Download Compositionality in Transparent Intensional Logic

Compositionality
in Transparent Intensional Logic
Pavel Materna
5. 5. 2017
Compositionality
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Compositionality: Definitions
Compositionality is a condition. It is
 “a condition on semantics for languages. A common
statement of it is that the meaning of a complex
expression is a function of the meanings of its
immediate constituents and the grammatical rule
which is used to combine them…”
(Hodges 2001, 7)
In other words, the meaning of a complex expression
 “is determined by the meanings of the component
expressions plus the way they are combined into the
complex expression.”
(Sandu, Hintikka 2001, 49)
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Formal definition of compositionality
“Consider now F, a k-ary syntactic operation on E.
m is F-compositional just in case there is a k-ary
partial function G on M such that whenever
F(e1,…,ek) is defined, then
m(F(e1,…,ek)) = G(m(e1),…,m(ek)).“
(Szabó 2005, 5)
E: set of expressions, m: a meaning-assignment,
M: set of ‘available’ meanings

(Cf. also Stechow, Wunderlich 1991, 107.)
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Two problems
A.

In which way does the syntactic function F
break the given expression into
(immediate) constituents (component
expressions)?
B.

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What do we mean by ‘meaning’?
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Component expressions: syntactic task?
“Autonomous syntax”:


“It would be…in vain to ask an autonomous syntactician what the term
‘constituent’ means. He certainly cannot say that a constituent is an
expression which is complete in that it refers all by itself to a definite
entity, in contrast to an incomplete expression which refers only in
combination with some other expressions. For that… would amount to
leaving the domain of autonomous syntax. The term ‘constituent’ (or
‘phrase’) is apparently not to be burdened with any pre-theoretical
meaning at all: a constituent is simply whatever the grammarians’ theory
brands as such in any particular case.
(Tichý 2004, 807)
“…logical grammar, with its principle of compositionality of meaning, goes
straight against the autonomy of syntax so cherished in the generative
tradition. …And that means, at least in principle, that semantic
considerations may influence the syntax, thus breaching the supposed
autonomy of syntax.”
(Gamut 1991, 141)
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Compositionality:
logical languages vs. natural language

“The principle of compositionality of meaning has important consequences for the
relationship between syntax and semantics. Usually in a logical system the definition
of the semantic interpretation of expressions closely follows the lead of their
syntactic construction. … The obvious way to proceed is to let the definition of the
semantics parallel the finite, recursive definition of the syntax. Succinctly put, logical
languages satisfy the following principle: the interpretation of a complex expression
is a function of the interpretations of its parts.… every syntactic rule should have a
semantic interpretation; and on the other hand, every aspect of the semantics which
is not related to the interpretation of basic expressions should be linked to a
syntactic operation. …
But a natural language is not something we construct; it comes as given.”
(Gamut 1991, 140)

Thus it seems that the Tarskian semantics of formal languages has got
compositionality gratis. Whatever can be called ‘meaning’ in such languages is
stipulated in such a way that the syntactic rules (determining, e.g., which
concatenation of symbols counts as ‘well-formed formula’) select just those
components of an expression that get due to the interpretation unambiguously a
‘meaning’, and derive unambiguously the ‘meaning’ of the whole expression from the
‘meanings’ of the components.
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Two consequences of compositionality
Two simple consequences of compositionality are
i.
a compositional semantics is Husserlian (see below),
ii.
synonyms are substitutable
Ad i): The term ‘Husserlian’ has been introduced by Hodges in his
(2001). Roughly, a semantics S is Husserlian if for every
expressions E, E’ it holds that if E is synonymous with E’ in S
then E belongs in S to the same (Tarskian) category as E’;
i.e., the expression containing E as its subexpression is
S-meaningful just if the expression containing E’ in the place
of E is S-meaningful.
Ad ii): If E is synonymous with E’ in S then so is any expression A
containing E as its subexpression with the expression B that
arises from A by substituting E’ for E
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Natural language: ambiguities

Simple examples show that both i) and ii) cannot be satisfied if
applied to expressions of a natural language. The reason is that
natural languages, which “come as given”, are replenished by
ambiguities.

