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Identity Of Indiscernibles




The principle is also known as Leibniz's law since a form of it is attributed to the German philosopher Gottfried Wilhelm Leibniz . It is one of his two great metaphysical principles, the other being the Principle Of Sufficient Reason . Both are famously used in his arguments with Newton and Clarke in the Leibniz-Clarke Correspondence .

Associated with this principle is also the question as to whether it is a Logical principle, or merely an Empirical principle.


IDENTITY AND INDISCERNIBILITY

There are two principles here that must be distinguished (two equivalent versions of each are given in the language of the predicate calculus).???

#The indiscernibility of identicals
  • For any x and y, if x is identical to y, then x and y have all the same properties.

  • : orall x orall y[x=y ightarrow orall P(Px \leftrightarrow Py)]

  • For any x and y, if x and y differ with respect to some property, then x is non-identical to y.

  • : orall x orall y[

  • eg orall P(Px \leftrightarrow Py) ightarrow x

eq y]
#The identity of indiscernibles
  • For any x and y, if x and y have all the same properties, then x is identical to y.

  • : orall x orall y[ orall P(Px \leftrightarrow Py) ightarrow x=y]

  • For any x and y, if x is non-identical to y, then x and y differ with respect to some property.

  • : orall x orall y [x

  • eq y ightarrow

eg orall P(Px \leftrightarrow Py)]

Principle 1. is taken to be a logical truth and (for the most part) uncontroversial. Principle 2. is controversial. Max Black famously argued against 2. (see Critique , below).

Note that these are all Second-order expressions. Neither of these principles can be expressed in First-order Logic .


CONTROVERSIAL APPLICATIONS

One famous application of the identity of indiscernibles was by René Descartes in his '' Meditations On First Philosophy ''. Descartes concluded that he could not doubt the existence of himself (the famous '' Cogito Ergo Sum '' argument), but that he ''could'' doubt the existence of his body. From this he inferred that the person Descartes must not be identical to his body, since one possessed a characteristic that the other did not: namely, it could be known to exist.

This argument is normally rejected by modern philosophers on the grounds that it derives a conclusion about what is true from a premise about what people know. What people know or believe about an entity, they argue, is not really a characteristic of that entity. Numerous counterexamples are given to debunk Descartes' reasoning via '' Reductio Ad Absurdum '', such as the following argument based on a Secret Identity :

#Entities ''x'' and ''y'' are identical if and only if any predicate possessed by ''x'' is also possessed by ''y'' and vice versa.
#Clark Kent is Superman's secret identity; that is, they're the same person (identical) but people don't know this fact.
# Lois Lane thinks that Clark Kent cannot fly.
#Lois Lane thinks that Superman can fly.
#Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly.
#Therefore, Superman is not identical to Clark Kent.
#Since in proposition 6 we come to a contradiction with proposition 2, we conclude that at least one of the premises is wrong. Either:
  • Leibniz's law is wrong; or else

  • A person's knowledge about ''x'' is not a predicate of ''x'', thus undermining Descartes' argument.



CRITIQUE

Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it is sufficient that one provide a Model in which there are two distinct (non-identical) things that have all the same properties. He claimed that in the symmetric universe where only two symmetrical spheres exist, the two spheres are two distinct objects, even though they have all the properties in common. ???


NOTES AND REFERENCES


  • ''Metaphysics an Anthology''. eds. J. Kim and E. Sosa, Blackwell Publishing, 1999

  • Lecture notes of Kevin Falvey / UCSB



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