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The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' is also possessed by ''y'' and vice versa; to suppose two things indiscernible is to suppose the same thing under two names. It states that no two distinct things (such as snowflakes) can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the indiscernibility of identicals, discussed below. A form of the principle 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. Because of its association with Leibniz, the principle is sometimes known as Leibniz's law. (However, the term "Leibniz's Law" is also commonly used for the converse of the principle, the indiscernibility of identicals (described below), which is logically distinct and not to be confused with the identity of indiscernibles.) Some philosophers have decided, however, that it is important to exclude certain predicates (or purported predicates) from the principle in order to avoid either triviality or contradiction. An example (detailed below) is the predicate that denotes whether an object is equal to ''x'' (often considered a valid predicate). As a consequence, there are a few different versions of the principle in the philosophical literature, of varying logical strength—and some of them are termed "the strong principle" or "the weak principle" by particular authors, in order to distinguish between them. Willard Van Orman Quine thought that the failure of substitutivity in intensional contexts (e.g., "Sally believes that ''p''" or "It is necessarily the case that ''q''") shows that modal logic is an impossible project.〔Quine, W. V. O. "Notes on Existence and Necessity." ''The Journal of Philosophy'', Vol. 40, No. 5 (March 4, 1943), pp. 113–127〕 Saul Kripke holds that this failure may be the result of the use of the disquotational principle implicit in these proofs, and not a failure of substitutivity as such.〔Kripke, Saul. "A Puzzle about Belief". First appeared in, ''Meaning and Use''. ed., A. Margalit. Dordrecht: D. Reidel, 1979. pp. 239–283〕 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 (equivalent versions of each are given in the language of the predicate calculus).〔 Note that these are all second-order expressions. Neither of these principles can be expressed in first-order logic. #The indiscernibility of identicals # *For any ''x'' and ''y'', if ''x'' is identical to ''y'', then ''x'' and ''y'' have all the same properties. # *: #The identity of indiscernibles # *For any ''x'' and ''y'', if ''x'' and ''y'' have all the same properties, then ''x'' is identical to ''y''. # *: |- | colspan=2 | |- | colspan=2 | |} Principle 1 doesn't entail reflexivity of = (or any other relation ''R'' substituted for it), but both properties together entail symmetry and transitivity (see proof box). Therefore, Principle 1 and reflexivity is sometimes used as a (second-order) axiomatization for the equality relation. Principle 1 is taken to be a logical truth and (for the most part) uncontroversial.〔 Principle 2, on the other hand, is controversial; Max Black famously argued against it. (see Critique, below). The above formulations are not satisfactory, however: the second principle should be read as having an implicit side-condition excluding any predicates that are equivalent (in some sense) to any of the following: #"is identical to ''x''" #"is identical to ''y''" #"is not identical to ''x''" #"is not identical to ''y''" If all such predicates are included, then the second principle as formulated above can be trivially and uncontroversially shown to be a logical tautology: if ''x'' is non-identical to ''y'', then there will always be a putative "property" that distinguishes them, namely "being identical to ''x''". On the other hand, it is incorrect to exclude all predicates that are materially equivalent (i.e., contingently equivalent) to one or more of the four given above. If this is done, the principle says that in a universe consisting of two non-identical objects, because all distinguishing predicates are materially equivalent to at least one of the four given above (in fact, they are each materially equivalent to two of them), the two non-identical objects are identical—which is a contradiction. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Identity of indiscernibles」の詳細全文を読む スポンサード リンク
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