In this paper we set out to understand the cognitive basis of Formal Concept Analysis used as an Exploratory Data Analysis framework under the guise of the Landscapes of Knowledge metaphor introduced by Wille. We show that it can be re-interpreted and extended in the framework of the Theory of Afordances from Ecological Psychology to provide not only diferent afordances for diferent flavours of formal analysis of the information captured by a formal context, but also a theory that sheds light on how we learn to do it, Perceptual Learning. This raises the issue of what it is that a formal analysis of a formal context provides. We introduce the concept of formal qualia as basic, incomparable, privative items of information aforded by each possible analysis and illustrate these concepts by the formal qualia provided by Formal Concept, Independence and Equivalence Analysis.
Exploratory Data Analysis (EDA) was introduced by Tukey1[
The theory of Ecological Psychology 1[0] would have otherwise: every inquisitive action carried out by an organism—indeed every instance of cognition—is of an interactive, exploratory nature. Predictive analysis can therefore only be a transient result in an endless cycle of interaction: as soon as we act on the prediction the cycle closes and we have again explored (just prior to a new prediction). The basis for this interaction is the notionaofofrdance in the original sense coined by James J. Gibson—the father of Ecological Psychology—in the context of animal psychology, “The afordances of the environment are what itofers the animal, what it provides or furnishes, either for good or ill1[0].”
Actually, Ecological Psychology can be taken as a philosophical foundation for Embodied,
Embedded, Extended and Enacted (4E) Cognition [
The consideration of Mathematics as the study of “possible realities” as well as the
consideration of the concept of a (mathematical) model, suggests that Maths as a whole is a mechanism
for extending reality 1[
Mathematics is a systematic way to build Extended Cognition in “alternate realities”: those that appear by the processing of abstracting models of the real world.
Recall that Wille introduced early on his “Landscapes of Knowledge” (LofK) metaphor22[] that Formal Concept Analysis (FCA) is eminently suited for the EDA of binary table data collected in the form offormal contexts. In this paper we try to elucidate the following question:
What are the afordances of applying hypothesis 1 to the particular instance of Mathematical
thinking and modelling eforts encapsulated within FCA?
We make a case below that LofK was embodied, embedded and extended as a matter of principle,
and, furthermore, we provide contrastive evidence in this paper that it was (implicitly) enacted,
by providing examples of diferent, complementary enaction in the form of Formal Independence
Analysis (FIA) [
For that purpose, we will briefly review the Ecological Theory of Afordances (section2.1), and in particular its operative interpretation of learning (secti2o.2n). To pave the way for our analysis of enactment we will briefly introduce the notion of FormalContext Analysis (FxA) as a generalisation of FCA (section2.4), that will provide the support for our analysis of the enactment aforded by Formal Concept Analysis.
In order to answer question1 we should have an answer to the following ones: 1. What is a satisfactory explanation of cognition for the human animal, that is, up to a certain extent, aTheory of Cognition? We suggests that the Theory of Afordances is one such. 2. What is an appropriateTheory of Learning within the above constraints? We suggest the theory of Perceptual Learning within the Ecological Approach to Perception might be one such theory. 3. What is a Theory of the praxis of EDA? For the case of boolean tables cast as formal contexts, we want to propose an adequate extension of LofK.
As mentioned above “... The afordances of the environment are what itofers the animal, what
it provides or furnishes, either for good or ill”1[0]. This basic operational definition has been
explored, deepened and complemented in the works of a number of researchers. The theory
here presented was first put forward in 2[] and later completed in [
Consider to be a set of abstract environments (yet to be fully determined), a set of organisms, and consider the three sets oofntological modes: abilities (of cognitive states of organisms) , behaviours (of organisms) , and features (of situations of environments ) .
By virtue of its dynamics, the environment is in saituation—for all purposes astate—in
which some of itsfeatures are salient. Note that situations are here understood as in Situation
Theory [
On the other hand, by virtue of its evolutionary development and, possibly, cultural history, the organism finds itself in a functional state ∶ → 2 described by its abilities, since such abilities allow it to carry out a catalogue of behaviours that interact with the environment.
Then an afordance () ∈ for a behaviour ∈ is an instance of a binary relation () = (, ) ∈ 2 × 2 , a tuple between a subset of abilities ∈ 2 and a subset of features of the environment ∈ 2 , paraphrased as “(In situation the environment for an organism having the ability set ) the behaviour is aforded.”
Then Perceives[, , () ≡ (, )] is the act of perception studied by ecological psychology. But typically the organism perceives the behaviour as being aforded, not the relation itself Perceives[, ()] and the environment is probablyperceptually transparent.
