Under-specification and contextual variability of abstract 
prepositions: a case study 
 
Alda MARI 
CNRS / ENST – INFRES 
46, rue Barrault 
75013 Paris, France 
mari@enst.fr 
 
 
 
 
Abstract 
 
In this paper we discuss some 
philosophical questions related to the 
treatment of abstract and underspecified 
prepositions. We consider three issues in 
particular: (i) the relation between sense 
and meanings, (ii) the privileged status of 
abstract meanings in the spectrum of 
contextual instantiations of basic senses, 
and finally (iii) the difference between 
prediction and inference. The discussion 
will be based on the study of avec (with) 
and the analysis of its abstract meaning of 
comitativity in particular. A model for 
avec semantic variability will also be 
suggested.  
1 Introduction 
Through the study of the preposition avec we 
discuss some major questions related to the 
analysis and model of abstract prepositions.  
Avec is traditionally (Spang-Hanssen, 1963) 
considered as a mixed preposition. On one side, 
its contextual behavior is very heterogeneous: 
contrary to the colored prepositions (e.g. 
spatial), it seems difficult to construct a unique 
sense from which all the meanings are derived; 
nevertheless, contrary to the uncolored 
prepositions such as de (of) or à (to) in French, 
avec seems to point to an identifiable set of 
constraints. It is our aim to investigate whether 
this set exists and to determine its status. This 
requires an answer to the following questions:  
1. How can the contextual variability of 
polysemous prepositions be mastered ? 
2. Does a privileged sense exists in the 
spectrum of this variability ?  
3. What is the relation between sense and 
meanings ?  
4. What criteria have to be filled by a model ?  
We discuss the first of these questions  in 
section 2 where we present the theoretical 
contention around the notion of polysemy of 
grammatical items and prepositions in 
particular, against which we will have to 
evaluate the results of our study. The second 
question will be illustrated in section 3 where 
we present the main meanings of avec and focus 
on comitativity, which, being more abstract then 
the others, seems to point directly to the under-
specified set of constraints defining avec in 
general. In section 4 we briefly introduce a 
model inspired from Channel Theory (Barwise 
and Seligman, 1997) and we test it on the 
problematic data related to the meaning of 
comitativity. In section 5 we come back to the 
two remaining questions: we present a model for 
the whole semantic domain of avec by 
introducing the distinction among notions and 
values and discuss the inferential power of our 
model.  
Following Pinkal's distinction between 
sense and meaning (Pinkal, 1985, Poesio, 1996), 
we consider that senses tell us under which 
circumstances in the world the sentence is true 
or false, and that meanings assigned to an 
expression behaves as functions from contexts 
to senses.  
2 Polysemy and contextual variability 
The notion of polysemy implies that (i) a word 
has many different meanings and that (ii) these 
meanings are in a certain way related to one 
another. A major issue is then to explain the 
phrase "in a certain way". By virtue of what 
principles are they related? Is it by virtue of a 
                     July 2002, pp. 17-24.  Association for Computational Linguistics.
                 Disambiguation: Recent Successes and Future Directions, Philadelphia,
                             Proceedings of the SIGLEX/SENSEVAL Workshop on Word Sense
unique sense that, together with a rule system 
allows to generate all of them, or simply by 
virtue of family resemblances ?  
More fundamentally, the notion of 
polysemy poses the major question of the 
relation between language and thought. It is not 
only a matter of discovering whether a unique 
sense exists, but also to determine whether this 
sense is cognitively grounded, or in other terms, 
whether it is pre-linguistic or perception based.  
Cognitive linguists such as Wierzbicka 
(1996) claim that all meanings are derived from 
a cognitive primitive-based dictionary by a 
cognitive grammar. Under her account, language 
is grounded in thought and polysemy is a 
consequence of the instantiation of thought into 
language, the cognitive dictionary being the glue 
of all the meanings of a polysemous word.  
Wittgenstein (1953/1961) introduces in his 
Philosophical Investigations (§68) a radically 
different point of view. He argues that it is 
clearly impossible to define words by abstract 
meanings on the account of pure observation. 
Provocative, this affirmation poses the problem 
of the existence of rules and opens a series of 
other questions: do definitions of polysemous 
word exist? And what do these definitions look 
like? Is it possible for an abstract definition to 
coexist with local values not caught by this 
definition? What is their status then?  
According to these antithetic philosophical 
positions, prepositions have undergone different 
treatments. Jackendoff (1987), Wierzbicka 
(1996) and Brøndal (1950) consider that they 
instantiate primitive atomic notions and that 
their meanings can be metaphorically and 
metonymically calculated; some other authors 
such as Cadiot (1997) consider instead that 
prepositions denote different properties on a 
variable spectrum that cannot be reduced to an 
atomic sense. 
These positions are more often grounded in 
philosophical considerations rather than in 
empirical observations. Our approach is 
fundamentally inductive and is focused on the 
meaning of comitativity which seems to trace a 
privileged way to the abstract notion of avec. 
3 The privileged status of comitativity 
Let us first consider a classification of avec 
meanings.  
Syntax: Semantics 
NP1 avec 
NP2 
Part-Whole: Un homme avec un chapeau 
/ A man with a hat 
NP1 (VP 
avec NP2) 
Reciprocity: Jean habite avec Anne / John 
lives with Ann 
Instrumental: Jean enfonce les clous avec 
un marteau / John pushes nails with a 
hammer 
Comitativity: Jean marche avec Anne / 
John is walking with Ann 
 