One kind of ambiguities comprises lexical ambiguities. These are
rather simple and can be thought of as ‘corrigible’. They make it
impossible to satisfy even the Husserlian condition. For example,
consider the word means. To be Husserlian the semantics would
have to admit that since “What Charles means is that…” is
meaningful and we can (roughly) claim that means and resources
are synonymous then “What Charles resources is that…” were
likewise meaningful.
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Natural language: non-lexical ambiguities

The non-lexical ambiguities are especially insidious. Here is an
example of a sentence that seems not to contain any lexical
ambiguity but does not admit an unambiguous interpretation
(cf. also the famous Montague’s example with seeking unicorns):
Charles wants to marry a princess.

“An expression may be ambiguous without having two distinct
constituent structures. But compositionality simply requires that
there be different ‘parts’ whenever there is non-lexical ambiguity,
and if none of the known notions will do, the parts have to be
‘invented’. … compositionality demands a disambiguated level of
representation in the syntax.”
(Gamut 1991, 218)
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Logical analysis of natural language
Summarising:
 Identifying the component expressions of a complex NL expression
presupposes a logical analysis of the natural language (LANL). This
is because of the fact that natural expressions develop
spontaneously so that the logical structure underlying the ‘natural
encoding’ given by the language (NB at the given stage of its
development) remains hidden and has to be discovered. The
degree of the adequateness of such discovery should be dependent
on the degree in which the respective LANL could ensure that
compositionality holds.

As for the disambiguation necessary according to the above
quotation, LANL should be capable to offer to each reading (i.e.,
to each analysis tree) of an ambiguous expression a separate
logical construction (indeed, under the assumption that the
component expressions occurring in the tree have been determined
as meaningful expressions and the dependencies between the
particular terminals are mediated via rules admitting semantic
interpretation).
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Horwich: compositionality independent of the
definition of meaning?



No great problems with compositionality are admitted from a
deflationary viewpoint. Horwich formulates the problem as
follows:
“The issue of how the meanings of sentences are built out of the
meanings of their constituents words”
(Horwich 1997, 503)
Further he claims that
“the compositionality of meaning imposes no constraint at all
on how the meaning properties of words are constituted.”
(Ibidem)
If this Horwich’s claim means that compositionality holds
independently on how the notion meaning is defined then it can
be easily demonstrated that this claim is wrong.
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Reference, denotation, meaning I:
TIL

The main principles of TIL can be found in the work of the founder of TIL
Pavel Tichý (see his Collected papers (2004), and his monograph (1988));
further information is offered, e.g., by Materna, P., Duží, M. (2005) and the
important definitions in Duží, M., Materna, P. (2005).
Since the apparatus is available in the literature just mentioned I will try to
get along without formal definitions, and since the principles have been
sufficiently defended in that literature I will only mention them and show
how they help to elucidate the problems of compositionality.

Let us return to various definitions of compositionality. In all of them what is
said to be ‘a function of’ or ‘determined by’ is called meaning. We can
therefore ask “what sort of stuff meaning is” (Horwich) or simply state that
meaning is in such definitions “a generic rather than specific term”
(Sandu, Hintikka).
In the latter text the authors propose to speak “of the different semantic
attributes of an expression”. Now we will examine three such semantic
attributes and show that only one of them can guarantee such a
disambiguation of a natural language which makes it possible to let
compositionality hold.
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Reference, denotation, meaning II:
Frege’s “sense”
Tichý’s criticism of Frege’s semantic schema:
a) “Sense”:
 Frege has characterized his sense as a mode of presentation of
the object denoted (“die Art des Gegebenseins”), which is, of
course, no definition. No wonder that, e. g., Bertrand Russell
refused to accept such an indefinite notion. In any case, the lack
of a satisfactory definition led to never ending discussions
concerning necessity and character of sense, or, as we use to
say nowadays, meaning. Alternatively, meaning is conceived of
as having “at least two components: the sense and the
reference” (Kirkham 1992/1997, 4).
 Since the term sense is no more frequently used we will use the
term meaning as one of the levels that are connected with
semantics and let it play the role that corresponds to Frege’s
characterization of sense as a mode of presentation.
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Reference, denotation, meaning III:
Frege’s denotation
b) Denotation:
 If the object denoted were unambiguously determined by the
sense then Frege’s notion of denotation (Bedeutung) as illustrated
by his famous example with morning star vs. evening star does
not satisfy this requirement.
 For Frege the expression morning star as well as evening star
denotes Venus, the particular celestial body. Imagine however a
well thinkable (in any case possible) situation when the role
played by the morning star (being the brightest celestial body in
the morning sky and suchlike) begins to be played by another
body, say, Mars. A natural way to state this change is to say “Now
Mars became the morning star”. The sense – be it anything –
connected with the term morning star did not change but the
Fregean denotation did. This is a consequence of what Tichý calls
– e. g., in (2004, 825) –
 “Frege’s Thesis”, viz. that “an expression … is not a name of the
determiner itself but rather of the object, if any, determined by it”.
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Reference, denotation, meaning IV:
Criticism of ‘Frege’s Thesis”






