Abstract
The paper provides an analysis of two traditionally accepted distinctions between nominal and real definitions. The historical background is Kazimierz Ajdukiewicz’s conceptions of definitions as well as remarks by other members of the Lvov-Warsaw School of philosophy. The aim of the paper is to show that the analysed distinction, at least in Ajdukiewicz’s version, cannot be sustained.
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Notes
He even attempted—in a somewhat unintuitive way in our opinion—to make this distinction for a formalized language (Ajdukiewicz 1936).
Twardowski wrote: ‘The characterization of an object […] [consists in] mentioning one or several of its distinctive features.’ (Twardowski 1901: 80) Distinctive features are ‘features that enable us to distinguish some objects from others’ (Twardowski 1901: 65). On the other hand, ‘a description means listing discernible parts and discernible features of an imagined object. […] So each description includes a partial statement of the contents of a concept of the object which we imagine; this statement is partial since it only concerns discernible features. And a description differs from a definition in that (1) it is limited to discernible features; therefore, when we describe a dog, we cannot say e.g. that it is a domesticated animal since we do not discern this feature when we imagine a dog; (2) an object of description must be imagined simultaneously; therefore, one cannot talk about a description of God or an atom.’ (Twardowski 1901: 82) .
Cf. Kotarbińska (1955: 35); Kotarbińska uses the term ‘essential feature’.
By saying ‘a sentence mentioning feature c reserved for object x’ we mean a sentence stating that feature c is possessed by object x.
This impression is toned down by Twardowski’s comment that ‘these are usually short definitions based on the relation between superior and subordinate concepts’ (cf. below, note 9). Consequently, the definition ‘A square is an equilateral rectangle’ is a brief way of saying that squares have all the features of rectangle and, additionally, they are equilateral.
For the sake of simplicity here and below we will reduce formulas like ‘name x signifies…’ to the form ‘x = …’.
We modify these conventions considerably since—unfortunately—their original formulation is burdened with different faults. Here is the most important statement of Twardowski on this subject. Twardowski believed that a certain object could be presented by means of different concepts. ‘And since a concept is a meaning of a word mentioning objects that are subordinate to this concept […], then different meanings can be related to a certain word.’ (Twardowski 1901: 75) ‘Definition serves to “establish” meanings of words and, consequently, concepts. Since different concepts of the same object (the same objects) are possible, a definition must choose among them and show the most appropriate concept which should always be used when presenting certain objects. Undoubtedly, the most appropriate concept of an object is the most precise one, that is, a concept by means of which one can present all features of an object. Therefore, a definition indicates those concepts, listing all the features of an object. Definition is a number of judgements stating what the features of an object (objects) are as represented by a certain name. Since all features of an object constitute the contents of the relevant concept […], one can say that a definition is a number of judgements stating what the features of the contents of a concept are as represented by a certain word. […] In many cases, […] definitions […] would have to be composed of a [considerable] number of judgements […], which would make them lengthy and difficult to remember. Therefore, usually definitions are given in a short form, based on a relation between the contents of superior and subordinate concepts.’ (Twardowski 1901: 76) Let us note that if (according to Twardowski) the concept of x is identified with a meaning of a name denoting (‘listing’) objects signified by (‘subordinate to’) the concept of x, this formula should be considered as burdened with a direct vicious circle.
Among others, he wrote: ‘If, however, we examine more closely the meaning of the terms “real definition” and “nominal definition”, we must come to the conclusion that it is not so, that the content of the concepts corresponding to those terms is not a specification of the content of some more general concept, and this means in turn that there is no general concept of definition of which the concepts of real definition and nominal definition would be specifications.’ (Ajdukiewicz 1958: 296) And further: ‘Hence it follows that the word “definition” which appears in the terms “real definition” and “nominal definition” has in isolation no meaning at all. If we use the word “definition” without any adjective, we use it elliptically and ambiguously, meaning either real definitions or nominal definitions […].’ (Ajdukiewicz 1958: 297).
Basically, here we mean normal definitions, that is, those with identity and equivalence as definitional functors. For equivalent definitions, an object definition would be of the structure: p, if and only if….
Other variants of language definitions—for identity definitions—would be of the structure: name ‘x’ means the same as name…. Equivalent language definitions would be of the structure: sentence ‘p’ states that… (or: sentence ‘p’ states the same as sentence…). Let us emphasize that in the above-mentioned variant, the language definitions, in fact, state a certain syntactic relation, namely, synonymy; using these definitions—replacing one expression (definiendum) with another (definiens) does not require knowing the meaning of either of these expressions. Therefore, we believe that the best variant of nominal definitions would be a variant that directly states the meaning of a defined name (as: connotation/denotation of name ‘x’ is…).
This not only concerns normal definitions but also—as we believe—e.g. axiomatic (pseudo-)definitions, at least as long as they can be paraphrased by means of normal definitions; we believe this is possible, although normal paraphrases would have a much more complicated form than their axiomatic originals.
Mortimerowa says here that definition is ‘a means of translation’ (1987: 80–81).
Here is how Ajdukiewicz phrased it: ‘One can formulate real definitions of objects of all types and orders: one can formulate clear-cut characterizations of singular objects, of classes, of properties (if they are distinguished from classes), of relations, etc. One can also formulate real definitions of words, since these too are objects which can be characterized univocally.’ (Ajdukiewicz 1958: 298).
Cf. comparable examples in (Ajdukiewicz 1963).
Cf. e.g. Ajdukiewicz (1959: 29).
References
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Acknowledgements
We would like to express our sincere thanks to all people involved in a discussion that developed after the text had been presented; and especially to Prof. Marek Lechniak and Prof. Jan Woleński, whose comments helped us to improve it.
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Brożek, A., Jadacki, J. Scrutiny of Ajdukiewicz’s concepts of definition. Stud East Eur Thought 68, 11–20 (2016). https://doi.org/10.1007/s11212-016-9244-y
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DOI: https://doi.org/10.1007/s11212-016-9244-y