The Reasons for Semantic Compositionality
Markus Werning
[email protected]
Department of Philosophy
Heinrich Heine University Düsseldorf
c Markus Werning – p.1/27
The Reasons for Semantic Compositionality Overview
Compositionality — the formal notion
Productivity
Systematicity
Interchangeability of synonyms salva significatione
c Markus Werning – p.2/27
The Reasons for Semantic Compositionality The Colloquial Notion
The meaning of an expression is
determined by the meanings of its
parts.
c Markus Werning – p.3/27
The Reasons for Semantic Compositionality Frege
Frege, in the only posthumously published manuscript Logic in
Mathematics, writes what later, to many, serves as the template for a
formal definition of the principle of Semantic Compositionality:
(...) thoughts have parts out of which they are built up. And these parts,
these building blocks, correspond to groups of sounds, out of which the
sentence expressing the thought is built up, so that the construction of
the sentence out of parts of a sentence corresponds to the construction
of a thought out of parts of a thought. And as we take a thought to be the
sense of a sentence, so we may call a part of a thought the sense of that
part of the sentence which corresponds to it.
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The Reasons for Semantic Compositionality The General Message
The quotation is, of course, worded in Frege’s jargon. A closer look,
however, reveals a message that can be put in more general terms and
might be convincing even to someone who refuses to adopt thoughts and
senses into his universe of discourse.
To uncover this message, I shall distinguish three aspects of Frege’s
statement:
(a) The part whole relation in the linguistic realm.
(b) The meaning function from the linguistic realm into the realm of
meanings.
(c) The correspondence between the linguistic part-whole relation and a
part-whole relation in the semantic realm.
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The Reasons for Semantic Compositionality The Part-Whole Relation between
expressions
First, he claims that there is a part-whole relation between sentences and
less complex items of language.
It has been common since to regard the part-whole relation in question
not as a relation of mereological constitution in the sense that a
mereological part of an object is tokened whenever and wherever the
object is tokened.
We will here conceive the part-whole relation in question as a relation of
syntactical constitution, instead.
Syntactical parts need not be mereological parts. In English the
expression ‘not’, e.g., is a syntactical part of the expression ‘can’t’,
although it’s not a mereological part thereof — one may even say that ‘go’
is a syntactical part of ‘went’.
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The Reasons for Semantic Compositionality Syntactic Constitution
Let us assume a language with grammar G. The grammar specifies a set of atomic
terms and a list of syntactic rules such that the set of terms T of the language is
determined. The syntactic part-whole relation can now be defined as follows:
A term s of a language with the set of terms T and grammar G is called a syntactic
part (or constituent) of a term t of T if and only if
(a) there is a Turing computable partial function α from the n-th Cartesian
product of the set of terms T into the set of terms T such that t is a value of
the function α with s as one of its arguments and
(b) there is a syntactic rule in the grammar G according to which α(s 0 , ..., sn−1 )
is a term of the language if α is defined for (s0 , ..., sn−1 ) and if s0 , ..., sn−1
are terms of the language.
In this case the function α is called a syntactic operation justified by the grammar
G.
In short, a term and any of its syntactic parts stand in the relation of value and
argument of a syntactic operation.
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The Reasons for Semantic Compositionality Partial Order
Since the set of rules of a language is closed under conjunction, every
iteration β(..., α(...), ...) of two syntactic operations α and β is itself a
syntactic operation. The transitivity of the relation of syntactic
constitution is thus warranted.
The reflexivity of the relation follows from the fact that the identity
mapping is a syntactic operation of every language.
Often it is also presumed that syntactic constitution be anti-symmetrical,
i.e.: a term s is identical with a term t if s is a syntactic part of t and t is a
syntactic part of s.
In the case of transitivity, reflexivity and anti-symmetry, syntactical
constitution would be a partial order in the set of terms.
