Sôritês paradoxon
Contextualism & borderline vagueness
Christopher B. Germann (Ph.D., M.Sc., B.Sc. / Marie Curie Alumnus)
URL: https://christopher-germann.de
One of the most widely cited arguments that motivates the application of
quantum probability (QP) to cognitive phenomena is the existence of inter-
ference effects in higher-order cognitive processes such as decision making
and logical reasoning (Aerts, 2009; Blutner, Pothos, & Bruza, 2013;
Busemeyer, Wang, & Lambert-Mogiliansky, 2009). A recent publication
entitled “a quantum probability perspective on borderline vagueness”
(Blutner et al., 2013) discusses the importance of the concept of noncom-
mutativity in the context of decisions involving natural concepts. Natural
concepts oftentimes lack precisely defined extensions, for instance, what is
the smallest size of a man called “tall”? The demarcating criterion which
differentiates between “tall” and “not tall” is not clearly defined (Karl Pop-
per struggled with the same “demarcation problemin the context of science
versus pseudo-science). The authors investigated the fuzziness of natural
everyday concepts and compare various approaches (e.g., fuzzy logic). We
argue that similar to semantic concepts, visual categorisation is oftentimes
ambiguous and vague. Specifically, we argue that the fuzzy boundaries of
natural concepts described in other quantum cognition models are particu-
larly applicable to visual judgments. For instance, what is the lowest lumi-
nance level of a stimulus categorised asbright”? The absence of a modulus
or “perceptual anchor” complicates the matter even further. As with natural
concepts, the demarcating boundaries between “bright” and “not bright” are
not clearly defined and it is often uncertain if the predicate applies to a given
visual stimulus (partly due to the imprecise definition of the predicate). It
follows that Sôritês paradox (also known as “the problem of the heap”) is
extendable to visual perception (and perception in general) especially in the
context of the “just noticeable difference, JND (Norwich & Wong, 1997).
Sôritês paradox (which has been ascribed to the Greek philosopher
Eubulides of Miletus) illustrates the vagueness of predicates (Blutner et al.,
2013). The paradox is based on the seemingly simple question: When does
a heap of sand become a heap?
The associated syllogistic argument can be
formulated as follows:
100000000000 grains of sand are a heap of sand.
A heap of sand minus one grain is still a heap.
Ergo, a single grain of sand is a heap.
Sôritês paradox as a syllogistic argument i.e., modus ponens
( ) ) ).
Repeated application of the minor premise (iterative removal of single grains
of sand, i.e., inferential “forward chaining”) leads to the paradoxical, but
deductively necessary (i.e., logically valid) conclusion that a single grain of
sand is a heap. Figure 1 illustrates Sôritês paradox applied to visual percep-
tion. Adjacent luminance differences (e.g., tick-mark 1 versus 2) are indis-
tinguishable by the human visual system while larger contrasts (e.g., tick
mark 2 versus 3) are easily distinguishable.
Figure 1. Sôritês paradox in visual brightness perception.
Conceptual vagueness has received a lot of attention from logicians, philos-
ophers, and psychologists (e.g., Eklund, 2011; Putnam, 1983; Serchuk,
Hargreaves, & Zach, 2011). Here we are particularly concerned with cases
of borderline contradictions such as “X is bright and not bright” where X
The “Bald Man (phalakros) paradox” is another allegory which illustrates the vagueness of predicates:
A amn with a full head of hair is not bald. The removal of a single hair will not turn him into a bold man. How-
ever, diachronically, continuous repeated removal of single hairs will necessarily result in baldness.
denotes a borderline case (Blutner et al., 2013). Specifically, the “superpo-
sition” of “bright” and “not bright” is relevant from a quantum cognition
perspective and it has been cogently argued in various psychological con-
texts that this kind of superposition introduces cognitive interference effects
(Aerts, 2009; Aerts, Broekaert, & Gabora, 2011; Blutner et al., 2013). The
postulated interference effects are analogous to those observed in quantum
mechanics (i.e., the principle of superposition). The mathematical similari-
ties have been discussed elsewhere (e.g., Busemeyer, Pothos, Franco, &
Trueblood, 2011) and go beyond the scope of this discussion.
