Comparability, Decision Theory and the AHP


IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

513 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

COMPARABILITY, DECISION THEORY AND THE AHP  

 

Sibs von Solms 

Department of Business Management 

University of Zululand 

Richards Bay, South Africa 

Email: sibsvonsolms@gmail.com  

 

ABSTRACT 

 

 

Saaty (2011) briefly discusses the three basic laws of Aristotelian logic and suggests a 

fourth, which he calls the Law of Comparisons.  He argues that comparison is both 

relevant and essential to the other three laws and, in fact, precedes them.  This view - 

comparativism - is however, not without criticism.  Here we present a more 

comprehensive discussion of various problems regarding comparability, focusing on 

three aspects; (i) the problem of a proper scale; (ii) the problem of a proper 

aggregation of conflicting criteria and (iii) the debate whether values are subjective or 

objective.  The debate regarding incomparability is varied and intense making a 

perfunctory or uncritical acceptance of comparativism wrong.  However, Saatian 

Comparativism will be shown to be a solution to the major issues raised by 

incomparativists.  Two conclusions are reached; (i) Saaty’s (2011) view is confirmed 

and (ii) the work of Saaty is not reflected in the incomparability or 

incommensurability literature and this debate stands to be enriched by seriously 

considering Saatian Comparativism. 

 

Keywords:  Analytic Hierarchy Process; covering value; incomparability; 

incommensurability; value realism 

 

 

1. Introduction 

In a recent article, Saaty (2011) briefly discusses the three basic laws of Aristotelian 

logic and suggests a fourth, which he calls the Law of Comparisons.  He argues that 

comparison is both relevant and essential to the other three laws and, in fact, precedes 

them.  Saaty (2011) states that rational thinking involves the basic assumption that 

things are implicitly comparable.  The view that things are implicitly and generally 

comparable is called comparativism, and a large amount of literature exists on the 

subject.  Authors argue both for and against comparativism from a philosophical as 

well as a decision theoretical perspective.  What is, however, missing from this 

debate is references to and arguments about AHP/ANP, while on the other hand there 

are also few references to comparability in the AHP/ANP literature. These two bodies 

of literature largely exist separately, without overlap or inter-literature critique.   

Although not explicit, Saaty (2011) seems to present the AHP/ANP as a decision-

making methodology that properly applies these four laws and, hence, 

comparativism.  Here we will come to the same conclusion after a more 

comprehensive discussion of various problems with comparability highlighted in the 

comparability literature.  The discussion will highlight general critiques of 

comparativism found in the comparability literature and point out how Saatian 

Comparativism answers these issues, in an attempt to stimulate useful debate between 

these two bodies of literature. 

 

mailto:sibsvonsolms@gmail.com


IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

514 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

2. Importance of the incomparability argument 

Saaty’s (2011) discussion purely states the importance of comparisons, but does not 

allude to the debate regarding the possibility of not always being able to compare 

items which is the view in opposition to comparativism (Chang, 1997).  The first 

question is whether a detailed discussion of this comparability/incomparability debate 

is of any real importance.  The consensus amongst pluralist philosophers is that 

pluralism entails value incomparability (Kekes, 1992; Mason, 2011).  Several authors 

point out that it is important to settle the argument regarding the comparability or 

incomparability of options because of the critical role comparisons play in practical 

decision-making, particularly in Rational Choice Theory (Pildes & Anderson, 1990; 

Broome, 1997, 2000; Eriksson, 2003; Gert, 2004; Hsieh, 2005b; Okapal, 2007, 2010; 

Kelly, 2008; da Silva, 2011 inter alia).  It is difficult to see how one can choose 

rationally if one cannot compare and say which of the available options are best, i.e. 

optimal.  The importance of comparability is not only explicit in Rational Choice and 

Decision Theory, but is also implicit in most forms of consequentialism (Chang, 

1998, 2013). Rauschmayer (2001) points out that due to the problem of 

incomparability – which he argues is much more prevalent than often thought – 

anybody involved in decision aiding has both a scientific as well as a social 

responsibility to make all assumptions, including the possibility of incomparability, 

clear when designing or using decision-making tools.  The importance of clarifying 

the incomparability issue is also clear when Aldred (2006) points out that 

comparability is so entrenched in economics and decision theory that it would shock 

(sic) economists and decision theorists to learn how widely disputed the 

comparability view is amongst philosophers. 
 

3. Introducing the incomparability argument 

The question is whether it is always possible to compare any two items a and b or 

whether there are cases when a and b are incomparable.  Chang (2002a) points out 

that philosophers typically have one of three reactions to a decision situation in which 

it is difficult to decide on comparisons, for example: Who is more creative Mozart or 

Michelangelo or which career is best, accounting or skydiving?  First, epistemicists 

insist that, although it may be difficult or even impossible to determine how the items 

compare, all things considered, one must be better than the other or they must be 

equally good.  Second, incomparabilists, however, insist that even omniscience will 

not yield a true comparison in terms of all the relevant considerations; hard cases are 

difficult precisely because there is no comparison of them – neither is better than the 

other nor are they equally good.  Third, the semantic indeterminists argue that it is 

indeterminate how the items compare all things considered – it is indeterminate 

whether one is better than the other or whether they are equally good.  On this view 

the indeterminacy arises because the terms, “better than” or “equal to” are vague and 

hard cases are borderline applications of one of these comparatives. Wasserman 

(2004) is one who relates indeterminism and vagueness.  Chang (1997) sums it up by 

pointing out that the epistemicists think that it is true that one item is always better 

than the other or they are equally good; incomparabilists think it is false that that one 

item is always better than the other or they are equally good; semantic indeterminists 

think it is neither true nor false that one item is always better than the other or they 

are equally good.   

 

Incidentally, a second distinction regards in-comparability and non-comparability of 

which Harris (2001) says that non-comparability relates to situations where the two 

items being compared do not share a covering value, e.g. comparing an apple and a 

concept as to taste – because concepts do not possess the value taste they cannot be 



IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

515 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

compared to apples in this regard.  Incomparability, on the other hand, relates to 

difficult comparisons where both items exhibit the covering value, e.g. comparing 

Mozart to Michelangelo as to creativity.  Non-comparability will not be dealt with 

here. 

 

Traditional Decision Theory bases preference modelling on two fundamental 

preference relations, i.e. preference and indifference.  If two items a and b are 

compared three possible relationships may be obtained; either a is preferred to b (aPb 

or a>b); or b is preferred to a (bPa or b>a); or neither is preferred – a situation of 

indifference - expressed as aIb or a~b.  These relationships are often modelled as 

three fundamental relations, i.e. “better than” (equivalent to aPb); “worse than” (bPa) 

and “equal to” (a~b).  The thesis, that if two items are evaluatively comparable, then 

a must be better or worse than b; or a and b must be equally good, is called the 

Trichotomy Thesis (Chang, 2002a).  According to this thesis the conceptual space of 

comparability between two items is spanned by the trichotomy of relations “better 

than”, “worse than” and “equal to”.  Part of the appeal of this thesis is in a tidy 

assimilation of evaluative (e.g. kindness, beauty) comparisons to non-evaluative (e.g. 

mass, length) comparisons that proceed in terms of “more”, “less” or “equal” 

amounts of some attribute.  The Trichotomy Thesis (TT) is almost universally 

accepted and is assumed by all the major writers on the topic of measurement, 

decision and game theory, but it is argued that in many cases none of these relations 

hold between two items a and b, and that, in such cases, the items should be deemed 

incomparable (Klocksiem, 2010; Harris, 2001; Aldred, 2002).  Chang (1997, 2002a) 

notices problems with the TT in its simple form but still rejects the possibility of 

incomparability.  She believes that two items can be evaluatively compared without 

any of the trichotomy relations necessarily holding between them, and suggests that 

there is conceptual space in the intuitive notion of evaluative comparability for a 

fourth relation to hold between two items, i.e. parity.  Providing a definition of parity 

or rough equality - these terms are often taken as similar - is not straightforward, 

given the persistent controversy over the notion (Griffin, 1997; Wasserman, 2004; 

Aldred, 2006).  Chang (1997, 2002a) maintains that parity is a positive value relation, 

that is, another relation that requires, or defines, comparability, in addition to “better”, 

“worse” and “equal”.  She terms parity the relation of options being ‘on a par’, and 

points to examples such as where an agent must choose between a career as a lawyer 

(l) and one as a clarinettist (c).  It may be that the two are comparable, but neither is 

better than the other, and yet a small improvement in one, a slight improved legal 

career (l
+
), does not make it better that the other (Chang, 1997).  In this formulation 

she does not specify which relation holds between l
+
 and c; but she simply denies one, 

thus it appears that l
+
Bl, lREc but not that l

+
Bc.  Wasserman (2004) says that Chang 

is wrong to reject the TT and argues that by acknowledging the vagueness inherent in 

some comparisons the need for relations like rough equality or parity disappears.  In 

Qizilbash’s (2000, 2005) view, vagueness of predicates is not only widespread but 

also leads inevitably to situations of difficult comparability, but accepts that parity is 

a comparative relation and, hence, incomparability does not follow if none of the 

standard trichotomy relations hold.  Gert (2004) argues that it is not necessary to 

define the fourth relation of parity if the terms “better than” or “worse than” are 

redefined in terms of majority consensus.  In his opinion, choosing a over b (ie a>b) 

would be justified if the majority would not be surprised by that choice.  If no 

majority exists, i.e. neither choosing a over b nor choosing b over a would be deemed 

irrational by a majority of observers, the items a and b should be seen as “equal”.  

The notion that community consensus represents truth is a general and long standing 

view in philosophy and is discussed elsewhere (von Solms, 2011).  Harris (2001), on 

the other hand, argues against Chang’s (1997, 2002a) views, but not by trying to 



IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

516 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

show how the TT can be defended but rather that restricting relations to comparable 

ones (be that three or four) is wrong and that the existence of incomparability must be 

accepted.  Aldred (2002) mentions five mutually exclusive value relations between 

two items a and b:  aBb  bBa  aEb  aREb  aICb  where B implies “better than”, E 

implies “equal to”, RE implies “roughly equal to” and IC implies “incomparability”.  