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The Reasons for Semantic Compositionality Some Technicalities
We allow terms to have variables ξ, ξ0 , ξ1 , etc. as syntactic parts. The set of
grammatical terms GT (G) is a set of terms such that the terms of the set do not
contain any variables. GT (G) is closed under syntactic constitution. The only
purpose for the introduction of variables is to specify the positions in which certain
grammatical terms occur as syntactic parts within other grammatical terms.
In our notation
t(p0, ..., pn−1 |ξ0 , ..., ξn−1 )
is the term one obtains if one simultaneously replaces each occurrence of the
variable ξi in the term t by the term pi , for each index i (0 ≤ i ≤ n − 1).
Given that a non-grammatical term t contains the variables ξ0 , ..., ξn−1 only, and
given that p0, ..., pn−1 are variable-free, the term t(p0, ..., pn−1 |ξ0 , ..., ξn−1 ) is
grammatical and the grammatical terms p0, ..., pn−1 figure in it at the positions
ξ0 , ..., ξn−1 as syntactic parts.
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The Reasons for Semantic Compositionality Meaning Functions
We assume that the introduction of terms and syntactic operations allows
us to disambiguated language with respect to
lexical ambiguities caused by homonymy (‘John went from bank to
bank’) and
syntactic ambiguities (‘John saw the girl with a telescope’).
We can now construe meaning as a surjective function from the set of
grammatical terms into a set of meanings:
µ : GT (G) → M.
A grammatical term of the language is called µ-meaningful if the term is
in the domain of the meaning function µ.
Any µ-meaningful terms p and q of the language are said to be
µ-synonymous (in symbols: p ≡µ q) if and only if they have the same
µ-meanings.
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The Reasons for Semantic Compositionality Semantic Compositionality — the Formal
Notion
Let µ be a meaning function for a language with grammar G. Then µ is
called compositional if and only if, for every syntactic operation α of the
language, there is a function µα (called a semantic operation) such that,
for every µ-meaningful term α(t0 , ..., tn−1 ) whose syntactic parts
t0 , ..., tn−1 are also µ-meaningful, the following equation holds:
µ(α(t0 , ..., tn−1 )) = µα (µ(t0 ), ..., µ(tn−1 )).
A language is called compositional just in case it has a compositional
meaning function.
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The Reasons for Semantic Compositionality The Equivalence of the Colloquial and the
Formal Notion of Compositionality
The formal notion of compositionality turns out to be equivalent to our
introductory formulation according to which the meaning of an expression
is determined by the meanings of its parts. We only have to translate ‘...
is determined by :::’ as ‘... is the value of some semantic operation with :::
as arguments’, to take ‘expression’ to mean term, and to specify ‘part’ as
meaning syntactic part.
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The Reasons for Semantic Compositionality Is Productivity a Reason for
Compositionality?
A language is called productive just in case
the grammar of the language comprises no more than finitely many
atomic terms,
the grammar of the language contains syntactic rule such that the
generation of potentially infinitely many terms is allowed,
the meaning function of the language is computable.
Many authors claim that only compositional languages are productive.
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The Reasons for Semantic Compositionality Counterexample: Holophrastic Quotation
Assume the grammar of a language justify the the following syntactic
operation of quotation:
q:T →T
s 7→ ‘s’
such that
µ(q(s)) = s.
The inclusion of this operation in a language warrants that the language
is productive.
This account of quotation is called holophrastic because it takes
well-formed phrases as unanalyzed wholes and sets them in quotation
marks.
Holophrastic quotation violates compositionality, provided the language to
be considered contains synonyms.
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The Reasons for Semantic Compositionality Proof
Assumption:
µ(pPaul and Peter are brothersq) = µ(pPeter and Paul are brothersq).
But: pPaul and Peter are brothersq
(1)
6= pPeter and Paul are brothersq.
(2)
=
µ(q(pPaul and Peter are brothersq))
(3)
=
pPaul and Peter are brothersq.
