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Applied Ontology 0 (0) 1 1

IOS Press

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The Semantics of Extensive Quantities

within Geographic Information

Eric Top a,∗, Simon Scheider a, Haiqi Xu a, Enkhbold Nyamsuren aand Niels Steenbergen a

aDepartment of Human Geography and Planning, Utrecht University, the Netherlands

E-mail: e.j.top@uu.nl

Abstract. The next generation of Geographic Information Systems (GIS) is anticipated to automate some of the reasoning

required for spatial analysis. An important step in the development of such systems is to gain a better understanding and

corresponding modeling practice of when to apply arithmetic operations to quantities. The concept of extensivity plays an

essential role in determining when quantities can be aggregated by summing them, and when this is not possible. This is of

particular importance to geographic information systems, which serve to quantify phenomena across space and time. However,

currently, multiple contrasting deﬁnitions of extensivity exist, and none of these sufﬁce for handling the different practical cases

occurring in geographic information. As a result, analysts predominantly rely on intuition and ad hoc reasoning to determine

whether two quantities are additive. In this paper, we present a novel approach to formalizing the concept of extensivity. Though

our notion as such is not restricted to quantiﬁcations occurring within geographic information, it is particularly useful for this

purpose. Following the idea of spatio-temporal controls by Sinton, we deﬁne extensivity as a property of measurements of

quantities with respect to a controlling quantity, such that a sum of the latter implies a sum of the former. In our algebraic

deﬁnition of amounts and other quantities, we do away with some of the constraints that limit the usability of older approaches.

By treating extensivity as a relation between amounts and other types of quantities, our deﬁnition offers the ﬂexibility to relate a

quantity to many domains of interest. We show how this new notion of extensivity can be used to classify the kinds of amounts

in various examples of geographic information.

Keywords: Extensive quantities, Deﬁnition, Geocomputation, Semantic labeling of geodata

1. Introduction

An important distinction in geographic analysis is between those quantities that can and those that

cannot be summed during spatial aggregation. These are known as, respectively, extensive and intensive

quantities. Human analysts can intuitively tell how a quantity should be processed when two regions are

merged. Two temperature values of two spatial regions, for example, should not be summed, although

they may be treated as a weighted sum when the regions are aggregated. In geographic information

systems (GIS), however, the values may be represented by the same concrete data types, and thus cannot

be systematically distinguished. Current GIS lack a method for automating aggregations because we lack

a theory of extensivity that can tell us under which circumstances we can sum up quantities in space,

time, and other kinds of domains.

One fruitful way to capture extensivity is in terms of a relation between different domains of measure-

ment (Scheider and Huisjes, 2019). This notion of extensivity entails that quantities can be aggregated

if they share domains of measurement by which they can be controlled and measured in a coordinated

manner. Controlling quantities need to be separated from each other, and both controlling and controlled

*Corresponding author. E-mail: e.j.top@uu.nl.

1570-5838/$35.00 © 0 – IOS Press and the authors. All rights reserved

2E.J. Top et al. / The Semantics of Extensive Quantities

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quantities need to be additive in a way that preserves sums. For example, the population of Europe can

be aggregated with the population of Africa, as both populations are part of the world population, but not

with the summed GDP of Africa, which does not share the same measurement domain. Here, the spatial

administrative units are controlling and population counts are measured. However, Europe’s population

should also not be aggregated with the population of Utrecht, because even though they share the same

measurement domains, Utrecht is already a part of Europe. While such observations may seem intuitive,

the sciences still lack a formalization of these kinds of considerations. Most existing deﬁnitions of ex-

tensivity that reached prominence are either too restrictive or too vague, leaving room for inadequate

interpretation (See section 2.1). Moreover, existing deﬁnitions of extensivity seem to misconceptualize

its relational character. For over a century, the prevalent idea was that extensivity is a ﬁxed property of

scales originating from the existence of a sum operator or the way they were derived from fundamental

measurement units. However, as our examples about population and temperature illustrate, though the

underlying measurement scales come with sum operations in all cases, it is not always meaningful to

sum up quantities when merging regions. Furthermore, in this article, we make the case for the view

that all quantities can be extensive with respect to some and intensive with respect to other quantities.

For example, temperature, which is regularly used as an example of an intensive quantity, turns out to

be extensive with respect to thermal energy. A measurement value of temperature can be obtained by

dividing a value of thermal energy by the product of mass and heat capacity. Imagine the temperature of

a heating system is measured before and after an amount of energy is added, and assume that all other

quantities are held constant1. When the mass and heat capacity are held constant, increases in thermal

energy translate into homomorphic increases in temperature.

A concise yet ﬂexible deﬁnition of extensivity would enable determining whether spatial arithmetic

is applicable or not based on classifying quantities accordingly. In previous work, we (Scheider and

Huisjes, 2019) have illustrated the merit of spatial extensivity in the context of geographic information

and mapping and managed to automatically distinguish extensive from intensive quantities with high

accuracy. However, though this work forms a basis for the current article, it was never formalized and

does not account for quantities that are additive in domains other than space, such as e.g. time. Also, if

we recognize there are multiple dimensions of extensivity, new ways to categorize quantities emerge.

A water ﬂow accumulation is extensive in space and time, the cost of a stay at a hotel is extensive in

time and some monetary currency, and the cost of rental cars is extensive in time (i.e. the duration of

renting), space (i.e. the amount of kilometers driven) and the amount of cars (i.e. renting two cars is more

expensive than one). Extensivity offers a new semantic dimension by which data can be discovered and

processed. A deﬁnition of extensivity would therefore also contribute to a data-driven science (Hey et al.,

2009; Gahegan, 2020) by determining which arithmetic operations can be applied to available quantities.

In this article, we suggest a ﬁrst-order formalization of quantity domains as a basis for a higher-

order, relational deﬁnition of extensivity using quantities as controls and measures. We then demonstrate

how this deﬁnition allows us to deﬁne various subclasses of extensive measurement across geographic

information examples in terms of an OWL2pattern with subsumption reasoning. Though our theory as

such is not restricted to geographic quantities, the concepts of control and measure on which it is based

are central for geographic information, as explained below. Our contribution is therefore threefold and

provides answers to the following questions:

1For convenience, also assume the heating system perfectly retains all energy (i.e. there is no loss of energy over time).

2Web Ontology Language, see e.g. Hitzler et al. (2009)

E.J. Top et al. / The Semantics of Extensive Quantities 3

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•In which way can extensivity be formalized in terms of measurement, control and summation of

quantities? Which classes of quantity domains need to be distinguished for this purpose?

•How can extensive quantities be measured across time, space and content, within the context of

geographic information?

•How can extensive geographic measurements be systematically categorized according to different

kinds of controls and measures, and how can this be used to automatically classify map examples?

Our answers contribute important formal distinctions to measurement ontology which are currently

lacking. While measurement theory formalizes levels of measurement scales in terms of the operations

they preserve on particular quantity domains (Suppes and Zinnes, 1962), ontologies like DOLCE (Ma-

solo et al., 2003), or, more recently, the FOUnt ontologies (Aameri et al., 2020), relate the underlying

quantity domains to each other and ontological ”background” phenomena like endurants, perdurants,

and their properties. However, so far, we do not know of any attempt at formalizing the notion of ex-

tensivity as a relational concept, and in the context of geographic information. Furthermore, while the

current ontological accounts provide useful insights and models about concepts of measurement, they

also come with certain restrictions that make it difﬁcult to account for the notion of extensivity.

First of all, most existing measurement ontologies lack the formal depth to specify the distinctions

needed for capturing the notion of extensivity. For example, lightweight ontologies such as the OM-

ontology (Rijgersberg et al., 2013) lack a rigorous formalization of the underlying concepts and are

instead primarily focused on terminological conventions (Balazs, 2008) in speciﬁc application areas

(Steinberg et al., 2016). Furthermore, foundational ontologies that would provide the formal depth come

with certain problematic assumptions and ontological commitments. For one, even though there are

notions that resemble our notion of amount in both DOLCE and the FOUnt ontologies, they are largely

centered around the idea of ‘physical matter’ or ‘stuff’ constituting endurants. Examples would be the

amount of clay constituting a statue or the amount of wine in a bottle, which are modeled as phenomena

that can be considered snapshots in time. However, for geographic information, we need to consider

the possibility of forming amounts across time as well as space. For example, quantiﬁcations of water

running through a waterfall (Galton and Mizoguchi, 2009) or of trafﬁc ﬂowing into a city presuppose

amounts in time as well as space. Second, even if including both endurants and perdurants as bearers

of amounts, the notion of extensivity requires also a degree of arbitrariness of forming amounts, which

stands in apparent contrast to the simple unity criteria for wholes within the boundaries of objects and

events that are underlying DOLCE (Guarino et al., 2000). Consider the amount of population living

close to a border or between two cities, the amount of water ﬂowing through the waterfall in between

two events, or the amount of space covered by an arbitrary circle around some point. These are relevant

geographic examples that illustrate that the unity criteria for amounts go beyond the constitution of

particular objects or events within the limits of their boundaries. Of course, we can always create objects

with unity for arbitrary portions of amounts (cf. Guizzardi (2010)), however, this appears redundant in

light of a theory of extensive amounts.