The first four are equivalent to the relationships of traditional decision theory and 

constitute the area of comparability.  Aldred (2002) uses IC as strict incomparability. 

 

Consequently, the possibility of incomparability, in contrast to Saaty’s (2011) general 

comparability view, warrants closer evaluation.  This evaluation will proceed via 

three routes: (i) the lack of a common measurement scale; (ii) the lack of a proper 

aggregation method when multiple criteria are involved and (iii) the debate whether 

values and scales are objectively given or subjectively derived. 

 

4. Comparability and commensurability: The problem of a scale 

The dictionary definitions of comparable and commensurable are confusing, but the 

preponderance of evidence indicates that comparable refers to the examination of 

items to determine how they may be similar, different, better or worse; while 

commensurable implies measurable by the same standard or scale of values in terms 

of size, degree, importance or quality.  However, Roget’s Thesaurus, a widely-used 

English Language Thesaurus, gives comparable and commensurable as alternatives 

(synonyms) of each other.  A wide ranging debate exists in the literature in which 

these two terms are defined as similar, related or different, making any discussion on 

the topic confusing.   

 

Aldred (2002) mentions incomparability along with incommensurability and indicates 

that these two terms are often confused and while he argues for keeping 

incomparability and incommensurability distinct, Qizilbash (2000, 2005), on the 

other hand, equates these two terms.  This begs the question as to how the 

relationship between incommensurability and incomparability should be understood. 

 

On one hand the two terms are conflated.  Raz (1986, 1997), for example, defines 

incommensurability in terms of incomparability when he says that A and B are 

incommensurate if it is neither true that one is better than the other nor true that they 

are of equal value.  Warner (1998) represents a view similar to Raz arguing that 

reasons are incommensurable when (and only when) they are not comparable as 

better, worse or equally good.  Heuer (2004), too, points out that the Razian account 

of incommensurability relies on the impossibility of comparison and that 

incommensurability itself is established by failure of comparison, by the fact that 

neither of two options is equal in value to, or better than the other.  Broome (2000) is 

another author who sees incommensurability as existing when none of the trichotomy 

relations are obtained between two items or values.  Grimm (2003) states that 

conflating incomparability and incommensurability is widespread and points to 

Griffin (1997) who states that what nearly everyone, on reflection, means by the 

incommensurability of values is their incomparability – that there are values that 

cannot be acquired on any scale, that they cannot even be compared as to greater, less 

or equal. 

 

On the other hand, some authors find the conflation of incomparability and 

incommensurability mistaken.  Grimm (2003), for instance, does not believe that 

incommensurability necessarily implies incomparability.  Chang (1997) states that 

things are incommensurable, when they cannot be precisely measured along some 



IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

517 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

common cardinal scale of units of value, and incomparable when no positive value 

relation holds between them – they cannot even be ranked on an ordinal scale.  She 

says that comparability is a necessary but not sufficient condition for 

commensurability.  Pildes & Anderson (1990) agree with this Changian definition of 

incommensurability, while da Silva (2011) states that incommensurability is not equal 

to incomparability and that it is specifically the latter that is very important in legal 

decision-making.  Sunstein (1997) makes a very clear distinction between the terms 

incomparability and incommensurability in opposition to the views of Raz (1986, 

1997).  Aldred (2006), too, defines incommensurability and incomparability in 

Changian terms, with the former as measurement and the latter as ranking.  Hsieh 

(2007) uses these terms in an interestingly different way.  For him, incomparability 

ordinarily describes two or more concrete bearers of value of which no positive 

comparative evaluative judgment is true, while he uses the term incommensurability 

to describe the way in which two or more abstract values stand in relation to one 

another.  Considering in greater detail the relationship between the incomparability of 

bearers of value and the incommensurability of abstract values, will be done later (in 

Section 5) because it is relevant to multi-criteria problems. 

 

It is often argued that commensurability is taken to imply measurement along a 

common cardinal scale - often monetary (Chang, 1997; Aldred, 2002, 2006; Gowdy 

& Erickson, 2005).  Sunstein (1997) makes it clear that commensurability requires 

cardinal metrics and that ordinal rankings – which he calls comparability – are not 

cases of commensurability, but nevertheless important in decision-making.  

Rauschmayer (2001) defines incomparability in terms of the relations of preference 

and indifference but quotes the definition of Sunstein (1997) for incommensurability.  

His view is, however, not exactly the same as that of Sunstein’s as Rauschmayer 

(2001) allows commensurability to be based on both ordinal and cardinal scalable 

values, while Sunstein (1997) requires cardinal metrics exclusively.  Chang (2013) 

also argues that commensurability requires a cardinal (interval or ratio) scale, while 

comparability requires only an ordinal scale.  Similarly, Luban (2001) defines two 

types of commensurabilities; R-commensurability and O-commensurability.  He says 

that formally items are commensurable with respect to a certain property (distance, 

fame, etc) when that property induces a function – more precisely, an isomorphism – 

placing the items in one-to-one correspondence with elements in some ordered set.  

This ordered set provides the common measure of the items.  The ordered set, in turn, 

can possess whatever mathematical structure one wishes to impose on it.  For 

purposes of economic analysis it is common to assume the set to possess all the 

properties of the real numbers and, along with the usual axioms, the function 

assigning a real number to each item of interest is regarded as a utility function.  

Luban (2001) defines R-commensurability as: items are R-commensurable if and 

only if they can be assigned real-valued utilities.  If no such utility function exists, the 

items are R-incommensurable.  R-commensurability is what is usually meant by 

commensurability and correspondingly, the incommensurability-of-values thesis is 

typically taken as the denial that disparate goods are R-commensurable (Luban, 

2001).  But he points out that it is important to realize that items may be 

commensurable in a weaker sense, if their common measure (musical complexity, 

creative talent, etc.) does not possess the rich mathematical structure of the real 

numbers.  Such goods may be ordinally commensurable, for example, if they are 

isomorphic to some set whose structure consists of an order- or partial-order relation.  

This can be called O-commensurability leading to an ordinal ranking of the items.  

Aldred (2002, 2006) clearly distinguishes commensurability as measurement by a 

cardinal scale from comparability as comparisons on an ordinal scale.  Mather (2002) 

distinguishes between cardinal measurement and ordinal ranking but labels both as 



IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

518 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

forms of commensurability – metrical and ordinal commensurability, respectively.  

Eriksson (2003) states that, to practical reason, (cardinal) incommensurability is bad 

enough, but worse is (ordinal) incomparability, since this possesses a clear threat to 

reason-based decision-making.  Griffin (1997) captures this idea when he says: 

 
“(W)hen many of us insist, for instance, that complex decisions about the environment 

cannot be reduced to cost-benefit analysis because some of the clashing values are 

incommensurable, we do not just mean that those values cannot be got on to additive 

cardinal scales, but they cannot be got even on to the ordinal scales that economists are by 

and large content to work with. … The serious threat to practical reason comes not from, 

say, a mere breakdown in addition or from the appearance of a lexicographical ordering, 

but from a breakdown in ranking. That threat is the most important one to confront.” 

 

In summary, the relationship between incommensurability and incomparability is 

complex with the literature either conflating or differentiating the terms. The 

consensus that is developing however seems to be that commensurability is cardinal 

measurement while comparability is ordinal ranking of alternatives.  If we accept 

these as working definitions, we are faced with the question of whether rational 

decision-making can be based on either.  From the arguments above it seems as if the 

answer may be yes;  preferred alternatives can be identified either through some form 

of cardinal measurement or through ordinal ranking and rational decision-making is 

only under threat if neither process is possible.   

 

This solution however, is only superficially valid because at least three objections 

threaten its validity; (i) a Measurement Theory objection that cardinal measurement is 

only available for selected variables and, hence, has a very limited application in 

decision-making; (ii) the Social Choice objection that ordinal aggregation – either 

across multiple criteria or multiple decision-makers – is subject to Arrowian 

impossibility and (iii) the philosophical argument that pluralism implies that no scales 

– be they cardinal or ordinal – exist to commensurate or compare diverse options.  

We return to these points in Sections 5, 6 and 7 respectively. 
 

5. Measurement theory: The problem of quantification 

The first threat, the Measurement Theory objection regarding cardinal measurement, 

can be dealt with briefly.  The view that only quantitatively measurable data is of real 

scientific value is promoted within the Traditional Measurement Theory (TMT) in 

opposition to the Representational Measurement Theory (RMT) (Barrett, 2003; 

Michell, 1997, 2003, 2008; Acton, 2003).  von Solms (2013) discusses this debate 

and reaches the conclusion that not only must the general critique of the TMT aimed 

at RMT be rejected but, more importantly, the insistence of the TMT proponents that 

only tangibles (quantitative variables) can be measured must be rejected because it 

excludes immediately the possibility of any real world decision-making in which 

intangibles always play an important role (Forman & Selly, 2001; Saaty, 2001, 2010, 

2013; Jackson, 2003; Saaty & Sagir, 2009). 
 