(4)
=
µ(q(pPeter and Paul are brothersq))
(5)
=
pPeter and Paul are brothersq.
(6)
In accordance with holophrastic quotation we get:
µ(p‘Paul and Peter are brothers’q)
µ(p‘Peter and Paul are brothers’q)
From (2), (4), (6) we get:
µ(q(pPaul and Peter are brothersq)) 6= µ(q(pPeter and Paul are brothersq)).
(7)
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The Reasons for Semantic Compositionality continued
Because of compositionality, there is a function µ q , such that
µ(q(pPaul and Peter are brothersq)) = µq (µ(pPeter and Paul are brothersq)).
(8)
Substitution of identicals according to (1):
µ(q(pPaul and Peter are brothersq)) = µq (µ(pPaul and Peter are brothersq)).
(9)
Another application of compositionality:
µ(q(pPaul and Peter are brothersq)) = µ(q(pPeter and Paul are brothersq)).
(10)
This contradicts (7).
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The Reasons for Semantic Compositionality Systematicity
Many authors cite the systematic correlation of linguistic capacities and
mental capacities as a reason for semantic compositionality.
Minds must have the capacity to compose contents/meanings, so it is
argued. Otherwise, they would not show a systematic correlation
between representational/linguisitc capacities:
If a mind is capable of certain intentional states in a certain intentional
mode, it most probably is also capable of other intentional states with
related contents in the same mode.
Mutatis mutandis: If a mind is capable of understanding certain linguistic
expressions, it most probable is also capable of understanding other
expressions with related meanings.
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The Reasons for Semantic Compositionality Examples of Correlated Capacities
the capacity to think that the dog is chasing the cat and the capacity to
think that he cat is chasing the dog.
the capacity to think that if the cat runs, then the dog will and the capacity
to think that if the dog runs then the cat will;
the capacity to see a visual stimulus as a square above a triangle and the
capacity to see a visual stimulus as a triangle above a square;
the capacity to prefer a green coat to a red shirt and the capacity to
prefer a red coat to a green shirt.
the capacity to understand the sentence ‘John loves Mary’ and the
capacity to understand the sentence ‘Mary loves John’.
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The Reasons for Semantic Compositionality Does Systematicity Presuppose
Compositionality?
The systematic correlation of both contents and meanings seem, indeed, to imply
more than mere syntactic recombination on the level of natural language or a
language of thought.
The capacity to think that a child with a red coat is distracted by an old herring is
not correlated with the capacity to think that a child with an old coat is distracted by
a red herring.
The thoughts ought to be correlated, though, if the fact that one is a syntactic
re-combination of the other was sufficient for systematic correlation.
Notice that both thoughts are syntactically combined from exactly the same
primitives by exactly the same operations.
One may, however, well have the capacity to think of red coats and old herrings
even though one lacks the capacity to think of red herrings.
The two thoughts fail to be correlated because r ed herring is idiomatic and — as a
consequence — semantic compositionality is violated.
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The Reasons for Semantic Compositionality Instability of Systematicity
Systematicity seem to be too instable a phenomenon as to justify a
general need for compositionality.
Consider the following examples:
(a) The man watches the show.
(b) (?) The show watches the man.
(c) The woman feels hatred.
(d) (?) Hatred feels the woman.
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The Reasons for Semantic Compositionality Interchangeability of Synonyms salva
significatione
The principle of interchangeability of synonyms salva significatione says
that the substitution of synonyms for expressions in any linguistic context
leaves unchanged the meaning of the context.
The principle can be regarded as the meaning (or intensional)
counterpart of the principle of extensionality, also called the principle of
interchangeability of co-extensionals salva veritate. It claims that the
substitution of co-extensional expressions for each other leaves
unchanged the truth value of the embedding linguistic context.
While the principle of extensionality is violated in intensional contexts —
contexts like ‘It is necessary that ...’, ‘S believes that ...’ — the principle of
interchangeability salva significatione even pertains to those cases.