 
 
NP1 VP 
avec NP2 
Manner: Jean accueille Anne avec la joie 
au cœur / John welcomes Ann joyfully 
NP1 VP 
avec N 
Manner: Jean accueille Anne avec joie / 
John welcomes Ann with joy 
NP1 VP 
avec NP2 S 
Influence: On a fait tout le voyage avec 
Pierre qui râlait sans arrêt / We did the 
entire trip with Peter who couldn’t stop 
complaining 
Avec NP2, 
S 
Thematic: Avec ce mauvais temps, il vaut 
mieux rester chez soi / By this bad 
weather, it is better to stay at home 
Table 1. Classification of avec meanings.  
The meaning of comitativity, as argued by 
Cadiot (1997), stays in the middle of the 
spectrum of the semantic domain of the 
preposition avec: it is less constrained than the 
meaning of reciprocity and more specific than 
the meaning thematic. Compared to 
instrumental and manner, it is characterized by a 
notion of symmetry as will be discussed at 
length in the next section. Its privileged status 
clearly emerges from the definition given for the 
first time by Guillaume (1919/1975) and 
adopted by Cadiot (ibid.) in a recent study. 
Guillaume (ibid.: 279) defines the comitativity 
as follows1: “ the preposition avec is an abstract 
image of parallelism: it expresses the relation 
holding between two entities that exist or act 
together, accomplish the same movements and 
follow the same directions. This image 
presupposes a certain equality … which is easily 
realized when the preposition links two nouns”. 
Cadiot (ibid.: 153) adds to this the notion of 
interaction: “avec creates … the conditions for 
an optimal interaction between two entities of 
the reality”.  
It is important to note that these definitions 
also apply to avec in general: according to these 
authors, they fit the meaning of comitativity and 
are modified into the other contextual 
possibilities enumerated in Table 1.  
                                                 
1 The translations are of the author of this paper.  
  
Three key notions emerge as fundamental 
from these citations: parallelism, interaction and 
finally association, which is strictly related to the 
first two. In the following section we show that the 
intuition grounding these definitions is correct, but 
that it is precisely the sense of association that 
needs an explanation: to define avec means to 
clearly formulate the constraints which need to be 
satisfied for two or more entities to be said 
"associated". We admit that the meaning of 
comitativity has a privileged status and focus on it.  
3.1 Cadiot's account of comitativity 
and problematic data 
On a phenomenological level, Cadiot (1997) 
analyzes the typical example of comitativity in 
(1), by a condition on symmetry (2) which 
implies a coordination (3): 
(1) Pierre marche avec Marie / Peter walks with 
Mary 
(2) a. If NP 1 VP avec NP2 then, NP2 VP avec 
NP1 
b. If “Pierre marche avec Marie”, then 
“Marie marche avec Pierre” 
(3) a. If NP1 VP avec NP2, then NP1 VP et NP2 GV 
b. If “Pierre marche avec Marie”, then 
“Pierre marche” and “Marie marche” 
These conditions properly describe the fact 
that in a situation where "Pierre walks with 
Mary" it is true that "Pierre walks", "Marie 
walks", "Pierre walks and Marie walks", "Marie 
walks with Pierre".  
Nevertheless, these conditions appear too 
vague: all the cases enumerated from (4) to (6) 
below are unproperly generated by (2) and (3).  
Consider the case of two animates who are 
both at the same time in the same place2.  
(4) a. Jean s’est retrouvé hier à la banque avec 
la voisine qu’il ne peut pas supporter / Tough he 
can’t stand her, John was at the bank with his 
neighbor  
b. (??)Jean est à Paris avec Chirac / John is 
in Paris with Chirac 
The condition (2) which properly generates 
(4a) does not explain why (4b) is difficult to 
interpret if Jean lives in Paris and Chirac is in 
Paris as the President of the Republic and there 
                                                 