Importantly for the experimental context at hand is the fact that the concept
“bright” is a vague concept because the exact demarcation from “not bright”
is arbitrary and imprecise. When making perceptual judgments on a scale
ranging from “bright” to “not bright”, the percipient is confronted with a
large degree of indeterminacy (especially when no absolute modulus is pro-
vided to anchor the judgment on the scale). It has been convincingly argued
that the logical principle of non-contradiction (i.e., the semantic principle of
) does not necessarily hold true in such situations (Blutner et al.,
2013). Epistemological accounts of vagueness (Sorensen, 1991; Wright,
1995) consider vagueness as the consequence of nescience on part of the
percipient and not a fundamentally ontological problem (but see Daniliuc &
Daniliuc, 2004). Ontological accounts (e.g., contextualism), on the other
hand, regard vagueness as a case of context-sensitivity (Åkerman &
Greenough, 2010; Greenough, 2003; Shapiro & Greenough, 2005), i.e., the
uncertainty associated with vagueness is regarded as a contextual phenom-
enon. This kind of context-dependence has been designated as “v-standards”
and it describes any contextual parameter that is responsible for the vague-
ness (Åkerman & Greenough, 2010; Blutner et al., 2013). Fuzzy set theo-
rists would agree with this ontological stance. They propose a form of logic
which allows for graded truth values (L. a. Zadeh, 1965; L. A. Zadeh,
2008). Alxatib & Pelletier (2011) concluded that such borderline cases pose
a serious problem for classical (Kolmogorovian/Boolean) logic. However,
Blutner et al., (2013) demonstrated that QP provides a powerful explana-
tory framework for borderline contradictions (Blutner et al., 2013). QP
For instance, as measured on a quasi-continuous psychophycial visual-analogue scale (Aitken, 1969).
The semantic principle (or law) of bivalence is closely related to the 3
Aristotelian law of thought, i.e., the law
of the excluded middle (principium tertii exclusi) which can be stated in symbolic notation as . ≡∼ ( ),
where ~ signifies negation (after Whitehead & Russell, 1910). We will discuss this logical principle in greater
detail in the context of quantum cognition in subsequent chapters because it plays a crucial role for superposi-
tional states (quantum logic).
utilises vectors in a Hilbert space and it defines a linear operator on .
Specifically, a projection operator
is a linear operator which projects vec-
tors to certain subspaces of
. The underlying algebraic logic is non-Bool-
ean in nature. Rather, it obeys the logic of orthoalgebra (Dalla Chiara &
Giuntini, 1995). A defining difference between orthoalgebra and Boolean
algebra is that the former does not obey the distributive law which form the
basis of the law of total probability. The law of total probability, in turn,
form the axiomatic basis for Bayes’ rule (Bayes & Price, 1763). Ergo, QP
is incompatible with Bayes’ rule. Crucially, in QP the order in which pro-
jection operators are combined can make a significant difference (Pothos &
Busemeyer, 2013). Two projection operators A and B in a given Hilbert
space do not necessarily have to commute. That is, QP allows for
(Blutner et al., 2013). However, if all projection operator com-
mute, QP is equivalent to Boolean algebra. Thus, Boolean algebra is a spe-
cial case of quantum probability theory which provides an overarching
(more generalisable) axiomatic framework. We would like to emphasize the
difference as it is crucial for the experimental investigation at hand: The
principle of commutativity (or the violation thereof) is a critical criterion to
differentiate between Boolean logic and quantum logic. We will discuss this
noncommutativity criteria in greater detail in the context of constructive
measurements of psychological observables. In QP notation, the term
∂(A,B) is called the interference term. If ∂(A,B) is zero, A and B commute
(Blutner et al., 2013) otherwise A and B are non-Abelian
. In the context
of psychological borderline vagueness (Alxatib & Pelletier, 2011), which is
notoriously difficult to explain in a classical probability framework, it has
been demonstrated that the QP model provides a parsimonious explanatory
model with an acceptable/good index of fit, χ
(4)=5.47; p=0.24 (Blutner
et al., 2013).
Most psychologists are familiar with the General Linear Model and specifically multiple regression. The
squared length of the projection in quantum probability theory is equivalent to the R
in multiple regression
analysis, i.e., the coefficient of multiple determination (Busemeyer & Bruza, 2012).