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The Reasons for Semantic Compositionality The Primary Reason for Compositionality
The following theorem proves the equivalence between the principle of
semantic compositionality and the principle of interchangeability of
synonyms salva significatione.
Here, meaning functions are called substitutional salva significatione if
they abide by the principle that the substitution of synonyms for
expressions in any linguistic context leaves unchanged the meaning of
the context.
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The Reasons for Semantic Compositionality Theorem
Let µ be a meaning function for a language with grammar G, and
suppose that every syntactic part of a µ-meaningful term is µ-meaningful.
Then the following are equivalent:
(a) µ is compositional.
(b) µ is substitutional salva significatione, i. e., if s is a term and
p0 , ..., pn−1 , q0 , ..., qn−1 are grammatical terms such that
s(p0 , ..., pn−1 |ξ0 , ..., ξn−1 ) and s(q0 , ..., qn−1 |ξ0 , ..., ξn−1 ) are both
µ-meaningful and, for all m < n,
p m ≡ µ qm ,
then
s(p0 , ..., pn−1 |ξ0 , ..., ξn−1 ) ≡µ s(q0 , ..., qn−1 |ξ0 , ..., ξn−1 ).
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The Reasons for Semantic Compositionality Proof
(a) ⇒ (b). Assuming (a), we prove (b) by induction on the complexity of s. In case n = 0,
s is a µ-meaningful term and the conclusion s ≡µ s is trivial. We now consider the case
where s is the term α(t0 , ..., tm−1 ). In this case we get s(p0 , ..., pn−1 |ξ0 , ..., ξn−1 ) by
substituting the terms pi for ξi , with 0 ≤ i < n, in all syntactic parts t0 , ..., tm−1 of the
term s. We analogously proceed with s(q0 , ..., qn−1 |ξ0 , ..., ξn−1 ) and thus have:
s(p0 , ..., pn−1 |ξ0 , ..., ξn−1 ) =
α(t0 (p0 , ..., pn−1 |ξ0 , ..., ξn−1 ), ..., tm−1 (p0 , ..., pn−1 |ξ0 , ..., ξn−1 ),
and
s(q0 , ..., qn−1 |ξ0 , ..., ξn−1 ) =
α(t0 (q0 , ..., qn−1 |ξ0 , ..., ξn−1 ), ..., tm−1 (q0 , ..., qn−1 |ξ0 , ..., ξn−1 ).
Since s(p0 , ...|ξ0 , ...) and s(q0 , ...|ξ0 , ...) are assumed to be µ-meaningful, their syntactic
parts ti (p0 , ...|ξ0 , ...) and ti (q0 , ...|ξ0 , ...), respectively,
are also µ-meaningful.
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The Reasons for Semantic Compositionality continued
By induction hypotheses we may, therefore, presume that
ti (p0 , ...|ξ0 , ...) ≡µ ti (q0 , ...|ξ0 , ...).
According to (a) the µ-meanings of s(p0 , ...|ξ0 , ...) and s(q0 , ...|ξ0 , ...), respectively, are a function of the meanings of their syntactic parts. Thus, the identity of
the µ-meanings of the parts of both terms implies the identity of the µ-meanings
of both terms.
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The Reasons for Semantic Compositionality continued
(b) ⇒ (a). (a) follows at once from the special case of (b) where s has the form
α(ξ0 , ..., ξn−1 ). For, in that case (b) just claims the functionality of the relation
µα = {h(µ(ξ0 ), ..., µ(ξn−1 )), µ(α(ξ0 , ..., ξn−1 ))i|(ξ0 , ..., ξn−1 ) ∈ dom(α)}.
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The Reasons for Semantic Compositionality Conclusion
Whereas productivity and
systematicity are unlikely to provide
sufficient justification for semantic
compositionality, the principle of
interchangeability of synonyms salva
significatione may well do.
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The Reasons for Semantic Compositionality