Definitions: the primitive concept of logics or The Leśniewski–Tarski legacy
E. G. K. López-Escobar, Francisco Miraglia · Dissertationes Mathematicae · 2002
Definitions, theses and hypotheses Definitions in antiquityDefinitions, in one form or another, have been with us since time immemorial.On the other hand it took the Socratic psyche to study them, classify them and make good use of them.Socrates and Plato first confronted the problem.For example in Laws X 895d we find 1 : ATHENIAN: In heaven's name, then, hold.You will grant, I presume, that there are three points to be noted about anything?CLINIAS: You mean?ATHENIAN: I mean, for one, the reality of the thing, what it is, for another the definition of this reality, for another its name.And thus you see that there are two questions we can ask about everything which is. CLINIAS: And what are the two?ATHENIAN: Sometimes a man propounds the bare name and demands the definition; sometimes, again, he propounds the definition by itself and asks for the corresponding name.In other words, we mean something to this effect, do we not? CLINIAS: To what effect?ATHENIAN: There is, as you know, bisection in numbers, as in other things.Well, in the case of a number, the name of the thing is 'even', and the definition, 'number divisible into two equal parts'. CLINIAS: Certainly.ATHENIAN: That is the sort of case I have in mind.We are denoting the same thing, are we not, in either case, whether we are asked about the definition and reply with the name, or about the name, and reply with the definition?It is the same thing we describe indifferently by the name 'even,' and the definition 'number divided into equal parts' ? CLINIAS: Identically the same.And what Socrates and Plato started, Aristotle made into a philosophical doctrine.For example consider the following passage from Anal.post.I. 2, 72a 14-24:Among immediate syllogistic principles, I call that a thesis which it is neither possible to prove nor essential for any one to hold who is to learn anything; but that which it is necessary for any one to hold who is to learn anything whatever is an axiom: for there are some principles of this kind, and that is the most usual name 1 From the translations given in [Hamilton and Cairns, 1996].