If a given system is in state Ψ, then a measurment will change the state of the system into a state which is an
eigenvector e of A and the observed value λ will be the corresponding eigenvalue of the equation A e = λ e. This
description implies that measurements are generally non-deterministic. The formulaic description for computing
the associated probability distribution Pr on the possible outcomes given the initial state of the system Ψ is as
follows: Pr () = E () ⟩, where E(λ) signifies the projection onto the space of eigenvectors of A with ei-
genvalue λ.
In group theory, Abelian groups denote a group in which the application of a group operation to two group
elements is independent on the order in which the operation is performed (viz a commutative group). In other
terms, Abelian Groups (eponymously named after the mathematician Niels Hendrik Abel) conform to the com-
mutativity axiom in abstract algebra (Durbin, 1967).
QP requires us to broaden and revise our conception of probability theory.
QP is a much more general axiomatic framework compared to classical prob-
ability theory because it is able to describes all real-world properties (both
at the micro and macro level). Some hardliners argue that reality as a whole
is quantum mechanical, i.e., the world (and the whole universe) are a quan-
tum system and that the underlying logical structure is based on the axioms
of quantum logic. However, every-day reality appears classical (i.e., Bool-
ean/Kolmogorovian) to the naïve percipient, but this is only the case be-
cause humans happened to almost exclusively perceive commuting observa-
bles (unless one discovers quantum mechanics or tests psychophysical com-
mutativity in controlled empirical experiments). This naturally reinforces
the “representativeness heuristic” which has been extensively studied in the
field of thinking and reasoning (Kahneman & Tversky, 1972). In other
words, numerous empirical encounters with commuting variables shaped
and moulded our representations, heuristics, and intuitions and created the
impression that commutativity is a constant nomological property of psy-
chological (and physical) observables. However, from a rationalist point of
view, insights derived from quantum mechanics require us to revise our
most fundamental concepts of logic and the associated mathematical models.
This empiricist position was also advocated by Quine, i.e., Quine argued that
logic and mathematics are also subject to revision in the light of novel ex-
periences and he explicitly employed “deviant quantum logic” as an example.
In other words, Quine adopted initially an empirical quasi-Bayesian updat-
ing approach to logic. However, Quine later changed his opinion on this topic
and argued that the revision of logic would be to essentially "change the
subject”. Hilary Putnam also participated in this fundamental debate about
the empirical status of logic and he argued that we are indeed living in a
quantum world in which quantum logic is applicable (Rorty, 2005). In the
same way as non-Euclidian space is a reality (which does not mean that
Euclidian geometry is wrong it just incomplete) quantum logic is a reality
with tangible real-world consequences (e.g., Qubits in quantum computa-
tion, logic gates according to von Neumann’s quantum logic, entanglement
in quantum encryption, superposition in macro-molecules like C60/”Bucky
balls”, quantum chemistry, quantum biology, quantum cognition, etc. pp.).
However, psychological factors like “the need for closure” might prevent
individuals with certain personality propensities to adopt quantum logic. For
instance, the personality trait “openness to experience” (McCrae, 1987)
might be significantly correlated with a propensity to seek deeper infor-
mation about the nature of reality, even in the light of seemingly paradoxical
data which might put a “conscientious” personality type at unease (McCrae
& Costa, 1997). We argue that quantum logic requires a great deal of di-
vergent thinking and that it is negatively correlated with personality trait
“need for closure”. Moreover, the evolutionary predisposition to rely on ha-
bitual mechanisms of thought (cf. dual-process theory) prevents deeper cog-
nitive reflection on the fundamental nature of basic cognitive concepts like
the 3
Aristotelian law of thought (which negates vagueness and multi-
modal logic). Likewise, existential phobias like metathesiophobia (fear of
change) and kainophobia (fear of novelty) are important psychological con-
cepts which are related to epistemological curiosity. The mere presentation
of empirical facts is not enough to change scientific attitudes, especially
when these facts dictate a revision of logic. Extralogical psychological fac-
tors need to be carefully considered because scientists are human beings
which are prone to fallacious reasoning and selectivity biases, that is, vari-
ous systematic errors of thinking an reasoning which lead to irrational de-
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