A Comparative Analysis for Resemblance
Summary: This paper contains a new semantic analysis for the verbal expression resemble. It
is argued that resemble is best conceived as a degree predicate, very much in analogy to
(transparent) gradable adjectives like close to (see Mador-Heim & Winter 2007). This move
can explain why resemble happily combines with the traditional positive, comparative, and
superlative operators, degree intensifiers and the like and it meets the philosophical tradition
that resemblance is a 4-place comparative relation (Lewis 1986, Williamson 1988).
Nevertheless, resemble is an intentional idiom (i.e., not transparent): it is well known that this
predicate (a) shows an ambiguity with respect to the specificity of its object: This horse
resembles a unicorn (a particular mythical object or just any unicorn), (b) it does not allow
for substitution of the object by extensionally equivalent expressions, and (c) it does not allow
for an existential generalization with respect to the object. These (and related) phenomena are
explained by combining Zimmermann’s property analysis for intentional predicates and the
theory of non-existent objects by Parson (1980) with the insights of von Stechow and Heim
for a comparative semantics (see von Stechow 2006, e.g.).
The Problem: The impossibility of a propositional analysis for predicates like resemble and
the restriction on quantifiers (i.e., only indefinites, definite descriptions and proper names
may get the unspecific reading) motivated the property analysis for intentional predicates.
Zimmermann (1993) proposed the following rule for resemble on the basis of prototypical
properties:
(1)
resemble(@)(x,P) = 1 iff there is a (possibly complex) property P* such that:
a. P*(@)(x) = 1 und
b. For all y, j: if y is a prototypical representative of objects that have P
then P*(j)(y) = 1
This rule explains the unspecific readings and it explains why genuine quantifiers cannot
occur in the object position of this predicate unless they get a specific reading, compare: This
horse resembles every unicorn.
However, this rule cannot capture (a) the subject opacity of constructions with resemble: A
unicorn resembles a horse. (b) The notion of prototypicality in the definition is problematic:
It is odd to say that a Ferrari resembles a tomato although both objects prototypically share
the color red. Zimmermann’s proposal cannot predict the oddity. Moreover (c), and most
importantly, it is not clear how do comparative expressions interact with the notion of
resemble in terms of prototypical properties.
The Data: I will mainly concentrate on the last phenomenon: comparative expressions of
resemblance. Consider the sentence in (2).
(2)
Mary resembles her mother more than Mary’s brother resembles his father.
This sentence expresses a comparison between differences of pairs of individuals with respect
to the looks, the character, the height, hair color or the like (some salient property). The
dimension of comparison is left unexpressed in this sentence but it could easily be added in
terms of a with-respect-to phrase. Assume for the ease of exposition that we compare those
family members in (2) with respect to their height. Height has the advantage of being
measurable. Empirical comparisons with respect to height are well understood. The resulting
orderings may be captured by a measurement structure. Assuming we are comparing heights
the sentence may be paraphrased as in (2’):
(2’)
The difference between Mary’s height and her mother’s height is smaller than
the difference between Mary’s brother’s height and their father’s height.
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The paraphrase has the following schema:
For any measurement function µ and any individuals a, b, c, d:
Diff(µ(a), µ(b)) < Diff(µ(c), µ(d))
Moreover, we might visualize the paraphrase as follows:
(2’’)
Diff(µ(a), µ(b))
––––––––––––––––––––––––––––––––|
–––––––––––––––––––––––––––––––––––––––––|
––––––––––––––––––––––––––––––––––––––|
––––––––––––––––––––––––––––––––––––––––––––––––––––––|
Diff(µ(c), µ(d))
Diff(µ(a), µ(b))
-------------|--------|---------------------------|--------|----|------------------|----------->
Diff(µ(c), µ(d))
a
c b
d
Not only the dimension of comparison may be left linguistically unexpressed. Consider the
elliptical constructions in (3) and (4). In (3) the second element of the compaired pairs is
implicitly invariant, in (4) the first element of the compaired pairs is implicitly kept invariant.
(3)
Mary resembles her mother more than Mary’s brother resembles her mother.
