Simplifying the Indiscernibility of Identicals

equals - greenI’ve posted quite a few times here before about identity, and about the principle often called “Leibniz’s Law” – the Indiscernibility of Identicals. This is often put:

Necessarily, for any x and any y, x is identical to y only if for any P, x has P if and only if y has P. (Compare, e.g. Colin McGinn, Logical Properties (New York: Oxford University Press, 2000), 4-7.

I agree with other philosophers that this has apparent counterexamples, if “P” can stand either for any predicate or for any property.

But in my view, the underlying metaphysical intuition – that a thing can’t at one time be and not be a certain way – is undeniable. (And of course, it has important theological implications.)

I would restrict the “P” above to intrinsic properties, if I believed in properties. But I don’t. So I’ve been putting the principle like this:

The Indiscernibility of Identicals: Necessarily, any A and B are identical only if they (1) never have differed, (2) don’t differ, (3) will not ever differ, and (4) could not differ.

This formula doesn’t import any assumptions about property-theory, either for or against. Rather, it uses only a primitive concept of differing, or being different – as to qualitative aspect or way of being, not as to number. I think this well captures the intuition and fundamental conviction noted above.

But it now strikes me that the formula is needlessly complicated. Why not just this?

Necessarily, any A and B are identical only if they could not simultaneously differ.

As with any version of Leibniz’s Law, this is supposed to give us a necessary condition for identity, and a sufficient condition for non-identity. If A and B really are numerically the same, they of course “they” (really: it) can’t at any time differ from how it is at that time. There’s the necessary condition. How about the sufficient condition? If A and B have, do, or will differ, then of course they could differ. Actuality implies possibility. So actually differing at some time is sufficient for non-identity. But of course, merely possibly differing at a time – being such that they could simultaneously differ – is enough to prove non-identity too. This is just what the prior formulation said, then, in fewer words.

To spell it all out, the claim is: Necessarily, for any x and any y, x and y are identical only if it is not possible that there is a z such that z is a time or a point in timeless eternity at which x and y differ.

Philosopher-friends: what say you?

 Update, in response to Anthony’s perceptive comment – I meant this:

Necessarily, any A and B are identical only if they could not simultaneously or timelessly differ.

 

 

 

About Dale

Dale Tuggy is a Professor of Philosophy at SUNY Fredonia, where he teaches courses in analytic theology, philosophy of religion, religious studies, and the history of philosophy.

16 Responses to Simplifying the Indiscernibility of Identicals

  1. Anthony Bigg says:

    I don’t believe that there are counter-examples to the original formulation of LL presented here (I assume that the proposed counter-examples relate to facts about things having different properties at different times – I don’t think that such facts are at all compelling counter-examples).

    So, given that I reject Quine’s criterion of ontological commitment, I’m happy to adopt the original formulation, in spite of its quantifying over properties.

    But moving on to your formulation, it seems to me that it presupposes a specific metaphysics of time, namely substantivalism (as opposed to relationalism). For if relationalism about time is correct, then it’s being a truth of metaphysics, at least plausibly, entails that it is a necessary truth, which seems to in turn entail that substantivalism is necessarily false. But if that is so, then there can be no “times” as such, in fact, necessarily so. Rather there are only temporal relations. But if there can be no times, then a fortiori there can be no time at which x and y differ. So if relationalism about time is right everything is identical, as any pair objects will satisfy the first disjunct of the formulation. And it seems that a formulation of LL shouldn’t commit us to a particular view of temporal metaphysics.

    Maybe the relationalist can have “times” if she thinks that they’re abstract objects, but I don’t believe in such things and it seems as though I don’t exist at some abstract object.

  2. villanovanus says:

    I frankly fail to see how Dale, with his various versions, would “simplify” the formulation of the principle of the indiscernibility of identicals.

    Also, he doesn’t seem aware that what he has provided, at the top of his post, is NOT the standard formulation of the principle of the indiscernibility of identicals, BUT its reverse, the principle of the the identity of indiscernibles. See here:

    1. The indiscernibility of identicals: for any x and y, if x is identical to y, then x and y have all the same properties (predicates).

