Neuropsychologia 47 (2009) 104–111
Contents lists available at ScienceDirect
Neuropsychologia
journal homepage: www.elsevier.com/loca
Is it logical to count on quantifiers? Dissociable n
numerical and logical quantifiers

Vanessa Gr
a
Department o
b
Department o
articl
Article history:
Received 1 Ap
Received in re
Accepted 11 A
Available onlin
Keywords:
fMRI
Number
Semantic mem
Comprehensio
ral su
gnitu
f per
m tw
atien
tha
ork, b
gulat
ge to
evalu
1. Introduction
Numeric
use langua
number wo
such as “mo
matter of m
cepts of ma
faculties are
study, we a
whose conc
magnitude.
The pro
received in
Cantlon &
Henik, & Go
Although re
with simple

Portionso
Washington, D
2007.
∗
Correspon
the University
United States.
E-mail add
(Dehaene, Piazza, Pinel, & Cohen, 2003), some complex numerical
manipulations appear to depend in part on linguistic mediation, as
0028-3932/$ –
doi:10.1016/j.nal knowledge and language are intricately related. We
ge to refer to numerical concepts, including cardinal
rds (i.e. “one”) and also using certain quantifier terms,
st”. However, the exact nature of this relationship is a
uch debate. Most previous analyses investigating con-
gnitude have examined the extent to which language
necessary for precise numerical understanding. In this
dopt an alternate approach; that is, we examine words
eptual representation depends in part on knowledge of
cesses underlying numerical comprehension have
creased attention recently (Ansari & Dhital, 2006;
Brannon, 2006; Cohen Kadosh, Cohen Kadosh, Kaas,
ebel, 2007; Piazza, Pinel, Le Bihan, & Dehaene, 2007).
gions near the intraparietal sulcus (IPS) are associated
magnitude judgments andmathematical computation
f thisworkwerepresentedat theAcademyofAphasiaAnnualMeeting,
C, 2007 and the Society for Neuroscience Annual Meeting, San Diego,
ding author at: Department of Neurology, 3 West Gates, Hospital of
of Pennsylvania, 3400 Spruce Street, Philadelphia, PA 19104-4283,
Tel.: +1 215 662 3361; fax: +1 215 349 8464.
ress:mgrossma@mail.med.upenn.edu (M. Grossman).
suggestedby their relianceonperisylvian language regions (Baldo&
Dronkers, 2007; Dehaene et al., 2003). However, language abilities
are not necessary for all aspects of number meaning; other aspects
of number knowledge, such as the appreciation of quantity, ormag-
nitude comprehension, can be demonstrated in preverbal infants
and primates, who lack advanced language abilities (Cantlon &
Brannon, 2007; Xu, Spelke, & Goddard, 2005).
Quantifier terms are noun phrases which we believe may also
rely in part on a magnitude comprehension system. These terms
are common in daily speech, yet we know little about their neural
basis. A quantifier is a noun phrase that asserts a property from
a set and maps this to a truth-value (Clark & Grossman, 2007;
Frege & van Heijenoort, 2000). There are several distinct classes
of generalized quantifiers defined in formal linguistics (Keenan &
Stavi, 1986; van Benthem, 1986), and we focus on two in the cur-
rent study: cardinal and Aristotelian. Cardinal quantifiers, which
we refer to as “numerical quantifiers” from this point forward, are
based in part on knowledge of magnitude. Consider the sentence
“at least three scientists drink coffee.” The comprehension of this
sentence relies on the magnitude expressed by “three”. If only two
coffee-drinking scientists can be identified, then the statement is
false. This can be contrastedwithAristotelian quantifiers,whichwe
will refer to as “logical quantifiers”. These do not depend on quan-
tity. Logical quantifiers like “some” and “all” are based instead on
an elementary logic system that detects the presence of a unique
see front matter © 2008 Elsevier Ltd. All rights reserved.
europsychologia.2008.08.015Troiani
a
, Jonathan E. Peelle
a
, Robin Clark
b
, Murray
f Neurology, University of Pennsylvania School of Medicine, United States
f Linguistics, University of Pennsylvania, United States
e info
ril 2008
vised form 4 August 2008
ugust 2008
e 22 August 2008
ory
n
abstract
The present study examined the neu
like “at least three” which require ma
be understood using a simple form o
fiers with converging observations fro
behavioral and structural data from p
findings are consistent with the claim
parietal-dorsolateral prefrontal netw
rostral medial prefrontal-posterior cin
tribution of abstract number knowled
and the potential role of a logic-basedte/neuropsychologia
eural networks underlying
ossman
a,∗
bstrate of two classes of quantifiers: numerical quantifiers
de processing, and logical quantifiers like “some” which can
ceptual logic. We assessed these distinct classes of quanti-
o sources: functional imaging data from healthy adults, and
ts with corticobasal degeneration who have acalculia. Our
t numerical quantifier comprehension depends on a lateral
ut logical quantifier comprehension depends instead on a
e network. These observations emphasize the important con-
the meaning of numerical quantifiers in semantic memory
ation in the service of non-numerical quantifiers.