Diff(µ(a), µ(b)) < Diff(µ(c), µ(b))
(4)
Mary resembles her mother more than Mary resembles her father.
Diff(µ(a), µ(b)) < Diff(µ(a), µ(d))
Sentence (5) can be viewed as a variant of the comparative construction that reminds us of the
positive constructions with respect to adjectival comparison. It means that the difference in
properties between a pair of individuals is smaller than some contextually supplied standard
of comparison c.
(5)
Mary resembles her mother more than Mary’s brother resembles his father.
Diff(µ(a), µ(b)) < c
The Semantics: In order to derive the paraphrases, I rely on the comparative semantics in the
tradition of von Stechow and Heim: see von Stechow (2006) for a new version of the positive
operator. The ontology for the semantics includes scales, degrees as points on a scale and
segments consisting of stretches on that scale. Segments on a scale come in three different
varieties: so-called positive extents (initial segments), negative extents (final segments) and
differentials (intermediate segments). In addition, my ontology provides measure functions in
two versions (see Heim-Mador & Winter’s two location functions): a measure function either
assigns an ordinary degree to an individual in the domain: x  µ(x). Or it assigns a degree to
a set of individuals in the domain: P  µ∪ (P). µ∪ may be defined on the basis of µ as in (6).
Let us call the second measure function unified measure function in accordance with MadorHeim & Winter (2007) although this operation is just some form of maximality operator well
known in comparative semantics. I will assume that all comparisons involve degrees and
therefore are representable by the means of measure functions.
(6)
For any properties P: µ∪ = MAX{µ(x): P(x)}
Maximality for degrees is defined as usual. Minimality is defined analogously.
(7)
If D is a set of degrees on a scale and ≥ is a suitable ordering relation then
MAX(D) = ιd[d ∈ D ∧ ∀d′∈D[d ≥ d′]]
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I propose the following semantics for resemble. As in Zimmermann’s version, the object
must be a predicate. The relation expressed is therefore a three-place relation that takes a
property, an individual and a degree as arguments and gives a truth value. Note that the
comparisons between the differences pattern with negative polar adjectives in general. The
differences compared are negative extents: resemble collects a set of degrees that are greater
than the difference between the compared items.
(8)
For any measure function µ:
resemble = λP.λx.λd. Diff[µ(x), µ∪(P)] < d
The measure functions µ and µ∪ contribute two degrees that can be viewed as an intermediate
segment on a scale. Diff is then an abbreviation for the function that measures the distance
between the two limits of that segment. It may be defined as follows:
(9)
Diff(D) = MAX(D) – MIN(D)
Let me note before passing on that be different from is (at least in one reading, see Beck
2000) just the positive polar variant of resemble:
(10) be different from = λP.λx.λd. Diff[µ(x), µ∪(P)] ≥ d
It is obvious how to combine theses semantics with the comparative and positive operator
introduced in adjectival semantics:
(11) more/-er = λD.λD*. {d: D*(d)} ⊃ {d: D(d)}
(12) POSN = λD.(∀d ∈ N(S)) D(d), where N is the neutral zone on the scale S.
Consider our sentence in (2) again. (2) is represented by the logical form in (2a) à la Bresnan
(1975). The truth conditions are stated in (2b).
(2)
Mary resembles her mother more than Mary’s brother resembles his father.
(2a) [more [λd.Mary’s brother [d [resembles his father]] λd.Mary [d [resembles her mother]]]
(2b) {d*: Diff[µ(Mary), µ∪(λy.y=Mary’s mother)] < d*} ⊃
{d: Diff[µ(Mary’s brother), µ∪(λy.y=Mary’s father)] < d*}
Further Applications: The restriction on quantifiers is readily derived since resemble only
takes predicates as internal arguments. In this respect I am following plainly the solution of
Zimmermann (1993). The proposed solution may also be suitable to solve the problem of
quantifier scope in complements of regular adjectival comparatives (see Heim 2006, e.g.) – at
least for the phrasal comparatives. I leave this thread of reasoning for further investigation.
The specificity ambiguity amounts to an ambiguity in scope of the indefinite with respect to
the comparative/positive operators. Consider for example the sentence in (15). (15a) and (15c)
are the logical forms for the unspecific and the specific reading, respectively.