The extensivity concept suggested in this article accounts for some of this ﬂexibility in terms of a

generalized notion of amounts that is used to control other measurable quantities, following earlier ideas

of Sinton (1978). With this approach, arbitrary divisions of amount portions can control arbitrary quan-

tities, not only by way of objects. This closely corresponds to the way how quantiﬁcation is done in a

GIS (cf. Chrisman (2002)). To this end, we formalize two classes of quantity domains (amounts and

magnitudes) which are used to deﬁne extensivity on a higher level and then introduce a simple design

pattern that can be used to classify various examples of extensive measurement across time and space

within geographic information.

4E.J. Top et al. / The Semantics of Extensive Quantities

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The rest of the paper is organized as follows. First, we review what is known about extensivity, quanti-

ties and measurement. Second, we reinterpret Sinton (1978)’s three roles of measurement (i.e. measure,

control, constant) and show in an informal manner how they can be used to specify extensivity relations.

Third, we present a formalized algebraic theory of extensivity as a relation between a measure and one

or more controls, including automatic proofs of theorems. Fourth, we translate this basic theory into a

lightweight OWL pattern adding classes speciﬁc for geographic information. We then propose twelve

categories of measurement of extensive quantities in the context of geographic information and show

how extensivity classes can be automatically inferred. Finally, we shortly discuss the implications of our

ﬁndings and conclude by answering the posed research questions.

2. Extensivity, quantities, and measurement

We start with reviewing existing literature on extensivity and quantities and scrutinize the underlying

approaches for our purpose. Furthermore, we critically examine Sinton’s notion of controlled measure-

ment, and discuss how it can be exploited for our purpose.

2.1. Extensivity and intensivity

The concept of extensivity originates from the ﬁelds of Physics and Chemistry where it is used to

describe the mathematical nature of properties. Its introduction and axiomatization can be accredited to

scholars in the ﬁrst half of the twentieth century (Hölder, 1901; Tolman, 1917; Campbell, 1920). Tolman

(1917) envisions extensivity as a way to describe phenomena whose measures are naturally additive. Of

all phenomena he identiﬁes only ﬁve as extensive in this sense, namely length, time interval, mass,

electric charge, and entropy. For a contemporary deﬁnition of extensivity, scholars often refer to the

green book of the International Union of Pure and Applied Chemistry (IUPAC), which describes an

extensive quantity as "a quantity that is additive for independent, non-interacting subsystems" (Cohen

et al., 2007). In practice, there seems to be an informal consensus that only properties like volume or mass

are considered extensive. Even within this consensus, disagreement exists about what physical properties

extensivity depends on. A number of papers from Physics and Chemistry try to address the confusion

surrounding the concept (Redlich, 1970; Canagaratna, 1992; Mannaerts, 2014). Mannaerts (2014) ﬁnds

that the expressions ’extensive quantity’ and ’extensive property’ are used interchangeably — He favours

the use of the term ’extensive quantity’ — and that some use additivity to deﬁne extensivity (i.e. the

sizes of two quantities can be added up during aggregation) while others use proportionality (i.e. a

quantity inextricably changes relative to changes of another quantity). Some scholars limit extensivity

to a relation of properties with respect to mass, while others relate them to the amount of substance

or volume (Mannaerts, 2014). Restricting extensivity to a speciﬁc kind of physical substance deviates

considerably from the original theory (Tolman, 1917), which holds that properties may be extensive

also with respect to time or entropy. Not only do scholars consider different properties as the source of

extensivity, they also disagree on the mechanisms of extensivity itself.

The concept of an extensive quantity is opposed to that of an intensive quantity, which has been deﬁned

as "a quantity that is independent of the extent of the system" (Cohen et al., 2007). Tolman (1917) argues

that, except for his ﬁve fundamental quantities, all quantities are intensive, because they are in some way

derived from the ﬁve fundamental quantities. A speed, for example, is found by dividing a length (i.e. the

distance) by a time interval. Some scholars hold that not all quantities are either extensive or intensive.

E.J. Top et al. / The Semantics of Extensive Quantities 5

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They argue that some quantities are expressed as conjugates (Alberty, 1997) or composites, which have

characteristics of both.

2.2. Quantities

Quantities are described as "...that by which a thing is said to be large or small, or to have part outside

of part, or to be divisible into parts" (Kocourek, 2018). Speciﬁcations of quantities are frequently present

in spoken language (Talmy, 1978). For example, the sentence ’The ﬂock of birds ﬂew over the wide

river’ not only speciﬁes two different entities (i.e. ’birds’ and ’river’), but also details their quantities

(i.e. ’ﬂock’ and ’wide’) and their interrelation (i.e. ’over’).

From a semantic viewpoint, quantities should be distinguished from numbers, which are mathematical

objects for representing measurement results, and measurement units, which indicate the measurement

system a quantity is measured in 3. In measurement theory, it is common to classify measurement sys-

tems using measurement levels, which range from nominal through ordinal and interval to ratio (Stevens

et al., 1946). These levels encode increasing amounts of information of a quantity, by preserving opera-

tions for class membership, order, relative position and absolute effect of a quantity (Suppes and Zinnes,

1962). Chrisman (1998) proposed to extend these levels with counts, degrees of class membership, cycli-

cal ratio, derived ratio, and immutable absolute measures, like probability.

Quantities can be negative and can be on a linear scale of measurement. For example, walking back-

wards for twenty meters can be seen as a negative quantity of forward movement associated with the

number -20 and the unit ’meters’. The term magnitude, also called impact or size, is used to measure a

quantity on a linear scale. Scholars sometimes distinguish multitudes from magnitudes (Lachmair et al.,

2018). Shortly put, multitudes refer to collections of discrete entities (e.g. a collection of cars), while

magnitudes capture linearly order-able phenomena (e.g. the length of a road). Plewe (2019) in addition

refers to continuously divisible quantities as ’geographic masses’ and illustrates the relevance of this

concept for Geography. Our approach (see below) can be used to make these notions more precise, by

formalizing what extensive quantities are in general, and how they are controlled in geographic infor-

mation.

Information about the extensivity of a quantity is closely related to its part-whole relations. Such

relations are commonly considered homeomerous with respect to its parts, meaning that all parts are

of the same kind of quantity as the whole (Gerstl and Pribbenow, 1993). For instance, sectioning a

portion of water results in sub-portions of water. According to Guizzardi (2010), homeomerous part-

whole relations can be modelled as maximally self-contained mereological sums (i.e. aggregations of

the subquantities) or by means of containment (e.g. a bottle of water). This approach implies that parts

of a quantity, also referred to as pieces (Lowe, 1998), are only instantiated if there is a need. For example,

a body of water may be subdivided into its parts to identify sweet water and salt water if necessary, but

this is not required for capturing the water concept. Guizzardi’s mereological approach also works for

universal properties and classes. For example, a car is a member of the collection of all cars (i.e. the

class of ’cars’), and the mass of said car is a part of the set of all mass in the universe (i.e. the ’mass’

property). The DOLCE ontology makes use of this principle of extensionality (Masolo et al., 2003;

Gangemi et al., 2001). Recently, work in the context of the FOUnt ontologies (Aameri et al., 2020) has

shown that formally adequate models of physical quantities need to incorporate formal relations between

property bearers, mereologies in different quantity dimensions (Ru and Gruninger, 2017), and quantities

3Different measurement systems (or reference systems) (Chrisman, 2002) can represent the same kind of quantity. For ex-

ample, the meter scale and the feet scale both represent the same quantity of lengths.

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represented by measurement scales. Going beyond existing ontologies of units of measure (Rijgersberg

et al., 2013), the deﬁnition of formal properties of quantities thus requires further concepts.

