Distance Distributivity in Polish:
Towards a Glue Semantics Approach
Adam Przepiórkowski
We propose a novel syntactico-semantic analysis of distance distributivity
in Polish and other languages, which is couched in Lexical Functional
Grammar coupled with Glue Semantics. We introduce and analyse
a troublesome construction, apparently not considered so far in
the distance distributivity literature, where the sorting key is syntactically
embedded in the distributive share. Worked-out examples are
provided with Glue Semantics proofs.
Keywords: distance distributivity, Glue Semantics, LFG, Polish
1 Introduction
The aim of this paper is to provide a semantic analysis of some distance distributivity facts in
Polish, including potentially problematic facts apparently not discussed previously either in the
context of Polish or on the basis of other languages. Distance distributivity may be illustrated
with the following examples from English, German, and Polish; their common feature is that
the distributive element (each, jeweils, po) combines directly with the distributed NP1 (e.g. two
sausages in (1)) and that the plural NP denoting the restriction of the distribution (e.g. boys
in (1)) may be expressed at some distance from the distributive element.
(1) The boys have bought two sausages each.
(2) Die
the
Jungen
boys
haben
have
jeweils
distr
zwei
two
Würstchen
sausages
gekauft.
bought
(German; Zimmermann 2002:37)
‘The boys have bought two sausages each.’
(3) Chłopcy
boys
kupili
bought
po
distr
dwie
two
kiełbaski.
sausages
(Polish)
‘The boys (have) bought two sausages each.’
Following Choe 1987, Zimmermann 2002 and subsequent literature, the phrase denoting the
distributed objects (two sausages here) will be called the distributive (or distributed) share, and
the phrase denoting the set over which distribution takes place (boys above) will be called the
sorting (or distributive) key.
Zimmermann 2002 – couched in transformational grammar and roughly following the approach
to semantics outlined in Heim and Kratzer 1998 – remains the most comprehensive
account of distance distributivity in German and cross-linguistically, but it is not without problems.2
Dotlačil 2012 notes that on Zimmermann’s account the relation between the distributive
Many thanks to Gianluca Giorgolo, Agnieszka Patejuk, Chris Piñón and – last but not least – an anonymous
reviewer; their comments led to numerous improvements in the form and content of this paper. (I only wish they
could also be blamed for the remaining errors.) The work reported here was partially financed within two projects:
NEKST (http://zil.ipipan.waw.pl/NEKST) and CLARIN-PL (http://clip.ipipan.waw.pl/CLARIN-PL).
1Polish is a determinerless language, hence the use of NP rather than DP here.
2See Przepiórkowski 2014b for extended discussion.
EISS 10
Empirical Issues in Syntax and Semantics 10, ed. Christopher Piñón, 107–124
http://www.cssp.cnrs.fr/eiss10/
© 2014 Adam Przepiórkowski 107
108 adam przepiórkowski
share and the sorting key must be expressed by a constituent in the syntactic tree (e.g. such
a constituent exists for have bought in (1)), but examples where no such constituent may be
posited are easily found, as in Alex and Sasha visited the capitals of three states each (there is
no constituent corresponding exactly to visited the capitals of ). Moreover, while Zimmermann
(2002) seeks to provide an account not relying on LF movement, he acknowledges (sect. 2.4.2 of
chap. V) that his analysis must assume such covert movement for some occurrences of jeweils,
e.g. in (4) (Zimmermann 2002:269):
(4) Jeweils
distr
zwei
two
Offiziere
officers
begleiteten
accompanied
die
the
Ballerinen
ballerinas
nach
to
Haus.
home
(German)
‘Each ballerina was accompanied home by two officers.’
Finally, his analysis does not handle inverse linking cases where the sorting key is syntactically
embedded in the distributive share, as in the Polish example (5) (whose schematic syntactic
structure is given in (6)) or the corresponding German example (7) (Malte Zimmermann, p.c.):3
(5) Przybyło
arrive.past
po
distr
3
3
przedstawicieli
representatives
25
25.gen
krajów.
countries.gen
(Polish)
‘3 representatives arrived from each of 25 countries.’
(6) Przybyło [po [3 [przedstawicieli [25 krajów]]]].
(7) Jeweils
distr
3
3
Abgeordnete
representatives
aus
from
25
25
Ländern
countries
trafen ein.
arrived
(German)
To the best of our knowledge such constructions – and the difficulties they cause – have not
been noticed in the distance distributivity literature so far.
We propose an analysis which is free from such problems: it does not assume that the relation
between the distributive share and the sorting key is expressed by a syntactic constituent,
it is uniformly formulated at the interface between the level of grammatical functions and the
semantic level, and it correctly handles constructions exemplified by (5) and (7).
The main idea of the account 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 (5), 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:
‘λY. for each elementy of Y, 3 representatives ofy arrived’. Finally, this new property combines
with the sorting key 25 krajów ‘25 countries’, resulting in the meaning: ‘for each of 25 countries,
3 representatives arrived’.
The remainder of this paper is structured as follows. Polish distance distributivity facts are
outlined in section 2. A brief introduction to Glue Semantics follows in section 3. The analysis,
together with some worked-out examples (including (5) above), is presented in section 4. Finally,
section 5 concludes the paper.
3In order to increase clarity and shorten the textual form of the examples, numbers are written as digits here;
the fully spelled-out form of (5) is: Przybyło po trzech przedstawicieli dwudziestu pięciu krajów.
distance distributivity in polish: towards a glue semantics approach 109
2 Distance Distributivity in Polish
The syntactic behaviour of the distributive po in Polish is complex. Przepiórkowski 2013 shows
that at least three morphosyntactically different distributive lexemes po exist in Polish, illustrated
below.4
(8) Dałem
gave-I
im
them.dat
po
distr
jabłku.
apple.loc
‘I gave them an apple each.’
(9) Dałem
gave-I
im
them.dat
po
distr
dwa
two.acc
jabłka.
apples.acc
‘I gave them two apples each.’
(10) ...nagroda
reward
należy się
is due to
po
distr
trzem
three.dat
osobom
person.dat.pl
z
from
każdej
each
klasy...
class
‘Three people from each class deserve a reward.’ (NKJP)
When po combines with a non-numeral nominal phrase, as in (8), this phrase must occur in the
locative case, which in Polish is reserved for complements of some prepositions. Such po+NP
phrases are restricted to so-called structural case positions (nominative, accusative, genitive
of negation). The situation is much more complex when the distributive po combines with a
numeral phrase. In some positions po behaves like a preposition assigning the accusative case;
this is illustrated in (9), where case would remain accusative even if the verb was negated,
cf. (11a) below. This shows that the NumP dwa jabłka ‘two apples’ receives its case from po, as
otherwise it would bear the genitive of negation, as in (11b).