2 The symbol (??) means that the sentence is 
interpretable under specific conditions.  
is no interaction between them. The spatio-
temporal association which seems sufficient for 
interpreting (4a) has to be reinforced by a 
stronger sense of interaction for (4b) to be 
interpretable.  
The condition in (2) also misses the 
difference between (5a) and (5b). Consider two 
inanimate entities:  
(5) a. Les verres sont dans le buffet avec les 
carafes / The glasses are in the cabinet with the 
pitchers 
b. (??)La porte est dans le salon avec la 
fenêtre / The door is in the living-room with the 
window 
The sentence (5b) cannot describe the 
relation existing between a window and a door 
of a living-room. For it to be interpretable, the 
door and the window have to be figured out as 
taken down.   
Finally, the condition in (2) misses the 
constraints related to the nature of the 
predicates: 
(6) a. Jean est gentil avec Marie / John is kind 
with Mary 
b. (??)Jean est triste avec Marie / John is sad 
with Mary 
The only possibility to interpret (6b) is that 
Mary has an influence on John’s sadness. The 
interpretation that John is sad, Mary is sad and 
that they are sad together at the same time, as 
foreseen by (2), is excluded.  
Of course these are all problematic data that 
a model of avec must explain. More abstractly, 
two major difficulties missed by (2) will have to 
be solved:  
(i) The spatio-temporal association 
problem. As discussed above (cf. (4a) / (4b)) the 
property of spatio-temporal location is not 
always sufficient to ensure the association of 
two entities.  
(ii) Regular association vs. accidental 
association. Consider two persons walking form 
point X to point Y. (7a) and (7b) describe this 
scene in a fundamentally different way: 
(7) a. Le passant A marche avec le passant B / 
Person A walks with person B 
b. Le passant A et le passant B marchent / 
Person A and person B walk 
(7a) means that the walk of the two persons 
is coordinated and that this coordination is not 
  
accidental. If one of the two persons turns 
around a corner, the other will do the same. This 
set of inferences is not enhanced by (7b), where 
et presents the coordination of the walks of the 
two persons as purely accidental.  
In the rest of the paper we will refer to (7a) 
as showing a togetherness effect. In the 
following section we introduce an intensional 
model which takes the mechanism of this effect 
into account.   
4 Togetherness effect 
The model we are about to present is an 
intentional model that takes into account the 
properties of the entities denoted by the NPs of 
the construction NP1 VP avec[+comitativity] NP2. 
This explanation contrasts with the extensional 
models that have been used to explain the 
togetherness effect (Lasersohn, 1998).  
Our representation serves the purpose of 
illustrating, by a concrete case, the speculative 
discussion in the next section. It can be 
questioned on many different formal aspects, 
and can be further elaborated or even differently 
expressed. Nevertheless the model is intuitive 
enough to ground the philosophical discussion 
underlying our case study.  
4.1 Shortcomings of extensional 
models 
Lasershon (ibid.) proposes a model of 
togetherness based on the notion of group:  
(8) Together: given the eventuality3 e, a 
property P and a group g, together is appropriate 
iff g ∈ P(e) and, for each part (proper or 
improper) e’ of e, if it exists x such that x ∈ 
P(e’), then P(e’) = P(e).  
Under this account, a group is thought of in 
terms of the minimal number of entities sharing 
a property in an event (or state) and all its parts. 
As such, this model fails to discard the cases of 
pure accidental association such as (7b): this 
notion of group can be applied to a scene where 
two (and only two) persons accidentally walk in 
the street from point A to point B and in all the 
parts (proper and improper) of this path. 
                                                 
