The cognitive situation is “enveloped” by a conceptual (notional) space, or a conceptual framework [Botha 1989; Gardenfors 2000; Efremov 2009; Pavlinov 2010b, 2011a, 2018]. It is formed by the basic thesaurus—those concepts and respective notions with which an object to be studied is defined; in our case, it is the taxonomic reality with all its properties and elements and occasionally causes. Metaphorically, the thesaurus can be called a conceptual (notional) model of this reality: the more complete and detailed the thesaurus is, the more adequately such a model reflects the reality in question, so if something in the latter is not reflected in the thesaurus by respective notions, this “something” does not exist in the cognitive situation [Kuraev and Lazarev 1988; Margolis and Laurence 2011]. From this, it follows by a “reverse reading” that each conceptual (notional) model shapes the studied reality in a specific way: what this model is, so also is the reality implied by it. This is just what is presumed by concepts of linguistic relativity and linguistic world picture developed by cognitive semantics [Gumperz and Levinson 1996; Talmy 2000].
The main elements of the thesaurus are concepts, notions, and terms [Voyshvillo 1989; Murphy 2002; Margolis and Laurence 2011]; the first two are semantic (substantive), whereas the third is semiotic (symbolic). They all relate in some way to an object being studied: the notion designates it, the concept provides a substantive interpretation of the notion, and the term denotes it with some symbol. A connection between a concept and an object is established by a definition of the notion; there are two main methods of definition. The intensional definition indicates the intrinsic properties of the object, while its extensional definition indicates its elements. In systematics, for example, the intensional definition of a taxon includes a list of the essential properties of the organisms allocated to it, while the extensional definition includes a list of these organisms.
The definitions of notions/concepts that compose a thesaurus should be complete, explicit, and strict for the conceptual space to be properly formatted: this is one of the key conditions of the principle of constructiveness [Voyshvillo 1989]. Completeness means that the definition provides exhaustive meaningful characteristics of the biological phenomenon reflected by the respective notion. Strictness presumes one-to-one correspondence between the notion and the object under a variety of possible conditions supposed by the cognitive situation. Explicitness means that the concept/notion should be not presumed but clearly defined, either directly in the given thesaurus or with reference to a more general thesaurus that includes this one. Indeed, in order to discuss effectively (constructively) the problems of species or homology, all concepts related to them must be defined in the manner just outlined; in ordinary language, this means to “agree on words.” Exceptions are undefined concepts introduced in the cognitive situation as basic; in systematics, for example, these are concepts of biota, biodiversity, organism, for which it is just assumed that researchers using the respective terms mean (approximately) the same natural phenomena.
However, in the natural sciences, these requirements are never met for several reasons, of which the main one is the fundamental inaccessibility of one-to-one correspondence between a natural phenomenon and its conceptual reflection in a cognitive situation. The boundaries of inaccessibility are established by the principle of the inverse relationship between rigor and meaningfulness of notion [Kuraev and Lazarev 1988; Voyshvillo 1989]: the more strictly the latter is defined, the less likely there is something in nature to which it may exactly correspond. The reason is that the rigor of definition is in close conjunction with formalization, and the latter is the reverse of substantiveness; mathematical notions are most strictly defined, but they are pure abstractions, which are not supposed to relate to something in Nature [Perminov 2001]. In systematics, this contradiction occurs, for example, in an aspiration to define the concept of monophyly or species as strictly as possible: intuitively more or less obvious, they appear less and less applicable in practice as they become more and more rigid [Pavlinov 1990, 2007b, 2018; Hull 1997; Holynski 2005].
One of the important causes of the non-rigid character of taxonomic notions/ concepts is that their definitions are context-dependent. This means that a concrete definition of a particular notion is given not for all “possible worlds,” but for a particular cognitive situation, and it depends on the entire contents of the basic thesaurus “enveloping” it. Therefore, some general notion (phylogeny, monophyly, species, homology, etc.) is split into particular interpretations whose meanings depend on the context of the particular thesauruses.