(11) a. Nie
neg
dałem
gave-I
im
them.dat
po
distr
dwa/*dwóch
two.acc/*gen
jabłka/*jabłek.
apples.acc/*gen
‘I didn’t give them two apples (each).’
b. Nie
neg
dałem
gave-I
im
them.dat
dwóch/*dwa
two.gen/*acc
jabłek/*jabłka.
apples.gen/*acc
‘I didn’t give them two apples.’
Finally, (10) illustrates that po sometimes does not assign case and may be transparent to
case assignment; the dative on trzem osobom ‘three people’ is assigned by the verb. While similar
examples may also be found for other morphological cases, including instrumental, genitive
and locative, they are often judged marginal or downright unacceptable, which shows that the
availability of this third lexeme po is restricted.
Despite such morphosyntactic idiosyncrasies, Przepiórkowski 2013 in the HPSG settings
and Przepiórkowski and Patejuk 2013 within LFG, provide a unified analysis of the three lexemes
po which treats all of them as heads. Hence, in the remainder of this paper we will not
distinguish them and we will assume that the phrase po combines with is its object.
Polish patterns with German rather than English in allowing the distributive share in the
subject position. In a classic paper on po, Łojasiewicz (1979:154) cites the following examples
4The first two examples, (8)–(9), are constructed but uncontroversial. As mentioned below, the acceptability
status of examples such as (10) is disputed, so this example is attested; NKJP stands for Narodowy Korpus Języka
Polskiego ‘National Corpus of Polish’ (http://nkjp.pl/; Przepiórkowski et al. 2012). Henceforth, Polish examples will
not be explicitly marked as such.
110 adam przepiórkowski
with (post-verbal) subjects:5
(12) Z
from
drzew
trees
spadło
fell
po
distr
jabłku.
apple.loc
‘An apple fell from each tree.’
(13) W
in
pokojach
rooms
będą
be.fut
po
distr
dwa
two
fotele.
armchairs
‘There will be two armchairs in each room.’
Such cases pose no problem for the analysis proposed below.
One aspect of distance distributivity in Polish that is not considered here is the possibility
of distribution over events. The argument that distributive elements like the German jeweils
may quantify over events comes from examples such as (14) adduced by Moltmann (1997) and
cited in Zimmermann 2002:28:
(14) Peter
Peter
hat
has
Maria
Maria
aus
for
jeweils
distr
zwei
two
Gründen
reasons
kritisiert
criticised
und
and
gelobt.
praised
(German)
‘Peter has criticised and praised Maria for two reasons respectively.’
This sentence means that for each of the two events involving Peter as an agent and Maria as a
patient, namely, that of criticising and that of praising, Peter had two reasons to be so involved
in them. Similarly, the only way to interpret (15), also from Zimmermann 2002:36, is to assume
a contextually given set of events of the Pope’s travels that jeweils quantifies over.
(15) Der
the
Papst
Pope
ist
has
in
to
jeweils
distr
drei
three
Länder
countries
gefahren.
travelled
(German)
‘The Pope has travelled to three countries each.’
Similar examples can be constructed in Polish:
(16) Piotr
Piotr
miał
had
po
distr
dwa
two
powody
reasons
by
to
chwalić
praise
i
and
krytykować
criticise
Marię.
Maria.
‘Peter had two reasons each to criticise and to praise Maria.’
(17) Papież
Pope
zwiedzał
visited
po
distr
trzy
three
kraje.
countries
‘The Pope visited three countries each time.’
Nevertheless, we assume simplistic eventless representations here and do not treat such cases
of distributivity over events.6
3 Glue Semantics
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 Glue Semantics (Dalrymple
1999, 2001) coupled with Lexical-Functional Grammar (Bresnan 2001, Dalrymple 2001),
5The case of dwa fotele ‘two armchairs’ is not given in (13), as it is not clear whether this phrase occurs in
the nominative or in the accusative here; Przepiórkowski 2013 and Przepiórkowski and Patejuk 2013 argue for the
accusative, despite appearances to the contrary.
6In Przepiórkowski 2014a, we show that the extension of the current analysis to distribution over events is
immediate.
distance distributivity in polish: towards a glue semantics approach 111
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:
(18) e((↑ subj)) t(↑)
We follow here the First Order approach to Glue Semantics (Kokkonidis 2008), where glue formulae
contain parameterised types, and assume two basic type constructors: e (for entity) and t
(for truth). The parameters of such basic type constructors are f-structures. As usual in LFG, the
up arrow ↑ in a lexical entry denotes the f-structure of the word, so (↑ subj) – with obligatory
parentheses, hence the double parentheses in the antecedent of (18) – denotes the f-structure
of the subject of this word. In effect, (18) says that by consuming the e type corresponding to
the subject of a form of yawn such as yawned, we may produce the t type corresponding to
yawned and, hence, to the whole clause headed by yawned (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 fstructure,
as the value of the subj attribute, so (18) is still relevant.
Glue Semantics is resource-sensitive: once a semantic resource – i.e., a glue formula –
is consumed, it cannot be reused. Dually, all semantic resources introduced by lexical items
(or otherwise; semantic resources may be introduced constructionally) must be consumed in a
derivation of the semantic resource of the whole sentence. For example, assuming that David
introduces a glue formula matching the antecedent of in (18), a proof rule analogous to
modus ponens (and introduced more formally below) consumes both formulae and produces the
formula t(↑) for the sentence David yawned. As this is the only resource left, the proof succeeds.
The other part of the meaning constructor is a formula in any language that allows application
and abstraction such as the language of the first-order predicate logic with lambda calculus.
For example, the meaning of David can be defined as a logical constant, David, and the meaning
of yawned can be defined as usual, as λX.yawn(X ) (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:
(19) David N (↑ pred) = ‘David’
David : e(↑)
(20) yawned V (↑ pred) = ‘yawn<subj>’
λX.yawn(X ) : e((↑ subj)) t(↑)
According to these lexical entries and standard LFG constituency rules, David yawned receives
the c-structure displayed in (21) and the f-structure in (22); moreover, given this f-structure,
meaning constructors are instantiated as in (23):7
7We adopt here the HPSG convention of naming feature structures with boxed numbers and of signalling
structure-sharing by the repeated occurrence of a boxed number (cf. 1 in (22)). Labels of meaning constructors
are written in [bold-in-square-brackets].