3 Eventualities are spatio-temporal entities such as 
states and processes (Binnick, 1991).  
4.2 Channel theory and intensionality 
To explain the togetherness effect, we have 
developed a modal model inspired by Channel 
Theory of Barwise and Seligman (1997). We 
claim that the notion of togetherness is 
interpretable in terms of channel or the linkage 
of properties of the parts of a whole.  
A whole regulating and coordinating the 
internal behavior of its parts is a distributed 
system or channel. A model considering 
properties and types is intensional: knowing the 
object type, one can predict its behavior.  
A channel is thus defined by constraints that 
universally quantify over types (and not entities 
as in the extensional accounts), guaranteeing 
that the linkage of the properties of the parts (i) 
is regular (vs. accidental) for a given object type 
and (ii) take place within the structure of the 
distributed system.  
Technically, a channel is defined as the 
combination of two infomorphisms. The notion 
of classification grounds the definitions in (10) 
and (11).  
(9) Classification. A classification is a triple 
(Objets, Types, �), where Objets is a set of objects, 
Types a set of categories or types, and � a relation 
between Objets and Types. If o ∈ Objets and σ ∈ 
Types, o � σ means that the o is of type σ. 
(10) Infomorphism. An infomorphism is a pair of 
classifications (Objects1, Types1, �1) and (Objects2, 
Types2, �2) associated with two total functions f : 
Objects1 → Objects2 et g : Types2 → Types1 
such that, for o ∈ Objects1 and σ ∈ Types2 :  
1
12
2
12()()
TypesTypes
fObjectsObjects
g
foiffog����g
→
σσ
←
MM
MM
Channel. A channel is a set of infomorphisms 
sharing a common classification called the core 
of the channel. 
12
’’’()’(
123
)fgfg
fgWholePartPart
αββα→←
←→
MMM
MMM
�
��� 
  
We claim that avec signals the presence of a 
distributed system regulating the properties of 
the entities it links, in a regular way. In other 
terms, the kind of association introduced by 
avec can be represented by a channel. We can 
thus formulate the abstract constraints defining 
the behavior of avec:  
(11) Under-specified sense of avec. Avec 
signals that the state of affairs it refers to is 
structured in a way that can be described by a 
channel.  
4.3 Avec-comitativity 
We can now come back to the explanation of the 
problematic data introduced in the discussion of 
avec-comitativity and test our model. 
The definition (12) establishes a clear 
distinction between (7a) and (7b): avec in (7a) 
signals the existence of an overall walk 
coordinating the two separate walks of the 
persons involved, where et in (7b) does not. It 
follows that only (7a) enhances a scenario that 
can be described by a channel. This under-
specified definition is specified as follows into 
the meaning of comitativity:  
(12) Avec-comitativity. In a structure NP1 VP  
avec NP2, avec signals the existence of a 
channel verifying the following conditions:  
(i) it links the phases describing the events 
involving the entities denoted by NP1 and 
NP2, 
(ii) each of the phases involving one of the 
two entities implies that the phase 
involving the other entity expresses an 
actual or potential influence, 
(iii) no other phases than the ones described in 
the sentence can be evoked4.  
GN1GN1GN2GN2
()(
123
;
)
,,,,
f’g’f’g’
fg
αββα→←
←→φφφφ
MMM
MMM
�
��� 
The following set of constraints further 
specify this representation (the operators F, P 
                                                 
4 This condition can be expressed by the notion of 
aggregation (Kratzer, 1989): the association has to be 
possible within the system itself and not by the 
intervention of external events.  
and � have their usual meaning of past, future 
and possibility):  
 
For each , it exists  such that
()’()
()’()
For each , it exists  it exists
’()()
’()()
ggF
gP
ggF
gP
α∈Σβ∈Θ
α◊β
◊β
β∈Θα∈Σ
β◊α
β◊α
�
�
�g
�g
 