The relation of denoting, if conceived of as a semantic relation (as it should
be, at least if semantics of NL is construed as LANL), is, of course, a
necessary relation.
Thus what is denoted by an empirical expression is always just a condition to
be fulfilled by an object of the given type. Such conditions (called
“determiners” in the above quotation) are best modeled as intensions in the
sense of P(ossible) W(orld) S(emantics). Intensions are in this sense
functions mapping possible worlds to objects of the given type, mostly to
chronologies of such objects, i.e., to functions from time moments.
Objects that are not intensions are extensions.
Empirical expressions denote intensions due to a linguistic convention (which
is given and which, therefore, enables us to say that from the viewpoint of
LANL the denotation of an empirical expression is given a priori).
Now whereas empirical expressions denote intensions independently of their
instantaneous actual ‘population’ and are so immune to empirical facts (so
that the relation of denoting is necessary) we can consider the value of any
such intension in the actual world-time. This is our opportunity to distinguish:
denoting as a necessary, independent of empirical facts relation, and
referring (or reference) as a contingent relation that is irrelevant – as being
contingent – for LANL.
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Reference, denotation, meaning: Examples




“the capital city of Poland” denotes the intension
called individual role (Church has called it individual
concept), which returns as its value in W at T at most
one individual; its reference in the actual world-time is
Warsaw, the town;
“The capital city of Poland is Warsaw” denotes the
proposition (a function from possible worlds to
chronologies of truth values); its reference in the
actual world-time is the truth-value True;
“star” denotes a property; its reference in the actual
world-time is the set of all actual stars;
etc. etc.
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Actual world. Non-empirical expressions


One misunderstanding should be avoided: we can
speak about the actual world but logically it is
inaccessible. Saying, e.g., “I am actually hungry” we
do not say anything more than “I am hungry”. A
systematic explanation of this frequently ignored fact
can be found in Jespersen (2005).
As for the non-empirical, in particular mathematical
expressions, to distinguish between denotation and
reference is inoperative:


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semantics of mathematical expressions does not need any
possible worlds or times, and
non-empirical (analytic) expressions containing empirical
subexpressions denote constant intensions, i.e., functions
whose value in all worlds-times is the same.
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‘Sense’ is not an intension

A frequent interpretation of Frege’s sense in the post-Fregean literature
consists in construing it as intension

At least two essential objections to this identification of senses
with intensions are:
Let the sense of an (empirical) expression be the intension of the
given type. Then the only object that would correspond to the
denotation could be the value of the intension in the actual world.
Then the link between the sense defined in this way and the
denotation would be contingent, not unambiguous.
For example, if the sense of the sentence “There are exoplanets
where there live mammals” were its truth-condition, i.e., the
respective proposition, then what would be denoted would be a
truth-value. But the proposition itself does not possess the force of
determining the truth-value. We need empirical methods to
verify/falsify empirical sentences.
Mathematical expressions would not possess any sense, since
they are independent of intensions.
a)
b)
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Meaning (‘sense’) as an abstract procedure

Let us consider some mathematical expression, say,
3 – 2  0.