112 adam przepiórkowski
(21) IP
¨¨¨
rrr
NP
N
David
I
VP
V
yawned
(22)
0

pred ‘yawn 1 ’
subj 1 pred ‘David’

(23) [David] David : e( 1 )
[yawned] λX.yawn(X ) : e( 1 ) t( 0 )
Now, using one of the proof rules of Glue Semantics, namely, the Implication Elimination
rule in (24), and performing the usual β-reduction, the meaning of David yawned may be derived
from the meaning constructors in (23) as shown in (25):
(24) a : A f : A B
E
f (a) : B
(25) David : e( 1 ) λX.yawn(X ) : e( 1 ) t( 0 )
E
yawn(David) : t( 0 )
Since both meaning resources introduced by lexical items, e( 1 ) and e( 1 ) t( 0 ), are consumed
in this proof, and the only meaning resource produced, t( 0 ), corresponds to the f-structure
of the whole sentence, this is a valid proof that the meaning side of the whole sentence is
yawn(David).
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. Early influential
papers are gathered in Dalrymple 1999, but they may be a little hard for an uninitiated
reader, as they use a different – perhaps less transparent – notation; the exception is Dalrymple
et al. 1999a, which introduces the notation adopted in subsequent work on Glue Semantics.
As mentioned above, in this paper we assume the First Order approach to Glue advocated in
Kokkonidis 2008, which allows quantification over e types, not just over t types, as in previous
versions of Glue Semantics – the analysis proposed below crucially relies on this type of
quantification.
The glue side of meaning constructors is a fragment of linear logic (Girard 1987). Resources
are understood here as types parameterised with functional structures, but that does not mean
that Glue Semantics is necessarily tightly coupled with LFG; versions of this approach have been
proposed for other grammatical formalisms, including Head-driven Phrase Structure Grammar
(Asudeh and Crouch 2002) and Lexicalized Tree-Adjoining Grammar (Frank and van Genabith
2001). Also, while the meaning side adopted here is a version of the language of predicate logic
with lambdas, this is not a necessity. Instead, Intensional Logic is employed in Dalrymple et al.
1999c and various derivatives of Discourse Representation Theory are used in Dalrymple et al.
1999b, Crouch and van Genabith 1999, and more recently in Haug 2013.
4 Analysis
4.1 Preliminaries
Let us first consider the two run-of-the-mill examples below:
distance distributivity in polish: towards a glue semantics approach 113
(26) Chłopcy
boys.nom
mają
have.pl
po
distr
dwa
two.acc
tatuaże.
tattoos.acc
‘(The/Some) boys have two tattoos each.’
(27) Piotr
Piotr.nom
kupił
bought.sg
dziewczynom
girls.dat
po
distr
róży.
rose.loc
‘Peter bought (the/some) girls a rose each.’
In both examples the po-phrase (the distributive share) occupies the position of the direct object
of the verb; the purely morphosyntactic difference between the accusative case of dwa
tatuaże ‘two tattoos’ in (26) and the locative case of róży ‘rose’ in (27) was explained in section
2. The sorting key is expressed by the subject Chłopcy ‘boys’ in (26) and by the indirect
object dziewczynom ‘girls’ in (27).
The intended meaning representations of these two examples are given below:
(28) Intended meaning representation of (26):
exists(Z, boys(Z) ∧ |Z | > 1,
all(X, |X | = 1 ∧ X ⊂ Z,
exists(V, |V | = 2 ∧ tattoos(V ), have(X,V ))))
(29) Intended meaning representation of (27):
exists(Z, girls(Z) ∧ |Z | > 1,
all(X, |X | = 1 ∧ X ⊂ Z,
exists(V, |V | = 1 ∧ roses(V ), bought(p,V,X ))))
In fact, both examples taken out of context are similarly ambiguous: the plural bare NPs (Chłopcy
‘boys’ and dziewczynom ‘girls’) may be interpreted either as indefinites or as definites. For reasons
of simplicity, both indefinites and definites are represented as generalised quantifiers in
the current paper; the former are approximated by the existential quantifier exists, as in the
representations above, and the latter will be represented below via the iota relation.
As common in LFG and Glue Semantics, generalised quantifiers are represented here as
relations between an individual and two propositions involving that individual, so that Everyone
yawned has the representation all(X, person(X ), yawn(X )) (Dalrymple 2001:227). Moreover, we
follow Dotlačil 2012 and earlier work on treating entities as sets,8 and properties – as sets of
such sets. For example,boys is the property of being a non-empty set of boys – either a singleton
or a set of higher cardinality (the superscript s indicates the possible plural) – and λZ. |Z | >
1∧boys(Z) is the property of being a set of at least two boys. On this view, the standard inclusion
relation ⊆ is defined on entities.
How do these meaning representations differ from meanings of corresponding examples
without the distributive element? The relevant examples and their intended collective meanings
(assuming the existential closure of all bare NPs) are given below.9
(30) a. Chłopcy
boys.nom
mają
have.pl
dwa
two.acc
tatuaże.
tattoos.acc
(Cf. (26))
‘(The/Some) boys have two tattoos.’
b. exists(Z, boys(Z) ∧ |Z | > 1, (Cf. (28))
exists(V, |V | = 2 ∧ tattoos(V ), have(Z,V ))))
8In particular, we do not distinguish between singleton sets and their elements.
9In case of (30), the collective meaning may be difficult to get, unless one understands tattoos as temporary
sticker tattoos (before they are applied).
114 adam przepiórkowski
(31) a. Piotr
Piotr.nom
kupił
bought.sg
dziewczynom
girls.dat
różę.
rose.acc
(Cf. (27))
‘Peter bought a rose for (the/some) girls.’
b. exists(Z, girls(Z) ∧ |Z | > 1, (Cf. (29))
exists(V, |V | = 1 ∧ roses(V ), bought(p,V,Z))))
The difference between the meaning representations in (30b) and (31b) above and the earlier
representations in (28) and (29) should make the impact of the distributive po clear: it takes
a property holding of some set and transforms it into an analogous property holding of each
singleton subset of the set. We formalise this observation in the following subsection.