    2. The identity of indiscernibles: for any x and y, if x and y have all the same properties (predicates), then x is identical to y.

    Dale (or, maybe, his source, Colin McGinn) has simply changed the order of no. 2, preceding it with the adverb “necessarily”.

  3. villanovanus says:

    P.S. If you want a genuine simplification of the principle of the identity of indiscernibles (and of its reverse, the principle of the indiscernibility of identicals), see the SEP article “The Identity of Indiscernibles” (http://plato.stanford.edu/archives/fall2008/entries/identity-indiscernible).

    There are also interesting sections on the “Ontological Implications” and on “The Impact of Quantum Mechanics”.

  4. Anthony Bigg says:

    Hello villanovanus,

    You write: “he [Dale] doesn’t seem aware that what he has provided, at the top of his post, is NOT the standard formulation of the principle of the indiscernibility of identicals, BUT its reverse, the principle of the the identity of indiscernibles.” Thus Dale did not, contra your allegation, misrepresent the indiscernibility of identicals as the identity of indiscernibles.

    I think that your allegation that Dale failed to formulate the indiscernibility of identicals correctly is in error.

    Dale initially formulates the thesis as “Necessarily, for any x and any y, x is identical to y only if for any P, x has P if and only if y has P.” Note that [A only if B] is equivalent to [if A then B]. Your formulation of the indiscernibility of identicals is: “for any x and y, if x is identical to y, then x and y have all the same properties ” This is the same as Dale’s original formulation, just a tad less fancy, as both formulations amount to IF x and y are = THEN they have all and only the same properties (Though Dale’s, unlike yours, adds a degree of modal strength by adding “Necessarily” at the beginning)

    Thus Dale did not, contra your allegation, misrepresent the indiscernibility of identicals as the identity of indiscernibles.

  5. Dale says:

    Hi Anthony,

    Slap me and call me a dogmatist, but I think it is obvious that things change. So no, I’m not worried about apparent counterexamples in terms of different times. But I am worried about these sorts, to the standard formulas: http://trinities.org/blog/archives/3011 (Laverne case)

    Notice that in both of my formulas I avoid quantifying over times; I only use tensed language. I don’t want to take a position on whether there are times.

  6. Dale says:

    Yes, I wasn’t confusing the Ind of Id with the Id of Ind – Anthony explains why.

    I take no stand on the Identity of Indiscernibles. And yes, it has been much more controversial – it is not obvious that there can’t be two things which are *in some sense* indiscernible.

  7. Anthony Bigg says:

    Hey Dale,

    Right, I remember those types of counter-examples, I’d forgotten about them (I really don’t like their name, talk about pretentious and verbose, these counter examples are said to be instances of “the failure of the doctrine of the the inter-substitutability of co-referential terms salva veritate in intentional contexts”). I’ll have to give them some more thought.

    On the issue about times though. I think that the matter is independent of whether or not you quantify over times (it seems clear that you did not quantify over times, as the variable which is a time ultimately lies within the scope of a modal operator). I still think that your formulation is committed to the existence of times, not because of any issue pertaining to quantifying over them, but rather because if there are no times, the formulation renders the moon and George Bush numerically identical, but that’s absurd, so in the absence of times, the formulation is false. But if we posit times, then we can save the re-formulation. In this sense, I think that the formulation is committed to times.

    What I was saying was this: Let’s say there are in fact no times. This being a metaphysical truth, it is plausible that it is necessary. Now, if there are necessarily no times, then it is not possible that there be a time. If it is not possible that there be a time, then it is not possible that there be a time at which Bush and the moon differ (I guess it’s like saying, if it’s not possible that humans exist, then it’s not possible that humans who wear blue hats exist). But then, if there is no possible time at which Bush and the moon differ, by the re-formulation they are identical, as they don’t satisfy the first disjunct (There is no possible time for them to differ) and they don’t satisfy the second disjunct (They’re obviously not in timeless eternity)

    So as I see it, there are 3 possible corollaries of the re-formulation,

    1) Bush = The Moon (Crazy!)
    2) There exist times (Temporal substantivalism)
    3) There are no times, BUT, times are possible, and hence there’s a possible time at which Bush differs from the moon (Thus metaphysical truths are not necessary truths)

    Of course, one could just use the re-formulation as a premise to argue for 2) or 3), but I think that a formulation of the indiscernibility of identicals should be as metaphysically neutral as possible. That’s why I like the original formulation at the top of the post.