© 2008 Elsevier Ltd. All rights reserved.

V. Troiani et al. / Neuropsychologia 47 (2009) 104–111 105
feature. In the sentence “some scientists drink coffee,” evaluation
involves the detection of a single scientist drinking coffee; knowl-
edge regarding precise numerosity is not required to understand
this sentenc
tists drink t
scientist fal
ments betw
may be sup
Evidence
nitive syste
from the id
with each c
appear to d
conceptual
(Cipolotti, B
Lehericy, &
(Simon,Ma
nitude know
exact functi
been fully c
strated that
magnitude
“at least thr
nitude infor
(“three” in
information
may also de
support wo
whennume
&Watanabe
Logical q
mechanism
and suppor
fiers thus m
(rmPFC), w
dichotomou
Frith, & Bur
occurrence
an attention
involves sel
al., 2003, 20
ing “some”
and a dicho
fied targets
To asses
cal quantifi
serial array
simple state
tation for s
us to model
confounds p
about the s
potential vi
(Hubbard, P
logical quan
are blue.” O
least three,”
Converg
fiers was ob
brain activi
formed this
each class o
for logical q
quantifiers.
with focal
the neuroanatomic distribution of disease with voxel-based mor-
phometry (VBM) analyses of high resolution structural MRI scans.
Corticobasal degeneration (CBD) is a rare neurodegenerative con-
that
in d
004
othe
ed in
abo
expe
be w
ods
icipan
assess
munit
, S.D.
nded
ion kn
lso ex
mean
.0±26
d in th
ylvani
ist w
are o
expe
osing
f lite
auto
progre
and/o
b rigid
mined
, we c
volum
ean±
tion.
tocol
ania.
cedure
ects an
ions c
ture o
resen
ntial v
any ob
as use
d in a
r type
rs tha
rs at
ven an
dgme
ntly sh
ose of
witho
nsure
contra
tion d
trial
tures
of stim
ities
e the
ber l
d con
um o
trial in
layed
eral pr
edure
ent.e. This is equally true for statements such as “all scien-
ea,” in which identification of a single non-tea drinking
sifies the statement. These differences in task require-
een numerical and logical quantifiers suggest that they
ported by dissociable cognitive mechanisms.
consistent with the hypothesis that two distinct cog-
ms support separate classes of quantifiers would come
entification of a distinct neural network associated
lass of quantifier. As noted above, numerical quantifiers
epend on a quantity-based system to determine their
accuracy. Much evidence from patient observations
utterworth, & Denes, 1991; Cohen, Dehaene, Chochon,
Naccache, 2000; Halpern et al., 2004b) and fMRI studies
ngin,Cohen, LeBihan,&Dehaene, 2002)associatesmag-
ledge with the intraparietal sulcus (IPS). Although the
on of the IPS in assessing numerical information has not
larified, previous neuroimaging studies have demon-
this region responds in a ratio-dependent manner to
judgments. Thus, for a quantified statement containing
ee” tobe true, IPS is important for evaluationof themag-
mation present in this statement. Because the criterion
the example) must be kept in mind while perceptual
is evaluated, comprehension of numerical quantifiers
pend on dorsolateral prefrontal (dlPFC) brain regions to
rking memory. This demand might be particularly high
ric computations require serialmaintenance (Botvinick
, 2007).
uantifiers appear to require a simple decision-making
adapted to interpret relatively constrained alternatives
t of a selective attentional system. Logical quanti-
ay depend in part on rostral medial prefrontal cortex
hich plays a role in simple decision-making about
s events in the environment (Gilbert, Spengler, Simons,
gess, 2006) such as attending to and interpreting the
of exceptional events. This areamaywork togetherwith
al mechanism in posterior cingulate cortex (PCC) that
ective attention to visual-perceptual stimuli (Small et
05). To determine that a quantified statement contain-
is true, for example, the set of items must be attended,
tomous decision about seeing at least one of the speci-
must be made.
s the neural networks supporting numerical and logi-
ers, we designed a task in which participants viewed
s of familiar objects and then judged the accuracy of a
ment containing a quantifier. We used a serial presen-
everal reasons: this design most appropriately allows
the processes involved in quantifier meaning, without
resented by potential differences in making decisions
timuli. Additionally, a serial presentation minimizes
sual-spatial processing associated with parietal cortex
iazza, Pinel, & Dehaene, 2005). Statements included
tifiers like “some” and “all,” such as “some of the balls
ther statements featured numerical quantifiers like “at
such as “at least three of the flowers are red.”