4.2 Semantics of po and Worked-out Example
The first version of the meaning constructor for po, labelled as [distr], is given below:10
(32) [distr] λS.λZ.all(X, |X | = 1 ∧ X ⊂ Z,S(X )) : ∀G,H. [e(G) t(H)] [e(G) t(H)]
The meaning part (on the left of the colon) directly reflects the considerations of the previous
subsection: 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. The glue part (on the right of the colon) says that po is
an identity function on semantic resources corresponding to properties: it consumes a resource
[e(G) t(H)] (for any G and H) in order to produce the same resource. Hence, po as construed
above may combine with just any e,t property in the sentence; as we will see below, this
analysis is too permissive and will be further constrained in section 4.4.
We will illustrate the analysis in detail on the basis of example (26), repeated below (with
the additional assumption that the subject is to be understood existentially):
(26 ) Chłopcy
boys.nom
mają
have.pl
po
distr
dwa
two.acc
tatuaże.
tattoos.acc
‘Some boys have two tattoos each.’
As usual in LFG and Glue Semantics, the two common nouns occurring in this sentence
have the following lexical entries (ignoring morphosyntactic features such as case or gender):
(33) chłopcy N (↑ pred) = ‘boys’
λX.boys
(X ) ∧ |X | > 1 : e(↑) t(↑)
(34) tatuaże N (↑ pred) = ‘tattoos’
λX.tattoos (X ) ∧ |X | > 1 : e(↑) t(↑)
Simplifying somewhat, we treat cardinals as existential quantifiers:
(35) dwa Num (↑ spec) = 2
λR.λS.exists(Y, |Y | = 2 ∧ R(Y ),S(Y )) :
∀H. [e(↑) t(↑)] [[e(↑) t(H)] t(H)]
While there are syntactic arguments that numerals take the following NPs as complements,
that is, that phrases of the form Num+NP are really headed by the numeral, we simplify here
10The meaning side is essentially the semantic representation of the abstract dist(ributivity) operator proposed
by Link 1991. The arguments given by Zimmermann 2002:68–69 that the German jeweils is not an overt realisation
of dist do not bear on the choice of this meaning representation here.
distance distributivity in polish: towards a glue semantics approach 115
by treating the numeral and the following noun as co-heads. Given the c-structure rule in (36),
we get the f-structure for dwa tatuaże ‘two tattoos’ shown in (37):
(36) NumP → Num N
↑=↓ ↑=↓
(37)
3

spec ‘2’
pred ‘tattoos’

Given this f-structure, all occurrences of ↑ in (34) and in (35) instantiate to 3 , so we can construct
the following proof for the meaning of dwa tatuaże ‘two tattoos’:11
(38) λR.λS.exists(Y, |Y | = 2 ∧ R(Y ),S(Y )) :
∀H. [e( 3 ) t( 3 )] [[e( 3 ) t(H)] t(H)]
λX.tattoos (X ) ∧ |X | > 1 :
e( 3 ) t( 3 )
E
λS.exists(Y, |Y | = 2 ∧ tattoos (Y ),S(Y )) :
∀H.[e( 3 ) t(H)] t(H)
The only missing lexical entries needed to analyse (26) are that of the main verb, mają
‘have’, as in (39), and that of po, as in (40):
(39) mają V (↑ pred) = ‘have<subj,obj>’
λX.λY.have(X,Y ) : e((↑ subj)) [e((↑ obj)) t(↑)]
(40) po P (↑ pred) = ‘po<obj>’
λP.P : ∀F. [e(↑) t(F)] [e((↑ obj)) t(F)]
λS.λZ.all(X, |X | = 1 ∧ X ⊂ Z,S(X )) : ∀G,H. [e(G) t(H)] [e(G) t(H)]
The lexical entry of the verb should be self-explanatory at this stage: the semantic resources
of the subject and the object must be consumed to produce a semantic resource corresponding
to the verb (and, hence, to the whole sentence headed by this verb). On the other hand, the
preposition po12 introduces two meaning constructors. The effect of the first one is that whatever
property P is specified elsewhere to hold of the meaning of the po-phrase, it must hold of
the meaning of the object of po instead. The other one is [distr] discussed above. These lexical
entries, together with standard c-structure rules, produce the following f-structure for the
complete sentence in (26):
(41)
0

pred ‘have 1 , 2 ’
subj 1 pred ‘boys’
obj 2

pred ‘po 3 ’
obj 3

spec ‘2’
pred ‘tattoos’



Given this f-structure, the meaning of mają ‘have’ instantiates to (42) and the first meaning
constructor of po instantiates to (43):
(42) [have] λX.λY.have(X,Y ) : e( 1 ) [e( 2 ) t( 0 )]
(43) [po] λP.P : ∀F.[e( 2 ) t(F)] [e( 3 ) t(F)]
11Each meaning constructor is broken into two lines for typographical reasons. We also drop the conjunct |Y | > 1
in the conclusion, as it follows from |Y | = 2.
12As discussed in section 2, there are three different lexemes po in Polish, but they are all analysed as heads, so
the lexical entry in (40) is a sufficiently good approximation of all of them.
116 adam przepiórkowski
At this point another Glue Semantics proof rule is needed, Implication Introduction, which
says that if the introduction of an assumption [x : A] leads to a proof of f : B then λx.f : A B
is proved:
(44) [x : A]1
...