(15)
(15a)
(15b)
(15c)
(15d)
Tom’s horse resembles a unicorn.
[POSN [λd.Tom’s horse [d [resembles a unicorn]]]
(∀d ∈ N(S)) λd.Diff[µ(Tom’s horse), µ∪(a unicorn)] < d}
a unicorn λx.[POSN [λd.Tom’s horse [d [resembles x]]
λx. unicorn(x) & (∀d ∈ N(S)) λd.Diff[µ(Tom’s horse), µ∪((λy.y=x)] < d}
(15b) states the truth conditions for the unspecific reading and (15d) the truth conditions for
the specific reading. (15d) may undergo existential closure or it may be interpreted within a
DRT-like framework. However, it should be noted that neither reading is intensional, contra
Zimmermann (1993). This view is only tenable if we assume that resemble is an extranuclear
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predicate in the sense of Parsons (like worship). Unicorns are existing though actually nonexistent, mythical objects. And resemble may be true of such objects. At this point I have to
refer the reader to Parson (1980) for the details.
The lack of existential generalisation and the lack of substitutivity may also be accounted for
in terms of extranuclearicity. The substitution of the expression a unicorn through a centaur
is not licit because the expressions refer to different sets of non-existent individuals.
(16) Tom’s horse resembles a unicorn.
(17) Tom’s horse resembles a centaur.
A generalization to the existence of unicorns or centaurs is not possible because they do not
exist in the actual world.
The subject opacity may be accounted for in terms of genericity, see Krifka et al (2003). The
genericity operator construes a tripartite structure. The restriction contains the indefinite and
the scope of the operator contains the property of resembling a horse, as in (18a).
(18) A unicorn resembles a horse.
(18a) GEN unicorn λx.[POSN [λd. x [d [resembles a horse]]]
(18b) GENx [unicorn x, (∀d ∈ N(S)) λd.Diff[µ(x), µ∪(a horse)] < d}]
Note that this correctly predicts that there is no restriction on genuine quantifiers for the
subject position and indefinites do not get a universal interpretation.
Furthermore, I assume that the value of the measure function is supplied contextually in most
cases. What counts as a salient measure function may depend on the context of utterance but
also on psychological parameters of stereotypical perception.
Conclusion: Zimmermann was right in postulating the property analysis for predicates like
resemble. Morever, I hope to have shown that a comparative semantics is mandatory for
resemble. The analysis is apt to explain the specificity ambiguity for these constructions
without regressing to a propositional analysis for the construction. An intensional analysis is
not necessary. And a propositional analysis is not necessary either. Resemble is a degree
predicate and resembles worship more than seek. Non-existent objects are needed in order to
analyse the case of worship. That is, the ontology does not get enriched by comparative
constructions. Worship does not show the specificity ambiguity since it is no comparative
construction. I conclude that the specificity ambiguity may have two reasons: either the
predicate is intensional or it is comparative.
References: Beck, Sigrid (2000): "The Semantics of different: Comparison Operator and
Relational Adjective". Linguistics and Philosophy 23, 101-139. Forbes, Graeme (2006):
Attitude Problems. Oxford University Press. Carlson, Gregory Frankcis Jeffrey Pelletier
(1995): The Generic Book. Chicago University Press. Mador-Haim, Sela & Yoad Winter
(2007) Non-Existential Indefinites and Semantic Incorporation of PP Complement, ms, to
appear in SALT 17 Proceedings. Parson, Terence (1980): Nonexistent objects. Yale
University Press. Von Stechow, Arnim (2006):”Times as Degrees: früh(er) “early(er)”,
spät(er) “late(r)”, and phase adverbs”, ms. Universität Tübingen. Williamson, Timothy
(1988): “First-order logics for comparative similarity“. In: Notre Dame J. Formal Logic,
Volume 29, Number 4 (1988), 457-481. Zimmermann, Th. Ede (1993): “On the Proper
Treatment of Opacity in Certain Verbs “. In: Natural Language Semantics 1 (1993), 149 - 179.
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