ing evidence to support two distinct classes of quanti-
tained from two sources. First, we monitored regional
ty with BOLD fMRI while healthy young adults per-
task. We expected distinct patterns of activation for
f quantifiers. This included rmPFC and PCC activation
uantifiers, and IPS and dlPFC activation for numerical
Secondly, we collected behavioral data from patients
cortical neurodegenerative disease, and we examined
dition
results
et al., 2
we hyp
impair
correct
would
would
2. Meth
2.1. Part
We
nia com
(M=24.4
right-ha
medicat
We a
years;
tion=49
identifie
of Penns
neurolog
are unaw
although
in diagn
review o
our own
gradual
rigidity,
and lim
as deter
patients
who had
years; m
participa
to a pro
Pennsylv
2.2. Pro
Subj
proposit
color fea
serially p
thepote
ing of m
designw
presente
particula
quantifie
quantifie
tifiers (e
Parity ju
consiste
the purp
stimulus
ing. To e
directly
no activa
Each
tifier. Pic
number
numeros
minimiz
their num
presente
a maxim
an inter-
was disp
with sev
the proc
experimcauses parietal lobe disease (Murray et al., 2007). This
eficits on tasks assessing number knowledge (Halpern
a,b). Even though these patients do not have aphasia,
sized that numerical quantifiers would be selectively
these patients due to their parietal disease. If we are
ut the distinct nature of the two quantifier systems, we
ct that their comprehension of numerical quantifiers
orse than their comprehension of logical quantifiers.
ts
ed 14 healthy adult participants from the University of Pennsylva-
y in the fMRI study. Participants ranged in age from 20 to 28 years
=2.70) and had an average of 15.6 years of education. All were
native English speakers, in good health, and none were taking any
own to affect cognitive function or brain activity.
amined 13 patients diagnosed with CBD (mean± S.D. age =64.1±9.6
± S.D. education=14.3±3.3 years; mean± S.D. disease dura-
.1 months; mean± S.D. MMSE=22.9±3.3). All patients were
e outpatient clinic of the Department of Neurology at the University
aMedical Center. The clinical diagnosiswasmadeby a board-certified
ith expertise in the diagnosis of dementing conditions (MG). We
f any published consensus criteria for the clinical diagnosis of CBD,
rts have suggested specific clinical features that may be important
CBD (Riley & Lang, 2000). The criteria we developed, based on a
rature concerned with clinical–pathological diagnosis of CBD and
psy series (Murray et al., 2007), include the insidious onset and
ssion of: apraxia, cortical sensory deficit, gait instability and axial
r asymmetric extrapyramidal features such as myoclonus, dystonia,
ity, but little resting tremor. These patients did not have aphasia
by clinical evaluation. To assess regional cortical atrophy in these
ompared the gray matter images of a subset of CBD patients (n=8)
etric MRI scans to eight healthy adults (mean± S.D. age =70.9±6.0
S.D. education=16.0±2.1 years). Participants were paid for their
Informed consent was obtained from all individuals according
approved by the Institutional Review Board at the University of
d patients determined the accuracy of grammatically simple,written
ontaining a quantifier (e.g. “some of the balls are blue”) that probed a
f a familiar object (e.g. balls, flowers, cars, shirts, hats and stars) in a
ted visual array (Fig. 1). Stimuli were presented serially to minimize
isual-spatial confoundassociatedwith scanningasinglearrayconsist-
jects presented at one time (Hubbard et al., 2005). An event-related
d, and 144 statements containing one of six different quantifierswere
fixed pseudorandom order where no more than three stimuli of a
occurred consecutively in the stimulus series. Statements contained
t were either logical (some and all), or numerical (consisting of the
least three, more than two, even, and odd). We included parity quan-
d odd) because we wanted to test a range of numerical quantifiers.
nt tasks are frequently used in numerical discrimination tasks, and
ow automatic access to numerical magnitude (Dehaene, 1997). For
this experiment, parity quantifiers provided an additional numerical
ut an increase of working memory demands beyond those of count-
that these were an acceptable stimulus to include as numerical, we
sted the main effects of parity and cardinal quantifiers. There were
ifferences between these two subtypes.
began with a 3 s presentation of the proposition containing a quan-
of the objects were then displayed, one at a time, for 1.5 s each. The
uli in the array varied between 4, 6, and 8 objects. We used small
in the numerical quantifiers and small numbers of total objects to
risk that CBD patients would not be able to perform the task due to
imitations. Following the serial array, the initial statement was again
currently with a “yes or no” probe. This remained on the screen for
f 3-s or until subjects responded using a keypad. Between each trial,
terval of 3, 6, 9, or 12 s was used, during which a white, blank screen
. Participantswere trained ahead of time on the experimentalmethod
actice items, and all participants appeared to understand the task and
for indicating their judgments during the practice session prior to the