It follows from the above that definitions of notions/concepts shaping the conceptual space in systematics are inevitably fuzzy, contextually dependent, and therefore metaphorical to an extent [Murphy 2002; Pavlinov 2018]; therefore, the notions elaborated by systematics cannot be considered rigid designators in the sense of S. Kripke [Kripke 1972]. Thus, the semantics of systematic thesaurus is adequately described by a probabilistic model of the natural science language [Nalimov 1979], with proper formalizations provided by fuzzy logic [Zadeh 1992; Kosko 1993]. This means that, for each concept, it is possible to fix more or less strictly only its “core” using the logical relation “A is B,” while its periphery remains fuzzy, presuming multiple contextual interpretations of the type “A may be Bl, B2, B3...” As a result, each notion/ concept in systematics occurs as a set of particular context-dependent interpretative definitions; with this, the latter should be strict only to the extent it is really needed within particular cognitive situations [Holynski 2005; Pavlinov 2010b, 201 la, 2018].
A fundamental property of the conceptual space is its hierarchical structure, which is determined by how concepts are introduced into the cognitive situation through their respective definitions [Hempel 1965; Quine 1996; Hacking 1983]. This hierarchy is explained by the epistemic principle of theory incompleteness, according to which no particular theory (as a conceptual system) can be exhaustively defined within its own thesaurus: for such a definition, a certain metatheory is needed, in terms of which the basic concepts of this particular theory are interpreted [Antipenko 1986; Perminov 2001]; in this case, theory and concept are equivalent as definable elements of the general conceptual space. This principle follows from the logical genus-species scheme, in which each particular concept is defined consistently as “species particular” in the context of the “generic common.”
For a formal representation of this hierarchy, the so-called conceptual pyramid can serve as a suitable metaphor; its forerunner is the “pyramid of concepts” of Medieval scholasticism [Makovelsky 2004; Pavlinov 2018]. The top of this “pyramid” corresponds to a framework theory (concept) of the most general order, lower levels correspond to more particular theories and concepts—and so on down to the lowermost levels of operational concepts. The “pyramidal” character of such a hierarchy is a consequence of the fact that, moving from initial to final links of the interpretation chain, the number of concepts gradually increases at each step. The reason is that the number of particular interpretations of each concept is always more than one at any level of generality; therefore, there are always less general concepts exceed in number more general ones. For example, in systematics, the general notion of affinity can be interpreted as either similarity or a kinship relation, while the latter can be defined in different ways depending on how the phylogeny is interpreted.
The metaphor under consideration presumes the following important features of the hierarchical structure of the conceptual space to be emphasized.
Firstly, concepts of lower levels can be meaningfully defined only in the context set at higher levels of the “pyramid” [Carrier 1994; Quine 1996]. This, in particular, is true for operational concepts and classification algorithms: neither they themselves nor the classifications elaborated with them have a fixed biological meaning, unless it is specified by some biologically sound concept (theory) of higher levels. This observation concerns classifications based on strictly empirical (in particular, phenetic) approximations, i.e., on similarity relations as such. This is because, outside of the context set by certain biologically sound concepts, in which these relations are substantively interpreted by indicating their biological causes, their specific biological meaning remains undefined. Indeed, operationally defined groups (phenons) can correspond to species or to biomorphs or to certain intraspecific groups (such as age stages or castes in insects). So these phenons as such can hardly be considered biologically meaningful without an indication of the particular elements of the structure of biota they correspond to, which requires prior substantive definitions of these elements.
Secondly, various particular (subordinate) theories and concepts that figure in systematics turn out to be interconnected through more general theories/concepts that unite them into a single conceptual system. This is provided by their meaningful cross-interpretations in terms of certain metatheories of higher levels of generality. For example, in phylogenetics, the possibility of cross-interpretation of particular meanings of kinship is provided by setting particular definitions of phylogeny, referring to the latter’s general understanding as one of the manifestations of the evolutionary development of biota.
-  The epistemic principle of theory incompleteness originates from the incompleteness theorem deduced within one of the versions of formal arithmetic by the mathematician Kurt Godel.