The most natural answer to the question what this expression
denotes is probably that it is the truth-value True. What would be
the way that links the expression with this truth-value, playing thus
the role of the Fregean sense?
The general form of the answer has been given by Tichý as early
as in (1968) and (1969). The Fregean sense is best construed as
an abstract procedure. Later – during developing the system of
TIL – Tichý has formulated exact definitions of such procedures:
they are what is called in TIL constructions and formally are
influenced by the ingenious Church’s idea that two
operations/procedures are the core of handling functions: creating
functions via abstraction and applying functions to arguments,
which led to -calculi. What the typed -calculus formulates on the
level of formal languages TIL interprets objectually:
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Constructions I.


Variables, compositions (-applications), closures (abstractions), trivializations (0X constructs X without any change)
and double executions (where C is a construction that constructs
a construction D the double execution 2C constructs what
constructs D) are extra-linguistic procedures, and the ‘language of
constructions’ is no formal language in the standard sense but
only a direct code that enables us to deal with the abstract
procedures.
Writing [X X1…Xm] for applying what is constructed (maybe
dependently on valuations v) by X to what is constructed by
X1,…,Xm, and
[x1…xm X] for constructing a function (technically just as it is
done in -calculus)
we can logically handle procedures ‘working’ in the area of
(partial) functions over some simple types
and even in the area of constructions themselves, which get their
types and become thus a kind of objects to be mentioned (not only
used) within a ramified hierarchy of types.
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Constructions II.




Constructions as abstract procedures embody an important property
which a meaning of an expression should possess:
they are (algorithmically) structured.
Tichý’s and Cresswell’s (see his (1985) ) idea of structured
meanings (see also the more recent Moschovakis (1994)) has been
realized in TIL in a systematic way:
meaning of an expression is a construction, and LANL tries to find
such a construction that would obey compositionality. (See Materna,
Duží (2005).)
A special kind of construction can be defined: roughly, a closed
construction, i.e., a construction not involving free variables.
Such a construction has been called concept (see Materna (1998,
2004)).
The meaning of any non-indexical expression is accordingly a
concept.
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Denotation, reference, meaning
Summarising:
 Fregean levels (sense, denotation/reference) have
been revised in TIL as follows:
i) Empirical expressions of a natural language denote
intensions, never their values in the actual world-time.
ii) Reference, as the value of the denotation in the actual
world-time, is logically (and thus for LANL)
inaccessible, hence it cannot be dealt with within LANL.
iii) Meaning of an expression E is a construction;
if E is a non-indexical expression its meaning is a
concept. Meaning in this sense constructs what the
expression denotes, which is as it should be.
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Constructions: Example I.

Returning to our example we can write down the
concept that is the meaning of this mathematical
sentence (whose denotation is True):
[0 [0– 03 02] 00]

i.e.: the function  constructed by trivialization is applied
to the pair M, N of procedures, where M is application
of the subtraction function (constructed by trivialization)
to the pair of numbers 3, 2 (constructed by trivialization)
and N is trivialization of 0.
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Constructions:
Example II.
To show an example of an empirical sentence let us consider the sentence
Charles believes that the Moon is bigger than the Earth.
Reading: “Charles has the property of believing that the Moon is bigger than the
Earth”.
‘Charles’, ‘Moon’ and ‘Earth’  names of definite individuals
(a simplification, of course),
 ‘bigger than’  name of a binary relation-in-intension between
individuals,
 ‘believe that’  a relation-in-intension between an individual and a
proposition.
 w is a variable ranging over the type of possible worlds,
 t ranges over the type of time moments (= real numbers),
 x ranges over the type of individuals.
Abbreviating [[Xw]t] as Xwt and omitting brackets where there is no danger
of confusion we analyze our sentence as follows:

