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The principle of the indiscernibility of identicals requires no restrictions


There is a certain argument against the principle of the indiscernibility of identicals (PInI), or the thesis that whatever is true of a thing is true of anything identical with that thing. In this argument, PInI is used together with the self-evident principle of the necessity of self-identity (“necessarily, a thing is identical with itself”) to reach the conclusion , which is held to be paradoxical and, thus, fatal to PInI (in its universal, unrestricted form). My purpose is to show that the argument in question does not have this consequence. Further, I argue that PInI is a universally valid principle which can be used to prove the necessity of identity (which in fact is how the argument in question is usually employed).

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  1. This is how the argument is standardly taken (especially in its Kripkean form), i.e., not as a refutation (by reductio) of PInI but as a validation of the necessity of identity. See, for instance, Maunu (forthcoming).

    Much of the material in this paper is familiar from the literature. My motivation for writing this paper is that even some renown philosophers seem to misconceive PInI (thus misleading students by their writings). Furthermore, I have noticed (via personal communication) that some philosophers find it hard to understand that the necessity of identity thesis has nothing to do with rigid designation for it can be expressed by means of definite descriptions (see the end of Sect. 5 below).

  2. For example, Morris (2006, p. 118): “It is a basic law of identity that if a is the same thing as b, whatever is true of a is true of b. That means that if we begin with a truth about an object, in which the object is referred to by one name, we should still have a truth if we refer to the same object by a different name.”

    Taylor (1998, pp. 45–46): “It is surely a reasonable principle that if a = b, then whatever property a has, b must have too. This is Leibniz’s principle of the indiscernibility of identicals. But, as soon as one states this seemingly unproblematic principle, there arise apparent counterexamples. ... [Russell’s example ‘George IV wished to know whether Scott was the author of Waverley’ ....] Frege’s approach attempts to preserve the logical sanctity of Leibniz’ law .... Russell too seeks to preserve the logical sanctity of Leibniz’ law ....” (Those who think, as Taylor apparently does, that PInI is in need of some kind of “saving”, must confuse PInI with PS.)

    Simons (1998, pp. 678–79): “[PInI] is uncontroversial, but needs careful formulation to exclude non-extensional contexts. For example, in ‘John believes that x defeated Mark Antony’, substituting the names ‘Octavian’ and ‘Augustus’ for x may yield different truth-values ...”.

  3. For example, Plantinga (1974, p. 15): “if x has P essentially, then the same claim must be made for anything identical with x. If 9 is essentially composite, so is Paul’s favourite number, that number being 9. This follows from the principle sometimes called ‘Leibniz’s Law’ or ‘The Indiscernibility of Identicals’:

    1. (3)

      For any property P and any objects x and y, if x is identical with y, then x has P if and only if y has P.

    Like Caesar’s wife Calpurnia, this principle is entirely above reproach. [Footnote:] Apparently Leibniz himself did not clearly distinguish (3) from:

    1. (3’)

      Singular terms denoting the same object can replace each other in any context salva veritate

    a ‘principle’ that does not hold for such excellent examples of language as English.”

  4. However, possibly Jacquette, despite his just-quoted statement, is confused, and is conflating PInI and PS; and he may be talking about logical necessity, understood in such a manner that only PS is relevant. But then his conclusion does not concern PInI but is only that co-referential terms cannot always be interchanged salva veritate, which is hardly news.

  5. This is Jacquette’s “\(\hbox {F}_{5}\)”.

  6. Jacquette may be seen as begging the question against the necessity of identity in his statements such as “the property is intuitively that of being logically necessarily identical to entity a, a property which certainly a has but b does not have, when it is only logically contingently true that \(a = b\)” (Jacquette 2011, p. 108).

  7. Cf. Kripke (1980, p. 3): “If ‘a’ and ‘b’ are rigid designators, it follows that ‘\(a = b\)’, if true, is a necessary truth. If ‘a’ and ‘b’ are not rigid designators, no such conclusion follows about the statement\(a = b\)’ (though the objects designated by ‘a’ and ‘b’ will be necessarily identical).” See also Maunu (forthcoming).

  8. I thank the anonymous referees of Synthese for useful suggestions that improved this paper.


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Maunu, A. The principle of the indiscernibility of identicals requires no restrictions. Synthese 196, 239–246 (2019).

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  • Indiscernibility
  • Leibniz’s law
  • Substitutivity
  • Intensionality
  • Necessity of identity