For extensivity, mereological relations are essential because they specify whether two quantities are

distinct, whether and how much they overlap and whether one quantity contains another. For example, a

university may host multiple lectures at once, meaning they share the same quantity of time. Summing

the total time of the lectures may indicate how long it would hypothetically take to attend all lectures

(e.g. 400 hours), but this does not correspond to the extent of time that is actually occupied by these

lectures (e.g. 3 hours). If two lectures with a duration of 2 hours each overlap for 1 hour, they together

occupy 3 hours in time. Claramunt and Jiang (2001) show that such relations are not limited to space or

time, but also exist between conjunctions of both.

2.3. Measurement of quantities

Sinton (1978) is well-known for his idea that the measurement of spatial information requires attribute

information about the space, time and theme components of the recording. Sinton argues that during any

measurement each one of these three components ﬁlls the role of the constant, the control or the measure:

•The constant component, also referred to as the support or the ﬁxed component, does not change

at any point in the measurement process.

•The control component is allowed to vary over its measurement scale at the observer’s discretion.

•The measure component is observed and its variation with respect to the control is recorded.

Take the example of a precipitation measurement. Precipitation is commonly measured with a rain

gauge. This rain gauge ﬁxes the spatial extent of the precipitation measure, e.g., to 1 dm². The amount

of water falling into the rain gauge is then measured in mm and converted to kg or liters over a variable

amount of time, e.g. an hour or a day (Chrisman, 2002). With an established constant (i.e. space) and

control (i.e. time), it is possible to measure an amount of rainfall in mm. Sinton’s work contains two

important messages: 1) a measurement of a phenomenon always requires other variables to be controlled,

and 2) geographic information always contains a combination of spatial information, obtained through

the measurement of locations and regions, temporal information, obtained through the measurement of

the progress of time, and thematic information, obtained through measuring some content of spatial or

temporal regions.

Chrisman (2002) argues that apart from space, time and theme, there is another kind of control, namely

control by relationship. For example, the measurement of a ﬂow of export products from countries to

one another ﬁrst requires establishing a relation between the countries (in the sense of a spatial network,

cf. (Kuhn, 2012)). Although this is a relevant case of extensivity (Scheider and de Jong, 2022), we leave

the study of network-controlled extensive quantities for future work.

The roles of measure, control and constant are essential for our purpose, because they aptly capture

how quantities can play different roles in deﬁning extensivity. Measured quantities are extensive if they

are controlled in a particular way by other quantities. However, Sinton’s idea requires some scrutiny

before it can be applied to quantities. For one, the ﬁllers of roles are by no means restricted to the three

components of geographic information. Many measurements ignore one or more of the components. For

example, when measuring the duration of a given lecture, there is no need to take the size of the lecture

room or the didactic ability of the lecturer into account. In fact, only the time interval at which the lecture

happens is required as a control to measure duration. Note also that in this example, the time component

appears as both measure and control at once. It is clear therefore that the components space, time and

E.J. Top et al. / The Semantics of Extensive Quantities 7

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theme are much too coarse to distinguish the relevant quantities, and thus for capturing extensivity. We

therefore adopt three alterations of Sinton’s idea. Firstly, we interpret Sinton’s components as classes of

quantities which might play a role or not in a given measurement. We thus allow for arbitrary combina-

tions of quantities ﬁlling the roles in a single measurement. Secondly, we assume that quantities exert no

inﬂuence on measurement (i.e. are kept constant) unless speciﬁed otherwise. This prevents the need for

explicitly ﬁlling the constant role. And third, we formally distinguish subclasses of quantity domains to

account for extensive measurements that can be made on a single one of Sinton’s dimensions.

3. A formal theory of extensive quantities

In the following, when we talk about quantities, we deviate from certain terminological habits in mea-

surement theory and ontology of measurement, simply because we believe our usage is closer to the

common understanding of a term. First, when we speak about a quantity, we mean an individual value

of measurement, such as the value represented by 15 kg. This is close to everyday usage, such as in

”the quantity of ﬂour used to make this bread”. Correspondingly, we use the term quantity domain to

talk about all elements of a domain of measurement, such as the kilogram scale (Probst, 2008). Second,

we do not assume quantities are necessarily on linear measurement scales. In the example above, the

quantity of ﬂour is not the same as its value in kilograms, though it can be measured on the kilogram

scale. This requires us to distinguish different kinds of quantity domains. In the following, we introduce

a formal theory in First Order Logic (FOL) about quantity domains, and measurement functions as map-

pings between these. FOL is sufﬁcient to reason about a single quantity domain, but strictly speaking,

we go beyond FOL when quantifying over different domains. Free variables in propositions are implic-

itly all-quantiﬁed over a quantity domain. Axiom sets of all (sub)theories are provably consistent, and

all theorems were automatically proven based on resolution using Prover94. The scripts are available

online56 . To get an overview of the following deﬁnitions, Table 1 provides a preliminary summary and

exempliﬁcation of the main formal concepts that we introduce below. Each of the concepts in the table

is also explained in the text.

3.1. Quantities, amounts and magnitudes

Certain kinds of quantities can be added up to or removed from each other, resulting in a new quantity

of which original quantities are parts. For example, a quantity of people can be added up to another

quantity of people to form a total sum of people, and the original quantities are parts of the whole.

We call quantities that can be added up in this way amounts. In our theory, this means that amounts

can be summed, be subtracted from and be part of each other. In the following, we will motivate and

illustrate the axioms with examples of amounts of space (E.g., spatial regions), amounts of time (E.g.,

time intervals), as well as amounts of matter and amounts of objects.7

4https://www.cs.unm.edu/~mccune/prover9/

5Amount theory: http://geographicknowledge.de/vocab/quantity_amount.txt

6Magnitude theory: http://geographicknowledge.de/vocab/quantity_magnitude.txt

7In the following, we use the terms amounts of space and (spatial) regions and the terms amounts of time and (time) intervals

interchangeably.

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Table 1

Preliminary summary of the main concepts

Concept Description Examples

Quantity An individual value of a measurement that can be

used to form sums.

A spatial region or its size or its proportion with re-

spect to another region’s size.

Amount An extensional mereological quantity whose domain

forms a lattice.

A spatial region (=amount of space).

Magnitude A linearly, monotonically ordered quantity. The size of a spatial region or its proportion with

respect to another region’s size.

Archimedean

magnitude

A magnitude whose domain forms a vector space

(can be summed and subtracted, but not multiplied)

The size of a spatial region.

Proportional

magnitude

A magnitude whose domain forms a mathematical

ﬁeld (allowing for products and ratios)

The proportion of a region’s size with respect to an-

other region’s size.

Quantity

domain

A set of quantities together with operations on them

(forming an algebra)

The set of all regions and the set of all region sizes

are domains of quantities of space.

Control The role of a quantity domain as a domain of mea-

surement, i.e., a domain of a measurement function.

When measuring the amount of population within a

region, the amount of space of that region is a con-

trol.

Measure The role of a quantity domain as a range of measure-

ment, i.e., a co-domain of a measurement function.

When measuring the amount of space occupied by

an amount of people, the amount of space is a mea-

sure.

3.1.1. Theory of amounts (extensional mereological quantities forming a Boolean lattice)

The parthood relations of amounts are captured by mereological axioms. We assume an amount do-

main8is a set with algebraic operations that satisfy the following partially ordered algebra:

Axiom 1. Partial order of parthood

x⊆xReﬂexivity

(x⊆y∧x⊇y) =⇒x=yAntisymmetry

(x⊆y∧y⊆z) =⇒x⊆zTransitivity

For example, if two amounts of sand are part of each other, they are identical, and parts of parts of

an amount of sand are also parts of the former amount of sand. Based on this, we deﬁne the following

predicates:

Deﬁnition 1. Strict order and overlap

x⊂y⇐⇒ (x⊆y∧ ¬(y⊆x)) Strict order

O(x,y)⇐⇒ ∃z(0 ⊂z∧z⊆x∧z⊆y)Overlap

y⊃x⇐⇒ x⊂yStrictly greater than

y⊇x⇐⇒ x⊆yGreater than or equal

8Many authors speak of dimensions of measurement rather than domains. We use the term ’domain’, because we want to

prevent the assumption of linearly ordered elements.

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Strictly ordered amounts are not identical, meaning one is a proper part of the other, and overlapping

amounts have a common part that is not empty (Casati et al., 1999).