f : B
I,1
λx.f : A B
Using this rule, (45) may be proved from (42) and (43) as shown in (46):
(45) [have-po] λX.λY.have(X,Y ) : e( 1 ) [e( 3 ) t( 0 )]
(46)
[X : e( 1 )]1
λX.λY.have(X,Y ) :
e( 1 ) [e( 2 ) t( 0 )]
E
λY.have(X,Y ) : e( 2 ) t( 0 ) λP.P : ∀F.[e( 2 ) t(F)] [e( 3 ) t(F)]
E
λY.have(X,Y ) : e( 3 ) t( 0 )
I,1
λX.λY.have(X,Y ) : e( 1 ) [e( 3 ) t( 0 )]
The conclusion may be combined with the conclusion of proof (38), repeated in (47), to render
the meaning of mają dwa tatuaże ‘have two tattoos’ in (48); the proof is shown in (49):
(47) [two-tattoos] λS.exists(Y, |Y | = 2 ∧ tattoos (Y ),S(Y )) : ∀H. [e( 3 ) t(H)] t(H)
(48) [have-po-two-tattoos] λX.exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y )) : e( 1 ) t( 0 )
(49)
[X : e( 1 )]1
λX.λY.have(X,Y ) :
e( 1 ) [e( 3 ) t( 0 )]
E
λY.have(X,Y ) : e( 3 ) t( 0 )
λS.exists(Y, |Y | = 2 ∧ tattoos (Y ),S(Y )) :
∀H. [e( 3 ) H] t(H)
E
exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y )) : t( 0 )
I,1
λX.exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y )) : e( 1 ) t( 0 )
The conclusion of proof (49) is of the form that may be combined with the second meaning
constructor for po given in (40):
(50) λX.exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y )) :
e( 1 ) t( 0 )
λS.λZ.all(X, |X | = 1 ∧ X ⊂ Z,S(X )) :
∀G,H. [e(G) t(H)] [e(G) t(H)]
E
λZ.all(X, |X | = 1 ∧ X ⊂ Z, exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y ))) :
e( 1 ) t( 0 )
Now we face an apparent problem, as – apart from the resource in the conclusion of
proof (50) – the only other resource left is that of chłopcy ‘boys’, introduced in (33) and instantiated
here to (51), and these two resources are incompatible (cannot be combined).
(51) [boys] λX.boys
(X ) ∧ |X | > 1 : e( 1 ) t( 1 )
However, as noted above, such bare NPs are understood as either indefinites or as definites, so
the grammar must provide appropriate meaning constructors completing the lexical meanings
of bare NPs. As it is not the aim of this paper to investigate the representation of (in)definites,
we approximate them via generalised quantifiers (even though it is well known that they have
distance distributivity in polish: towards a glue semantics approach 117
different scopal properties than usual quantifiers). In the case at hand, the meaning constructor
that is needed is (compare this to the meaning of dwa ‘two’ in (35)):
(52) [existential] λR.λS.exists(Z,R(Z),S(Z)) :
∀H. [e( 1 ) t( 1 )] [[e( 1 ) t(H)] t(H)]
Once this constructor is available, the existential meaning of chłopcy ‘boys’ may be derived
using the Implication Elimination proof rule:
(53) λX.boys
(X ) ∧ |X | > 1 :
e( 1 ) t( 1 )
λR.λS.exists(Z,R(Z),S(Z)) :
∀H. [e( 1 ) t( 1 )] [[e( 1 ) t(H)] t(H)]
E
λS.exists(Z, boys
(Z) ∧ |Z | > 1,S(Z)) :
∀H.[e( 1 ) t(H)] t(H)
Applying the same proof rule to the conclusions of (50) and (53), we obtain the same (up
to variable names) meaning side as the intended meaning representation of (26), given in (28):
(54) λZ.all(X, |X | = 1 ∧ X ⊂ Z,
exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y ))) :
e( 1 ) t( 0 )
λS.exists(Z, boys
(Z) ∧ |Z | > 1,S(Z)) :
∀H.[e( 1 ) t(H)] t(H)
E
exists(Z, boys
(Z) ∧ |Z | > 1,
all(X, |X | = 1 ∧ X ⊂ Z,
exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y )))) : t( 0 )
The schematic structure of the whole proof is given below, with references to subproofs:
(55) [have] [po]
(46)
[have-po]
[two] [tattoos]
(38)
[two-tattoos]
(49)
[have-po-two-tattoos] [distr]
(50)
[distr-have-po-two-tattoos]
[boys] [existential]
(53)
[boys-existential]
(54)
[boys-existential-distr-have-po-two-tattoos]
Note that all resources introduced by lexical items have been consumed in the process and
that the only resource left is t( 0 ), which corresponds to the complete sentence; hence, this is a
linguistically valid proof (Asudeh 2012:chap. 5).
An analogous proof could be constructed for the definite reading of chłopcy ‘boys’, using
the following meaning constructor instead of [existential] of (52):
(56) [definite] λR.λS.iota(Z,R(Z),S(Z)) :
∀H. [e( 1 ) t( 1 )] [[e( 1 ) t(H)] t(H)]
Such meaning constructors must be optionally available for any common noun. If the noun
contributes to the restriction of a lexical quantifier, as in case of tatuaże ‘tattoos’ restricting the
quantifier dwa ‘two’, optional meaning constructors of this kind cannot be used – the lexical
quantifier consumes the resources necessary to activate such constructors. On the other hand,
when there is no appropriate lexical quantifier, either the existential closure or the definiteness
meaning constructor may activate and combine with the bare noun.13
13We assume that such optional meaning constructors are introduced in lexical entries of common nouns, as
part of a common noun template, so as to avoid missing generalisations (Asudeh et al. 2013); another option would
118 adam przepiórkowski
4.3 Sorting Key within Distributive Share
Let us now turn to (5), repeated below as (5 ), where the sorting key, 25 krajów ‘25 countries’, is
syntactically embedded within the phrase expressing the distributive share, po 3 przedstawicieli
25 krajów ‘3 representatives of (each of) 25 countries’; the schematic constituent structure is
repeated as (6 ).
(5 ) Przybyło
arrive.past
po
distr
3
3
przedstawicieli
representatives
25
25.gen
krajów.
countries.gen
‘3 representatives arrived from each of 25 countries.’
(6 ) Przybyło [po [3 [przedstawicieli [25 krajów]]]].
Lexical entries for 3 and 25 parallel that for dwa ‘two’ given in (35):
(57) 3 Num (↑ spec) = 3
λR.λS.exists(Y, |Y | = 3 ∧ R(Y ),S(Y )) :
∀H. [e(↑) t(↑)] [[e(↑) t(H)] t(H)]
(58) 25 Num (↑ spec) = 25
λR.λS.exists(Y, |Y | = 25 ∧ R(Y ),S(Y )) :
∀H. [e(↑) t(↑)] [[e(↑) t(H)] t(H)]
Similarly, the lexical entry for krajów ‘countries’ is analogous to those for chłopcy ‘boys’ and
tatuaże ‘tattoos’ in (33) and (34), and the entry for przybyło ‘arrived’ is simpler than that for
mają ‘have’ in (39), as it only takes one argument:
(59) krajów N (↑ pred) = ‘countries’
λX.countrys (X ) ∧ |X | > 1 : e(↑) t(↑)
(60) przybyło V (↑ pred) = ‘arrive<subj>’
λX.arrive(X ) : e((↑ subj)) t(↑)
What is new in this example is a relational noun, przedstawicieli ‘representatives’:14
(61) przedstawicieli N (↑ pred) = ‘representatives<obj>’
λY.λX.representatives(X,Y ) ∧ |X | > 1 :
e((↑ obj)) [e(↑) t(↑)]
The meaning constructor of (61) differs from that of (59) and other non-relational nouns in the
additional requirement of the argument of the noun.
