FASL 23, 262–281
Michigan Slavic Publications
2015
A Weakly Compositional Analysis of
Distance Distributivity in Polish
Adam Przepiórkowski
Institute of Computer Science of the Polish Academy of Sciences
1 Introduction
Distance distributivity is a phenomenon where a distributive element such
as each occurs at some structural distance from the nominal phrase that
restricts it, as in The boys have two apples each.1 In this sentence, the
distributive element each occurs in the object position, while its restriction,
the boys, is the subject of the sentence. This should be contrasted with
the determiner uses of each, as in Each boy has two apples, where each
combines directly with its restriction, as other ad-nominal quantifiers do.
There are various terminological conventions in the literature, e.g.,
Choe 1987 calls such uses of each “anti-quantifiers”, and Safir and Stowell
1988 call them “binominal”. Both terms are suboptimal: much subsequent
literature attempts to describe such distributive elements (DEs) as
more-or-less ordinary quantifiers (not as special “anti-quantifiers”) and it
is clear now that DEs in other languages, including German and Polish, do
not need two nominal expressions (the boys and two apples above) but –
as shown by Moltmann 1991, 1997 – may quantify over events expressed
by verbal constituents (hence, they are not “binominal”). In this paper
we adopt the terminology of Zimmermann 2002, who introduced the term
1 I am grateful for comments from the audience of FASL 23 and, especially, to the two
anonymous reviewers. Also the acknowledgements of the accompanying papers Przepiórkowski
2014a,b carry over here. Needles to say, all remaining errors are my own.
DISTANCE DISTRIBUTIVITY IN POLISH 263
“distance distributivity” (DD), and call the nominal phrase DE attaches to
(two apples above) “distributive share” (DS), and the phrase expressing the
set of entities which restrict DE (the boys above) – “distributive key” (DK).
It is clear that DD is not a completely uniform phenomenon and constraints
on structural relations between DSs (and, hence, DEs which attach
to them) and DKs differ across languages. Zimmermann 2002, working
within the transformational paradigm of late 1990s, explains these
differences in terms of inherent features of DEs and distinguishes two
classes of DEs: those that have determiner features and, hence, must be
c-commanded by a DP for these features to be licensed, and those that do
not have such determiner features. The c-command requirement prevents
the former from occurring in the (underlying) subject position, as in the unacceptable
English *One student each gave presents to the teachers (Safir
and Stowell 1988: 436, (26a)), while no such restriction is observed in case
of the German DE jeweils or the Polish DE po:2
(1) Jeweils
DISTR
ein
one
Offizier
officerNOM
begleitete
accompanied
die
the
Ballerinen
ballerinasACC
nach
to
Haus.
home
(German)
‘Each ballerina was accompanied home by one officer.’
(Zimmermann 2002: 27, (16))
(2) Z
from
drzew
trees
spadło
fell
po
DISTR
jabłku.
appleLOC
(Polish)
‘An apple fell from each tree.’ (Łojasiewicz 1979: 154)
To the best of our knowledge, Zimmermann 2002 remains the only
comprehensive syntactico-semantic analysis of DD of the kind observed in
German and Polish. The aim of this paper is to show that Polish data do not
comfortably fit the account of Zimmermann 2002 (§ 2) and to introduce a
construction which that analysis cannot account for (§ 3). Due to lack of
space, an alternative analysis is only sketched here (§ 4), but it is presented
in gory technical detail in accompanying papers (Przepiórkowski 2014a,b).
2 Unlike the binominal each, which always follows the DS, jeweils usually precedes the
DS (Zimmermann 2002: § III.5.3), while po always occurs immediately before it.
264 ADAM PRZEPIÓRKOWSKI
2 Zimmermann 2002
Various problems, both empirical and theoretical, with earlier accounts of
DD such as Choe 1987, Safir and Stowell 1988, Moltmann 1991, 1997 and
Link 1998, are discussed and criticised in Zimmermann 2002, so here we
only refer to Zimmermann’s approach.
While Zimmermann 2002 remains the most comprehensive account of
distance distributivity in German and cross-linguistically, it is not without
problems. First, as noted by Dotlaˇcil 2012, Zimmermann’s assumption
that the relation between DS and DK is expressed by a syntactic constituent
(e.g., have in Each boy have two apples) does not always hold. For
example, in Alex and Sasha visited the capitals of three states each there is
no constituent corresponding exactly to visited the capitals of.
