Let’s make a conjecture: the Universal Object (the UO) with respect to a group of individuals (a, b, c...) possesses only those properties which are common to the all corresponding individuals (a, b, c...) (subordinated to the UO). If any property does not belong to all 'individual' objects of a certain group but only to some of them, then the 'general' object corresponding to a given group of 'individual' objects cannot possess this property.

If an object a possesses the property P (but objects b, c,... do not possess that property), then the UO cannot possess the property P, because the property P is not common to the all individuals subordinated to the UO.

If the UO does not possess the property P then the UO has to possess the property of not possessing the property P (on the basis of the Principle of the Excluded Middle). But the property of not possessing the property P is not common to the all individuals subordinated to the UO too, because the object a does not possess that property. Therefore the UO cannot possess the property of not possessing the property P, because - according to the „conjecture” - the UO can possess only those properties which are common to the all individuals subordinated to the UO.

Then the UO possesses the property of not possessing the property P and - simultaneously - the UO does not possess that property. Here is a contradiction !

Lesniewski wrote in 1927: "I devoted a passage of the work entitled 'The Critique of the Logical Principle of the Excluded Middle' (see op. cit., pp. 317-320) to a critique of the concepts of the 'general objects'[...]. Attempting to prove that 'no object is a 'general object', I stated in that passage that regardless of the specific forms which the 'general object' takes according to various thinkers[...] - those objects possess, for such authors, a certain characteristic peculiarity consisting in this, that 'the object', which is allegedly a 'general object' with respect to some group of 'individual' objects, can possess only those features which are common to all the 'individual' objects corresponding to them.[...] In connection with that passage and with reference to all those who, by reason of the meaning they give to expressions of the type 'general object with respect to objects a', are inclined to state the proposition 'if X is a general object with respect to objects a, X is b, and Y is a, then Y is b', I wish to state here that this proposition entails the proposition 'if there exist at least two different a, then a general object with respect to objects a does not exist', in accordance with the following schema:

1. if X is a general object with respect to objects a, X is b, also Y is a, then Y is b. (assumption) from 1. it result, that
2. if X is a general object with respect to objects a, X is different from Z, and Z is a, then Z is different from Z, and
3. if X is a general object with respect to objects a, X is identical with Z, and Y is a, then Y is identical with Z; from 2. it follows, that
4. if X is a general object with respect to objects a, and Z is a, then X is identical with Z, from 4. however, that,
5. if X is a general object with respect to objects a, Z is a, and Y is a, then (if X is a general object with respect to objects a, X is identical with Z, and Y is a); from 5. and 3. it follows, that,
6. if X is a general object with respect to objects a, Z is a, and Y is a, then Y is identical with Z, from 6. however, that,
7. if there exist at least two different a, then a general object with respect to objects a, does not exist.[...]

St. Lesniewski, "O podstawach matematyki", Przeglad Filozoficzny 1927 XXX, z 2/3

To download a short fragment of E. Luschei's The Logical Systems of Lesniewski (A Reconstruction of Lesniewski's Argument against Universals): looschei.pdf

Mariusz Grygianiec

Mariusz Grygianiec - Institute of Philosophy, Warsaw University, POLAND.

mariuszg@gazeta.pl