With these lexical entries, as well as the lexical entry for po given in (40) above, the fstructure
of (5) is as shown in (62).
be to add them to appropriate c-structure rules.
14We remain agnostic as to whether obj, assumed in (61), is really the right grammatical function for the complement
of przedstawicieli ‘representatives’. Dalrymple et al. 1999c:57 and Dalrymple 2001:249 analyse arguments
of English nouns rumor and relative, introduced by the prepositional markers about and of, as values of oblabout
and oblof, respectively.
distance distributivity in polish: towards a glue semantics approach 119
(62)
0

pred ‘arrived 1 ’
subj 1

pred ‘po 2 ’
obj 2

spec ‘3’
pred ‘representative 3 ’
obj 3

spec ‘25’
pred ‘country’




The intended meaning of (5), given in (63), may be attained via the proof schematically
shown in (64), where the particular meaning constructors, as instantiated for (62), are given
in (65)–(76).15
(63) exists(Z, |Z | = 25 ∧ countrys(Z),
all(X, |X | = 1 ∧ X ⊂ Z,
exists(V, |V | = 3 ∧ representatives(V,X ), arrived(V )))) : t( 0 )
(64) [arrived] [po]
E
[arrived-po]
[3] [representatives]
EEI
[3-representatives]
EEI
[arrived-po-3-representatives] [distr]
E
[distr-arrived-po-3-representatives]
[25] [countries]
E
[25-countries]
E
[25-countries-distr-arrived-po-3-representatives]
(65) [25]
λR.λS.exists(X, |X | = 25 ∧ R(X ),S(X )) : ∀H. [e( 3 ) t( 3 )] [[e( 3 ) t(H)] t(H)]
(66) [countries]
λX.countrys(X ) ∧ |X | > 1 : e( 3 ) t( 3 )
(67) [25-countries]
λS.exists(X, |X | = 25 ∧ countrys(X ),S(X )) : ∀H.[e( 3 ) t(H)] t(H)
(68) [3]
λR.λS.exists(X, |X | = 3 ∧ R(X ),S(X )) : [∀H. [e( 2 ) t( 2 )] [[e( 2 ) t(H)] t(H)]
(69) [representatives]
λY.λXrepresentatives(X,Y ) ∧ |X | > 1 : e( 3 ) [e( 2 ) t( 2 )]
(70) [3-representatives]
λY.λS.exists(X, |X | = 3 ∧ representatives(X,Y ),S(X )) : ∀H. e( 3 ) [[e( 2 ) t(H)] t(H)]
(71) [po]
λP.P : ∀F. [e( 1 ) t(F)] [e( 2 ) t(F)]
(72) [arrived]
λX.arrived(X ) : e( 1 ) t( 0 )
(73) [arrived-po]
λX.arrived(X ) : e( 2 ) t( 0 )
(74) [arrived-po-3-representatives]
λY.exists(X, |X | = 3 ∧ representatives(X,Y ), arrived(X )) : e( 3 ) t( 0 )
15The parts of the proof marked with EEI consist of three steps analogous to subproofs given in (46) and
in (49). Again, we omit |X | > 1 once it follows from particular cardinalities contributed by the numerals.
120 adam przepiórkowski
(75) [distr-arrived-po-3-representatives] (see (32) for [distr])
λZ.all(X, |X | = 1 ∧ X ⊂ Z,
exists(V, |V | = 3 ∧ representatives(V,X ), arrived(V ))) : e( 3 ) t( 0 )
(76) [25-countries-distr-arrived-po-3-representatives]
exists(Z, |Z | = 25 ∧ countrys(Z),
all(X, |X | = 1 ∧ X ⊂ Z,
exists(V, |V | = 3 ∧ representatives(V,X ), arrived(V )))) : t( 0 )
This proof shows that the analysis proposed in the previous subsection provides a correct meaning
representation for troublesome cases when the sorting key is embedded within the phrase
expressing the distributive share.
4.4 Constraining Analysis
Unfortunately, as it stands, the analysis heavily overgenerates. For example, apart from (64),
there are other proofs for the same sentence, leading to nonsensical or wrong meaning representations.
The problem is that the meaning of po, as given in (32) and (40), may combine
with any (appropriately typed) property available in the derivation, e.g., with [countries] in
(66), with [arrived] in (72) or with the property derived from [representatives] in (69) by
introducing the assumption Y : e( 3 ) and using the Implication Elimination rule (24).
We will illustrate this problem with a simpler example, by showing that the sentence
Chłopcy mają po dwa tatuaże ‘(Some/The) boys have two tattoos each’, given as (26) in section
4.2, has another proof, leading to the incorrect meaning in (77), paraphrased as “for each
of some two tattoos, there are some boys that have it.”