Second, the careful reader of footnotes will note that Polish (and Slavic
in general) fits rather uncomfortably into Zimmermann’s account.3 In particular,
it seems unexpected on Zimmermann’s analysis that the po DE
obligatorily precedes DS in Slavic. While a cross-linguistically valid analysis
is highly desired, we feel that it should be guided by more detailed
investigations into particular languages.
Third, although Zimmermann (2002) seeks to provide an account not
relying on LF movement (and gives good arguments against the LF-based
analysis of Safir and Stowell 1988), he acknowledges that his analysis must
assume such covert movement for some occurrences of jeweils, including
(1) above (see his § 2.4.2 in ch. V, pp. 271ff.).
Fourth, in the course of providing the details of the syntactico-semantic
analysis of DD across languages, Zimmermann (2002) is forced to introduce
some non-standard mechanisms and make a number of assumptions
contradicting the majority view in the framework hosting the analysis.
One such mechanism is the “Type-Triggered λ-Abstraction” (p. 219), a
very specific composition rule supplementing the more run-of-the-mill
(Bittner 1994, Heim and Kratzer 1998) “Index-Triggered λ-Abstraction”
and triggered in some contexts as “a last resort mechanism that only applies
if all else fails”. Among unusual assumptions there is also one about head
3 See, e.g., fn. 86 on p. 131, fn. 87 on p. 132 (together with fn. 76 on p. 119). Also aspects
of Korean seem problematic, e.g., fn. 83 on p. 134, fn. 98 on p. 143, fn. 21 on p. 276, as well
as main text on p. 140.
DISTANCE DISTRIBUTIVITY IN POLISH 265
movement out of adjuncts (fn. 76 on p. 119, but see also fn. 87 on p. 132),
and another about event binding within VP (p. 226). Moreover, while putting
much emphasis on the compositionality of the proposed analysis, some
of its elements are not fully compositional, e.g., the context-driven insertion
of various restrictions into the representation of the German DE jeweils,
e.g., in (176) on p. 232 (with the introduced relation P), in (184) on
p. 234 (with the relation Ď), and in (219) on p. 247 (with a new set variable).
Such ad hoc mechanisms result in rather different representations of
similar sentences (e.g., (171e) on p. 230 vs. (177) on p. 232).
Fifth, despite all this additional machinery, there are attested constructions
that – as far as we see – cannot be handled in the approach of Zimmermann
2002. We introduce one such construction below.
3 Inverse Linking Distance Distributivity Construction
There is a construction problematic for previous analyses of DD, bearing
certain resemblance to the inverse linking construction discussed in May
1985: 68ff. and Heim and Kratzer 1998: § 8.6, among others. In this construction
– exemplified with the Polish sentence (3) (whose schematic syntactic
structure is given in (4)) and the corresponding German sentence (5)
– the distributive key is syntactically embedded within the distributive
share:
(3) Przybyło
arrivePAST
po
DISTR
3
3
przedstawicieli
representatives
25
25GEN
krajów.
countriesGEN
(Polish)
‘3 representatives arrived from each of 25 countries.’
(4) Przybyło [po [3 [przedstawicieli [25 krajów]]]].
(5) Jeweils
DISTR
3
3
Abgeordnete
representatives
aus
from
25
25
Ländern
countries
trafen ein.
arrived
(German)
‘3 representatives arrived from each of 25 countries.’
(Malte Zimmermann, p.c.)
The structure given in (4) is not controversial. The Polish DE po is analysed
as – or simply assumed to be – a preposition (Łojasiewicz 1979, Franks
1995) which combines with the following nominal phrase.4 Numerals are
4 While there are reasons to postulate more than one DE po in Polish, they are all best
analysed as heads (Przepiórkowski 2006, 2010, 2013, Przepiórkowski and Patejuk 2013),
266 ADAM PRZEPIÓRKOWSKI
also analysed as heads of numeral phrases in Polish on the basis of substitution
tests and case assignment (Saloni and ´Swidzi´nski 1998, Przepiórkowski
1999; but see also Franks 1995). In any case, whether the numeral
phrase 3 przedstawicieli... ‘three representatives...’ is taken to be headed
by the numeral or by the noun, 25 krajów ‘25 countries’ is an argument of
przedstawicieli ‘representatives’, so – at least at the surface – it must be
contained in the maximal projection of this noun.5 Hence, the DK 25 krajów
‘25 countries’ is contained within the DS 3 przedstawicieli 25 krajów
‘3 representatives of 25 countries’.