For Chemero [
Afordances may fail to be detected. This classical observation is due to 8[] in the context of technological afordances, but applies equally to any other type. Figure2 presents the phenomenology of afordances in terms of classical detection theory.
One of the tenets of E. Gibson’s Ecological Theory of perception9][ is that detection can be improved throughperceptual learning that involves long-term changes to perception due to practice or experience16[]. In this view perceptual learning is “any relatively permanent and consistent change in the perception of a stimulus array, following practice or experience with this array” 9[]. Confirmatory and exploratory are both modes of operation in Perceptual Learning, hence both CDA and EDA are supported by this framework.
A further development of this demands that this change must: 1. involve a change in perception, e.g. must betruly perceptual, that is, only if it afects how the stimulus appears to the subject in her perceptual experience. 2. be long term, e.g. short term adaptations will not count. 3. resultfrom practice or experience, not contingent on whether the perceptual organ changes (e.g. in blindness or correction).
The ecological theory of perception asserts that developmental perceptual learning—in infants—is carried out seamlessly into adulthood, where it is properly conceived as perceptual learning. In their first year of life infants learn(a) to communicate and interact with other people, (b) to reach, grasp and interact with objects, an(dc) to locomote and ambulate in the real world. This process of perceptual learning is always on as “Potential new afordances never stop becoming available, nor do people of any age stop learning to perceive them (…) as reaching, handling and locomotion develop, encounters with the environment broaden and potential afordances multiply (…). Increasing experience provides a backlog of knowledge that breeds new opportunities.”9[, Ch.10].
Critically, since it tries to understand human behaviour, Ecological Perception Theory considers Perceptual Learning the means to achieve the four distinctive emergent hallmarks of human behaviour [9, Ch.9]: 1. agency, the control over your own behaviour. 2. prospectivity, the forward-looking1 character of behaviour that results in plans. 3. perceiving order, because it is all around and we need to detect and use it. 4. flexibility , that is, accommodating to new circumstances with alternatives that are not random, but appropriate to the task.
Landscapes of Knowledge (LofK, [22]) is a collection of metaphors on how to use FCA 7[] for
EDA. A partial list of the metaphors elicited in1[
The metaphors are supported on the “canonical behaviours” oarctivities to carry out when
examining boolean data with FCA. Valverde-Albacete et. al.1[
Note that, although these activities are aligned with the purposes of EDA, from the point of 1Sometimes called “predictive”
Activity Description Aliases Focusing Selecting what one wants to look into. contextualizing Exploring v“…aglouoekidinega…fo”r something of which one only has a browsing Searching “l…eslsosopkeicnigfyfobrutsonmotetlhoicnaglwizeh…ic”h one can more or Recognizing “s…hippesr…c”eiving clearly circumstances and relationC- larifying Identifying j“e…cdtewteitrhmiinnainggivtehne ctlaaxsosnifoicmaictiopons…i”tion of an ob- icnataegtaoxriozninogm,yplacing “…examining data in their relationships while Analyzing guided by the theoretical views and declared pur
poses…” Investigating “q…uisrtyu…d”y by close examination and systematic enDeciding “d…err…es”2olving a situation of uncertainty by an orImproving “…enhancement in quality and value. …” distilling, valorizing “…to reshape a given structure, which, within the Restructuring scope of our discussion, is conceptual in its na
ture…” Memorizing “h…asa bpereoncelsesaorfncedomamnidttrientgaainneddr.e…p”roducing what coorymmitting to memHypothesizing
Abducing facts and relationships based on data. forming an opinion Indexing
Indexing other knowledge resources using con-interfacing resources cepts, objects or attributes view of Perceptual Learning, of which EDA is certainly an instance, CDA is just an incomplete exploration, so these activities should also be relevant to it, despite the fact that FCA is seldom used for CDA.