Axiom 2. Sums and differences

x+y=y+xCommutativity

(x+y) + z=x+ (y+z)Associativity

x+ 0 = xIdentity +

x\x= 0 Inverse

y\0 = yIdentity Minus

Axiom 2 introduces operations for adding and subtracting amounts. Note the identity (empty) element

0 which can be added without changing anything and which results from subtractions of amounts from

themselves. To give some motivation for these axioms, it is apparently irrelevant in which order we sum

up amounts of sand. Furthermore, if we add/subtract no sand to/from an amount of sand, the amount is

left unchanged. Furthermore, if we remove the entire amount of sand, nothing is left over. This apparently

applies also in the case of spatial regions as well as time intervals.

In addition, we introduce a product operation for amounts, which is interpreted in terms of an intersec-

tion of two amounts. Intersection distributes over sums of amounts, and 1 is an identity element which

corresponds to the largest (supremum) amount:

Axiom 3. Products

x∗y=y∗xProduct Commutativity

(x∗y)∗z=x∗(y∗z)Product Associativity

x∗1 = xProduct Identity

x∗0 = 0 Product Neutrality

x∗(y+z)=(x∗y)+(x∗z)Distributivity

Intersection means that, for instance, the intervals of 25 to 29 minutes and 28 to 31 minutes intersect

in the range of 28 to 29 minutes. It apparently does not matter in which order we perform this intersec-

tion. Furthermore, while intersecting a time interval with the zero interval results in the zero interval,

intersecting it with the largest interval leaves it unchanged. Finally, intersecting intervals with interval

sums is like intersecting with each summand ﬁrst and then summing up the result.

The algebra so far forms a ground mereology with sums, differences and products9. It is so far not

in any way different from ordinary algebra. However, the mereology of amounts comes with further

characteristics that differ substantially from ordinary algebra and are more similar to the algebra of

sets. First, the largest (supremum) and smallest (inﬁmum) amounts and their complements interact with

addition and intersection in speciﬁc ways:

9More precisely, the partial order axioms form a ground mereology. The following lattice axioms establish extensionality

and introduce a closure principle, see Casati et al. (1999).

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Axiom 4. Supremum, Inﬁmum, additive distributivity and complements

x\1 = 0 Inﬁmum

x+ 1 = 1 Supremum

x+ (y∗z)=(x+y)∗(x+z)Additive distributivity

x+−x= 1 Additive complement

x∗ −x= 0 Product complement

−(−x) = xInvolution

−(x+y) = −x∗ −ydeMorgan 1

−(x∗y) = −x+−ydeMorgan 2

The zero (inﬁmum) amount can be obtained by subtracting the largest (supremum) amount from any

other amount, and adding the largest amount to any other amount returns the largest amount. Also, addi-

tion apparently distributes over products. For example, adding an amount of space to an intersection of

two regions is the same as adding it to each region separately and then intersecting the result. Further-

more, adding an amount of space to its complement generates the largest region, and intersecting both

produces the empty region. For example, the complement of the region where it rains is the region where

it does not rain. Involution means that it should be the case that the complement of the latter region is

the region where it rains, i.e., double complements lead back to the same region. The deMorgan axioms

do the same for sums and products.

In addition, amounts are not linearly ordered and instead form a Boolean lattice in the mathematical

sense, similar to the one depicted in the Hasse diagram in Fig. 1. To express this, we require the existence

of joins and meets10, as well as a non-total partonomy:

Axiom 5. Lattice axioms

x⊆x+yExistence of joins

x⊆z∧y⊆z=⇒x+y⊆zExistence of joins 2

x∗y⊆xExistence of meets

z⊆x∧z⊆y=⇒z⊆x∗yExistence of meets 2

∃x,y(¬(x⊆y∨y⊆x)) Non totality

To illustrate the existence of joins, it is apparent that a spatial region is always part of its sum with

another spatial region, and the sum of any parts of a spatial region is always part of that spatial region.

The existence of meets implies analogously that intersections of regions are always parts of those re-

gions, and a part of two regions is also part of their intersection. Note that these lattice properties may

10The following lattice and relative complement axioms are analogous to the algebra of sets (inclusion axioms and relative

complements).

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Fig. 1. Hasse diagram of the power set of a set of three elements. Set {x} is independent of the set {y,z}.

not apply to other spatial concepts that are not amounts.11 Finally, there always exist amounts that are

autonomous in the sense that they are not a part of each other. The last part of the axiom gives rise to

amount hierarchies that are independent from each other. For example, in a map there are always two

spatial regions that are not part of each other.

Axioms 1-5 are logically equivalent to a Boolean lattice (with at least two non-ordered elements), and

thus our theory bears similarity to the mereology of the FOUnt ontologies (Aameri et al., 2020). This can

be seen by the fact that idempotence and absorption laws become provable theorems, which, together

with axioms 1,2,3,4, constitute a standard axiomatization (Padmanabhan and Rudeanu, 2008):

Theorem 1.

x+ (x∗z) = xAbsorption 1

x∗(x+z) = xAbsorption 2

x+x=xAdditive idempotence

x∗x=xProduct idempotence

It can now be proven that if you sum up two amounts, where one is part of the other, this will always

generate the greater one of the two as a result of the operation. From this, the reﬂexivity of sums follows,

which is in apparent contrast to the number line12, and similarly for products. It follows also that the

empty amount 0is part of every other amount, and that a non-zero product of two amounts makes these

amounts overlap. Furthermore, a well known fact of algebra is provable, namely translation invariance:

i.e. adding an amount to two amounts that are part of each other preserves parthood:

11For example, the pair of two islands of Japan is not itself an island, but the pair of two amounts of islands is an amount of

islands. Furthermore, when we talk about spatial regions we do not make any assumptions about the connectedness of spatial

regions, and thus two unconnected regions can form a region.

12This would mean e.g. that 4+4 = 4

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Theorem 2.

(x⊆y) =⇒x+y=yReﬂexivity of sums

x⊆y=⇒x∗y=xReﬂexivity of products

0⊆xEmpty amount

(x∗y=z∧0<z) =⇒O(x,y)Product overlaps

(x⊆y) =⇒(x+z⊆y+z)Translation invariance

Finally, subtractions of amounts can be deﬁned simply as the intersection of an amount with the

complement of its intersection with the amount that is to be subtracted:

Axiom 6. Amount differences

x\y=x∗(−(y∗x)) Def subtraction

Based on this deﬁnition, many theorems about relative complements can be proven13. For example,

subtracting an amount from any of its parts generates the empty amount. In particular, based on Axioms

1-6 we can prove that amounts can always be composed and decomposed into non-overlapping parts:

Theorem 3.

x+y=z∧ −O(x,y) =⇒z\x=yDecomposability 1

x⊂y=⇒(y\x=z=⇒y\z=x)Decomposability 2

Amounts therefore satisfy the strong supplementation principle of extensional mereology, i.e., if two

amounts are not part of one another, then there exists some non-overlapping part. A known logical

consequence is that non-zero amounts with the same proper parts are equal (Casati et al., 1999, cf. ch.

3.3). This makes the mereology of amounts extensional:

Theorem 4.

¬(y⊆x) =⇒(∃v(v⊆y∧ ¬O(v,x))) Strong supplementation

The amount theory speciﬁed above contains the most important elements for characterizing sets in

terms of set intersection and union. Note, however, that set theory is only a particular interpretation of

amounts. There are also other important interpretations, such as amounts of matter, or else in terms of

intervals in time or portions of space. We do not want to make any further ontological commitments at

this stage (e.g. about discreteness or atomicity), as our goal is to deﬁne extensivity in general.

13We leave away the details for lack of space. See our documentation of proofs.

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3.1.2. Magnitudes as linearly ordered monotonic quantities

Each amount can be measured and thus compared to others on a linear scale of measurement. For

example, two amounts of water can be measured on a common scale for liters. The elements of such a

scale are also called quantities, although they are quantities of a different kind. To distinguish the two,

we call the latter magnitudes. Intuitively, magnitudes allow us to measure amounts and to put them in

relation even if they are not part of each other: we can order them, compute differences, and we can

measure their proportions. To make this notion precise, we hold that magnitudes are also quantities,

thus having an order operation (⩽), which, like the parthood relation of amounts, also satisﬁes Axioms

1 (partial order), and basic axioms for sums (+) and differences (\) (Axiom 2). Furthermore, just like

amounts, magnitudes are translation invariant, or monotonic (so that adding the same magnitude on each

side of a balance preserves the order). We illustrate magnitude axioms with examples about lengths, sizes

and weights.