Note that although (3) is a constructed example, analogous attested
examples may easily be found in the National Corpus of Polish (NKJP;
Przepiórkowski et al. 2012; http://nkjp.pl/) and in the Internet, e.g. (constraining
our search to the same relational noun):6
(6) ...proponował
proposedM1
po
DISTR
dwóch
twoACC.M1
przedstawicieli
representativesACC.M1
miasta
cityGEN.N
i
and
ComArchu...
ComArchGEN.M3
‘...he proposed two representatives for each of the city and
ComArch.’ (NKJP)
(7) W
into
skład
make-upACC
jury
juryGEN
wchodzi
enters
po
DISTR
2
twoACC.M1
przedstawicieli
representativesACC.M1
organizatorów
organisersGEN.M1
konkursu.
competitionGEN.M3
‘2 representatives of each of the organisers of the competition belong
to / constitute the jury.’
(http://zporuszcza.polaniec.pl/index_pliki/bezpieczna_skola.pdf)
so treating them all as prepositions is a reasonable first approximation. Note that they differ
from the prefix po- (Bogusławski 1993, http://pinon.sdf-eu.org/covers/dpp.html), which
has a related but different distributive meaning.
5 Note that this is an island, presumably also for covert movement:
(i) *Czego
whatGEN
przybyło
arrivePAST
po
DISTR
3
3
przedstawicieli?
representatives
6 In the glosses, M1 stands for the human-masculine gender and M3 – for inanimatemasculine,
assuming the 5 Polish genders proposed in Ma´nczak 1956. Other morphosyntactic
symbols follow the Leipzig Glossing Rules.
DISTANCE DISTRIBUTIVITY IN POLISH 267
In the examples above, the DK is an argument of the relational noun
przedstawiciel ‘representative’, which heads the DS. Examples of DK adjuncts
to heads of DSs are also easy to find, but they are not discussed here,
as they do not pose a particular problem for Zimmermann’s analysis.
Let us attempt to analyse such constructions. The cross-linguistic denotation
of DEs proposed by Zimmermann 2002: 122 is given below:
(8) vDEw “ λP.@zrpz P Ziq Ñ DxrPpxq^Rjpz,xqss
In this representation, P stands for the property expressed by the DS. For
example, in The boys have two apples each, P would be the property of
being a set of two apples; let us schematically represent this property as
λx.2applespxq. Given the representation of each in (8), two apples each
receive the following representation (via functional application):
(9) vtwo apples eachw “ @zrpz P Ziq Ñ Dxr2applespxq^Rjpz,xqss
Zi and Rj are variables which are coindexed with, respectively, the DK (the
boys) and the relation between the DK and the DS (have). Via the “IndexTriggered
λ-Abstraction” (Zimmermann 2002: 217), when the phrase two
apples each with the representation in (9) is a constituent-tree sister of a
node with index j, expressing a 2-place relation such as λzλx.havepz,xq,
(9) can be transformed to (10) below and then be applied to the haverelation
to render (11) for the verbal phrase (VP) have two apples each.
(10) vtwo apples eachw “ λRj.@zrpz P Ziq Ñ Dxr2applespxq^Rjpz,xqss
(11) vhave two apples eachw “ @zrpz P Ziq Ñ Dxr2applespxq^havepz,xqss
Similarly, when the VP is a sister to the boys indexed with i, λ-abstraction
is licensed again (see (12)) resulting in a function that can be applied to the
meaning of the boys, giving the meaning of the sentence in (13).
(12) vhave two apples eachw “
λZi.@zrpz P Ziq Ñ Dxr2applespxq^havepz,xqss
(13) vthe boys have two apples eachw “
@zrpz P vthe boyswq Ñ Dxr2applespxq^havepz,xqss
Returning to (3), its analogous desired representation is given in (14):7
7 We ignore here the event variable introduced by the verb and bound via existential closure
at the end of the derivation, but see below.
268 ADAM PRZEPIÓRKOWSKI
(14) vprzybyło po 3 przedstawicieli 25 krajóww “
@zrpz P v25 countrieswq Ñ Dxr3representativespx,zq^arrivedpxqss
How can it be derived, assuming the representation of the DE po as in (8)?
For the sake of the argument, let us give as much leeway to Zimmermann’s
approach as possible and assume that any kind of LF-movement is allowed,
even in violation of island constraints.