6. All-things-considered judgments: The problem of aggregation 

The Social Choice objection requires a more comprehensive discussion.  Over and 

above the problem of a scale there is a second problem in the incomparability debate 

relating to the complexity of choice situations which manifests in two aspects (Ellis, 

2008).  First, difficult situations are characterized by comparisons where the 

evaluations of options need to be based on multiple aspects of evaluation.  Second, a 

related insight is that evaluations, on such multiple criteria, require some form of 

aggregation to achieve a single overarching relation amongst alternatives.  Although 



IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

519 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

both these themes are found in the comparability literature, solutions are either not 

discussed or those offered are not satisfactory.  Some philosophers are skeptical of 

the integration (i.e. to aggregate multiple criteria evaluations into an overarching 

result) and the best developed form of this argument is that some values are 

incommensurable, they resist integration and so they cannot be brought together to 

form a single assessment (Ellis, 2008, 2012).  Ellis (2008) acknowledges that in 

principle incomparability may have nothing to do with composite (multi-criteria) 

values, but that in practice the existence of multiple irreducible ends is at the heart of 

the incomparability issue. Rauschmayer (2001), too, links the problem of 

incomparability to that of MCDM when he says that generally speaking, the value 

system of the deciding individual will contain different values (criteria, points of 

view), which cannot be reduced to a single measure and, hence, is one of the reasons 

for the incomparability of decision options.  Several authors argue that, particularly in 

environmental and sustainability decisions, the evaluation procedure cannot rely on a 

single numeraire (commensurability) but must be based on weak comparability 

operationalized through multi-criteria evaluations because the world is characterized 

by deep complexity and that neither the problem situation description nor the 

viewpoints of decision-makers can be expressed using a single perspective (O’Neill, 

1997; Martinez-Alier et al, 1998; Geldermann et al, 2000; Munda, 2004).  O’Neill 

(1997) provides a clear definition of weak comparability by saying that it is based on 

the idea that the same value can, at the same time, be a good X and a bad Y, i.e. 

different descriptions of the issue may lead to different outcomes because a unique 

single comparative term is not available for the comparison.  His example of weak 

comparability is that the same area can be described both as a pretty scene and as a 

bad habitat because the invader plants growing there may make the area pretty to look 

at but not a suitable habitat because of the replacement of natural indigenous 

vegetation. 

 

Any comparison must proceed with respect to an evaluative covering consideration.  

X cannot be better than Y, tout court, but can be better than Y only with respect to, 

say, well-being, beauty or morality (Martinez-Alier et al, 1998; Chang, 2004a; Hsieh, 

2005a).  Just as it makes no sense to say that one stick is greater than another, tout 

court, it makes no sense to say that one item is better than another, tout court.  One 

stick can be greater than another with respect to length but not mass, and one item be 

can better than another with respect to beauty but not morality.  Comparability is a 

three-place relation: X is comparable with Y with respect to C, where C is the 

covering consideration.  This covering consideration can be simple (uni-dimensional) 

or complex (multi-dimensional).  In the complex case X is better than Y, all things 

considered, if there is some set of values (c1,c2,...,cn), which together comprises C, 

that are the things to be considered (Chang, 2013).  The complex case includes the 

view that values can form a hierarchical structure.  For example, Grimm (2003, 2007) 

defines two different values, general and contributory – general values represent 

overarching values with several contributory values which together constitute each 

general value.  He presents two examples.  First, in selecting a basketball team the 

general value could be ‘excellence at basketball’ with contributory values ‘defence’, 

‘shooting’ and ‘passing’.  Second, for awarding a prize in painting, the general value 

could be ‘artistic ability’ with its contributing values ‘mastery of perspective’, ‘use of 

colour’ and ‘expressiveness’.  Boot (2007) distinguishes between covering values and 

contributing values as follows: covering value is a value with respect to which 

options are compared while a contributory value is a value contributing to a covering 

value (e.g. ‘analytical skill’ or ‘originality’ contributing to the covering value 

‘philosophical talent’).   

 



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Covering values often contain more than one contributory value (Harris, 2001).   In 

such complex (multi-criteria) cases, the fact that one alternative (say A) can be better 

than another (say B) in terms of some and worse in terms of other criteria 

(contributing values), is used to identify the alternatives A and B as incomparable 

(Munda et al, 1994; Chang, 1997; Geldermann et al, 2000; Rabinowicz, 2008).  

Rauschmayer (2001) calls this the problem of bi-directionality and points out that it is 

well known in the literature of the MCDA (Brans & Mareschal, 2005), while Ellis 

(2008) refers to such cases as conatively mixed.  Boot (2007) argues that not all cases 

of multi-criteria comparisons are problematic, and indicates that if (i) the covering 

value is one-dimensional or (ii) the covering value is multi-dimensional provided 

there is no bi-directionality of contributory values, multi-dimensionality does not 

pose a problem.  Seung & Bonevac (1992) state that when two items are compared 

using a single value-measure (e.g. comparing apples and oranges according to 

sweetness) the resultant ranking is an algorithmic ranking.  If, on the other hand, two 

items need to be compared using multiple criteria (e.g. comparing apples and oranges 

on sweetness, juiciness and nutritional content) an algorithmic ranking can only be 

obtained in the case that one alternative is ranked above the other on all criteria.  If, 

however, apples are ranked above oranges on sweetness; oranges above apples on 

juiciness and apples and oranges are rank similar on nutrition no algorithmic ranking 

can be obtained because no super-measure exists that can be used to aggregate the 

disparate rankings and the only alternative in such cases is indeterminate ranking, 

which is based on intuition (Seung & Bonevac, 1992).  The question is now, how can 

the problem of bi-directionality be resolved or avoided? 

 

The first approach to address the aggregation problem is philosophical in nature.  A 

necessary feature of comparativist theories is to describe how all the factors are put 

together because this unity is demanded by the notion of an All-Things-Considered 

(ATC) judgment.  For ATC judgments to be possible they have to have a unity in 

virtue of which their components (the various factors) have the normative relations 

they do (Okapal, 2007, 2010).  Chang (2004a) clearly feels the need for some form of 

aggregation to move from individual Covering Values to an ATC judgment and 

suggests the More Comprehensive Value (MCV) as the integrative factor.  She, 

however, does not discuss or present any practical ways in which the ATC can be 

obtained (Okapal, 2007, 2010).  On the contrary, she identifies the unifying attribute 

of MCVs as a profound mystery.  Chang (2004a) explains the nature of 

comprehensive factors using a jigsaw puzzle metaphor according to which the correct 

solution relies on the picture that is created once all the pieces are in place.  Her MCV 

view is like a jigsaw puzzle where the picture is what tells how the pieces go together 

and Comprehensive Values would be, metaphorically, this picture (Chang 2004b).  It 

is insightful to listen to Chang’s (2004a) own presentation: 

 
“The ‘picture’ that puts values together is the unity of a more comprehensive value.  

Although it is hard to explain just what this ‘picture’ is, it is important to emphasize 

that the mystery of what makes values hang together is not peculiar to the more 

comprehensive values approach.  Any normative theory that recognizes values is 

saddled with the problem of explaining their unity…. We have no account of what it is 

about such values in virtue of which their components (sic) values hang together in the 

way that they do.”  

 

MCV judgments are multi-criteria judgments in which several values together make 

up the final judgment, which is not found via an aggregation across criteria but via 

the intrinsic normative structure of the covering value.  Chang (2004a) argues that 

there seems to be no practical way to establish the unifying principle except that she 



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explicitly argues against it being in subjective criteria weighting
1
.  Okapal (2007, 

2010) agrees that the Changian approach lacks clarity in the details and presents – 

what he calls – a sophisticated orthodox approach.  But this approach also provides 

little practical guidance as to how aggregation is achieved via normative-level 

criteria, interaction principles and judgment.  Mason (2011) argues that there is a 

problem in understanding quite what a synthesizing category or covering value is, and 

how the covering value determines the relative weightings of the constituent values.  

One possibility is that it does it by pure stipulation - as a martini just is a certain 

proportion of gin and vermouth.  However, stipulation does not have the right sort of 

explanatory power (Mason, 2011).  Tiberius (2008), too, criticizes the More-

Comprehensive-Value approach by indicating that it is unclear what kind of 

explanation the MCV can offer for the respective weights of different conflicting 

values.  Although such mysterious attributes may be philosophically interesting, this 

is an extremely unsatisfactory notion for practical real world decision-making and 

relies implicitly on the assumption that covering values are objectively given and not 

subjectively derived by the decision-makers, a theme we return to in Section 7.  

 

The second approach lies in the attempt at aggregating ordinal rankings.  As 

mentioned above, the suggestion was that the threat of (cardinal) incommensurability 

could be avoided by using ordinal ranking to achieve comparability.  This is, 

however, problematic.  

 

First, ordinal comparability also runs afoul of bi-directionality as demonstrated by 

examples in Roy & Vincke (1984) and Seung & Bonevac (1992).  One approach to 

decision aiding, trying to accommodate incomparability rather than avoid it, is 

outranking (Bisdorff, 2004).  The American MAUT tradition is based on the principle 

of general comparability, thus representing an epistemicist tradition while the 

outranking methods of the French school of MCDM accept that not all items are 

necessarily comparable, hence, representing a more incomparabilist tradition, within 

which incomparability is accepted as inevitable  and honoured as representing 

decision situations more realistically (Simpson, 1996; Coello Coello, 2000; Brans & 

Mareschal, 2005; Geldermann et al, 2000; Vincke, 2000; Bouyssou, 2001).  Within 

the outranking methods two candidates A and B are deemed incomparable in cases 

where the problems are multi-criteria comparisons and contradictory (bi-directional) 

evidence of which candidate is preferred emanates from different criteria (Roy & 

Vincke, 1984).  The significance of the notion of incomparability lies – as in the 

Seung & Bonevac (1992) case - not in the multi-criteria aspects of the problem, but 

rather in the aggregation aspect, i.e. in trying to find an overall preference from the 

conflicting information.  The infrequent existence of a dominance relationship and 

the resulting high incidence of incomparability necessitate the search for additional 

information to help reduce the number of incomparabilities.  Different outranking 

methods use different sets of additional information to achieve this (Brans & 

Mareschal, 2005).  Aggregation of ordinal rankings can be attempted either using a 

dominance relationship as in the outranking methods, or a different approach is to 

attempt to integrate diverse options based on ordinal rankings and some associated 

decision rules (Greco et al, 2005; Dembczyński et al, 2007). 

 

Second, ordinal aggregation has serious problems of its own.  In a multi-criteria 

environment, Rauschmayer (2001) argues, incommensurability can occur at two 

levels; first, when comparing options in terms of criteria – called criterial 

                                                           
1
 Here the focus is on the lack of clarity as to how the covering value facilitates aggregation, 

the objection to subjective criteria weighing will be discussed below in Section 7. 



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incommensurability and second, aggregate incommensurability when attempting to 

find the relative importance of the criteria.  If incommensurability is assumed to be a 

serious issue in rational decision-making, and the decision maker can only make use 

of ordinal data at both these levels, decision between options will not be possible.  