Axiom 7.

(x⩽y) =⇒(x+z⩽y+z)Translation invariance (monotonicity)

However, in contrast to amounts, magnitudes do not have lattice properties, but instead are linearly

ordered (no two magnitudes of the same magnitude domain are not ordered in some way):

Axiom 8.

x⩽y∨y⩽xTotality

We furthermore need to distinguish two subclasses of magnitude domains, based on how they serve

to compare amounts: Either in terms of measuring sizes (Archimedean magnitude domains, denoted

by the class ArchimedeanMagnitudeD), or in terms of measuring proportions (proportional magnitude

domains, denoted by the class ProportionalMagnitudeD):

Archimedean magnitudes (totally ordered vectors). The ﬁrst kind of magnitude can be used to compare

the sizes of amounts, but not proportions. We call these Archimedean magnitudes. Examples are the

kilogram scale for measuring weight or the meter scale for measuring length.

An important but rather subtle issue is that the quantities of an Archimedean magnitude domain can

only be used to build orders, sums and differences among themselves, but not products or ratios. As was

argued by Simons (2013), it is nonsense to multiply or divide two weights and expect another weight

as an outcome. The latter ”divisions” should therefore not be regarded as algebraic operations within a

domain, but really relations among different domains of measurement (cf. Aameri et al. (2020)). Thus,

while it is possible to compute a proportion of two Archimedean magnitudes coming from the same

domain, such proportions are not in this domain anymore. For example, a proportion of 10 kg and 5 kg

weights is not itself a kg weight, yet it is possible to say that 10 kg is double the amount of 5 kg. Note

how this is equivalent to the impossibility of multiplying two vectors in a vector space with each other

to obtain another vector, and yet there is the possibility of comparing two vectors by some scalar value.

We agree with Aameri et al. (2020) that this is more than just a superﬁcial similarity. Correspondingly,

we specify a domain of Archimedean magnitudes ArchimedeanMagnitudeD(MArch), with x,y∈MArch,

in terms of a totally ordered vector space, using elements of a separate domain of proportional magni-

tudes a,b∈MProp ,ProportionalMagnitudeD(MProp)(deﬁned below) as scalars, which can form scalar

products with these vectors:

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Axiom 9.

(a+b)∗x= (a∗x)+(b∗x)Scalar distributivity

a∗(x+y) = (a∗x)+(a∗y)Vector distributivity

a∗(b∗x)=(a∗b)∗xScalar associativity

(x⩽y∧0⩽a) =⇒a∗x⩽a∗yScalar translation invariance

To illustrate distributivity of scalars, it is enough to realize that the way how scalars extend vectors

is also the way how we can increase lengths or weights: It does not matter whether we double two

weights separately and then sum them up or whether we ﬁrst sum them up and then double the result.

Scalar products have furthermore an Archimedean property, which requires that we can always ﬁnd a

positive proportional magnitude that makes some positive Archimedean magnitude as big as another

given positive Archimedean magnitude. This uniquely identiﬁes a proportional magnitude, which can

also be expressed as a ”ratio”14:

Axiom 10.

(0 <x∧0<y) =⇒ ∃a(a∈P∧0<a∧a∗x=y)Archimedean axiom

(0 <x∧0<y) =⇒a∗x=y↔a= (y/x)Def Archimedean Ratio

For example, it is always possible to ﬁnd a unique multiple that describes how far we need to extend

a given length to match another given length. This multiple can be regarded as the proportion of the

two lengths. Based on these axioms, it can be proven that doubling of a positive magnitude results

in a magnitude always greater than the original one (positivity), which stands in direct contradiction

to the principle of reﬂexivity of sums for amounts, and which can be used to inﬁnitely extend any

domain of magnitudes. Furthermore, building a proportion of one and the same Archimedean magnitude

yields 1 (the neutral element of proportional magnitudes), and multiplying a proportion of Archimedean

magnitudes with its denominator retrieves its numerator magnitude:

Theorem 5.

0<x=⇒x+x>xPositivity

0<x=⇒x/x= 1

(0 <x∧0<y) =⇒(x/y)∗y=x

These axioms make our magnitude theory similar to Luce and Suppes’ (Luce and Suppes, 2002;

Suppes and Zinnes, 1962) theory of ”extensive measurement”15, except that we dismiss the solvability

axiom16, and that we treat proportions as a domain separate from an Archimedean magnitude domain.

Luce and Suppes (2002) use their theory to formalize mass or weight measurements on a pan balance.

14Note that the ratio symbol used in Axiom 10 does not mean that a division operation exists on Archimedean magnitude

domains. It is rather a rewriting of the scalar product.

15Note: this notion is not to be confused with our notion of extensivity.

16The latter would enforce inﬁnitely dense magnitudes, which would exclude (discrete) count scales.

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We use our theory to talk more generally about quantities such as size, duration, or the count of a

collection. These can be compared on a linear scale, yet are not proportions, and we also do not assume

their domains are inﬁnitely dense (which would exclude the possibility of discrete scales such as count

scales).

Proportional magnitudes. Proportional magnitudes can be used to express proportions of Archimedean

magnitudes. We assume there is such a magnitude scale at least for every Archimedean magnitude

scale17. For example, we can say that if the birth weight of a baby is 3 kg and now is 6 kg, then the

baby’s weight has doubled, i.e., the weights stand in the weight proportion 2. To axiomatize proportional

magnitude domains, we amend the general magnitude Axioms 1, 2, 7 and 8 with the product Axiom 3

and the following product ordering axiom:

Axiom 11.

(0 <x∧0<y) =⇒0<x∗yproduct order

Together, these axioms specify a totally ordered mathematical ﬁeld. In distinction to an Archimedean

magnitude, we can now form products and ratios in the usual (unrestricted) manner to form new propor-

tions. For example, if the birth weight of another baby is 2 kg and now is 3 kg, then its weight gain rate

(proportion of its two weights) is 11

2. In a proportional domain, we can always compare the two pro-

portions 2and 11

2with each other by forming another proportion 3

4. This new proportion is meaningful

because it tells us that the growth rate of the second baby is 3

4of the growth rate of the ﬁrst one.

3.1.3. Quantities

As we explained at the beginning of Sect. 3, our theory of extensive measurement of quantities reﬂects

a kind of usage of the term quantity which is very common, yet has not been adopted by measurement

theory. It can be illustrated by ”the quantity of sand in this box” vs. ”a quantity of 4 kg of sand”. These

two sentences stand for two different meanings of quantity that are captured by our distinction of amount

and magnitude. Although this usage of the term is different from its technical use in measurement theory,

it precisely allows us to measure extensivity along a single one of Sinton’s dimensions, as in ”this region

has a size of 10km2”. Here, both the region and the size can be considered spatial quantities, yet quantities

of a different kind. In consequence, quantity cannot be an independent notion anymore. It rather needs

to be regarded as a super-category of both the notions of amounts and magnitudes18. When we talk about

quantities, we therefore either talk about quantiﬁable amounts or results of quantifying those amounts on

a linear scale. The notion of quantity preserves only a core algebra common to both theories, namely the

Axioms 1 (partial order), basic axioms for sums (+), differences (\) (Axiom 2), as well as translation

invariance. In the following, for quantities we simply reuse symbols +,\etc. If we generalize over

amount partonomies and magnitude orderings, we use a generalized order symbol ≼.

3.2. Measure and control

How are quantities related to each other? Sinton’s roles (Sinton, 1978) illustrate how quantities can

control measurements. In our theory, we assume that the role of control is always played by amount

17To capture proportions among different Archimedean scales (e.g. spatial density of counts with respect to areas), we

need a proportional scale for every possible pair of Archimedean scales, as well as an isomorphic mapping between the two

Archimedean scales that allows comparing them. We leave this to future work.

18Whether there are further sub-theories for quantities is a question we leave open to future work.

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quantities whereas the role of measure can be played by any kind of quantity19. Furthermore, measuring

a quantity means that the partonomy of controls is preserved in the measures. For example, bakers

may want to measure how much ﬂour they use per day in kg. Here, we measure a magnitude of ﬂour

controlled by an amount of ﬂour, which in turn is controlled by an amount of time. This measurement

can be done by dividing the day into different baking periods, and this implies that for every part of the

day, the amount of ﬂour must be smaller than or equal to the amount of the full day.

