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Definition I
P is the abstract object for individuals denoted by the term "N" iff P is the object possessing only those properties which are connoted by the term "N"
Definition II
P is the abstract object for individuals dennoted by the term "N" iff P is the object possessing only those properties which are common to the individuals dennoted by the term "N"
Let the term "N" connote properties: a, b, c. (Each term can only connote limited number of properties).
If an object possesses properties a, b, c it must possess the others, for instance m, n, r ...etc. (Each object possesses unlimited number of properties).
The abstract object for objects dennoted by the term "N" possesses only those properties which are connoted by the term "N". But the term "N" connotes only limited number of properties. Then the abstract object possesses only limited number of properties. Unfortunately, each object must possess unlimited number of properties. From it follows that there isn’t any object which possesses only limited number of properties. So there isn’t any object which is the abstract object.
Let the object X belong to dennotation of the term "N". The object X must possess the property t which isn’t possessed by any other object dennoted by the term "N". So, other object dennoted by the term "N", for instance, the object Y, doesn’t possess the property t. We can say that the object Y possesses a lack of the property t. This lack can be named the property t’. Then we can say that the object Y possesses the property t’.
There is a question: does the abstract object possess the property t or not?
If the abstract object possesses the property t then there occures a contradiction, because the property t isn’t a property which is common to the individuals dennoted by the term "N", so the abstract object can’t possess that property.
If the abstract object doesn’t possess the property t then it possesses a lack of the property t. Then it possesses the property t’. But the property t’ isn’t common to the individuals dennoted by the term "N". So the abstract object can’t possess the property t’ too. So there isn’t any object which is the abstract object.
Let the sign "u" be the sign of the property of the generality of properties which are common to the individuals dennoted by the term "N".Let the object Z be the object which belongs to the dennotation of the term "N".
There is a question: does the object Z possess the property u or not? Obviously not, the object Z cannot possess the property u (it means: it must possess not only those properties which are common to the individuals dennoted by the term "N"); it must be different from the other objects dennoted by the term "N".
But the abstract object for individuals which belong to the dennotation of the term "N" possesses the property u (according to the definition II). Then the abstract object possesses the property which isn’t common to the individuals dennoted by the term "N" (For instance, the object Z doesn’t possess that property). Then the abstract object possesses and - simulaneously - doesn’t possess the property u. So there’s a contradiction.
Let the property c be the property of not possessing itself.
There is a question: does the property c possess the property c or not?
If the property c possesses the property c then it possesses the property of not possesing itself, so it possesses the propery of not possessing of property c therefore it doesn't possess the propery c.
If the property c doesn't possess the property c then it doesn't possess itself, so it possesses the property of not possessing itself, therefore it possesses the property c.
There's a contradiction in two ways.
Mariusz Grygianiec
- Institute of Philosophy, Warsaw University, POLAND.