From this Rauschmayer (2001) emphasizes that the way in which MCDM 

aggregation is done critically affects the success of decision outcomes and concludes 

that (i) both criterial as well as aggregate incommensurability leads to 

incomparability; (ii) ordinal aggregate commensurability leads - at best - to limited 

comparability and (iii) only cardinal criterial and aggregate commensurability 

guarantee general comparability.  Because the weighting of criteria before comparing 

options on the different criteria is necessary, but an ordinal weighting process is 

problematic - ordinal weighting makes no sense other that providing a ranking of the 

criteria (Vargas, 1994).  Ordinal rankings are informationally poor and cannot 

provide an adequate basis for a proper comparison process (Sen, 1995).  These 

problems emanate from Social Choice Theory and include Arrow’s impossibility 

theorem and the Condorcet and Borda voting paradoxes (List, 2013; Arrow, 1951).  

Both outranking methods and methods using ordinal ranking plus decision rules are 

linked to Social Choice Theory (Greco et al, 2005; Munda, 2012).  These problems 

are exacerbated in the Group Decision-making (GDM) environment.  Nooteboom 

(1984) shows that intransitivity flows from preference orderings when multiple 

(conflicting) criteria are involved and points out that this is similar to Arrowian 

Impossibility in Social Choice Theory.  The latter deals with diverging inter-personal 

preference orderings while the former relates to intra-personal preference conflicts.  

May (1954) suggests an intra-personal analogue to Arrow's impossibility theorem 

showing that an individual's response to a plurality of values will, given certain 

additional assumptions, lead to intransitive preference orderings.  Hurley (1985) 

challenges May's (1954) assumptions – specifically, universal domain and 

independence of irrelevant alternatives - as implausibly strong in the intra-personal 

case; but her work does not exclude the possibility that values may disobey the canon 

of rationality that insists on transitivity (Edmundson, 2009).  Pildes & Anderson 

(1990) identify radical incommensurability as resulting when individual or collective 

value judgments fail to converge on a confident, complete ranking of the options at 

stake.  This indicates that incommensurability is not only a phenomenon – as it 

commonly appears in the incomparabilist literature – of an individual that cannot 

compare items or options, but clearly also a problem due to incomparability being 

caused by multiple judges not agreeing.  So, even if individuals could compare 

options but as a group cannot come to a substantive consensus – either on which 

values to apply or how options rate on these values (criteria) – this could be seen as 

instances of incomparability.  Kornhauser (1998) states that arguments concerning 

incommensurability have a formal structure that parallels the structure of arguments 

concerning the appropriate aggregation of interests of different individuals and, 

incidentally, calls the inability of an individual to come to a consolidated all-things-

considered judgment in a multi-criteria situation radical incommensurability.  The 

structure of the problem of incommensurability thus parallels the problem of 

collective choice.  Bouyssou et al. (2000) agree saying that aggregating the opinion or 

the preferences of voters or individuals of a community into collective or social 

preferences is a problem quite similar to devising comprehensive preferences of a 

decision-maker from a set of conflicting criteria in MCDA.  The discord amongst the 

views of different group members is analogous to the discord amongst the 

perceptions of a single decision maker and similarly leads to indecision – only in 

GDM both intra- and well as inter-member indecision could be present 

simultaneously.  If ordinal ranking is applied to multi-criteria problems the Arrowian-

type inconsistencies are magnified (Arrow & Raynaud, 1986).  The complication in 



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Group Decision-Making is that both the aggregation of group members’ individual 

ordinal rankings and the aggregation of different criteria rankings (if done ordinally) 

are subject to Arrowian inconsistencies (Hurley, 1985, 1989; Coello Coello, 2000; 

Boot, 2009).  As long as ordinal scales are used in multi-criteria or GDM situations 

the demon of incomparability will remain. 

 

An attempt to avoid the problem of ordinal ranking is to use a lexicographic 

approach.  Assigning lexical priorities means that one value gets absolute weight in 

relation to another and always receives priority in case of value conflicts (Boot, 

2007).  The Lexical Priority Approach avoids the problem of determining 

equivalence relations between values because a lexically prior value, however small 

its amount, always gets priority to another value, however large its amount.  Boot 

(2007) argues that a scheme of lexical priorities seems more capable of resolving 

value conflicts than the Relative Weight Approach because the former does not 

require a (subjective) equivalence relation which is problematic in cases of 

heterogeneous values.  However, the Lexical Priority Approach has its limitations 

because, as Boot (2007) demonstrates, there are few, if any, values that earn an 

unconditional lexical priority to other values and we have to fall back on balancing 

the relevant values by assigning relative weights in order to be capable of rationally 

resolving value conflicts.  Chapman (1998) states that a Lexical Priorities Approach 

honours incommensurability better because it does not try to commensurate criteria 

that do not have a common measure.  He suggests a method where the criteria are 

used singly and in a particular order, called the conceptually sequenced argument.  

This method, however, results in different selections depending on the order in which 

the criteria are used, indicating that although no relative weight is explicitly allotted 

to criteria, the order acts as if the first criterion is allocated more weight than the 

criteria used subsequently.  Chang (2001) points out that the view of Raz (1986) on 

constitutive incommensurability and that of Anderson (1997) on hierarchical 

incommensurability are similar to each other.  Both views are similar to Boot’s 

(2007) lexical incommensurability, arguing that certain values are, by nature, of a 

higher status than others and that this makes any attempt at comparison between 

higher and lower values wrong.  The paradigm example involves friendship and 

money.  Comparing friends to monetary gain bars the comparer from being a (true) 

friend because it is constitutive of friendship that friends are to be considered more 

important than mere monetary gain (Raz, 1986; Anderson, 1997).  Chang (2001) says 

that the arguments of many authors (including Raz and Anderson) in favour of 

constitutive incommensurability can be answered with an account of emphatic 

comparability in place.  She concludes that if a certain token of one value is seen as 

lexically (or constitutively) higher than a token of another value, instead of regarding 

this as a case of incommensurability, it could more accurately be seen as a case of 

comparability with the higher value regarded as emphatically better than the lower 

value (Chang, 2001, 2013). 

 

In summary, the impracticality of the MCV approach ; the limited applicability of the 

Lexical Priority Approach  and the Arrowian impossibility problem in ordinal 

aggregation approaches brings us back full circle to the use of cardinal scales and the 

incommensurability issue (Tiberius, 2008; Okapal, 2007, 2010; Mason, 2011; Chang, 

2001, 2013; Boot, 2007; Hurley, 1985, 1989; Vargas, 1994; Coello Coello, 2000; 

Forman & Selly, 2001; Saaty, 2001, 2010). 

 

 

 

 



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7. Incomparability, objective or subjective values and scales? 

We have come to an impasse.  On the one hand, using cardinal scales to 

commensurate objects or actions was found to run afoul of the incommensurability 

thesis; while on the other hand, ordinal comparisons were shown to falter on 

aggregation into all-things-considered analysis due to Arrowian impossibility.  This 

leaves rational decision-making in a quandary because without some form of rating or 

ranking the best alternatives cannot be identified making rational choice impossible.  

One aspect of the incommensurability debate that is relevant here but not normally 

explicitly discussed is whether values are to be seen as objectively given or 

subjectively found.  The answer significantly impacts if and how comparisons are 

performed and comparison scales developed.  In our earlier discussions, questions of 

what incommensurability or incomparability involves were discussed along with 

questions as to how these issues affect choice, but exactly why incommensurability 

and incomparability exist was not evaluated.  In the debate regarding objectivism or 

subjectivism in value, the possibility of identifying a source for the problems of 

incommensurability and incomparability presents itself.  Before turning to a possible 

solution for the incommensurability/incomparability impasse some discussion of the 

objectivism/subjectivism debate will prove fruitful. 

 

In general ethics, philosophers actively and explicitly debate whether good is 

objective or subjective.  On the one hand are arguments for value anti-realism, i.e. 

subjectivism about values (Mackie, 1977; Thomson, 1997; Fehige, 2006; Heathwood, 

2014), while on the other hand strong arguments are raised for value realism (Railton, 

1986; Arneson, 1999, 2010; Oddie, 2005; Bradley, 2014).  Arneson (1999) says that 

in this philosophical debate, different questions have been asked under these two 

descriptions, that of good as objective or subjective.  Subjective theories of human 

good are sometimes taken to be those that make welfare depend at least in part on 

some mental state.  The intended contrast is with objective theories of well-being 

which make the well-being of an agent depend entirely on states of the world apart 

from the state of mind of the agent whose well-being is under review. This, Arneson 

(1999), claims is a coherent usage, but potentially confusing. He prefers to let the 

contrast between objective and subjective mark the contrast between (i) views which 

hold that claims about what is good can be correct or incorrect and that the 

correctness of a claim about a person's good is determined independently of that 

person's volition, attitudes, and opinions and (ii) views which deny this.  Oddie 

(2005) defines value realism as the thesis that value claims;  (i) can be literally true or 

false; (ii) that some such claims are indeed true; (iii) that their truth is not simply a 

matter of any individual's subjective attitudes or even of the attitudes of some larger 

collective.  The Subjective Theory of Value, on the other hand, is a theory 

of value which advances the idea that the value of a good is not determined by any 

inherent property of the good, nor by the amount of labour required to produce the 

good, but instead value is determined by the importance an acting individual places 

on a good for the achievement of their desired ends (Rorty, 1991; von Mises, 1998; 

Heathwood, 2014). 

 

Linking this debate with the one regarding incomparability, Rauschmayer (2001) 

indicates that the philosophical debate about Incomparability is marked by three 

oppositions of which the second is about the origin of value(s); i.e. whether they are 

objective or subjective - values are somehow given and not subject to conscious 

change (objective), or they are constructed by practical reason in the decision process 

(subjective).  Mather ([2002) distinguishes between real and rational 

incommensurability where real incommensurability results due to objective 



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evaluative facts in reality outside of (and beyond) human knowledge and belief, while 

rational Incommensurability implies that the ranking of items cannot be made by 

human judges according to the rational principles.  Boot (2007) concludes that it is 

often impossible to rationally and objectively (instead of intuitively and subjectively) 

weigh the different types of values involved in a decision situation and to rationally 

and objectively resolve their conflicts.  He argues that this impossibility is due to 

incomplete comparability of the relevant options, which in its turn, is largely caused 

by incommensurability of the relevant values. 