wt [wt x [0Belwt x [wt [0Biggerwt 0Moon 0Earth]]]wt 0Charles]
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Compositionality w.r.t.
reference, denotation and meaning I.
Once more, let compositionality be defined like in Szabó:
m(F(e1,…,ek)) = G(m(e1),…,m(ek)).
We will try to examine what happens if m is interpreted as:
a) reference, b) denotation, c) meaning (defined as above)
 Let us recollect that synonymy is defined as follows:
The expression E is synonymous with the expression E’ iff
m(E) = m(E’).
 Finally, an easily derivable consequence of compositionality is:
If e is a constituent of E, and if E’ is like E with the only distinction
that e is replaced by e’, then
m(e) = m(e’) implies m(E) = m(E’).
(Principle of substitutability, PS)
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Compositionality w.r.t. reference, denotation and
meaning II:
a) m is reference
Obviously, if reference is defined as above then we cannot expect that
the m in definition of compositionality were reference. Indeed, we can
easily prove that the PS does not hold for reference.
Let our Charles believe that the following sentence is true:
The Moon is bigger than the Earth.

Now the truth-value of the proposition denoted by this sentence in the
actual world-time and so the reference of the sentence  is False.
Hence the sentence is synonymous (w.r.t. reference) with the sentence
Every woman has two wings.

If PS held for reference Charles would have to believe that this
sentence too were true. But he does not have to, of course.
In the semantics for which the meaning of E equals the reference of E
the PS does not hold so that compositionality does not hold either.
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Compositionality w.r.t. reference, denotation and
meaning III:
b) m is denotation.
We can immediately show that in this case doxastic or epistemic contexts make PS
fail. Indeed, consider the sentences
A. Some dogs are dangerous.
B. Some dogs are dangerous and the only even prime is two.


The sentence A. is synonymous with the sentence B. w.r.t.
denotation: The point is that the proposition denoted by “The only
even prime is two” is true in all worlds-times. Thus the conjunction of
the sentence A. with the sentence B. denotes one and the same
proposition. If PS held for m = denotation then the following two
sentences would be synonymous:
Charles believes that some dogs are dangerous.
Charles believes that some dogs are dangerous and (that) the only
even prime is two.
This is however evidently not the case.
In the semantics for which the meaning of E equals the denotation of E
PS does not hold so that compositionality does not hold either.
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Compositionality w.r.t. reference, denotation and
meaning IV:
c) m is a concept
First of all, this case is immune to counterexamples like those ones that were adduced in a) or
b).
For example, the sentences A. and B. above are not synonymous w.r.t. meaning: the
construction (concept) underlying A. differs from the construction (concept) underlying B.
This does not mean, however, that other counterexamples cannot be found. Consider the
following ones:
0yellowness constructs the same property as 0yellow.
i)
(Properly speaking, these are two ways of encoding in TIL one and the same
construction.)
Yet we cannot say ”This house is yellowness” – it seems that our semantics is not even
Husserlian!
Solution: The distinction between yellowness and yellow is in fact no distinction of
meaning. Yet a distinction is present: the ness in the first expression
signalizes that this name of a property can be used only if the property is
mentioned (i.e., is in the supposition de dicto) rather than used to be
predicated of something. (Thus there is a constraint to compositionality, which
could be classified as compositionality in the sense iii) in Sandu, Hintikka
(2001, 50).) There are more such signals in natural languages, cf. bravery vs.
brave, in German Tugend vs. tugendhaft, schön vs. Schönheit etc. As soon as
such a signal occurs in the respective analysis tree compositionality is no more
jeopardized in this respect.
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Compositionality w.r.t. reference, denotation and
meaning IV:
c) m is a concept
ii)
Without any lexical change the sentence Charles wants to marry a princess.
allows for two readings:
a) There is a princess and Charles wants to marry he
b) Charles wants that there were a princess whom he would marry.
So the expression “to want to marry a princess” as if denoted two properties.
Indeed, let Charles have the property according the reading a).
Peter might have the property according to reading b).
A simple substitution of the expression “wants to marry a princess” into both the
statements that claim the possessing of a property by Charles and Peter would
not result in the claim that Peter has the same property as Charles.
Solution: The possible readings a) and b) show that
1. disambiguation is possible,
2. the ambiguous sentence is something like an abbreviation (where semantic
distinctions are lost). There are two meanings (concepts) belonging to the
sentence, each of them is the meaning of one of the two readings. (See
Appendix.)
No problem with compositionality arises after the disambiguation has been realized.
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Compositionality w.r.t. reference, denotation and
meaning V:
c) m is a concept
iii)
Suppose that Charles seeks the murderer of his father.
Analyzing simple words as expressing trivialized objects (‘simple
concepts’, Materna (2004)) can we really say that in any context
0seek is sufficient?
Consider the situation when Charles says: “Success, it is Peter X!” and
another situation when he says: “Success, he was in his flat!”
Solution: This is again a problem of disambiguation.
In a disambiguated language we would have:
seek1 as look for the place where X occurs…and
seek2 as investigate who plays the role of… or so.
The resulting ambiguity is, as a matter of fact, a lexical ambiguity
and should be signalized in the linguistic resources on which
the analysis trees are based.
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Appendix
We will show the two meanings that correspond to the two readings of the sentence
Charles wants to marry a princess.
Simple types: atomic
 … {True, False}
 … individuals
 … time moments / real numbers
 …possible worlds (logical space)
functional
(1…m) … partial functions from 1  …  m to 
Types of objects: Charles … 
Want … ((((())))) abbr.: (  ())
(relates an individual X with a property, viz. which X wants to possess)
Marry … ()
Princess … ()
in general:  abbreviates (())
Types of logical functions:
 … ()
 … (())
Variables:
w  , t  , x, y  
First reading: There is a princess and Charles wants to marry her.
wt [0x [0 [0Princesswt x] [0Wantwt0Charles wt y[0Marrywty x]]]]
Second reading: Charles wants that there were a princess and he would marry her.
wt [0Wantwt0Charles wty [0 x [0 [0Princesswt x] [0Marrywty x]]]]
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References I.