Ever since Wille himself cautioned againsotnly reading hierarchical knowledge from FCA,
there have been attempts at “other readings” from the information collected in a formal context,
e.g. [
In previous works we have been calling collectively these three kinds of analyses emanating from a formal context FormalContext Analysis (FxA): • FCA, the analysis in terms of upper and lower bounds of the order imposed by the polars of the context on the object extents and attribute intents, • FIA, the analysis in terms of maximal antichains of that order, and • FEA, the analysis in terms of the equivalence relations which are refinements of the standard congruences on objects and attributes imposed by the polars of the context.
of Fx A Our results adopt the form of an argument clarifying the afordance-theoretic bases of the LofK paradigm as an EDA technique. For that purpose, with respect to the questions stated in the previous section we hypothesize that: 1. 4E cognition (Embodied, Embedded, Extended and Enacted) is an adequate frame theory for cognition. 2. Perceptual Learning as interpreted by the Ecological Approach to Psychology is an adequate theory to approach/describe 4E learning, and in particular the EDA process. 3. Afordances also stem from Extended cognition environments and abilities, e.g. Mathematical “worlds”. By increasing/extending/expanding the inventory of features of the environment, we expand its describable situations. For instance, we can extend the afordances of FCA to the new features and abilities provided by FxA.
In the following subsections we further argue these points.
A strong case of why LofK complies with 4E cognition are the metaphors made evident in the
analysis of Table 1. Specifically, recall that metaphorical interpretation eases the comprehension
of an unfamiliar domain of knowledge (concept lattices) in terms of an already-known domain
(terrain navigation, object grasping): the entities and issues in the new domain are mapped
onto the well-known domain to make them more intellectually apprehensible1[
First, this and our previous positing Mathematics as a type of Extended Cognition would entail that LofK is also of that type since it is based on the Mathematical concepts and abstractions of FCA.
Second, note how the Extension forcibly makes approaching concept lattices as embodied (graspable) and embedded (navigable). Therefore, this metaphorical transfer essentially pretends that LofK is an embodied and embedded approach to the EDA of formal contexts, mainly using (bodily) navigation and manipulation.
Third, seeing LofK through the lens of Perceptual Learning further supports this extension hypothesis: the metaphors in Table1—the extension mechanism—suggest that what is being exported towards the extended environment are the capabilities for “navigation” around a lattice, “manipulating” it and communicating its description, which—we found in Se2c..2—are the basic abilities developed and accrued by humans on the physical world.
Finally, to argue that LofK is enacted, we use Afordance Theory. After Chemero’s proposal [
Note that an afordance may fail to be evinced by an organism in a particular environment and that not all afordances are efective , e.g. despite the fact that a drought may induce me to dance to the Rain God, this is not going to bring about any rain. Instead of using the heavily-connoted concept ofsuperstition we have been using that ofmirage to refer to these non-efective afordances in the context of FCA. For instance, the presence of a top—most abstract—and bottom—most specific—concepts in a concept lattice is a mirage demanded by the technique of analysis.
In the case of FCA, the information provided by standard concept lattices refers to the two processes ofconcept formation and concept hierarchisation. It is not immediate, or indeed possible, to act jointly on extents and intents based on the information provided by the formal context. Neither are the hierarchical relationships between concepts readily available as features of the (extended) environment.
Note that an afordance may fail to be evinced by an organism in a particular environment, e.g. because of a failure in its senses, and that not all afordances areefective , e.g. despite the fact that a drought may induce me to dance to the Rain God, this is not going to bring about any rain.
Instead of using the heavily-connoted concept ofsuperstition we have been using that of mirage to refer to these non-efective afordances in the context of FCA. For instance, the presence of a top—most abstract—and bottom—most specific—concepts in a concept lattice is a mirage demanded by the technique of analysis. Likewise, FCA fails to distinguish between objects that have identical intensions—an example of a non-evinced afordance from the formal context—in which case we have elsewhere asserted that is hasbalind spot for such phenomenon. Indeed, the technique of Conceptual Exploration was invented to address this limitation6][.
However, we know that FCA is not the only possible analysis that can be carried out for a
particular formal context: both FIA and FEA shed light on whattype of information is carried
by it (Section2.4). Can we formalize suchtypes of information?
1. In the previous section we saw that LofK is Enacted through the conformation of
information provided by FCA based on formal concept formation and hierarchisation.
2. It stands to reason that FIA will provide a diferent conformation of information to LofK
based on tomoi—the analogue of formal concepts but based on pairs of antichains—and
independence [
Note that these diferent conformations of information are incomparable—formal concepts vs. tomoi vs. pairs of partitions, or hierarchisation vs. independence vs. indistinguishability— and we apprehend intuitions on what knowledge they convey based on our mathematical pre-conceptions of (order-theory) bounds, anti-chains and partitions, and their typical order relation.
We introduce the following definition of a formal quale ( plural qualia) as the basic, incomparable, privative conformation of information aforded by each possible analysis on a formal context. Here we suggest thatit is productive to think of FCA as providing formal qualia on hierarchisation, FIA on independence and FEA on indistinguishability, and this is consistent with the Ecological approach to defining information Enactively .