 

However, different authors’ objectivist/subjectivist stances are not always explicit in 

the incommensurability/incomparability literature.  Often the case for (or against) 

commensurability is phrased in terms that imply an objective view of value.  So, for 

example, is incommensurability often defined as the case where two values (A and B) 

are incommensurable because no measure exists by which both A and B can be 

measured (Raz, 1986, 1997), or it is also seen in, for example, the question ‘Was 

Mozart more creative than Michelangelo?’ (Laitenin, 2008; Klocksiem, 2010).  In 

both these cases the impression is created (albeit not explicitly stated) that the 

measurement (or comparison) of objects must rely on a pre-existing objective fact as 

to their relative value - a fact that must be discovered from the structure of reality 

rather than established via the preferences of decision-makers.  Boot (2007) is more 

explicit than most, indicating that comparisons can be done either in a subjective or 

objective way, but identifies subjective comparisons as cases of incomplete 

comparison.  To the extent that judgments are not merely based on objective amount 

comparisons but also on subjective importance comparisons, they are not the result of 

a completely rational comparison. 

 

That any plausible theory of practical reason must be comparativist in form is a 

frequent point made, over a number of years, by Chang (1997, 2004c, 2012, 2013).  It 

is no wonder then that Okapal (2010) believes Chang to have developed the most 

detailed comparativist view.  Okapal (2007, 2010) criticizes this view, not because of 

its comparativism, but rather in terms of its insistence on covering values at the 

expense of other more orthodox views.  For our purposes, however, it is the light that 

this critique sheds on the debate about the objectivity or subjectivity of value that is 

of importance.  Ruth Chang not only publishes widely within the 

incommensurability/incomparability field but she is also frequently critiqued by other 

authors (and her views can fruitfully be considered as a starting point when 

discussing the objectivity/subjectivity issue (Chang, 1997, 2001, 2012, 2013; 

Anderson, 1997; Broome, 1997, 2000;  Gert, 2004; Boot, 2007, 2009; Okapal, 2007, 

2010) .  Okapal (2007, 2010) reads Chang (1997, 2002b, 2004c) as arguing for a 

situation that values are somehow given and fixed in decision-making situations.  The 

Changian MCV approach is a case in point as aggregation is found via the intrinsic 

normative structure of the covering value and not via any subjective criteria 

weighting because the criteria (covering values) possess an inherent, objective 

unifying principle even if individual judges do not know what it is (Chang, 2004a).  

Okapal (2007, 2010) characterizes the core idea in the Changian system as that for 

any given choice situation a single, comprehensive factor exists that determines 

choice.  Interpreting the jigsaw puzzle metaphor of Chang (2004a, 2004b) in a literal 

(strong) way, Okapal (2007, 2010) reaches the conclusion that this implies that the 

Changian Comprehensive Value Approach accepts these factors as static, complete 

and given to the decision-maker, and that this precludes any genuine disagreement, 

because in any disagreement (at least) one of the parties must be wrong.  The 

impression that Chang (2002b) proposes an objectivist approach also seems clear 

from her own statement that by ‘the justification of choice’ she has in mind the all-in, 



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normative, undefeated, objective ground of or reason or warrant for choice.  This, 

however, remains obscure because, a few paragraphs later, she says that although she 

focuses on the objective justification of choice the same arguments can be adjusted to 

apply to justification subject to an agent’s foibles (sic).  She mixes statements of 

objects having value with statements regarding agents having reasons to value these 

objects in a way that makes it difficult to see whether she is arguing for intrinsic 

(objective) value or more subjective value instilled by the agent’s valuation (Chang, 

2004c).  For example, she often refers to ‘evaluative facts’ about objects and actions 

but it is not clear as to where the facts are located.  An example she uses is the choice 

to eat ice cream – it is not clear whether she sees the value of this as based on the 

physical and chemical properties of the ice cream (objective) or on the agent’s 

(subjective) valuing the taste and pleasure of enjoying an ice cream.  Chang (2004c) 

identifies desire-based accounts, in which all practical reasons are grounded in the 

present desires of the agent; justification has its source in the fact that an agent wants 

it.  In opposition to these she identifies value-based accounts, reasons for acting are 

provided by facts about the value of something, where being valuable is not simply a 

matter of being desired.  It is not the fact that an agent wants something that makes 

having it valuable – it would be valuable even if the agent does not desire it.
2
  Chang 

(2004c) feels that the conflict between these two stories is striking because it is so 

stark; desire-based theorists think that all reasons are grounded in desires while their 

value-based opponents tend to think that none are, and as with many such conflicts, 

the truth may lie somewhere in the middle – the position she argues for.  Chang 

(2013) is concerned that evaluation is often done in the abstract rather than in terms 

of substantive considerations.  She says that there is good reason to think that values 

do not rank themselves in the abstract but are rather ranked by substantive covering 

values.  This, however, still sounds as if the covering values do the ranking 

independent of the decision-maker’s preferences or goals and it seems that Grimm 

(2007) reads the Changian view as objective, i.e. that measures exist and rank options 

vis-á-vis humans measuring and ranking. 

 

In a more explicit discussion, Hsieh (2007) states that values are incommensurable if 

and only if there is no true general overall ranking of the realization of one value 

against the realization of the other value.  This (as mentioned above) is the definition 

of value realism - the thesis that value claims can be literally true or false and that 

their truth is not simply a matter of any individual's subjective attitudes (Arneson, 

1999, 2010; Oddie, 2005).  Richardson (1994) defines strong commensurability as the 

thesis that there is a true ranking of the realization of one value against the realization 

of the other value in terms of one common value across all conflicts of value.  A 

denial of such a singular common value, however, does not rule out what Richardson 

(1994) calls weak commensurability - the thesis that in any given conflict of values, 

there is a true ranking of the realization of one value against the realization of the 

other value in terms of some value.  Another conception of value incommensurability 

denies both strong and weak commensurability claiming that in some conflicts of 

values – in cases where the gain in one factor cannot compensate for the loss in 

another factor - there is no true ranking of values.  Hsieh (2007) points out that 

several authors argue for different external (non-comparativist) sources of 

commensurating value.  He concludes, however, that even if these are external 

                                                           
2
 Chang (2004c) identifies versions of the Value-Based view, according to which it is not 

strictly the evaluative fact that provides the reason but the facts upon which the evaluative fact 

supervenes.  The Supervenience debate is vast and complex (Dorsey, 2012) but will be 

ignored here, because our argument stands whether Objectivism is based on Evaluative facts 

or on the facts on which they supervene. 



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sources of commensurating value, it does not follow that they provide a systematic 

resolution to value conflicts because two individuals can still resolve the same value 

conflict differently.  The fact that it may be rational to resolve the same value conflict 

in different ways, points to the possibility of value incommensurability, i.e. that no 

true ranking of options exists.  The aspect of truth is not always clear within the 

incommensurability/incomparability discussions but is central to the general ethics 

debate.  Value realism relies on the idea of truth, specifically, truth as correspondence 

with facts.  Truth, however, is an elusive concept and particularly the realist 

interpretation is widely criticized (Rescher, 1973; Checkland & Scholes, 1990; Rorty, 

1991; Jackson, 2003; Bowden & Swartz, 2004; Mingers, 2008).  Mather (2002) 

points out that the concept of truth is ambiguous, and - particularly the realist version 

- is probably beyond the reach of human decision-makers.  Mather (2002) tries to 

circumvent the problems the realist version of truth presents to decision-making by 

distinguishing between real and rational truth; real truth resides in moral realism 

while rational truth implies that the truth of propositions are determined by human 

judges in deliberative processes applying rational principles.  Arneson (1999) 

mentions that one criticism of the objective list theories is a skeptical doubt that there 

is no rational way to determine what putative goods qualify as entries on such a list.  

Although he does not discuss it any further, he acknowledges that this doubt is a 

genuine worry.  Crisp (2013), on the other hand, does criticize Objective List 

Theories for, inter alia, this exact reason.  The problem of how to understand truth 

leads to the view that mistaken decisions are often based on a lack of full information 

or the presence of cognitive bias and several authors, consequently, argue for a full 

information account of the good (Railton, 1986; Arneson, 1999; Carson, 2013).  

Railton (1986) defines a person’s non-moral good as consisting in what he would 

want himself to want, or to pursue, were he to contemplate his present situation from 

a standpoint fully and vividly informed about himself and his circumstances, and 

entirely free of cognitive error.  These full information accounts are not without 

critics (Rosati, 1995; Murphy, 1999).  The criticisms vary but what must be accepted 

here is that such an idealized decision-maker is of no use in practical decision-making 

because, given that the real truth is guaranteed only by ideal cognitive circumstances 

– by optimal coherence with a perfect data base that we do not have, rather than by 

apparent coherence with the sub-optimal data base we actually do have, we have no 

categorical assurance of the actual correctness of our inquiries, and no unqualified 

guarantee that they provide the real truth (Rescher, 1973).  The problem and 

limitations of a realist view of truth in the context of decision-making were identified 

and discussed elsewhere (von Solms, 2009, 2011). 

 

The debate regarding objective/subjective incommensurability or incomparability is 

important also regarding the establishment of scales.  Are scales given by the 

structure of reality, i.e. do different values exhibit fixed, predetermined amounts of 

value derived from a universal hierarchy of values?  Can values be measured on 

fixed, predetermined scales or failing that must they be deemed incommensurable - in 

principle - because no universal scales exist in nature?  Alternatively, the subjective 

side of the debate argues for measurement or comparisons that rely on human 

preference and choice.  The implication of this is that scales must be developed by 

decision-makers using logical and rational principles – Mather’s (2002) rational 

commensurability.  