Using our basic theory of quantities, we can specify this idea by introducing a measurement function

which maps controlled amount quantities to measured quantities such that the ordering is preserved:

Deﬁnition 2. Measurement of quantities

Let Xbe a domain of amounts, and Ybe any domain of quantities. Let mbe a function X→Y. Then m

is called a measurement function iff for all x1,x2∈X,x1≼x2=⇒m(x1)≼m(x2). All x∈Xare

called controls and all y∈Ywith y=m(x)for some xare called measures.

3.3. Deﬁning extensivity

In this section, we deﬁne extensivity as a property of a measurement function between two quantities.

In our example of the baker, regular recordings of the used ﬂour give the baker the ability to calculate

the total amount of ﬂour during a day or a week by adding up partial recordings. This can only be

done because time intervals as well as amounts of ﬂour both can be added up and subtracted in a

coordinated manner. In particular, time intervals of the partial recordings should not overlap, otherwise

the calculation will be wrong.

We say that a domain of quantities is additive/subtractive with respect to a measurement function m,

iff the following holds:

Deﬁnition 3. Additivity and subtractivity of m measurements in quantity domain X

∀x,x′∈X(¬O(x,x′) =⇒m(x) + m(x′) = m(x+x′)) Additivity

∀x,x′∈X(x≼y=⇒m(y)\m(x) = m(y\x)Subtractivity

Additivity in Def. 3 requires that the measurement of the sum of any pair of quantities of a control

domain should be the same as the sum of their measures, given that control quantities do not overlap.

To illustrate, consider the weight of the contents of two buckets of ice. Piling up the contents of the two

buckets results in a quantity of ice that has the same weight as the sum of the individual weights of each

of the buckets of ice (minus the buckets themselves). However, this is only the case if the amounts of ice

do not overlap.

Subtractivity in Def. 3 likewise requires that if we remove an amount xfrom another one y of which

xis a part, then the measure of the resulting amount will be the same as when subtracting the measure

of xfrom the measure of y. If we remove e.g. one third of the pile and measure the rest, then the result

should be the same as when subtracting the measure of this third from the measure of the entire pile.

Whether a quantity domain is extensive depends on the additivity and subtractivity of all its elements

in the context of a measurement function:

19When we measure some magnitude, for example 5 kg, we can only identify that measurement using an amount, for

example an amount of ﬂour. Reversely, magnitudes cannot uniquely identify amounts. We suspect extensive measurements

between magnitudes are just a shorthand for an underlying controlling amount.

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Deﬁnition 4. Extensivity of quantity domain Y w.r.t. domain X under m:

A quantity domain Y is extensive with respect to a control domain X under a measurement function m iff

m is on X and its range is in Y and m is additive and subtractive in the control domain X.

Note that extensive measurements are always homeomerous in the sense that every mereological part

of a controlling amount can be measured within the same quantity domain Y. If a quantity domain Y

is extensive with respect to an amount domain Xunder measurement m, additional theorems can be

proven. For example, if there is an amount zthat is not part of x, then this implies there must be a non-

zero supplement wthat is part of zand which joins with xin an additive manner (follows from strong

supplementation and additivity):

Theorem 6.

¬(z⊆x) =⇒ ∃w(w⊆z∧0⊂w∧ ¬O(x,w)∧m(x) + m(w) = m(x+w))

Additive supplementation

This formalized notion of extensivity applies to many examples of quantities. For example, in the

speciﬁed sense, an amount of sand is extensive with respect to a given volumetric space. In addition,

a weight of sand (in kg) is extensive with respect to the corresponding amount of sand. Note that ex-

tensivity can also apply in the opposite direction: the volumetric space that sand occupies is extensive

with respect to the amount of sand. And a volume of sand is extensive with respect to the corresponding

volumetric space it occupies. While it happens to be the case that volumetric space and mass of sand are

both extensive with respect to each other, it should be stressed that extensive relations are not necessarily

bi-directional. This depends on whether mis bijective or not (and thus whether there exists an inverse

function). In our theory, it can e.g. be proven that mneeds to be non-injective in case mmaps into a

magnitude, under the additional assumption of domain closure (such that there always exist amounts

with equal magnitude).

There is also the possibility that a single measure is extensive with respect to multiple controls. For

example, a measure of total precipitation is controlled by space (e.g. m2) and time (e.g. days). At this

point only a theory of relations between a measure and a single control has been established. However,

the deﬁnition can be easily adapted. In the case of multiple controls of a measure, let mbe a function

X,A,B,... →Y, where X,A,B,... are all domains of controls. We deﬁne additivity with respect to one

of these controls keeping the others ﬁxed:

Deﬁnition 5. Partial additivity of measurement m with respect to domain X

∀a∈A,∀b∈B,... ,∀x,x′∈X.

(¬O(x,x′) =⇒m(x,a,b,...) + m(x′,a,b,...) = m(x+x′,a,b,...))

For example, precipitation can be considered extensive with respect to its spatial control when its

temporal control is ﬁxed. If a measure is partially additive with respect to a single control, we can

also say the measure is partially extensive. For example, the measure of total precipitation is partially

extensive with respect to its spatial control. If and only if the deﬁnition holds for all inputs we can

speak of a fully extensive measure. For example, total precipitation is partially extensive with respect

to all spatial and temporal controls, thus is fully extensive. However, partial extensivity does not always

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imply full extensivity. For example, imagine two three-dimensional cubes and two two-dimensional,

horizontal areas which the cubes occupy. If the cubes are placed side-by-side horizontally, then the areas

are extensive with respect to the volumes of the cubes. However, if they are stacked on top of one another,

the increase of volume is not accompanied by an increase of area.

3.4. Non-extensive quantities

Quantities are not necessarily extensive in the sense deﬁned above, even though, as quantities, they

can always be used in sum operations. For example, if we measure the temperature of body mass on a

ratio scale (in Kelvin), then it is clear we can build meaningful sums (e.g., in order to compute aver-

ages) and even ratios of the temperatures of two bodies. However it is not necessarily the case that the

temperature of the merger mass of these bodies will correspond to the sum of their temperatures. Thus

the temperature quantity cannot be considered extensive with respect to mass. There seems to be a cor-

responding fundamental misunderstanding in past theories about extensivity: authors have been calling

measurement scales ”extensive” whenever a suitable sum operator was available on that scale (such as

in the case of "extensive measurement" in Luce and Suppes (2002)), but apparently without fully realiz-

ing that the concept of extensivity cannot be deﬁned as a property of a scale alone. Instead, it needs to

be deﬁned as a relation between domains of measurement. For the same reason, extensivity must be a

concept different from a particular level of measurement (such as Ratio, Interval or Ordinal). The latter

is nothing but a class of automorphisms on a single domain of measurement (Suppes and Zinnes, 1962),

cf. Scheider and Huisjes (2019).

Cohen et al. (2007) deﬁned intensivity based on ”independence” of a measure from an extent. If

we understand the latter in terms of a spatial control quantity, we can deﬁne intensivity as the lack of

extensivity of a controlling function: Iff extensivity does not hold for this function, the measured domain

is intensively-related to the control domain. For example, population density is intensive with respect

to the controlling amount of space, and so are many other derived quantities (e.g., average income,

proportion of green space).

Note however that intensivity as a concept is relative to a control, and thus not the same thing as the

concept of quantities derived from others. To see this, consider again the same example. The measure

of population density is derived from a measure of population size and an area size. And in fact, it is

intensive with respect to both space and time as control. However, population density is also controlled

by migration ﬂow balance, i.e., the sum of migration inﬂow minus outﬂow. If we keep areas and time

intervals constant, population density becomes extensive with respect to ﬂow balance, since adding some

ﬂow surplus corresponds to a density increase which satisﬁes the additivity and subtractivity conditions.

4. Extensive measurement in a lightweight ontology for classifying data examples

To make the logical theory developed in Sect. 3 usable for automated classiﬁcation of data examples,

we have translated it into a lightweight ontology, which we call the Amounts and Magnitudes Measure-

ment Ontology (AMMO)20, and which is speciﬁed in the Web Ontology Language (OWL)21. Extensivity

of measurements cannot be deﬁned in OWL due to the inherent expressivity limitations of description

20http://geographicknowledge.de/vocab/AMMO.ttl

21https://www.w3.org/OWL/

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logic (DL). However, though axiomatizations in FOL or of higher order do not carry over, it is still pos-

sible to model extensive measurements by class subsumption in OWL. Fig. 2 presents a schematic view

of our ontology pattern. In this ontology classes are deﬁned for quantities, their domains (characterized

by the sufﬁx -D), as well as measurement functions (with sufﬁx -MF) and measurements (with sufﬁx

-M). The latter simply denote results of measurement, i.e., tuples of controls and associated measures.