 
Let us analyze this definition:  
(i) Avec links the eventualities which 
involve the entities denoted by NP1 and NP25.  
(ii) The types describing these eventualities 
are phases i.e. descriptions including the 
possible previous and past developments of the 
actual eventuality (Penczek, 1995).  
Moreover, we have to note that the 
constraints characterizing the channel present 
modal types. This is licit within Channel Theory 
and is particularly useful for our purposes.  
We can conclude that the channel signaled 
by avec-comitativity links the present, past and 
future developments of the eventualities which 
involve the denotations of NP1 and NP2.  
4.3.1 Solution of the spatio-temporal 
juxtaposition problem 
Let us come back to the paradox of spatio-
temporal co-localisation. In most cases, avec is 
used when two entities interact. Cadiot (1997) 
considers this condition as necessary. 
Nevertheless some cases of pure spatio-temporal 
co-localisation are supported by avec, such as 
(4a). They can now be easily explained. 
Consider an example:  
(13) A son insu, Jean s'est retrouvé sur la 
montagne avec un ours / Without knowing it, 
John was on the mountain with a bear 
In the referred state of affairs there is no 
interaction between John and the bear. 
Nevertheless, according to our experience, we 
know that the particular location of a person and 
a bear being on a mountain can evolve toward 
an interaction by virtue of them being in the 
same place. It is because John and the bear are 
in the same place at the same time, that they 
could interact. According to (13), this scenario 
can be described by a channel that takes the 
                                                 
5 From now on, we will refer to the denotations of NP
1 
and NP2 by X and Y respectively.  
  
form of potential interaction and links their 
mutual positions.  
This is not the case for the spatio-temporal 
location of John and Chirac in (4b). In the case 
where they would meet, it would not be by 
virtue of their being both in Paris, at least not in 
a default context; an extra eventuality should 
intervene for John et Chirac to meet and this is 
not allowed by (13) nor by the definition of 
channel in (11). There is then no distributed 
system regulating their particular spatio-
temporal properties. 
The definition (13) also explains why in 
some other cases avec is impossible. Consider 
the window and the door of a living-room (5b): 
the properties of their spatio-temporal locations 
are not mutually regulated. This is why the 
interpretation of the sentence enhances a 
scenario in which someone has taken them down 
in the course of a remaking of the house, for 
instance.  
The same explanation holds for (6b). Avec 
forces an interpretation where the sadness (or 
another property) of Mary influences the 
sadness of John: avec signals a coordination of 
the properties of the entities it links. The 
definition in (12) forbids the interpretation that 
"Jean is sad, Marie is sad, they are sad 
independently from one another and they are sad 
in the same spatio-temporal location".  
5 A model for avec: notions, values, 
inferences 
Let us recall the argumentation we have been 
pursuing until now. Avec shows a very wide 
contextual variability and the challenge is to 
reconstruct a unique set of constraints 
explaining the coherence of its semantic 
spectrum. The meaning of comitativity has 
offered a privileged way toward this set defined 
in (12). This definition is quite abstract and is 
differently instantiated into the other contextual 
possibilities. The table 2 summarizes the 
definitions that can be given to each of the 
meanings listed in table 1.  
Association  
NOTIONS Influenceb Spatio-temporal 
taceg 
 
VALUES 
Comi
tativit
yd 
Influe
ncee 
Inter-
proposi
tionalz 
Part-
whole h 
Instrume
ntal / 
Mannerq 
Table 2. Model for avec semantic domain. 
This table suggests that the meanings of 
avec can be classified into two families: the 
spatio-temporal trace and influence. The 
observable meanings, or values, differently 
instantiate the abstract constraints that we call 
notions.  
NOTIONS 
� αTwo entities are thought of as acting (or taking 
place) within the same scene in such a way that a 
connection exists between them. 
� βWith connects phase by phase the two 
eventualities in which X and Y are involved 
without the intervention of eventualities other 
than the ones described in the sentence. 
� γThe spatio-temporal trace of X gives an access 
to the spatio-temporal trace of Y without the 
intervention of entities other than the ones 
referred to in the sentence. 
VALUES 
� δThe potential for the eventuality in which X (or 
Y) is involved to influence (or have influenced) 
the eventuality in which Y (or X) is involved. 
� εThe potential for the eventuality in which Y is 
involved to influence the way in which X 
controls the eventuality in which is involved. 
� ζThe potential for the eventuality described by 
the PP to influence the eventuality described in 
the main proposition. 
� ηThe spatio-temporal trace of Y is accessible 
from X, without the intervention of entities 
exteriors to the ones described in the sentence. 
� θThe description of the eventuality in which X is 
acting implies the entity denoted by Y in its 
same spatio-temporal trace. 
The comitativity meaning belongs to the 
influence family. The meanings instantiating the 
spatio-temporal trace notion can be described as 
signaling a channel as well, and do not have to 
be confused with spatio-temporal juxtaposition. 
The regulation of spatio-temporal traces can 
take different forms according to the possible 
meanings: instrumental, manner of part-whole 
relation.  
In the case of instrumental, the regulation of 
the spatio-temporal traces takes the form of a 
control of X over Y. In the following scenario, 
John "reads" the manual: 
(14) Jean apprend l'histoire avec un nouveau 
manuel / John learns history with a new manual 
For manner, the regulation takes the form of 
the relation source / feeling: John is the source 
of the joy:  
  