 

In summary, incommensurability or incomparability based on an objectivist 

comparativism leads to serious problems because accepting that measurement or 

comparisons must proceed via existing scales or hierarchies and accepting that, in 

many cases, such scales or hierarchies are not available, practical decision-making is 



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seriously compromised.  This problem is arguably attested to by authors in the 

incommensurability/incomparability literature when attempts are made to escape the 

dilemma.  The first line of argument is to avoid comparativism by substituting other 

bases of choice, e.g. agency or practical reason and a second line of argument is to 

include indeterminate choices into comparativism via terms like parity; rough 

equality; clumpiness or vagueness  (Raz, 1986, 1997; Anderson, 1997; Chang, 2002a; 

Griffin, 1997; Hsieh, 2005a; Broome, 1997, 2000; Harris, 2001).  A subjectivist 

solution, discussed above, is Social Multi-Criteria Evaluation (but this is based on 

weak comparability which involves ordinal multi-criteria evaluations which was 

shown to falter on Arrowian impossibility.  In the light of these problems, Saaty’s 

(2010, 2011) form of comparativism should now be considered as an alternative 

solution (Munda, 2004; O’Neill, 1997). 

 

8. A solution to the problems of incomparability – Saaty’s 
comparativism 

The first problem identified above was that of scale; the debate regarding 

incommensurability vs. incomparability. Working definitions emanating from this 

debate resulted in using incommensurability in cases where no cardinal scale exists to 

measure options while incomparability exists when no ordinal ranking can be 

achieved amongst options.  By using cardinal scales (ratio/absolute) based on 

comparisons, Saatian Comparativism bridges this 

incommensurability/incomparability gap.   By realizing the possibility of establishing 

cardinal scales through a process of pairwise comparisons and appreciating the 

mathematical requirements of this process, AHP achieves commensurability via 

comparability.  Saaty (2010) argues that many people think that measurement needs a 

physical scale with a zero and a unit to apply to objects or phenomena.  He points out 

that this is not true.  Surprisingly, he says, accurate and reliable relative scales - that 

do not have a zero or a unit – can be derived by using human understanding and 

judgments. Decision-makers can apply their minds and understanding to make 

comparisons; and these comparisons can be made on a meaningful scale.  

 

The second issue discussed was the argument – mainly from Traditional 

Measurement Theory – that we cannot measure intangibles and that true measurement 

is only possible for objects that can be shown to be quantitative.  Saatian 

Comparability does not only argue against the view that intangibles cannot be 

measured but also provides a mathematically sound alternative in which intangibles 

are included along with tangible variables in measurement via comparisons and, 

hence, in decision-making (Saaty & Sagir, 2009; Saaty, 2010, 2013). 

 

The third problem was that of aggregation in culti-criteria decision-making situations.  

The problem of multiple criteria decisions was shown to emanate from attempts to 

combine evaluations on diverse (conflicting) criteria.  This problem, named bi-

directionality, is a result of the use of ordinal scales where proper weighing of criteria 

importance is not done adequately.  The problem – Arrowian impossibility - 

identified in Social Choice Theory presented by voting-style ordinal evaluations, is 

conclusively removed in Saatian Comparativism by the use of absolute scale values 

(Saaty & Vargas, 2005; Saaty & Peniwati, 2008).  GDM presents a similar problem 

when preferences of a single decision-maker are combined into an overall group 

preference structure, but here too Saatian comparativism provides for aggregation via 

consensus or geometric mean calculations or via a combination approach in the 3-

phase application of the AHP (Forman & Peniwati, 1998; Saaty & Peniwati, 2008; 

Saaty, 2010; von Solms & Peniwati, 2001). 



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The last, and possibly the most important, aspect of the incomparability debate was 

that of whether values and preferences are subjective or objective.  The search for an 

objective, eternal and universal hierarchy of values applicable to all decision-makers 

in all decision situations leads inevitably to an acknowledgement of incomparability 

as we find that such a hierarchy does not exist, or is at least not available to human 

decision-makers.  In contrast, Saatian Comparativism is based on the acceptance of 

the futility of searching for ‘pre-determined’ cardinal scales, ordinal hierarchies or 

lexical priorities as a general method.  On the contrary, the human need for and 

ability to compare is not taken seriously enough and we must accept the innate ability 

of human agents to express preferences through comparisons.  Instead of seeking for 

universal ranking of values the AHP focuses on providing decision-makers with a 

way of prioritizing issues in a specific decision situation relevant to the specific 

environment of the particular situation, the objectives of the people involved in that 

decision and the issues at the heart of this decision (Saaty & Peniwati, 2008).  The 

AHP depends on the knowledge and experience of people and varies from problem to 

problem.  Saaty (2010) says that the AHP relies on judgments and is therefore 

subjective because judgments can differ from one person to another, while Saaty & 

Vargas (2012) say that the AHP provides the objective mathematics to process the 

inescapably subjective and personal preferences of an individual or a group in making 

a decision.  In Saaty’s (2000) own words:  

 
“What does it mean to be objective?  If all reality is a matter of interpreting stimuli and 

information according to our needs and goals, then it is meaningless to speak of universal 

objectivity.  Facts, numbers and other stimuli are regarded by each person according to a 

certain purpose.  Unless purpose and the experience behind it are identical for all people, it 

is futile to insist that they all look at a datum with the same sense of priority or urgency.  

What we need is to persuade each other of the usefulness for all concerned to see things 

from the standpoint of some interpretation.  ...  We are all conditioned and biased by our 

family, environment, the teachers who teach us and the books we read.” 

 

No wonder then that Vargas (1994), when comparing AHP to MAUT and outranking 

methods, specifically mentions that the composition principles are just as subjective 

as the concept of independence on which they are based.  He claims that the basic 

distinction with the AHP is that from the start, it assumes that all is relative and 

subjective.  

 

9. Conclusion 

The debate regarding incomparability was seen to be varied and intense making a 

perfunctory or uncritical acceptance of comparativism wrong.  Many reasons for 

incommensurability or incomparability were highlighted.  The first conclusion 

reached from our discussion is that Saaty’s (2011) view - that comparison forms an 

important aspect of logic - is confirmed and although incomparability seems a serious 

problem, a solution seems to exist in Saatian Comparativism.  A second issue that 

became apparent during our review of the literature is that the work of Tom Saaty is 

not reflected in the incomparability or incommensurability literature as far as the 

current author could establish.  The only references to AHP are found in (i) the 

outranking literature arguing that the AHP is based on general comparability while 

the ORMs honour and incorporate incomparability and (ii) the decision theoretical 

literature where mostly the mathematical or axiomatic aspects of AHP that are 

discussed. The incommensurability/incomparability literature is void of any mention 

of the philosophical and logical contributions made by Saaty to the comparability 

debate and to the sound mathematical methodology underpinning his views.  The 



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comparativism debate can only be enriched if, on one hand, the AHP/ANP literature 

recognizes the problems and concerns regarding comparativism raised by the 

incommensurability/incomparability authors.  On the other hand, the comparability 

literature should seriously consider Saatian Comparativism and the solutions it 

proposes.  This paper is an attempt to help initiate a fruitful dialogue between these 

groups.           

 

  



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REFERENCES 

 

Acton, G.S. (2003). What is good about Rasch measurement? Rasch Measurement 

Transactions, 16(4), 902-903. 

 

Aldred, J. (2002). Cost-benefit analysis, incommensurability and rough equality. 

Environmental Values, 11(1), 27-47. doi: 10.3197/096327102129340966 

  

Aldred, J. (2006). Incommensurability and monetary valuation. Land Economics, 

82(2), 141-16. 

 

Anderson, E. (1997). Practical reason and incommensurable goods. In Chang, R. 

(Ed), Incommensurability, incomparability and practical reason. Cambridge, MA: 

Harvard University Press. 

 

Arneson, RJ. (1999). Human flourishing versus desire satisfaction. Social Philosophy 

and Policy, 16(1), 113-142. 

 

Arneson, RJ. (2010). Good, period. Analysis, 70(4), 731-744. 

doi: 10.1093/analys/anq019 

  

Arrow, KJ. (1951). Social choice and individual values. New York, NY: John Wiley 

& Sons. 

 

Arrow, KJ & Raynaud, H. (1986). Social choice and multicriterion decision-making. 

Cambridge, MA: MIT Press. 

 

Barrett, PT. (2003). Beyond psychometrics: Measurement non-quantitative structure 

and applied numerics. Journal of Managerial Psychology, 18(5), 421-439. doi: 

10.1108/02683940310484026 

 

Bisdorff, R. (2004). Concordant outranking with multiple criteria of ordinal 

significance. 4OR, 2(4), 293-308. doi: 10.1007/s10288-004-0053-7 

 

Boot, M. (2007). Incommensurability, incomplete comparability and the scales of 

justice. DPhil Dissertation; Balliol College, Oxford, UK: University of Oxford. 

 

Boot, M. (2009). Parity, incomparability and rationally justified choice. Philosophical 

Studies, 146 (1), 75-92. doi: 10.1007/s11098-008-9245-x 

 

Bouyssou, D. (2001). Outranking methods. In Floudas, C.A. & Pardalos, P.M. (Eds), 

Encyclopedia of optimization. London, UK: Kluwer Academic Publishers. 

 

Bouyssou, D., Marchant, T., Pirlot, M., et al. (2000). Evaluation and decision model: 

A critical perspective. Dordrecht, The Netherlands: Kluwer Academic Publishers. 

 

Bowden, B & Swartz, N., (2004). Truth. The internet encyclopedia of philosophy. on-

line at http://www.iep.utm.edu/t/truth.htm 

 

Bradley, B. (2014). Objective theories of well-being. In Eggleston, B. & Miller, D.E. 

(Eds), The Cambridge companion to utilitarianism. Cambridge, UK : Cambridge 

University Press. 

http://philpapers.org/s/Richard%20J.%20Arneson
http://www.informatik.uni-trier.de/~ley/db/journals/4or/4or2.html#Bisdorff04
http://www.iep.utm.edu/t/truth.htm


IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

532 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

 

Brans, J-P. & Mareschal, B. (2005). PROMETHEE methods. In Figueira, J. Greco, & 

Ehrgott, M. (Eds), Multiple Criteria Decision Analysis: State of the art surveys. New 

York, NY: Springer Science & Business Media Inc. 

