THE FORMALIZATION OF IDENTITY
If there are two or several occurrences of the symbols a or A in the same formula, this does not mean that they necessarily denote one and the same thing. On the other hand, different subformulae can denote one and the same thing. I was born in a town, and you were born in a town. But probably we were born in different places. In 1994 I was at Asilomar, and in 1994 you were at the conference center in California. We were at the same place.
In those cases, when it is known that various occurrences of the same or of different subformulae denote the same object, this fact should be expressed with the aid of additional symbols. There is no need to include those symbols in the list of the WFF's types and to increase their number, since formulae with identifying symbols can be formally defined through WFF listed above as abbreviations.
Defining identity we should take the direction of the identification into account. This may seem strange, since identity is a symmetric relation. However, it very often happens that it is not entirely a matter of indifference from which side we approach identity. Because of pollution Odessa newspapers identify the odor of the Black Sea with the smell of the toilet. It is bad of course. Zwanetskii has changed the direction of identification. He insists that in Odessa the odor of the toilet is the same as the smell of the sea. It is good! For us it is
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particularly important that without distinguishing directions of identification we could not distinguish the operations of synthesis and analysis from one another.
Let us use the small Latin letter j (jay) in italics for denotation of an object with which the identification is being carried out: jA . That bold faced letter in front of the formula denotes an object being identified: jA . The assertion of the identity of any object to any object will have the form: jAjA . In particular: {jAja}, {jaja}, {jajt}, etc.
We shall construct a formal definition of identity if we find as a definiens such a WFF which corresponds to the idea of identity according to some methodologically acceptable identity principle.
As such a principle, let us take the principle which was formulated by Aristotle and is usually called Leibniz's principle: "That which is said about one thing should be said about the other" (Aristotle).
According to that principle we can write: (jA)[(jA*)A].
(a) This is a necessary but not a sufficient condition of identity. Another necessary condition is (jA)[(jA*)A].
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(b) Let us combine the conditions (a) and (b) by closing (b) and putting [(jA )[(jA*)A]] instead of jA into (a). We receive:
(jA)[([(jA)[(jA*)A]]*A] (3.1)
If it is known that the formula (3.1) is true, we can omit in it the symbols j, j because that formula can be true only if A in the first, second and third occurrences denotes one and the same object. Regarding jAjA as the definiendum we receive:
jAjA =def (A)[([(A)[(A*)A]]*)A] (3.2)
The definiens of our definition is a WFF. It expresses the concept of identity by Aristotle.*
In particular we have:
jajt =def(a)[([(t)[(a*)A]]*)A (3.2’)
jaja=def(a)[([(a)[(a*)A]]*)A (3.2’’)
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* Note that there are great difficulties in expressing the identity principle in the Predicate Calculus (Tondl, 1973).
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Formulae with jay operators are open. But we will often have to deal with a closed identification, when we are concerned with concepts about objects which are identical to other objects. In this paper we shall restrict ourselves to the case of an undirected closed identity. We shall express it with the help of the Greek letter ι (iota) in front of formulae denoting identified objects. For various types of WFF we can introduce a series of definitions, using jay operators which have already been defined, e.g.:
(ι A)ι A =def([jAjA])[jAjA] (3.3)
(ι A*)ι A =def ([jAjA]*)[jAjA] (3.4)
In some cases in order to avoid ambiguity in definitions we make use of marks, e.g., the formula ((ia)a)ia can be marked out as ((iax)a + )tao-It's definition should be:
((ι ax)a+)ι a0 =def ({[jaxja0])a+)[jaojax} (3.5)
Note that marks can be eliminated with the aid of WFF constructed in a special way which are called Localizers. They fix the place of the given sub-formula in a formula.
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Not one, but many identifications can be postulated in one and the same formula. In the case when there are two or more groups of occurrences of subformulae in a given formula, such that every occurrence of a single group denotes the same object, but occurrences of different groups may denote different objects, several identifying operators must be used. In order to obtain the necessary variety in these operators, the letter iota can have various subscripts, be doubled, tripled, etc. For example, the following formulae can occur:
([(ιA)ιιA])[(ιA)ιιA]
([(ιιιA*)ιιA]){ιA , ιιA, ιιA, ιιιA, ιa}
The elimination of double, triple and so on iota operators is carried out in such cases just as for single ones. It follows from the definitions that the total number of iota operators of one kind (including direct and inverse operators of this kind) in one and the same formula must
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be at least two. e.g. ιa is not a WFF, since it contains only one iota-operator.
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