(15) Jean parle à Marie avec joie / John talks to 
Mary with joy 
Finally, for the part-whole relation, the 
regulation of the spatio-temporal traces of the 
entities linked by avec is inscribed at the 
referent level: 
(16) Une maison avec une terrasse / A house 
with a terrace 
5.1 Prediction and pragmatic inference 
Given this configuration of senses and 
meanings, we can now come back to the more 
general and speculative discussion. Avec shows 
a clear variability on the horizontal axis, and a 
complex organisation on the vertical axis. First 
of all, the distinction between notions and values 
roughly corresponds to the sense / meaning. The 
term notion, clearly implies that the definition 
has a cognitive reality and is probably learned 
and stored as such.  
On the vertical level there is a difference in 
nature between notions and values: notions can 
be considered as models for values. The overall 
abstract notion defined in (12) is a model for 
avec.  
The issue related to the predictive power of 
our model is still open.  
To affirm that a model is predictive means 
to recognize that it is possible to generate all and 
only the observed meanings by an appropriate 
rule system. In spite of the fact that we have 
reconstructed an under-specified and general 
model for avec, we must recognize that it is 
impossible by virtue of the definition in (12) to 
generate all (and only) its meanings.  
The main reason is philosophically evident: 
to generate such meanings one should 
previously know them and because we do not 
have an access to all the existing corpora, it is 
impossible to affirm that the model generates all 
the possible meanings. This argument has been 
well known since Wittgenstein (1953/1961).  
Nevertheless, we believe that we can find 
more than pure family resemblances. First of all, 
even if very abstract, the polysemous domain of 
prepositions seems well structured: the 
definitions of the notions that we have 
inductively found show that it is possible to 
identify with a certain precision the semantic 
domain of avec. Moreover, our model constrains 
the licit interpretations.  
It follows that in spite of being predictive, 
our model allows to formulate inferences.  
Consider the interface between semantics 
and pragmatics. On a semantic level we have 
admitted that avec creates a scenario that can be 
described in terms of a distributed system 
regularly linking the properties of two or more 
entities / events. On a pragmatic level we can 
conclude that when using avec, the speaker 
wants to signal the existence of such a 
correlation. The hearer will then infer that a 
regular association exists between the entities / 
events linked by avec in the sentence uttered by 
the speaker, and that such an association must be 
sought. Let us look at an example.  
(17) Avec les stylos dans le verre, la police 
donne des PV / With the pens in the glass the 
police give parking tickets 
Consider a situation where a speaker and a 
hearer trust each other, that is to say, a situation 
where pragmatic irrelevance is not a matter. If 
the speaker utters a sentence such as (18), the 
hearer will be looking for a non accidental 
association between the events (as kind of 
entities) associated by avec: namely the fact that 
the pens are in the glass and that the police give 
tickets. 
This is exactly what the model we have 
built for avec allows us to infer: the two events 
of the pens being in the glass and the police 
giving parking tickets are thought, said and 
interpreted as regularly (vs. non accidentally) 
associated.  
This is the case for many familiar examples, 
occurring in different syntactic environments:  
(18) Avec la pluie, je suis de mauvais humeur / 
With this rain, I am in a bad mood 
(19) J’ai mis les verres avec les carafes / I put 
the glasses with the pitchers 
This observation leads us to conclude that 
the model is expressive enough to be 
generalized and to constrain the interpretations 
of apparently very different meanings. A clearer 
formalization of these speculative conclusions 
on the contextual interpretation of under-
specified representations is currently under 
construction.  
 
Acknowledgments Many thanks to Patrick 
Saint-Dizier and Jacques Jayez for their careful 
readings and useful suggestions. 
  
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