(77) exists(Y, |Y | = 2 ∧ tattoos (Y ),
all(X, |X | = 1 ∧ X ⊂ Y,
exists(Z, boys
(Z) ∧ |Z | > 1, have(Z,X ))))
The proof is analogous to (55), and it is given in (78) below, with references to subproofs:
(78) [have] [po]
(46)
[have-po]
[boys] [existential]
(53)
[boys-existential]
(79)
[boys-existential-have-po] [distr]
(80)
[distr-boys-existential-have-po]
[two] [tattoos]
(38)
[two-tattoos]
(81)
[two-tattoos-distr-boys-existential-have-po]
(79)
[Y : e( 3 )]2
[X : e( 1 )]1
λX.λY.have(X,Y ) :
e( 1 ) [e( 3 ) t( 0 )]
E
λY.have(X,Y ) : e( 3 ) t( 0 )
E
have(X,Y ) : t( 0 )
I,1
λX.have(X,Y ) : e( 1 ) t( 0 )
λS.exists(Z, boys
(Z) ∧ |Z | > 1,S(Z)) :
∀H.[e( 1 ) t(H)] t(H)
E
exists(Z, boys
(Z) ∧ |Z | > 1, have(Z,Y )) : t( 0 )
I,2
λY.exists(Z, boys
(Z) ∧ |Z | > 1, have(Z,Y )) : e( 3 ) t( 0 )
distance distributivity in polish: towards a glue semantics approach 121
(80) λY.exists(Z, boys
(Z) ∧ |Z | > 1, have(Z,Y )) :
e( 3 ) t( 0 )
λS.λY.all(X, |X | = 1 ∧ X ⊂ Y,S(X )) :
∀G,H. [e(G) t(H)] [e(G) t(H)]
E
λY.all(X, |X | = 1 ∧ X ⊂ Y, exists(Z, boys
(Z) ∧ |Z | > 1, have(Z,X ))) :
e( 3 ) t( 0 )
(81) λY.all(X, |X | = 1 ∧ X ⊂ Y,
exists(Z, boys
(Z) ∧ |Z | > 1, have(Z,X ))) :
e( 3 ) t( 0 )
λS.exists(Y, |Y | = 2 ∧ tattoos (Y ),S(Y )) :
∀H.[e( 3 ) t(H)] t(H)
E
exists(Y, |Y | = 2 ∧ tattoos (Y ),
all(X, |X | = 1 ∧ X ⊂ Y,
exists(Z, boys
(Z) ∧ |Z | > 1, have(Z,X )))) : t( 0 )
A preliminary solution to this problem – presented in greater detail and further refined
in Przepiórkowski 2014a – is inspired by the Glue Semantics approach to Negative Polarity
Licensing proposed by Fry 1999. The original intuition behind this approach is that a Negative
Polarity Item (NPI) “attaches” to its usual meaning a marker which is transferred during the
semantic derivation until it meets a licensor which discharges (i.e. consumes) it. In the case at
hand, the distributive share acts as an NPI and the marker is discharged when the distributive
meaning of po combines with a meaning containing the contribution of this distributive share.
Technically, we introduce the “marked” type td , modify the distributive meaning constructor
so that it eliminates the marking (we will call it [distr-E]), and add another meaning constructor
in the lexical entry of po which introduces the marking (we will call it [distr-I]); compare
the lexical entry (82) for po below with (40) above:
(82) po P (↑ pred) = ‘po<obj>’
[po] = λP.P : ∀F. [e(↑) t(F)] [e((↑ obj)) t(F)]
[distr-E] = λS.λZ.all(X, |X | = 1 ∧ X ⊂ Z,S(X )) :
∀G,H. [e(G) td (H)] [e(G) t(H)]
[distr-I] = λQ.Q :
∀H. [[e((↑ obj)) t(H)] t(H)] [[e((↑ obj)) t(H)] td (H)]
In the running example, given the f-structure (41), the three meaning constructors in the lexical
entry of po instantiate to:
(83) [po]
λP.P : ∀F.[e( 2 ) t(F)] [e( 3 ) t(F)]
(84) [distr-E]
λS.λZ.all(X, |X | = 1 ∧ X ⊂ Z,S(X )) : ∀G,H. [e(G) td (H)] [e(G) t(H)]
(85) [distr-I]
λQ.Q : ∀H. [[e( 3 ) t(H)] t(H)] [[e( 3 ) t(H)] td (H)]
With these meaning constructors, the proof of the correct meaning in the running example is
similar to that in (55), with [distr] replaced by [distr-E] and with [distr-I] combining with
the meaning of dwa tatuaże ‘two tattoos’. Modified partial conclusions are presented below
(unchanged constructors are repeated for convenience):
(45 ) [have-po]
λX.λY.have(X,Y ) : e( 1 ) [e( 3 ) t( 0 )]
122 adam przepiórkowski
(47 ) [two-tattoos]
λS.exists(Y, |Y | = 2 ∧ tattoos (Y ),S(Y )) : ∀H. [e( 3 ) t(H)] t(H)
(86) [distr-I-two-tattoos]
λS.exists(Y, |Y | = 2 ∧ tattoos (Y ),S(Y )) : ∀H. [e( 3 ) t(H)] td (H)
(87) [have-po-distr-I-two-tattoos]
λX.exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y )) : e( 1 ) td ( 0 )
(88) [distr-E-have-po-distr-I-two-tattoos] (= [distr-have-po-two-tattoos] in proof (55))
λZ.all(X, |X | = 1 ∧ X ⊂ Z, exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y ))) : e( 1 ) t( 0 )
(89) [boys-existential] (= conclusion in subproof (53) = [boys-existential] in proof (55))
λS.exists(Z, boys
(Z) ∧ |Z | > 1,S(Z)) : ∀H.[e( 1 ) t(H)] t(H)
(90) [boys-existential-distr-E-have-po-distr-I-two-tattoos] (= conclusion in proof (55))
exists(Z, boys
(Z) ∧ |Z | > 1,
all(X, |X | = 1 ∧ X ⊂ Z,
exists(Y, |Y | = 2 ∧ tattoos (Y ), have(X,Y )))) : t( 0 )
Note how the marking d is introduced by [distr-I] on the quantifier two tattoos in (86), how it
is transferred to the predicate in (87) and how it is eliminated by [distr-E], which now expects
its semantic argument to be so marked, in (88). The proof is summarised below.
(91)
[have][po]
EEI
[have-po]
[two][tattoos]
E
[two-tattoos] [distr-I]
E
[distr-I-two-tattoos]
EEI
[have-po-distr-I-two-tattoos] [distr-E]
E
[distr-E-have-po-distr-I-two-tattoos]
[boys][existential]
E
[boys-existential]
E
[boys-existential-distr-E-have-po-distr-I-two-tattoos]
At the same time, the unwanted proof (78) for the same sentence (26) is blocked now. Since
the constructor [distr-I] may only combine with the constructor of a quantifier whose restriction
is expressed by the object of po, it cannot combine with the existential chłopcy ‘boys’,
whose restriction on the glue side containse( 1 ) instead of thee( 3 ) expected by [distr-I]. Hence,
[boys-existential] in a putative analogue of proof (78) cannot contain the marker d , so it cannot
pass it to [boys-existential-have-po], and so [distr-E] cannot combine with it. While
[distr-I] may still combine with [two-tattoos], neither the resulting [distr-I-two-tattoos]
nor [distr-E] may enter the proof now.