The DE must first combine with its sister – either the DS or a trace
resulting from movement. But since traces are of the semantic type <e>,
and DE expects a property of type <e,t>, no movement of the whole DS is
possible.8 On the other hand, we have to assume the LF-movement of 25
krajów ‘25 countries’; otherwise, if the whole 3 przedstawicieli 25 krajów
‘3 representatives of 25 countries’ is consumed as P, there would be no
DK to subsequently provide the meaning of Zi. So the only way to proceed
with the analysis is to assume the following schematic structure at LF:
(15) [25 krajów]i [przybyło [po 3 przedstawicieli ti]]
The DE po expects a property, so let us assume the representation of the
argument of po as in (16) and the result of its combination with the representation
of po given in (8) – as in (17):
(16) v3 przedstawicieli tiw “ λx.3representativespx,ziq
(17) vpo 3 przedstawicieli tiw “
@zrpz P Ziq Ñ Dxr3representativespx,ziq^Rjpz,xqss
This representation is already getting incoherent, as it now involves two
variables coindexed with 25 krajów ‘countries’ – zi of type <e> and Zi of
type <e,t>. Obviously, instead of the variable zi, the second argument of
3representatives should be the variable z bound by the universal quantifier.
Hence, the existential closure over zi in (16) would not help here either.
Even if this problem could somehow be solved, there is no binary relation
that could provide the meaning of the binary Rj – przybyło ‘arrived’ is a
unary predicate.9
8 Also, the “Index-Triggered λ-Abstraction” is not applicable in this configuration.
9 Zimmermann 2002: 226, fn. 67, considers the possibility of a family of denotations for
DE, with Rj of different arities greater or equal to 2. Perhaps this idea could be extended
even further, to Rj of arity 1, but this would not solve the problem of the incoherent
DISTANCE DISTRIBUTIVITY IN POLISH 269
Since there is no constituent in the representation of (3) that expresses
a binary relation needed to provide the denotation of Rj, let us attempt to
analyse this sentence in a way analogous to the German (18), which also
involves a unary predicate (the idiomatic keep watch).
(18) Jeweils
DISTR
zwei
two
Jungen
boysNOM
standen
kept
Wache.
guard
(German)
‘Two boys kept watch at a time.’ (Zimmermann 2002: 249)
Here, the distribution is over events; the target denotation can be paraphrased
as “for all elements z of a contextually salient set (of events) Zi,
there is a set of two boys x, and an event e, such that the elements of x kept
watch in e, and event e is related to event z by a temporal, causal, subpart,
or other contextual relation R” (Zimmermann 2002: 261):
(19) vjeweils zwei Jungen standen Wachew “
@zrpz P Ziq Ñ Dxr2boyspxq^Derkept_watchpx,eq^Rpe,zqsss
In order to derive this representation, Zimmermann 2002: 259 assumes
the standard representation of jeweils in (8), which gives rise to the following
representation of jeweils zwei Jungen:
(20) vjeweils zwei Jungenw “ @zrpz P Ziq Ñ Dxr2boyspxq^Rjpz,xqsss
The meaning of the verbal component t1 standen Wache, with t1 representing
the trace of the subject jeweils zwei Jungen, is less obvious (here, after
applying λ-abstractions):
(21) vt1 standen Wachew “ λx1λei.Derkept_watchpx1,eq^Rpe,eiqs
The variable x1 in (21) represents the subject of the predicate, while R represents
a contextually given relation between the event e predicated by the
verb and an event ei in the preceding discourse (Zimmermann 2002: 260).
This way the denotation of standen Wache ‘stood guard’ is a 2-place predicate,
as expected. With this representation of the verbal predicate, the
result of λ-abstraction of Rj in (20) applied to (21) is (19) above.
How can this analysis be carried over to (3)? First of all, let us assume
the representation of przybyło ‘arrived’ analogous to that in (21):
representation in (17). Also, the analysis proposed below does not have to assume a family
of representations of the DE po.
270 ADAM PRZEPIÓRKOWSKI
(22) vt1 przybyłow “ λeiλx1.Derarrivedpx1,eq^Rpe,eiqs
Assuming λ-abstraction over Rj in (17) above and subsequent application
of the resulting function to the denotation in (22), the denotation in (23)
below results. Combining this representation with that of 25 krajów ‘25
countries’, we get (24).
(23) vprzybyło po 3 przedstawicieli tiw “
@zrpz P Ziq Ñ Dxr3representativespx,ziq^Derarrivedpx,eq^Rpe,zqsss
(24) vprzybyło po 3 przedstawicieli 25 krajóww “
@zrpz P v25 krajówwq Ñ
Dxr3representativespx,ziq^Derarrivedpx,eq^Rpe,zqsss
This representation is close to the correct one but – unfortunately – it again
contains the free variable zi which should really be bound by the universal
quantifier @z.
