Saturday, January 5, 2013

Leibniz's Law


G.W. Leibniz (1646-1716) says in Section IX of his Discourse on Metaphysics (Discours de Métaphysique, 1686) that no two substances can be exactly alike. This is known as Leibniz's Law. Another way of expressing this is: No two substances can be exactly the same and yet be numerically different. If two substances were exactly the same, then they would be the same substance and would not be two separate substances. 
      Leibniz's Law (that no two things can share all their properties in common) can be expressed in a positive way as follows: if two things are identical, then they share all their properties in common (this metaphysical principle is called the indiscernibility of identicals), and conversely, if two things share all their properties in common, then they are identical (this metaphysical principle is called the identity of indiscernibles). According to the indiscernibility of identicals, if two things are identical, then no difference between them is discernible, and according to the identity of indiscernibles, if no difference is discernible between two things, then they are identical.
      An objection that might be raised to the identity of indiscernibles is that if two things are superimposed on each other, then they might temporarily share all their properties in common and yet not be the same. It might not be possible to discern that they are two things, rather than one. An objection that might be raised to the indiscernibility of identicals is that if two things are, to any means of detection, identical, then they might still differ from each other in a property that is undetectable. It might not be possible in practice to detect any difference between them. 
      Thus, a distinction may need to be made between theoretical and practical discernibility. The difference between two nonidentical things may in some cases not be practically discernible or verifiable. Similarly, the sameness of two things may in some cases not be practically discernible or verifiable.             
      Leibniz's Law can be expressed symbolically as (x)(y) [x=y → (F)(Fx ↔ Fy)], which may be read as "for every x and for every y, if x is identical to y, then every property F that is possessed by x is also possessed by y, and every property F that is possessed by y is also possessed by x" (this is the indiscernibility of identicals), and conversely as (x)(y) [(F )(Fx ↔ Fy) → x=y], which may be read as "for every x and for every y, if every property F that is possessed by x is also possessed by y, and every property F that is possessed by y is also possessed by x, then x is identical to y" (this is the identity of indiscernibles).
      The philosopher Max Black (1952) offers several arguments against the principle that if no difference is discernible between two things, then they are identical. He argues that two things cannot be identical, since if they were, then they would be only one thing, and not two. If we say that a is identical to b, then we are merely using two different names to refer to the same thing. And if a and b are merely two different names for the same thing, then when we say that "a is identical to b," we are merely saying that "a is a," which is a tautology. The principle that "If there is no difference between a and b, then they are the same" is trivial. And if there were a universe consisting of two exactly similar spheres, then conceivably two things could share the same properties and still not be the same, and thus the identity of indiscernibles would again be put into question.
1
      It may, however, be worth noting that two things may be similar to, or the same as, each other in possessing many distinct kinds of properties. Identity between two things may involve material, formal, spatial, temporal, relational, and other kinds of properties.
      Can an exact duplicate or replica of something be properly called "identical to" or "the same as" that thing? If so, why may there still be some doubt or uncertainty about whether the two things are alike in every respect? What may happen to the identity of the two things as they change over a period of time?
      Do changes in the properties of things always change the natures of those things? Moreover, do changes in the physical, intellectual, emotional, or social attributes of a person always change the nature of that person?  Are you the same person that you were 5 minutes ago? 7 days ago? 5 years ago?
      Surely, there must be some properties that are relevant to sameness, and some that are irrelevant. Should we then relativize or qualify the indiscernibility of identicals by saying that in order for two things to be the same, they must share all properties that are essential or relevant to their sameness? Some properties may be essential to the identity of two things, while other properties may be unessential. 

1Max Black, "The Identity of Indiscernibles," in Mind, Vol. 61, No. 242 (April, 1952) pp. 153-164.

No comments:

Post a Comment