meaning of the embedded clause is then the construction
wt [0Immortalwt 0Charles].
If we analyzed (L) as a hyper-intensional attitude to the construction of a
proposition, we would simply omit the second step (after the substitution); hence we
would not use a double execution, but a simple execution
(L1)
wt [0Believehwt [0I x [[0Boywt x]  [pwt x]]]
[0Sub [0Tr [0I x [[0Boywt x]  [pwt x]]]] 0y 0[wt [0Immortalwt y]]]],
where Believeh / (  *1).
5.8.2.5 Subjunctive conditionals (contra-factual)
The problem of conditional statements, or “contra-factuals”, can be illustrated be
the following example:
(JC)
“If Jacques Clousseau owned something, then he would be a sticker for it”
The problem of an analysis of such statements is a broad one. Here we just outline
the solution based on Tichý’s tacit-premise theory (for details see Tichý 2004, pp. 543576). Tichý proposes an amendment of the Mill-Ramsey-Chisholm theory, which is
simply the following: a subjunctively-conditional statement expresses a construction of
the form wt [A  B], where variables w, t can be free in the construction A and/or in
the construction B, A v , B v , and the implication function  is of type
(  ); its arguments are propositions. This function takes the value True if in all the
world-time couples in which the proposition v-constructed by A is true it holds that also
the consequent proposition v-constructed by B is true. Moreover, parts of the construction
A are often tacitly understood rather than explicitly spelled out in the antecedent of the
conditional statement. Reason for using the implication function  (instead of the
common material implication ) is according to Tichý the fact that arguments of this
function can often be v-constructed by open propositional constructions or picked out by
a propositional office / (), as for instance in the sentence “If the most John’s
favourite proposition were true then he would weight more than 1000 kg”.
Hence a conditional statement is not an analytical statement, it does not express a
construction of the proposition TRUE (true in all the world-time couples w,t), it is rather
an empirical statement. Informally explicated, the explicitly stated antecedent proposition
is itself too week to imply the consequent proposition. However, the conditional
statement is nevertheless true, if in all those world-times w*, t* that differ from the actual
w, t only in some intuitively obvious aspect, the antecedent proposition implies the
consequent one. The antecedent proposition is then v-constructed in such a way that in
w*, t* that tacit assumption is true (therefore “contra-factual”).
In our sentence (JC) the tacit premise is the proposition that Jacques Clousseau
does not own anything. Hence the statement can be reformulated in a rather metaphoric
way: Jacques Clousseau does not own anything, but in all the worlds-times that are the
same as the actual one (in particular Jacques Clousseau is the same as he actually is)
except of the fact that Jacques Clousseau owns something it is true that Jacques
Clousseau cares for his ownership.
Let us analyse first a simpler case of the conditional statement without taking into
account the anaphoric reference:
(JC1) “Jacques Clousseau does not own anything, but if he owned something then he
would care for his ownership (anything he’d own) carefully.”
Types: J(acques)C(lousseau) / , Own(something) / (), Care(for something) / (),
x  , y  , z  ).
(JC1’) wt [w*t* [x [0Ownwt 0JC x]  x [0Ownw*t* 0JC x]] 
w*t* z [[0Ownw*t* 0JC z]  [0Carew*t* 0JC z]]]
Now we have to take into account that the meaning of the consequent clause is an open
construction w*t* z [[0Ownw*t* y z]  [0Carew*t* y z]] that is to be completed by the
substitution function Sub:
2 0
[ Sub 00JC 0y 0[w*t*
z [[0Ownw*t* y z]  [0Carew*t* y z]]]].
The double execution is indispensable here, because the result of the substitution is a
construction of the proposition, whereas the second argument of the implication function
 is a proposition. The analysis of the sentence (JC) is thus obtained as follows:
(JC1’) wt [w*t* [x [0Ownwt 0JC x]  x [0Ownw*t* 0JC x]] 
2 0
[ Sub 00JC 0y 0[w*t*
z [[0Ownw*t* y z]  [0Carew*t* y z]]]]].
5.8.2.6 Concluding remarks.
Let C(y) be a construction with the free variable y. The meaning of an expression E with
an anaphoric reference y is the construction C(y), y/*n, y  α. Let the construction A/*m
(A v α, or 0A v , or possibly also Awt v α), be the meaning of the antecedent
expression to which the anaphoric pronoun refers. Then the analysis of the anaphoric
clause with the sub-expression E is a construction of one of the following forms:
[(2)[0SUB 00A 0y 0C(y)]](wt),
[(2)[0SUB 0A 0y 0C(y)]](wt),
[(2)[0SUB [0Tr A] 0y 0C(y)]](wt),
[(2)[0SUB [0Tr Awt] 0y 0C(y)]](wt).
The function Tr has to be used when substituting a lower-order entity to a higher-order
context, i.e., an intension / extension to a hyperintensional context or an extension to an
intensional/hyperintensional context.
The double execution (2[SUB…]) and the
intensional descent (wt) are used according to the type conditions associated with the
expression E in the whole sentence.
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