Note that the term “quale” comes from the Philosophy of the Mind, and is both heavily defended and contested even in the context of Neuroscience14[]. Formal qualia transcend this debate by adhering to the interpretation aforded by mathematical training about the basic units of analysis: that is, what an order relation is, equivalently a partition, an anti-chain, an so on. Of course the basic challenge next is to find whether there are any more means of obtaining formal qualia from a context. This is left for further work.
What are the afordances of the environment for an organism at a definite time point ? Clearly, at least, those afordances for any possible behaviour ∈ whose abilities are encompassed by the present cognitive state o f and whose situation is “compatible” with the present state of the environment:
(, ) = {(, ) ∣ ∃ ∈ , () = (, ), ⊆ (), ⊆ ()} If we want to model learning we should extend this to every possible cognitive state too, so we rather provide the time variable to the afordance functions: where the new(, ) and (, ) have the obvious nuance of fixing the time instant where the states are being observed.
In this paper we have argued how not only standard FCA but our long-sustained efort to extend its modelling capabilities that—called xFA—can be better understood in the framework Note that we have posited here two diferent order relations between sets of abilities of the organism and sets of features of the environment.
After Gibson [
We started from previous metaphorical analyses of the flavour of EDA provided by standard FCA, that Wille himself called the “Landscapes of Knowledge” approach. These lend support to the hypothesis that LofK is Embodied and Embedded, but also argue for the Extendedness of those types of model-based EDA.
In order to make Enaction apparent, we recollected the extensive list of data-exploring activities licenced by the LofK approach. Such data-exploring behaviours make abilities appear in the functional state (for EDA) of humans which are, in the Perceptual Learning tradition, the basic components for afordances to be available for behaviour. The “features of the environment” that support such abilities are data to be analysed and encapsulated in the formal context. Finally we argued that each “flavour” of formal analysis in FxA brings a diferent kind of qualitative information to the EDA: FCA brings hierarchy, FIA independence and FEA indistinguishability. In providing diferent types of qualitative information through the behaviour aforded by the data in the context, FxA brings in a rich Enaction.
To start the discussion, note that the afordances that emerge from mathematical constructs
like FCA are beyond those ofered by the LofK paradigm or metaphor theory. A previous but
more extensive analysis of the “afordances” of LofK vis-à-vis other metaphors in Data Mining
can be found in [18, § 4.1]. But the “afordances” there are more those of interface design8[]
than those of the ecological approach to perception1[
Also, the embodiment and embedding of lattices in the extended environment seems to be a matter of physical complexity scale: when the lattice has few nodes we prefer to consider it a thing to be grasped and manipulated, while if the lattice is moderately to really complex the preferred metaphor is that of navigation. Further investigation of this issue should be carried out.
It may be construed that formal concepts and their counterparts, concept lattices, are equally “physical” and identically informative—of so the Basic Theorem of FCA would have us believe— so that lattices may not ofer more afordances than contexts. These dual domains of information are readily seen in science and engineering: signals in time are oftetnransformed into a dual domain—think of their frequency description. Despite signals having unique representations in either domain—up to a set of points of measure zero—the time and frequency representations enable diferent e.g. perceptual afordances that the human auditory system is only too eager to capitalise on. The investigation of what this concept of a(domain) transform brings to the consideration of concept lattices vs. formal contexts in the Enaction phenomenon is also left for future work.
To drop formal qualia in the midst of the (perceptual, philosophical) qualia debate, are the former more like glimpsing at the information in the formal context with glasses of diferentlycoloured lenses, or rather like using diferent senses on it? This latter metaphor makes more sense, for the time being, since we do not know if there is an analogue of the concept of “spectrum of light” for the gamut of possible conformations of the information in a formal context. Indeed, the theory of afordances assures that as long as we find new opportunities for operating on formal concepts, new information will be born out of the Enaction of such actions. This seems an open question for the time being. [21] Valverde-Albacete, F.J., Peláez-Moreno, C., Cordero, P., Ojeda-Aciego, M.: Formal equivalence analysis. In: Proc. Conf. Internat. Fuzzy Syst. Assoc. and European Soc. Fuzzy Logic and Technol. (EUSFLAT 2019). Atlantis Press (2019) [22] Wille, R.: Conceptual Landscapes of Knowledge: A Pragmatic Paradigm for Knowledge Processing. In: Gaul, W., Locarek-Junge, H. (eds.) Classification in the Information Age. pp. 344–356. Springer Berlin Heidelberg (1999)