Compositionality in Transparent Intensional Logic

1
Compositionality in Transparent Intensional Logic

• Pavel Materna
• Institute of Philosophy,
• Academy of Sciences of Czech Republic
• Prague
• materna_at_lorien.site.cas.cz

2
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)

3
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.)

4
Two problems

• A.
• In which way does the syntactic function F break
  the given expression into (immediate)
  constituents (component expressions)?
• B.
• What do we mean by meaning?

5
Component expressions syntactic task?

• Autonomous syntax
• It would bein 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)

6
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.

7
Two consequences of compositionality

• Two simple consequences of compositionality are
• a compositional semantics is Husserlian (see
  below),
• 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

8
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.

9
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 Montagues 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)

10
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).

11
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 Horwichs claim means that
  compositionality holds independently on how the
  notion meaning is defined then it can be easily
  demonstrated that this claim is wrong.

12
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.

13
Reference, denotation, meaning II Freges sense

• Tichýs criticism of Freges 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 Freges
  characterization of sense as a mode of
  presentation.

14
Reference, denotation, meaning III Freges
denotation

• b) Denotation
• If the object denoted were unambiguously
  determined by the sense then Freges 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)
• Freges Thesis, viz. that an expression is
  not a name of the determiner itself but rather of
  the object, if any, determined by it.

15
Reference, denotation, meaning IV Criticism
of Freges 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.

16
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.

17
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
• 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.

18
Sense is not an intension

• A frequent interpretation of Freges 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.

19
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 Churchs 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

20
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 X1Xm for applying what is
  constructed (maybe dependently on valuations v)
  by X to what is constructed by X1,,Xm, and
  ?x1xm 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.

21
Constructions II.

• Constructions as abstract procedures embody an
  important property which a meaning of an
  expression should possess they are
  (algorithmically) structured. Tichýs and
  Cresswells (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.

22
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.

23
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.

24
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 Xwt 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
  0Earthwt 0Charles

25
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)

26
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.

27
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.

28
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.
• (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.

29
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.

30
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 occursand
• 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.

31
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 ?y0Marrywty 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

32
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
  Smiths 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

33
References II.

• Materna, P. (2004) Conceptual Systems. Logos
  Verlag, Berlin
• Materna, P., Duží, M. (2005) Parmenides
  Principle (The analysis of aboutness)
  Philosophia 32, Nos 1-4, 155-180
• Moschovakis, Y. N.(1994) Sense and denotation as
  algorithm and value, Postscript file, 39 pages
  Published in Lecture Notes in Logic, 2 (1994),
  Springer J. Oikkonen and J. Vaananen, eds. pages
  210-249.
• Sandu, G., Hintikka, J. (2001) Aspects of
  Compositionality, . Journal of Logic, Language,
  Information 10, 49-61
• Szabó, Z.G. (2005) Compositionality. The
  Stanford Encyclopedia of Philosophy
  lthttp//plato.stanford.edu/archives/spr2005/entrie
  s/compositionality/gt.
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34
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) Charles believes that the Moon is bigger
  than the Earth. .t
• b) Charles..t1
• c) believe..... t21
• d) the Moon.t221
• e) (is) bigger than..t222
• f) the Earth.t223
• g) that the Moon is bigger than the
  Eartht22
• h) believe that the Moon is bigger than the
  Eartht2
• A probable structure of the respective analysis
  tree can be suggested as follows
• a) t(t1, t2(t21, t22 (t221, t222,
  t223)))
• 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 0Earthwt
  0Charles
• C3(e) t222 C4(d) t221
  C5(f) t223
• 
  
• C1(c)t21 C6(g) t22
• 
  C8(h) t2 C2(b) t1