Note that the target representation in (14) could be derived from the DE
denotation in (8), but such a derivation would violate Zimmermann’s basic
assumptions about constituency and surface compositionality: the DE po
would first have to combine with przybyło ‘arrived’ in (25) rendering the
denotation in (26), then with 3 przedstawicieli ‘3 representatives’ treated
as a binary relation λzλx.3representativespx,zq, resulting in (27), and then
with 25 krajów ‘25 countries’, resulting in (28):
(25) vprzybyłow “ λx.arrivedpxq
(26) vprzybyło pow “ @zrpz P Ziq Ñ Dxrarrivedpxqs^Rjpx,zqs
(27) vprzybyło po trzech przedstawicieliw “
@zrpz P Ziq Ñ Dxrarrivedpxq^3representativespx,zqss
(28) vprzybyło po trzech przedstawicieli 25 krajóww “
@zrpz P v25 countrieswq Ñ Dxrarrivedpxq^3representativespx,zqss
In summary, whether treating (3) as an instance of distribution over entities
(25 countries) or over events (arrivals), we do not see a way to derive
an acceptable meaning of this sentence, given the approach of Zimmermann
2002. This, combined with the reservations expressed in § 2, calls
for a new approach to distance distributivity in Polish; such an approach is
sketched below.
DISTANCE DISTRIBUTIVITY IN POLISH 271
4 An Outline of an Alternative Account
The main idea of an alternative account, more fully described in Przepiórkowski
2014a, from which this section draws heavily, is this: the semantic
impact of po activates only once the distributive share combines semantically
with the verb and creates a property. For example, in case of (3), the
meaning of przybyło 3 przedstawicieli, ‘λY. 3 representatives of Y arrived’,
is derived first. Then, the meaning of po combines with this property, let
us call it S, holding of some set Y, and produces a new property, which
is just like S but holds of each element of Y individually: ‘λY. for each
element y of Y, 3 representatives of y arrived’. Finally, this new property
combines with the distributive key 25 krajów ‘25 countries’, resulting in
the meaning: ‘for each of 25 countries, 3 representatives arrived’.
This idea relies on the possibility to combine the meaning of po with
the property ‘λY. 3 representatives of Y arrived’ expressed by przybyło 3
przedstawicieli, rather than with the meaning of the syntactic sister of po.
It would be difficult to implement this idea in a framework that understands
compositionality narrowly, as in these two recent formulations:
• The meaning of a complex expression functionally depends on the
meanings of its immediate parts and the way in which they are
combined. (Zimmermann 2012: 82)
• The meaning of a complex expression is determined by its immediate
structure and the meanings of its immediate constituents.
(Szabó 2012: 79)
This is the usual understanding of compositionality – unquestioned in
transformational approaches – but it is not the only one. In fact, as discussed
in detail in Janssen 2012 and Szabó 2012, the provenance of this
– originally massively ambiguous – principle is murky (it should probably
not be attributed to Frege, but rather to his student, Carnap 1947), there
are no strong fundamental – as opposed to methodological – arguments for
adopting it, and the reasons for its widespread use are mostly technical.
As noted already in 1987 (see the reprint, Halvorsen 1995: 295), compositionality
should be replaced in constraint-based theories by systematicity,
a method of automatic derivation of utterance interpretations from
the lexical information and any rules of the interpretation scheme.10 The
10 This emphasis on the meanings of utterances rather than the meanings of arbitrary
272 ADAM PRZEPIÓRKOWSKI
alternative analysis of DD in Polish is couched in just such a constraintbased
theory, namely, Lexical Functional Grammar (LFG; Bresnan 2001,
Dalrymple 2001), coupled with a resource-based approach to meaning
composition, namely, Glue Semantics (Dalrymple 1999, 2001). The latter
explicitly adopts this weak notion of compositionality, where the meaning
of a sentence depends on the meanings of its words and the way these
are combined, but where syntactic structure and lexical semantics may not
fully specify either (Crouch and van Genabith 1999: 122).
In traditional approaches to compositionality (e.g., Heim and Kratzer
1998), meanings combine when they are expressed by siblings in a constituency
tree. By contrast, in LFG + Glue, meanings combine based on
f(unctional)-structures, rather than on c(onstituent)-structures, and meaning
representations are paired with glue formulae specifying how these
meanings combine with which other meanings. Any pair consisting of a
meaning representation and a glue formula is called a meaning constructor.