 

Broome, J. (1997). Is incommensurability vagueness? In Chang, R (Ed), 

Incommensurability, incomparability and practical reason. Cambridge, MA: Harvard 

University Press.  

Broome, J. (2000). Incommensurable values. In Crisp, R. & Hooker, B. (Eds); Well-

being and morality: Essays in honour of James Griffin. Oxford, UK: Clarendon Press. 

 

Chang, R.E. (1997). Introduction. In Chang, R.E. (Ed), Incommensurability, 

incomparability and practical reason. Cambridge, MA: Harvard University Press. 

 

Chang, R.E. (1998). Comparison and the justification of choice. University of 

Pennsylvania Law Review, 146(5), 1569-1598. 

 

Chang, R.E. (2001). Against constitutive incommensurability or buying and selling 

friends. Philosophical Issues, 11(1), 33-60. doi: 10.1111/0029-4624.35.s1.2 
 

Chang, R.E. (2002a). The possibility of parity. Ethics, 112(4), 659-688. doi: 

10.1086/339673 

 

Chang, R.E. (2002b). Making comparisons count. London, UK: Routledge. 

 

Chang, R.E. (2004a). All things considered. Philosophical Perspectives, 18(1), 1-22. 

 

Chang, R.E. (2004b). Putting together morality and well-being. In Baumann, P. & 

Betzler, M. (Eds), Practical conflicts. Cambridge, UK: Cambridge University Press. 

 

Chang, R.E. (2004c). Can desires provide reasons for action?  In Wallace, R.J., Pettit, 

P., Scheffler, S. & Smith, M. (Eds), Reasons and value: Themes from the philosophy 

of Joseph Raz. Oxford, UK: Oxford University Press. 

 

Chang, R.E. (2012). Value pluralism. In Wright, J.D. (Ed), International 

encyclopedia of the social and behavioral sciences, 2
nd

 Edition. Oxford, UK: Elsevier 

Publishers. 

  

Chang, R.E. (2013). Incommensurability (and incomparability). In LaFollette, H. 

(Ed); International encyclopedia of ethics. Hoboken, NJ: Blackwell-Wiley.  

 

Checkland, P.B. & Scholes, J. (1990). Soft systems methodology in action. 

Chichester, UK: John Wiley & Sons. 

 

Coello Coello, C.A. (2000). Handling preferences in evolutionary multiobjective 

optimization: A survey. 2000 Congress on Evolutionary Computation, 30-27. 

Piscataway, NJ; IEEE Service Center. 

 

Crisp, Rb (2013)b Well-being; In Zalta, E.N. (Ed); The Stanford encyclopedia of 

philosophy. on-Line at http://plato.stanford.edu/archives/sum2013/entries/well-being  

 

http://philpapers.org/go.pl?id=CHATPO-5&proxyId=&u=http%3A%2F%2Fdx.doi.org%2F10.1086%2F339673
http://plato.stanford.edu/archives/sum2013/entries/well-being


IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

533 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

da Silva, VA. (2011). Comparing the incommensurable: Constitutional principles, 

balancing and rational decision. Oxford Journal of Legal Studies, 31(2), 273-301. 

doi: 10.1093/ojls/gqr004 

 

Dembczyński, K., Kotlowski, W. & Slowiński, R. (2007). Ordinal classification with 

decision rules. In Raś, Z.W., Tsumoto, S. & Zighed, D. (Eds), Proceedings of the 3
rd

 

International Workshop, Mining Complex Data; Warsaw, Poland.  

 

Dorsey, D. (2012). Intrinsic value and the supervenience principle. Philosophical 

Studies, 157(2), 267-285. doi: 10.1007/s11098-010-9636-7 

 

Edmundson, W.A. (2009). Pluralism, intransitivity, incoherence. In White, M.D. 

(Ed), Theoretical foundations of law and economics. Cambridge, UK: Cambridge 

University Press. 

  

Ellis, S. (2008). The main argument for value incommensurability (and why it fails). 

Southern Journal of Philosophy, 46(1), 27-43. doi: 10.1111/j.2041-

6962.2008.tb00068.x 

 

Ellis, S. (2012). Prioritizing multiple objectives: Feeling our way. Working Paper; 

Department of Philosophy, Norman, OK: University of Oklahoma. available On-Line 

at www.ou.edu/ouphil/faculty/ellis/Prioritizing.pdf  

 

Eriksson, L., (2003). Completeness and incomparability: Limits of RCT. Annual 

Meeting of the American Political Science Association, Philadelphia, PA. on-line at 

www.allacademic.com/meta/p62882_index.html 

 

Forman, E.H. & Peniwati, K. (1998). Aggregating individual judgments and priorities 

with the Analytic Hierarchy Process. European Journal of Operational Research, 

108(1), 165-169. 

 

Forman, E.H. & Selly M.A. (2001). Decision by objectives: How to confine others 

that you are right. Singapore: World Scientific Publications. 

 

Geldermann, J., Spengler, T. & Rentz, O. (2000). Fuzzy outranking for environmental 

assessment: case study: Iron and steel making industry. Fuzzy Sets and Systems, 

115(1), 45-65. doi:10.1016/S0165-0114(99)00021-4 

 

Gert, J. (2004). Value and parity. Ethics, 114(3), 492-510. 

 

Gowdy, J. & Erickson, J.D. (2005). The approach of ecological economics. 

Cambridge Journal of Economics, 29(2), 207-222. doi: 10.1093/cje/bei033 

 

Greco, S., Matarazzo, B. & Slowiński, R. (2005). Decision rule approach. In Figueira, 

J., Greco, S. & Ehrgott, M. (Eds), Multiple Criteria Decision Analysis: State of the 

art surveys. New York, NY: Springer Science & Business Media Inc. 

 

Griffin, J. (1997). Incommensurability: What’s the problem?. In Chang, R. (Ed),  

Incommensurability, incomparability, and practical reason. 35 Cambridge, MA: 

Harvard University Press.  

  

http://ojls.oxfordjournals.org/search?author1=Virg%C3%ADlio+Afonso+da+Silva&sortspec=date&submit=Submit
http://philpapers.org/s/Dale%20Dorsey
http://philpapers.org/go.pl?id=DORIVA&proxyId=&u=http%3A%2F%2Fdx.doi.org%2F10.1007%2Fs11098-010-9636-7
http://www.ou.edu/ouphil/faculty/ellis/Prioritizing.pdf
http://www.allacademic.com/meta/p62882_index.html
http://dx.doi.org/10.1016/S0165-0114(99)00021-4


IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

534 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

Grimm, S.R. (2003). Is incommensurability a problem for practical reason? Paper 

Presented at Columbia/NYU Graduate Conference in Philosophy, New York, NY: 

New York University.  

 

Grimm, S.R. (2007). Easy cases and value incommensurability. Ratio, 20(1), 26-44. 

doi: 10.1111/j.1467-9329.2007.00344.x 

 

Harris, G.W. (2001). Value vagueness, zones of incomparability and tragedy. 

American Philosophical Quarterly, 38(2), 155-176. 

 

Heathwood, C. (2014). Subjective theories of well-being. In Eggleston, B. & Miller, 

D.E. (Eds), The Cambridge companion to utilitarianism. Cambridge, UK: Cambridge 

University Press. 

 

Hsieh, N-H.; (2005a). Equality, clumpiness and incomparability. Utilitas, 17(2), 180-

204. doi: http://dx.doi.org/10.1017/S0953820805001512  

 

Hsieh, N-H. (2005b). Value maximization, incomparability and managerial choice. 

3
rd

 Biennial Global Conference on Business Ethics. Santa Clara, CA: Santa Clara 

University. 

 

Hsieh, N-H. (2007). Incommensurable values. In Zalta, E.N. (Ed), The Stanford 

encyclopedia of philosophy; on-line at http://plato.stanford.edu/entries/value-

incommensurable 

 

Hurley, S.L. (1985). Supervenience and the possibility of coherence, Mind, 94(376), 

501-525. doi:10.1093/mind/XCIV.376.501 

 

Hurley, S.L. (1989). Natural reasons, personality and polity. New York, NY: Oxford 

University Press. 

  

Jackson, M.C. (2003). Systems thinking: Creative holism for managers. Chichester, 

UK: John Wiley & Sons.  

 

Kekes, J. (1992). Pluralism and conflict in morality. Journal of Value Inquiry, 26(1), 

37-50. doi: 10.1007/BF00136589 

 

Kelly, C. (2008). The impossibility of incommensurable values. Philosophical 

Studies, 137(3), 369-382. doi: 10.1007/s11098-006-0005-5 

 

Klocksiem, J. (2010). In defense of the trichotomy thesis. Acta Analytica, 25(3), 317-

327. doi: 10.1007/s12136-009-0067-z 

 

Kornhauser, L.A. (1998). No best answer?, University of Pennsylvania Law Review, 

146(5), 1599-1637. 

 

Laitenin, A. (2008). Strong evaluation without moral sources. Berlin, Germany: 

Walter de Gruyter GmbH & Co. 

 

List, C. (2013). Social choice theory. In Zalta, E.N. (Ed), The Stanford encyclopedia 

of philosophy. on-line at http://plato.stanford.edu/archives/win2013/entries/social-

choice  

 

http://dx.doi.org/10.1017/S0953820805001512
http://plato.stanford.edu/entries/value-incommensurable
http://plato.stanford.edu/entries/value-incommensurable
http://philpapers.org/go.pl?id=HURSAT&proxyId=&u=http%3A%2F%2Fdx.doi.org%2F10.1093%2Fmind%2FXCIV.376.501
http://plato.stanford.edu/archives/win2013/entries/social-choice
http://plato.stanford.edu/archives/win2013/entries/social-choice


IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

535 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

Luban, D. (2001). Value pluralism and rational choice, Georgetown University Law 

Center Working Paper No 264335, Washington, DC: Georgetown University. 

 

Martinez-Alier, J. Munda, G. & O’Neill, J. (1998). Weak comparability of values as a 

foundation for ecological economics. Ecological Economics, 26(3), 277-286. 