5 Conclusion
Analyses of distance distributivity, such as Choe 1987, Safir and Stowell 1988, Moltmann 1997,
Zimmermann 2002 or Dotlačil 2012, have so far been formulated mainly within the transformational
paradigm. In contrast, the current paper provides a non-transformational analysis,
couched within Lexical Functional Grammar and coupled with the morphosyntactic account of
Przepiórkowski and Patejuk 2013. On the semantic side, we employed the resource-sensitive
approach of Glue Semantics. Empirically, the main point of this paper is the introduction – and
successful analysis – of a construction troublesome for previous analyses, where the sorting
distance distributivity in polish: towards a glue semantics approach 123
key is syntactically embedded in the phrase expressing the distributive share.
The account proposed here is still at a relatively early stage of development. It remains to be
seen whether the mechanism employed to harness overgeneration, introduced in section 4.4,
is sufficiently general and robust. Moreover, we had nothing to say about distribution over
events, witnessed in Polish and German, among other languages. (Both points are addressed in
Przepiórkowski 2014a.) Nevertheless, we hope that the current proposal provides a reasonable
backbone to flesh out a more exhaustive constraint-based and resource-sensitive analysis of
distance distributivity in Polish and other languages.
References
Asudeh, Ash. 2012. The logic of pronominal resumption. Oxford: Oxford University Press.
Asudeh, Ash, and Richard Crouch. 2002. Glue semantics for HPSG. In Proceedings of the HPSG 2001
Conference, ed. Frank van Eynde, Lars Hellan, and Dorothee Beermann, 1–19. Stanford, CA: CSLI
Publications.
Asudeh, Ash, Mary Dalrymple, and Ida Toivonen. 2013. Constructions with Lexical Integrity. Journal of
Language Modelling 1:1–54.
Bresnan, Joan. 2001. Lexical-functional syntax. Blackwell Textbooks in Linguistics. Malden, MA: Black-
well.
Choe, Jae-Woong. 1987. Anti-quantifiers and a theory of distributivity. Ph. D. dissertation, University of
Massachusetts, Amherst.
Crouch, Richard, and Josef van Genabith. 1999. Context change, underspecification, and the structure of
glue language derivations. In Dalrymple (1999), 117–189.
Dalrymple, Mary, ed. 1999. Semantics and syntax in Lexical Functional Grammar: The resource logic approach.
Cambridge, MA: The MIT Press.
Dalrymple, Mary. 2001. Lexical Functional Grammar. San Diego, CA: Academic Press.
Dalrymple, Mary, Vineet Gupta, John Lamping, and Vijay Saraswat. 1999a. Relating resource-based
semantics to categorial semantics. In Dalrymple (1999), 261–280.
Dalrymple, Mary, John Lamping, Fernando Pereira, and Vijay Saraswat. 1999b. Overview and introduction.
In Dalrymple (1999), 1–37.
Dalrymple, Mary, John Lamping, Fernando Pereira, and Vijay Saraswat. 1999c. Quantification, anaphora,
and intensionality. In Dalrymple (1999), 39–89.
Dotlačil, Jakub. 2012. Binominal each as an anaphoric determiner: Compositional analysis. In Proceedings
of Sinn und Bedeutung 16: Volume 1, ed. Ana Aguilar Guevara, Anna Chernilovskaya, and Rick
Nouwen, volume 16 of MIT Working Papers in Linguistics, 211–224.
Frank, Anette, and Josef van Genabith. 2001. GlueTag: Linear logic based semantics for LTAG – and what
it teaches us about LFG and LTAG. In The Proceedings of the LFG’01 Conference, ed. Miriam Butt and
Tracy Holloway King, 104–126. University of Hong Kong: CSLI Publications.
Fry, John. 1999. Proof nets and negative polarity licensing. In Dalrymple (1999), 91–116.
Girard, Jean-Yves. 1987. Linear Logic. Theoretical Computer Science 50:1–102.
Haug, Dag. 2013. Partial control and anaphoric control in LFG. In The Proceedings of the LFG’13 Conference,
ed. Miriam Butt and Tracy Holloway King, 274–294. Stanford, CA: CSLI Publications.
Heim, Irene, and Angelika Kratzer. 1998. Semantics in generative grammar. Blackwell Textbooks in
Linguistics. Malden, MA: Blackwell.
Kokkonidis, Miltiadis. 2008. First-order glue. Journal of Logic, Language and Information 17:43–68.
Link, Godehard. 1991. Plural. In Semantik. Ein internationales Handbuch der zeitgenösischen Forschung,
ed. Arnim von Stechow and Dieter Wunderlich, 418–440. Berlin: Mouton de Gruyter.
Łojasiewicz, Anna. 1979. O budowie wyrażeń z przyimkiem po dystrybutywnym. Polonica V:153–160.
Moltmann, Friederike. 1997. Parts and wholes in semantics. Oxford: Oxford University Press.
Przepiórkowski, Adam. 2013. The syntax of distance distributivity in Polish: Preserving generalisations
with weak heads. In Proceedings of the HPSG 2013 Conference, ed. Stefan Müller, 161–181. Stanford,
124 adam przepiórkowski
CA: CSLI Publications.
Przepiórkowski, Adam. 2014a. Syntactic and semantic constraints in a Glue Semantics approach to distance
distributivity. In The Proceedings of the LFG’14 Conference, ed. Miriam Butt and Tracy Holloway
King. Stanford, CA: CSLI Publications. To appear.
Przepiórkowski, Adam. 2014b. A weakly compositional analysis of distance distributivity in Polish.
Submitted to Formal Approaches to Slavic Linguistics 23 Proceedings.
Przepiórkowski, Adam, and Agnieszka Patejuk. 2013. The syntax of distance distributivity in Polish:
Weak heads in LFG via restriction. In The Proceedings of the LFG’13 Conference, ed. Miriam Butt and
Tracy Holloway King, 482–502. Stanford, CA: CSLI Publications.
Przepiórkowski, Adam, Mirosław Bańko, Rafał L. Górski, and Barbara Lewandowska-Tomaszczyk, ed.
2012. Narodowy korpus języka polskiego. Warsaw: Wydawnictwo Naukowe PWN.
Safir, Ken, and Tim Stowell. 1988. Binominal each. In Proceedings of the North East Linguistic Society
(NELS), Vol. 18, 426–450.
Zimmermann, Malte. 2002. ‘Boys buying two sausages each’ – on the syntax and semantics of distance
distributivity. Ph. D. dissertation, Universiteit van Amsterdam.
Institute of Computer Science, Polish Academy of Sciences
http://nlp.ipipan.waw.pl/~adamp/