For example, the glue part of the meaning constructor for various forms
of yawn is:
(29) pÒ SUBJqσ Òσ
As usual in LFG, the up arrow Ò in a lexical entry denotes the f-structure of
the word, pÒ SUBJq denotes the f-structure of the subject of this word, and
σ is a function from f-structures to s(emantic)-structures. In effect, (29)
says that, by consuming the s-structure corresponding to the subject of
yawn, we may produce the s-structure corresponding to yawn and, hence,
to the whole clause headed by yawn (in LFG heads normally share their
f-structure with their projections).
This mode of composition remains true regardless of specific tree configurations.
For example, when yawn is a complement of a control verb, its
covert subject is never realised in the c(onstituent)-structure, according to
standard LFG analyses, but it is still present in its f-structure, as the value
of the SUBJ attribute, so (29) is still relevant.
The other part of the meaning constructor is a formula in any language
that allows application and abstraction, e.g., the language of the first-order
predicate logic with lambda calculus. For example, the meaning of David
syntactic components is strongly related to the principle of contextuality, a postulate that
does deserve to be called Frege’s principle; see Janssen 2012 for discussion.
DISTANCE DISTRIBUTIVITY IN POLISH 273
can be defined as a logical constant, David, and the meaning of yawned can
be defined as usual, as λX.yawnpXq (ignoring event variables, semantic
roles, tense and aspect, etc.). In complete meaning constructors, the meaning
part is separated from the glue part by the uninterpreted colon character
(:), so the complete meaning constructors for David and yawned are as in
the second lines of the following lexical entries:
(30) David N pÒ PREDq “ ‘DAVID’
David :Òσ
(31) yawned V pÒ PREDq “ ‘YAWN<SUBJ>’
λX.yawnpXq : pÒ SUBJqσ Òσ
According to these lexical entries and standard LFG constituency rules,
David yawned receives the c-structure displayed in (32) and the f-structure
in (33); moreover, given this f-structure, meaning constructors are instantiated
as in (34):
(32) IP
¨¨¨
rrr
NP
N
David
I1
VP
V
yawned
(33)
0
»
–
PRED ‘YAWNx 1 y’
SUBJ 1
”
PRED ‘DAVID’
ı
fi
fl
(34) [David] David : 1 σ
[yawned] λX.yawnpXq : 1 σ 0 σ
Now, using one of the proof rules of Glue Semantics, namely, the Implication
Elimination rule in (35), and performing the usual β-reduction,
the meaning of David yawned may be derived from the meaning constructors
in (34) as shown in (36):
(35) a : A f : A B
E
fpaq : B
(36) David : 1 σ λX.yawnpXq : 1 σ 0 σ
E
yawnpDavidq : 0 σ
274 ADAM PRZEPIÓRKOWSKI
Since both meaning resources introduced by lexical items, 1 σ and
1 σ 0 σ , were consumed in this proof, and the only meaning resource
produced, 0 σ , corresponds to the f-structure of the whole sentence, this is
a valid proof that the meaning side of the whole sentence is yawnpDavidq.
Obviously, we cannot do justice to Glue Semantics within the confines
of this paper; the above is only meant to make the analysis below more
accessible to motivated readers not familiar with this approach. The best
introduction to Glue Semantics may still be found in the classical LFG
textbook of Dalrymple 2001, on which the above exposition is based.
Let us now return to the problematic distance distributivity construction
exemplified by (3), repeated below:
(3) Przybyło
arrivePAST
po
DISTR
3
3
przedstawicieli
representatives
25
25GEN
krajów.
countriesGEN
‘3 representatives arrived from each of 25 countries.’
The lexical entry for przybyło ‘arrived’ matches that of yawned given
in (31) above (note that we ignore the event variable again, solely for reasons
of simplicity):
(37) przybyło V pÒ PREDq “ ‘ARRIVE<SUBJ>’
λX.arrivepXq : pÒ SUBJqσ Òσ
The meaning constructors of common nouns are a little less obvious:
(38) krajów N pÒ PREDq “ ‘COUNTRIES’
λX.countryspXq^|X| ą 1 : pÒσ VARq pÒσ RESTRq
First, we follow Dotlaˇcil 2012 and earlier work on treating type e objects
as sets, and properties – as sets of such sets. For example, countrys is the
property of being a non-empty set of countries – either a singleton or a
set of higher cardinality (the superscript s indicates the possible plural) –
and λX.|X| ą 1^countryspXq is the property of being a set of at least two
countries. On this view, the standard inclusion relation Ď is defined on
type e objects. Second, the glue side shows that semantic structures may
have some internal structure: s-structures of common nouns, which are
of type xe,ty, have the attributes VAR and RESTR, representing a variable
(of type e) and a restriction on that variable (of type t); cf. Dalrymple
2001: 250–253.