Additionally, we use the classes Additive,Subtractive and Extensive for corresponding notions of our

theory. The Quantity class has subclasses for amounts and magnitudes, the latter of which has in turn

two subclasses for archimedean- and proportional magnitudes. The QuantityDomain class has the cor-

responding kinds of domains as subclasses. The MeasurementFunction class has four subclasses. Two

of these are the AmountOfAmountMF class, which has amounts as both the control and measure, and

MagnitudeOfAmountMF, which has an amount control and a magnitude measure. The Measurement-

Funtion class also has the ExtensiveMF and IntensiveMF subclasses, which represent respectively ex-

tensive and intensive measurements, where the latter is deﬁned as the logical complement of the former.

Though all measurement functions can have speciﬁc measurements as elements, the formal properties

of quantities are deﬁned on the domain level and not the elemental level. The class membership of

quantities and measurements may be thus be inferred from their relations to quantity domains and mea-

surement functions. For example, an entity is an ExtensiveM if it is an element of an ExtensiveMF, and an

entity is an ExtensiveMF iff it is a measurement function, and additive as well as subtractive. Two OWL

Fig. 2. Extensivity measurement concepts

properties (hasMeasureD,hasControlD) link from measurement functions to quantity domains and al-

low to specify which domain contains the control quantity and which contains the measure quantity.

Two similar properties are deﬁned between single quantities and measurements. Two more properties

hasElement and isElementOf link between quantities and their domains and between measurements and

measurement functions.

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4.1. Classiﬁcation of geographic measurements by means of extensivity

Extensive measurement functions between quantity domains are central for tasks in geographic in-

formation. We introduce quantity domain classes based on the categories of time, space and content

introduced by Sinton (1978) and Chrisman (2002), who refer to them as space, time and theme, or by

Wright (1955) who uses the terms space, time and substance. The GeoAMMO22 ontology is speciﬁc

for geographic quantities and inherits from the AMMO ontology. In GeoAMMO, we introduce a set

of subclasses for spatial, temporal and content quantities. This includes the SpaceAmountD class of re-

gion domains and the SizeMagnitudeD class of spatial magnitude domains. For example, the domains

of ’country areas’ and ’country area sizes’ are both spatial, while the former is an amount domain and

the latter is a magnitude domain. Similarly, we consider two classes of domains of temporal quantities,

where the TimeAmountD class denotes domains of amounts of time, and their durations correspond to

the DurationMagnitudeD class. Finally, we consider ContentAmountD and ValueMagnitudeD for quan-

tity domains not represented by temporal or spatial reference systems. Again, all these domain classes

have equivalents on the level of elements where the -D sufﬁx is dropped.

Different classes of extensive geographic measurement functions are obtained by distinguishing

the categories of the quantity domains that act as controls and measures. Using the two triads of

SpaceAmountD,TimeAmountD and ContentAmountD, and SizeMagnitudeD,DurationMagnitudeD, and

ValueMagnitudeD, a total of twelve measurement function classes can be distinguished, where each

measurement function class is represented as an arrow between domain categories in Fig. 3. Three mea-

surement function classes map from amount domains to magnitude domains within the category time,

space, or content, six map between amount domains of different categories and three functions are auto-

morphisms on three types of amount domains.

Fig. 3. Extensivity triangle, showing possibilities of extensive measurement functions between three categories of quantity

domains.

In the following, we discuss each of the measurement function classes using examples of geographic

maps, nine of which are assembled in Fig. 4 and Fig. 5. These maps are all univariate, but since ex-

tensivity is induced at the measurement function level, the same principles apply to multivariate maps

if for each variable a different function is assumed. We do not provide separate examples for the auto-

morphisms since these can be explained by means of the examples for the other measurement function

classes. Table 2 gives a preliminary overview of the twelve measurement function classes.

22http://geographicknowledge.de/vocab/GeoAMMO.ttl

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Table 2

Overview of geographic measurement function classes

Superclass Class Control Measure Example

MagnitudeOf-

AmountMF

SizeMF SpaceAmount SizeMagnitude Europe area →Europe area size

DurationMF TimeAmount DurationMagnitude Length of a day in minutes

ValueMF ContentAmount ValueMagnitude Temperature in Celsius

AmountOf-

AmountMF

(different cat.)

CapacityMF SpaceAmount ContentAmount People in Europe

OccupancyMF ContentAmount SpaceAmount Area owned by local farmers

AccumulationMF TimeAmount ContentAmount Total precipitation in an hour

DynamicMF ContentAmount TimeAmount Years with olympic games

SpacetimeMF SpaceAmount TimeAmount Time of a train trip

TimespaceMF TimeAmount SpaceAmount Space traversed during a ﬂight

AmountOf-

AmountMF

(same cat.)

SpaceMF SpaceAmount SpaceAmount Deforested area in a natural reserve

TimeMF TimeAmount TimeAmount Time of the day spent awake

ContentMF ContentAmount ContentAmount Amount of ﬂour in a stack of pancakes

4.2. Magnitude-of-amount measurements

AMagnitudeOfAmountMF is a function that retrieves a magnitude from some amount. We distinguish

three of these, namely SizeMF which measures from regions (amounts of space) to sizes (spatial mag-

nitudes), DurationMF which measures from amounts of time to durations (temporal magnitudes), and

ValueMF which measures from some other content amounts to other magnitudes, such as a count of

objects, a monetary value, or a weight.

Functions in the SizeMF class yield spatial magnitudes from amounts of space. Figure 4a provides an

example of size measurements. The map depicts the spatial sizes of the provinces of the Netherlands.

Clearly the regions of the provinces do not overlap and are partially ordered. They form a lattice with

an extensive mereology (regions can be part of one another). The amounts are related to their size

magnitudes, which in turn are totally ordered. According to our deﬁnition of additivity, the sizes of the

regions can be directly summed to infer the sizes of mergers, because the regions do not overlap.

Functions in the DurationMF class yield temporal magnitudes from amounts of time. Figure 4b shows

the age of churches in the Netherlands that exist for at least 500 years. In this example, the periods of

existence of each church overlap for at least the last 500 years, meaning for some time the churches exist

at the same time. Just like with sizes, durations can be compared and be added up to derive the duration

of existence of all churches. However, when summing up, overlaps need to be taken into account.

Functions in the ValueMF class yield magnitudes from amounts of content. For example, in Figure

4c, each bubble represents a magnitude of energy of an amount (a discrete collection) of wind turbines.

Note that each bubble may contain multiple wind turbines which are implicit here. Another possible

value measure would be the number of wind turbines in each cluster.

4.3. Amount-of-amount measurements

An AmountOfAmountMF measures an amount by using another amount as a control. For example, a

population can be measured by controlling space and counting the individuals within this space. Also,

the space they occupy can be found by measuring the spatial extents of the individuals. Note that the

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(a) Size measurements (b) Duration measurements (c) Value measurements

Fig. 4. Examples of magnitude-of-amount measurements

former and latter measurements are opposed to each other23. We distinguish nine amount-of-amount

measurement functions. Six of these are mappings between different amount categories, while three of

these, namely SpaceMF,TimeMF, and ContentMF, are functions from amount domains to other domains

in the same category (e.g. from hours to minutes).

Based on this, we deﬁne six subclasses of measurement functions, namely CapacityMF, Occupan-

cyMF, AccumulationMF, DynamicMF, SpacetimeMF, and TimespaceMF, where an amount domain is

extensive with respect to an amount domain of a different category.

ACapacityMF maps from a spatial amount to a content amount. Figure 5a shows the population

amounts of each province (e.g. the ’population of Utrecht’) which has a certain magnitude (e.g. 500,000).

The population amounts themselves are measured with the regions as controls. For example, the popu-

lation of Utrecht is measured with the region of the province of Utrecht as control. An OccupancyMF

is the converse in the sense that it maps from a domain of content amounts to a domain of amounts of

space that these contents ’occupy’. Figure 5b shows e.g. the living areas of European pine martens in the

Netherlands, which is the the space these animals occupy.

An AccumulationMF maps from a domain of time amounts to a content amount domain. Resulting

measurements are accumulations of content within an amount of time. Figure 5c shows the net gain of

long-wave radiation over one day. For each point in the Netherlands, a magnitude is given of the net

radiation gain or loss accumulation over a day. These magnitudes are understood as mappings from radi-

ation content, which is controlled by some time period. The converse of the accumulation measurement

is the DynamicMF, which maps from content amounts to temporal amounts. The example in Figure 5d

shows the amounts of days per region that have exceeded a threshold of >14 mm precipitation in a year.

