¹ Russell, B., Human Knowledge, Its Scope and Limits, London 1948, p. 78.

² If the other view were assumed concerning individual names, the definition sentence would have to run “N = A”, e.g., “John Smith is identical with this man” (and not, as before, “This man is John Smith”). But it must be borne in mind that the problem of ostensive definability of individual names is a controversial issue. Cf., for instance, Xenakis, J., “Function and Meaning of Names,” Theoria, vol. XXII, 1956.

³ Of these two schemata, (2) and (3), type (2) is considered to be the basic one. Let it also be noted that the symbol “sim” is an abbreviation of “similar”.

⁴ This statement might be called a definitional assumption since ostensive definition performs its function, i.e., establishes a relation between the term defined and the scope of its applicability, on the condition only that statement (4) is true. Usually, ostensive definition is resorted to only by those who adopt the definitional assumption on which a given definition is based.

⁵ Cf. Ajdukiewicz, K., Zarys Logiki (Outline of Logic), Warsaw, 1955, p. 192 ff. It must be noted that this book discusses a solution of the problem in general, without paying any special attention to problems of defining.

⁶ A complete solution of the definition problem would consist in finding a condition that would be sufficient and at the same time necessary. A complete and a partial solution of the definition problem could be distinguished from the point of view of the first interpretation, too. The characteristic given on p. 5 refers to a complete solution. A partial solution would consist in finding correct substitutions for variables not in the definition statement itself, but in some of its consequences, namely those which differ from the definition statement only in this, that the symbol of equivalence is replaced by the symbol of implication in either direction. This means that it would consist in finding sentences determining partial criteria.

⁷ This condition is treated, as the usage is, not only as a condition of efficacy of the defining operation, but also as a condition without which ostensive definition would not be “what it is”, i.e., would not be an ostensive definition at all.

⁸ It is perhaps worth mentioning that the lesser the scope, so to speak, of the operation recommended by the given instruction, the greater the eliminating force of that step. For instance, the instruction “Smell it” eliminates all sense qualities except those which belong to the scope of the olfactory, since such an operation usually serves only the purpose of perceiving the various smells. On the other hand, the instruction “Touch it” has a much lesser eliminating value, since by touch we perceive temperature of objects, their consistence, shape, type of surface, etc. Further elimination is in such cases usually carried out by an inductive method, to be discussed below.

⁹ Johnson, W. E., Logic, Part I, Cambridge, 1921, p. 4.

¹⁰ The notion of induction requires here comments analogous to those given before in connection with the notion of a solution of the definition problem (p. 5-6). To avoid misunderstandings we must explain that induction must here be interpreted broadly enough so as to cover those cases, too, in which reasoning is based not on verbal premises but on perception, and the conclusion need not necessarily be a general sentence. As an example of induction interpreted so broadly we could give a mental process which leads the recipient of the definition to a correct solution of the definition problem even if neither clearly formulated premises nor a clearly formulated conclusion come in question. The very fact of a correct use of the term defined seems to prove that we have to do with some generalization: the recipient of the definition behaves as if he based himself on a general statement and applied it to particular cases, as if he understood clearly the criteria of applicability of the term in question. In view of that similarity to induction in the narrower sense it seems legitimate to extend to notion of induction so as to cover the cases mentioned here; the more so since such an interpretation of induction is in conformity with the common practice of using that term. It can easily be seen that the terms “reasoning”, “premises” and “conclusion” have been used here in sense broader that those to be found usually in school textbooks of logic. This broadening of meaning follows the same direction as in the case of induction and solution of problems.

We, of course, fully realize the vagueness of notions characterized as they have here been. But it is difficult to abstain, when discussing the problems under consideration, from referring to the pragmatic aspect of language, and, as is known, that pragmatic aspect defies attempts to be formulated clearly and with precision. We prefer, however, rather to risk being blamed for vagueness than to pass over in silence those matters which seem essential.

¹¹ Since the statement that a certain degree of similarity is not sufficient for an object to be N results from the statement that the given degree of similarity is sufficient for an object not to be N, that is, to be non-N, therefore the reasoning here is analogous to the former case. The assumption on which that reasoning is based, and which is a consequence of the definition statement on the strength of meaning of the terms appearing in that statement, can briefly be formulated as follows: if a greater similarity is not sufficient, a lesser one is also not sufficient.

¹² The general concepts of the efficacy and economical character of action are taken from Traktat o dobrej robocie (Treatise on Good Work) by T. Kotarbinski (Łódź 1957).

¹³ These problems have been dealt with in our paper “Definicja” (“Definition”), Studia Logica, vol. II, Warszawa, 1955.

¹⁴ Or a similar form, corresponding to a formulation in a metalanguage, in which definition directly refers to the term being defined, names it, and ascribes it a definite semantic function. These differences in formulating definitions do not seem essential at all.

¹⁵ If, despite this fact, it is difficult to resist the impression that by formulating an ostensive definition we after all state something by means of such a definition, then the reason must be sought in our taking into consideration not only the definition statement, but also the definitional assumption on which it is based and which, therefore, is associated with it. Let us recall that the definitional assumption differs from the definition statement only in this, that the variables which are free in the definition statement are in the definitional assumption bound by the existential quantifier (see p. 4) E.g., if the definition statement is of the type “” then the statement which it assumes is a sentence of the type “”

¹⁶ If this condition is not satisfied, we can speak, following Johnson's example, of denotative definition. This would be broader than ostensive definition, regardless of whether they refer to standards that are known to the recipient from his experience or not.

¹⁷ The symbol “... x ...” can, of course, stand in each of these implications for a different sentential function including “x” as the only free variable.

The notion of conditional definition was introduced by R. Carnap in Testability and Meaning, Philosophy of Science, vols. III & IV, 1936-7.

¹⁸ This schema is a generalization of schemata (2) and (3) on p. 4.

¹⁹ Robinson, R., Definition, Oxford 1950.