DISTANCE DISTRIBUTIVITY IN POLISH 275
Entries of relational nouns are just like those of common nouns, but
they add a specification of an internal argument:
(39) przedstawicieli N pÒ PREDq “ ‘REPRESENTATIVES<OBJ>’
λY.λXrepresentativespX,Yq^|X| ą 1 :
pÒ OBJqσ rpÒσ VARq pÒσ RESTRqs
The meaning constructor of (39) differs from that of (38) and other nonrelational
nouns in the additional requirement of the semantic resource corresponding
to the argument of the noun.
Further, simplifying somewhat, we treat cardinals as existential quan-
tifiers:
(40) 3 Num pÒ SPECq “ 3
λR.λS.existspY,|Y| “ 3^RpYq,SpYqq :
rpÒσ VARq pÒσ RESTRqs r@H.rÒσ Hs Hs
(41) 25 Num pÒ SPECq “ 25
λR.λS.existspY,|Y| “ 25^RpYq,SpYqq :
rpÒσ VARq pÒσ RESTRqs r@H.rÒσ Hs Hs
As common in LFG and Glue Semantics, generalised quantifiers are represented
here as pair quantifiers, that is, as relations between an individual
and two propositions involving that individual, so that Someone yawned
has the basic representation existspX,personpXq,yawnpXqq (Dalrymple
2001: 227). In our setup, cardinality is additionally specified, so – for
example – Two people yawned will have the following representation:
existspX, personspXq^|X| “ 2, yawnpXqq.
Finally, we assume the following lexical entry of po:
(42) po P pÒ PREDq “ ‘PO<OBJ>’
pÒ OBJqσ “ Òσ
λS.λZ.allpX,|X| “ 1^X Ă Z,SpXqq :
@G,H. rG Hs rG Hs
Observe that po is analysed as a preposition here (but see fn. 4). The import
of the second line, pÒ OBJqσ “ Òσ , will be explained below. The third line –
the meaning part of the meaning constructor – says that po takes a property
S and returns a property that holds of Z if and only if S holds of all singleton
(proper) subsets of Z. Finally, the glue part in the fourth line says that po
276 ADAM PRZEPIÓRKOWSKI
is an identity function on semantic resources corresponding to properties:
it consumes any resource rG Hs in order to produce the same resource.
Since G and H may be any semantic resources (of appropriate types), this
analysis is much too permissive as it stands – it is appropriately constrained
in Przepiórkowski 2014b.
We do not present here syntactic rules which serve to build the constituency
structure of the running example, as they are trivial and of secondary
importance to the current analysis. Crucially, we assume that these rules
– together with the lexical entries above – lead to the following functional
structure for the complete sentence in (3):
(43)
0
»
—
—
—
—
—
—
—
—
—
—
–
PRED ‘ARRIVEDx 1 y’
SUBJ 1
»
—
—
—
—
—
—
—
–
PRED ‘POx 2 y’
OBJ 2
»
—
—
—
—
–
SPEC ‘3’
PRED ‘REPRESENTATIVEx 3 y’
OBJ 3
«
SPEC ‘25’
PRED ‘COUNTRY’
ff
fi
ffi
ffi
ffi
ffi
fl
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
While there are syntactic reasons to assume that numerals take the following
NPs as their arguments, we simplify here by treating the numeral
and the following noun as co-heads. Hence, both the lexical entry for krajów
in (38) and the lexical entry for 25 in (41) contribute to the innermost
feature structure in (43), marked with the label 3 . In other words, the Ò
variable in these lexical entries instantiates to 3 , so the meaning constructors
instantiate, respectively, to:
(44) [countries]
λX.countryspXq^|X| ą 1 : p 3 σ VARq p 3 σ RESTRq
(45) [25]
λR.λS.existspX,|X| “ 25^RpXq,SpXqq :
rp 3 σ VARq p 3 σ RESTRqs r@H.r 3 σ Hs Hs
Using the Implication Elimination rule in (35), and performing the usual
β-reduction, these meanings combine to:11
11 In (35), substitute “p 3 σ VARq p 3 σ RESTRq” for A, “@H.r 3 σ Hs H” for B,
“λX.countryspXq^|X| ą 1” for a and “λR.λS.existspX,|X| “ 25^RpXq,SpXqq” for f.