ASpacetimeMF maps, as the name suggests, from a domain of amounts of space to some domain of

amounts of time. Figure 5e shows the route from Utrecht University to Groningen University, along with

an indication of how long traveling this route takes by car. Note that this indication is not just a duration

magnitude, but also implies a ﬁnite interval in time in which someone actually traveled. A longer path

23They correspond to the opposing arrows ”capacity” and ”occupancy” in Fig. 3.

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implies a larger time interval. This notion of time is key to Hägerstrand’s Time Geography, which tells us

that space accessibility is limited by temporal constraints (Hägerstrand, 1970). A TimespaceMF maps

from amounts of time to amounts of space. Figure 5f shows the magnitudes of the amounts of trafﬁc

activity in 2019 in traveled kilometers. Note that these amounts are extensive with respect to amounts of

time, so they can be summed up with the amounts of trafﬁc activity in 2018 to result in the amount for

two years.

(a) Capacity measurement (b) Occupancy measurement (c) Accumulation measurement

(d) Dynamic measurement (e) Space-time measurement (f) Time-space measurement

Fig. 5. Examples of amount-of-amount measurements

An amount may need to be measured with another amount from the same category as control. For

example, the churches in Figure 4b are selected from a bigger set of churches based on whether they

have existed over 500 years. Only the resulting sub-selection is shown in the map with corresponding

duration magnitudes. It is also possible to measure with semantically different controls within the same

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category. For example, an amount of pets can be measured with an amount of households as control.

In turn, the amount of pets can be a control for measuring the amount of mice caught by each pet. We

refer to all these options as SpaceMF,TimeMF and ContentMF for measurement functions from and to

respectively spatial, temporal and content amounts.

4.4. Automatically inferring measurement function classes

The membership of measurements to the twelve measurement function classes can be inferred by

means of subsumption reasoning. We exemplify this with a subset of the classes in the GeoAMMO ontol-

ogy. Speciﬁcally, these are the classes SpaceAmountD for domains of amounts of space, TimeAmountD

for domains of amounts of time and ContentAmountD for content domains and classes for their corre-

sponding amount elements, and the AccumulationMF and CapacityMF classes for measurement func-

tions that map from respectively amounts of time and and amounts of space to contents, also with corre-

sponding measurement classes.

We illustrate the inference steps with a measurement scenario of the amount of births per year in

Dutch municipalities. For this purpose, we deﬁned data instances that encode the information given by

the corresponding map data. To show how measurement function classes can be inferred from quantity

classes, we did not specify the measurement function classes of measurement instances. We manually

speciﬁed only the classes of quantities (based on their domains) as well as hasMeasure and hasControl

relations between measurements and quantities. Using Protégé’s HermiT 1.4.3.456 reasoner, it becomes

possible to automatically infer the class membership of measurement instances, as illustrated with the

blue lines in Fig. 6. For example, since the function AmountsofBirthsMF is both additive and subtractive,

it is also ExtensiveMF. Furthermore, since its element tuple Ams2021BirthsM (denoting the amount of

births in Amsterdam in 2021) is controlled by a space amount and extensively measures some content

amount, it can be classiﬁed as CapacityM. Furthermore, since it is also controlled by some time amount,

it is also an AccumulationM.

Fig. 6. Births in Amsterdam in 2021 as an example of inference of measurement function classes. Ovals are instances and

rectangles are classes. Dotted lines denote class membership and blue lines denote automatically-inferred relations.

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5. Discussion and conclusion

To better understand and to automate decisions on the applicability of arithmetic operations to spatial

information, we have suggested that the concept of extensivity should be redeﬁned as a formal property

of a measurement function from one or more control quantity domains to a measure quantity domain.

We have proposed an algebraic formalization of the underlying notions quantity, amount, magnitude,

and additivity of a measurement function, and have proven theorems that correspond with our intuition

about these concepts. In our theory, amounts appear as distinct standalone entities with an extensional

mereology and with sum, difference and product operators in the form of a mathematical lattice. Magni-

tudes, in contrast, are linearly ordered scales that can be used to measure amounts. The notion of quantity

is considered merely a generalization of these notions. In distinction to previous measurement theories,

extensivity is deﬁned as a relation between quantity domains and a measurement function. Furthermore,

while extensivity is currently primarily used to describe the behavior of physical properties, like mass

and volume, our model can be used to generalize the applicability of this concept across various domains

of measurement and different cases of information aggregation relevant for geographic information.

Our deﬁnition of extensivity not only lifts the restrictions of a ﬁxed range of properties that can be

considered extensive, but also the reliance on system theory. We exchanged the notion of quantities ex-

tensive with respect to systems (Cohen et al., 2007) for a simpler notion of quantities with an extensional

mereology, similar to Guizzardi (2010) and Gangemi et al. (2001). Furthermore, extending on our pre-

vious ideas (Scheider and Huisjes, 2019), we reused Sinton (1978)’s notion of measure and control to

formally deﬁne extensivity with respect to various control domains within the categories time, space and

content. This gives rise to an extensivity triangle as a more versatile and succinct model of extensivity

that is directly applicable to various forms of geographic information. We deliberately limited our scope

to geographic information in this article. We have tested our model by applying it to a range of map

examples, which allowed us to systematically categorize measurements relevant for GIS into twelve

classes of extensivity that can be distinguished in principle. The GeoAMMO ontology is usable for se-

mantic enrichment of any form of quantity in geographic information, although its classes may need be

made more speciﬁc for some purposes, e.g. when different kinds of content amounts or different kinds

of intensive measurements need to be distinguished.

Regarding our formal model, future work should concentrate on three issues. One strand of research

is about the formal properties of a measurement function. Intuitively, we would always expect that

two different controls can exist that have the same measure. This means the measurement function is

expected to be non-injective in the case of a magnitude measure. An example would be two different

piles of sand with the same weight, or two lectures of the same duration. However, if the measure is

also an amount, we would expect, in contrast, that the measurement function is injective, at least in

many cases. So every quantity in the range of the function has only a single quantity in its domain. For

example, two distinct amounts of sand are always contained in different amounts of space, even if they

have the same weight magnitude. However, we also have counter examples (measurements of amounts

that are not injective), like the measurement of a projected area given some region in three dimensions.

Second, we have not investigated ‘derived’ (composite and conjugate) quantities in the context of

extensivity. These quantities are related to others via certain operations, captured by formal relations be-

tween quantity domains (Aameri et al., 2020). We have argued in this paper that extensive and intensive

quantities should not be confused with such derived quantities. However, this warrants further research.

We suspect that, while extensivity can be deﬁned as a property of a measurement function, compositivity

and conjugativity may be deﬁned in terms of transformations between magnitude domains of particular

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kinds of amounts that are related themselves. An example of the former would be a transformation from

population count to population density using the size of the region that is occupied by the population

amount. The latter is about reverting this transformation by multiplying the density with area size.

And third, regarding our categorization of classes of extensivity, the content class is still a coarse con-

tainer for many different kinds of geographic amounts. It could thus be further differentiated according

to the core concept model of spatial information (Kuhn, 2012). For example, one can distinguish count-

able object-based amounts from measurable integrals of ﬁelds. It would be beneﬁcial to study amounts

and magnitudes in relation to Kuhn’s core concepts. For example, an amount of objects is bound to

be atomic, which is not the case for a ﬁeld amount. Furthermore, to make the model practically us-

able, the dependency of various arithmetic operations (like weighted average and sum) on the form of

extensive and intensive quantities needs to be investigated (Scheider and Huisjes, 2019). Another area

of research is to investigate the role of amounts in natural language processing, such as geo-analytical

question answering (Xu et al., 2020). Future research should focus on developing conceptual modelling

practices involving extensivity relations, testing our notion of extensivity on empirical data involving

analyst behavior, and on exploring intensive and alternative types of quantities and relations.

Although we limited our focus to the domain of geographic information, we do not exclude the pos-

sibility that our theory can be used for, or at least adapted to, other domains as well. As mentioned by

Simons (2013), there is a tendency to confuse quantities with numbers in the context of measurements,

and this tendency is not limited to the domain of geographic information. Similarly, we believe our

relational notion of extensive measurement may help clarify when statistical uses of sums are valid in

arbitrary domains.

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