Cresswell, M.J. (1985) Structured Meanings, MIT Press, Mass.
Duží, M., Materna, P. (2005): “Logical Form”. In: Sica, G., ed.: Essays
on the Foundations Of Mathematics and Logic, Polimetrica International
Scientific Publisher Monza/Italy
Gamut, L.T.F. (1991): Logic, Language and Meaning II., Intensional
Logic and Logical Grammar. Chicago University Press.
Hodges, W. (2001): “Formal Features of Compositionality”. Journal of
Logic, Language, and Information 10, 7-28
Horwich, P. (1997): “The Composition of Meanings”, The Philosophical
Review, Vol 106 No 4, 503-532
Jespersen, B. (2005): “Explicit Intensionalization, Anti-Actualism, and
How Smith’s Murderer Might Not Have Murdered Smith” Dialectica
59/3, 285-314
Kirkham, Richard L. (1992/1997): Theories of Truth. The MIT Press,
Cambridge, Mass., London 1992, 4th printing 1997.
Materna, P. (1998): Concepts and Objects. Acta Philosophica Fennica
63, Helsinki
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References II.


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Appendix II.
Let us return to the sentence:
Charles believes that the Moon is bigger than the Earth.
In any analysis tree the following components (constituents) will occur:
a)
b)
c)
d)
e)
f)
g)
h)
”Charles believes that the Moon is bigger than the Earth.”….t
“Charles”…………………………………………………………..t’1
“believe”………………………………………………………..... t’21
“the Moon”……………………………………………………….t’221
“(is) bigger than”………………………………………………..t’222
“the Earth”……………………………………………………….t’223
“that the Moon is bigger than the Earth”………………………t’22
“believe that the Moon is bigger than the Earth”………………t’2
A probable structure of the respective analysis tree can be suggested as follows:
a) = t(t’1, t’2(t’21, t’22 (t’221, t’222, t’223)))
We can see that the particular steps (subconstructions, subconcepts) of the meaning of the sentence
correspond to the constituents of the suggested tree above and that the syntactic dependencies that
make up this tree correspond to the objectual functions constructed by the particular subconstructions.
w t [w t x [0Belwt x [w t [0Biggerwt
0Moon
0Earth]]]
C3(e) t’222 C4(d) t’221
wt
0Charles]
C5(f) t’223
C6(g) t’22
C1(c)t’21
C8(h) t’2
C2(b) t’1
C7 (a) t
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