DISTANCE DISTRIBUTIVITY IN POLISH 277
(46) [25-countries]
λS.existspX,|X| “ 25^countryspXq,SpXqq : @H.r 3 σ Hs H
Similarly, lexical entries (39) (for przedstawicieli) and (40) (for 3) contribute
to the construction of f-structure 2 , so Ò in those entries instantiates
to 2 and, hence, pÒ OBJq instantiates to 3 :
(47) [representatives]
λY.λX.representativespX,Yq^|X| ą 1 :
3 σ rp 2 σ VARq p 2 σ RESTRqs
(48) [3]
λR.λS.existspX,|X| “ 3^RpXq,SpXqq :
rp 2 σ VARq p 2 σ RESTRqs r@H.r 2 σ Hs Hs
(49) [3-representatives]
λY.λS.existspX,|X| “ 3^representativespX,Yq,SpXqq :
@H. 3 σ rr 2 σ Hs Hs
In fact, in order to derive [3-representatives] from [3] and [representatives],
another standard proof rule is needed, Implication Introduction
(Dalrymple 2001: 236, Asudeh 2012: 79), which we will not cite here for
lack of space. Instead we note that the proof captures the intuition behind
the function composition in Categorial Grammar (cf., e.g., Steedman
2000: 40), where functions X{Y and Y{Z may compose into X{Z.
Given the f-structure (43), the meaning constructor of przybyło in (37)
instantiates to [arrived], as Ò instantiates to 0 and, hence, pÒ SUBJq – to 1 :
(50) [arrived]
λX.arrivedpXq : 1 σ 0 σ
Finally, the meaning constructor of po in (42) contains no Ò symbols,
only variables G and H matching any (appropriately typed) resource, but
there is another line in this lexical entry, pÒ OBJqσ “ Òσ , which – given (43)
– instantiates to 2 σ “ 1 σ . The intuition behind this meaning constructor
and this constraint is that po makes no semantic impact where it occurs –
it equates its semantic resource 1 σ with that of its argument 2 σ – but it
contributes the distributive meaning constructor which activates elsewhere
in the semantic derivation.
278 ADAM PRZEPIÓRKOWSKI
Given that 2 σ “ 1 σ , and substituting 0 σ for H in the meaning constructor
[3-representatives] (49), this constructor may be combined with
[arrived] (50) (again, via a few proof steps, including Implication Introduction),
rendering:
(51) [arrived-3-representatives]
λY.existspX,|X| “ 3^representativespX,Yq,arrivedpXqq : 3 σ 0 σ
After substituting G and H with, respectively, 3 σ and 0 σ in the meaning
constructor for po in (42), [arrived-3-representatives] combines with this
meaning constructor directly, resulting in:
(52) [distr-arrived-3-representatives]
λZ.allpY,|Y| “ 1^Y Ă Z,
existspX,|X| “ 3^representativespX,Yq,arrivedpXqqq : 3 σ 0 σ
Finally, substituting 0 σ for H in the meaning constructor for the quantifier
phrase 25 krajów, given in (46), it combines directly with the above
meaning constructor (52), rendering the intended meaning of the whole
functional structure 0 :
(53) [25-countries-distr-arrived-3-representatives]
existspZ,|Z| “ 25^countryspZq,
allpY,|Y| “ 1^Y Ă Z,
existspX,|X| “ 3^representativespX,Yq,arrivedpXqqqq : 0 σ
5 Conclusion
One of the first influential analyses of distance distributivity, Choe 1987,
is not compositional. Further work – of which Zimmermann 2002 is a
premiere example – tried to provide compositional analyses of the phenomenon
at the syntax-semantics interface. While it remains the most
comprehensive analysis of DD of the kind also observed in Slavic languages,
it is not without problems and limitations, discussed in § 2 and
§ 3. The alternative analysis, outlined in § 4, is compositional in a rather
weak sense, but it is systematic: the meaning of an utterance is derived
from the meanings of lexical items and the way they combine. Even if not
all technical details of the presented analysis are transparent to readers not
previously exposed to LFG and Glue Semantics, it should be clear that the
advantage of this relaxed approach to compositionality is a much simpler
DISTANCE DISTRIBUTIVITY IN POLISH 279
syntax: no ad hoc (covert movement, etc.) rules are needed to account for
the semantic complexity. Instead, the complexity resides exactly where it
should: in the lexical entries of semantically complex items.
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adamp@ipipan.waw.pl