 

Mason, E. (2011). Value pluralism. In Zalta, E.N. (Ed); The Stanford encyclopedia of 

philosophy, published on-line at http://plato.stanford.edu/entries/value-pluralism 

 

Mather, H.S. (2002). Law-making and incommensurability. McGill Law Journal, 

47(2), 345-388. 

 

May, K.O. (1954). Intransitivity, utility and the aggregation of preference patterns. 

Econometrica, 22(1), 1-13. 

 

Michell, J. (1997). Quantitative science and the definition of measurement in 

psychology. British Journal of Psychology, 88(3), 355-383. 

 

Michell, J. (2003). The quantitative imperative - Positivism, naive realism and the 

place of qualitative methods in psychology. Theory and Psychology, 13(1), 5-

31.doi: 10.1177/0959354303013001758 

 

Michell, J. (2008). Is psychometrics pathological science?, Measurement, 6(1-2), 7-

24. doi: 10.1080/15366360802035489 

 

Mingers, J.C. (2008). Management knowledge and knowledge management: Realism 

and forms of truth. Knowledge Management Research & Practice, 6(1), 62-76. doi: 

10.1057/palgrave.kmrp.8500161 

 

Munda, G. (2004). Social Multi-Criteria Evaluation (SMCE): Methodological 

foundations and operational consequences. European Journal of Operational 

Research, 158(3), 662-677. doi:10.1016/S0377-2217(03)00369-2 

 

Munda, G. (2012). Choosing aggregation rules for composite indicators. Social 

Indicators Research, 109(3), 337-354. doi:10.1007/s11205-011-9911-9 

 

Munda, G. Nijkamp, P. & Rietveld, P. (1994). Qualitative multicriteria evaluation for 

environmental management, Ecological Economics 10(2), 97-112. 

doi: 10.1016/0921-8009(94)90002-7 

 

Murphy, M.C. (1999). The simple desire-fulfillment theory. Nous, 33(2), 247–272. 

 

Nooteboom, B. (1984). Intransitive preferences in retailing. Services Industries 

Journal, 4(1), 82-92. 

 

Okapal, J.M. (2007). Comparative choice without comprehensive factors. 104
th
 

Annual Meeting of the American Philosophical Association, Chicago, IL Central 

Division. 

 

Okapal, J.M. (2010). Problematic arguments for comprehensive values. Florida 

Philosophical Review, 10(1), 43-74. 

 

http://plato.stanford.edu/entries/value-pluralism


IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

536 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

O’Neill, J. (1997). Value pluralism, incommensurability and institutions. In Foster, J. 

(Ed), Valuing nature? Ethics, economics and the environment. London, UK: 

Routledge. 

  

Pildes, R.H. & Anderson, E.S. (1990). Slinging arrows at democracy: Social Choice 

Theory, value pluralism and democratic politics. Columbia Law Review, 90(8), 2121-

2214. 

 

Qizilbash, M. (2000). Comparability of values, rough equality and vagueness: Griffin 

and Broome on incommensurability. Utilitas, 12(2), 223-240. 

 

Qizilbash, M. (2005). The mere addition paradox, incompleteness and vagueness, 

Conference on Broome’s Weighing Lives. London School of Economics; University 

of London. 

 

Rabinowicz, W. (2008). Value relations. Theoria, 74(1), 18-49. doi: 10.1111/j.1755-

2567.2008.00008.x 

 

Railton, P. (1986). Moral realism. Philosophical Review, 95(2), 163-207. doi: 

10.2307/2185589 

 

Rauschmayer, F. (2001). Philosophical aspects of incommensurability and 

incomparability, Informatica, 12(1), 119-132. 

 

Raz, J. (1986). The morality of freedom. Oxford, UK: Clarendon Press. 

  

Raz, J. (1997). Incommensurability and agency. In Chang, R.E. 

(Ed), Incommensurability, incomparability and practical reason. Cambridge, MA: 

Harvard University Press. 

  

Rescher, N. (1973). The coherence theory of truth. Oxford, UK: Clarendon Press. 

 

Richardson, H.S. (1994). Practical reasoning about final ends. Cambridge, UK: 

Cambridge University Press. 

 

Rorty, R. (1991). Objectivity, relativism and truth: Philosophical papers. Cambridge, 

UK: Cambridge University Press. 

  

Rosati, C.S. (1995). Persons, perspectives and full information accounts of the good. 

Ethics, 105(1), 296-325. doi: 10.1086/293702 

 

Roy, B. & Vincke, P. (1984). Relational systems of preference with one or more 

pseudo-criteria: Some new concepts and results. Management Science, 30(11), 1323-

1335. http://dx.doi.org/10.1287/mnsc.30.11.1323 

 

Saaty, T.L. (2000). Fundamentals of decision making and priority theory with the 

Analytic Hierarchy Process. Pittsburgh, PA: RWS Publications. 

 

Saaty, T.L. (2001).  Decision-making with dependence and feedback: The Analytic 

Network Process; 2
nd

 Edition. Pittsburgh, PA: RWS Publications. 

  

Saaty, T.L. (2010). Principia mathematica decernendi: Mathematical principles of 

decision-making. Pittsburgh, PA: RWS Publications. 

http://philpapers.org/go.pl?id=RAIMR&proxyId=&u=http%3A%2F%2Fdx.doi.org%2F10.2307%2F2185589
http://philpapers.org/go.pl?id=ROSPPA-4&proxyId=&u=http%3A%2F%2Fdx.doi.org%2F10.1086%2F293702
http://dx.doi.org/10.1287/mnsc.30.11.1323


IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

537 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

  

Saaty, T.L. (2011). The serious omission of comparisons in Aristotle’s laws of 

thought. International Journal of the Analytic Hierarchy Process, 3(2), 180-181. doi: 

http://dx.doi.org/10.13033/ijahp.v3i2.124 

 

Saaty, T.L. (2013). On the measurement of intangibles: A principal eigenvector 

approach to relative measurement derived from paired comparisons. Notices of the 

American Mathematical Society, 60(2), 192-208. 

 

Saaty, T.L. & Peniwati, K. (2008). Group decision making: Drawing out and 

reconciling differences. Pittsburgh, PA: RWS Publications. 

  

Saaty, T.L. & Sagir, M. (2009). Extending the measurement of tangibles to 

intangibles, International Journal of Information Technology and Decision Making, 

8(1), 7-27. doi: 10.1155/2012/873710  
 

Saaty, T.L. & Vargas, L.G. (2005). The possibility of group welfare functions. 

International Journal of Information Technology & Decision Making, 4(2), 167-176. 

doi:10.1007/s00355-011-0541-6 

 

Saaty, T.L. & Vargas, L.G. (2009). Decision-making with the Analytic Network 

Process: Economical, political, social and technological applications with benefits, 

opportunities, cost and risks, 2
nd

 Edition; New York, NY: Springer Science & 

Business Media. 

 

Saaty, T.L. & Vargas, L.G. (2012). Models, methods, concepts and applications of 

the Analytic Hierarchy Process, 2
nd

 Edition. New York, NY: Springer Science & 

Business Media. 

  

Sen, A.K. (1995). Rationality and social choice. American Economic Review, 85(1), 

1-24. 

 

Seung, T.K. & Bonevac, D. (1992). Plural values and indeterminate rankings. Ethics, 

102(4), 799-813. 

 

Simpson, L. (1996). Do decision makers know what they prefer?: MAVT and 

ELECTRE II. Journal of the Operational Research Society, 47(7), 919-929. 

 

Sunstein, C.R. (1997). Incommensurability and kinds of valuation: Some applications 

in law. In Chang, R.E. (Ed), Incommensurability, incomparability and practical 

reason. Cambridge, MA: Harvard University Press.  

 

Thomson, J.J. (1997). The right and the good. Journal of Philosophy 94(6), 273-298. 

doi: jphil199794637 

Tiberius, V. (2008). The reflective life: Living wisely with our limits. Oxford, UK: 

Oxford University Press. 

 

Vargas, L.G. (1994). Comparison of three multicriteria decision making 

methodologies: The Analytic Hierarchy Process, Multiattribute Utility Theory and 

Outranking Methods, Proceedings of the 3
rd

 International Symposium on the Analytic 

Hierarchy Process, Washington, DC. 

 

http://dx.doi.org/10.1155/2012/873710
http://philpapers.org/go.pl?id=THOTRA&proxyId=&u=http%3A%2F%2Fdx.doi.org%2Fjphil199794637


IJAHP Article: von Solms/Comparability, decision theory and the AHP 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

538 Vol. 7 Issue 3 2015 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v7i3.313 

Vincke, P. (2000). {P,Q,I,J} – Preference Structures. In Fodor, J. De Baets, B. & 

Perny, P. (Eds), Preference and decisions under incomplete knowledge. New York, 

NY: Physica-Verlag.  

 

von Mises, L. (1998). Human sction: A treatise on rconomics, Scholar’s Edition. 

Auburn, AL: Ludwig von Mises Institute.  

 

von Solms, S.H. (2009). Homogeneity and choice aggregation in the Analytic 

Hierarchy Process. Proceedings 10
th
 International Symposium on the Analytic 

Hierarchy Process, Pittsburgh, PA: University of Pittsburgh. 

 

von Solms, S.H. (2011). Validity of the AHP/ANP: Comparing apples and oranges, 

International Journal of the Analytic Hierarchy Process, 3(1), 2-27. doi: 

http://dx.doi.org/10.13033/ijahp.v3i1.60 

 

von Solms, S.H. (2013). The fiction of a factual approach to decision-making. 

Proceedings 12
th
 International Symposium on the Analytic Hierarchy Process, Kuala 

Lumpur, Malaysia. 

 

von Solms, S.H. & Peniwati, K. (2001). To agree or not to agree, that is the question: 

Choice aggregation in the AHP. Proceedings 6
th
 International Symposium on the 

Analytic Hierarchy Process, Bern, Switzerland. 

 

Wasserman, R. (2004). Indeterminacy, ignorance and the possibility of parity. 

Philosophical Perspectives, 14(1), 391-403.