  8. Dale says:

    Another thing philosophers want Ind of Id to do, is to imply the transitivity of =. I think my principle still does this.

    Informal proof: Suppose x = y and y = z. We need to infer that x = z. My principle says things are identical only if it is not possible that they differ. Then, it is not possible that x and y differ. Neither is it possibly for y and z to differ. But differing from is transitive – if x can’t differ from y, and y can’t differ from z, then x can’t differ from z. Suppose now that this is false: x = z. It would follaw that x CAN differ from z, for they DO differ; x is identical to x, and z is not. But then, we must deny our supposition that it is false that x = z. So, it is true that x = z. In sum, my principle does imply that = is transitive.

  9. villanovanus says:

    @ Anthony.

    Sorry to insist, but you, Dale and (possibly) Colin McGinn have ALL got it upside-down.

    Here is McGinn’s presumed formulation (that is Dale’s, endorsed by you), itemized according to its “relevant bits”:

    [Necessarily] [A] for any x and any y, [C] x is identical to y [only] [B] if for any P, x has P if and only if y has P.

    “Necessarily” and “only” have been put in [square brackets] because they are obviously redundant, for the logical validity.

    Here is my formulation (actually, Wikipedia > Identity of indiscernibles, source SEP) for no.2 (identity of indiscernibles):

    [A] for any x and y, [B] if x and y have all the same properties (predicates), [C] then x is identical to y.

    I trust there is no doubt that the second formulation of [B] (“if x and y have all the same properties/predicates”) is logically … indiscernible from the first (“if for any P, x has P if and only if y has P”).

    MdS

  10. Anthony Bigg says:

    Oh, I think I see your issue villanovanus.

    You think that the “only” goes with the “x and y” thus making it “x is identical with y only (I.e and nothing else besides y), if blah blah blah about properties. This formulation would be making us guilty of the inversion you suggest. However, the “only” goes with the “if” to form “x is identical to y, only-if blah blah blah. And, as I said earlier, A only-if B = If A then B, thus we have the identity of x and y as the antecedent and the part about properties as the consequent. Thus, we do in fact, despite your suggestion, present the indiscernibility of identicals.

    If what I’ve said here about your understanding is incorrect, I don’t know what to say, because what Dale has posted here is clearly and undeniably the indiscernibility of identicals and NOT the identity of imdiscernibles.

  11. villanovanus says:

    [Dale, #6, February 13, 2013 at 10:24 am][a]… I wasn’t confusing the Ind of Id with the Id of Ind – Anthony explains why.

    [b]I take no stand on the Identity of Indiscernibles. And yes, it has been much more controversial – it is not obvious that there can’t be two things which are *in some sense* indiscernible.

    [a]Yes you (both) were, and I trust, in the meantime, I have made my explanation sufficiently foolproof …

    [b] My opinion is that, while the Identity of Indiscernibles (i.e. the reverse of what you have formulated) may indeed be “controversial”, the TRUE Indiscernibility of Identicals is trivial, because it is a direct consequence of the assumption of the x = y identity. (Of course I consider the expression “partial identity” —other than as a technical biological expression— an amusing oxymoron — actually, altogether moron …).

    As this website is called “trinities”, it is worth noting that the “persons of the trinity” (at least, according to the traditional orthodox doctrine) are neither identical nor indiscernible …

    MdS

  12. villanovanus says:

    @ Anthony

    You are simply wrong. As already said, that “Necessarily” and that “only” are utterly redundant and pleonastic, from a logical POV (IOW, they are there only for the rhetoric effect), and have nothing to do with your repeated and confirmed blunder.

    Sorry … obviously my proof wasn’t … foolproof … ;)

    MdS

  13. Dale says:

    villanovanus,

    If Anthony has not diagnosed your mistake, then I’m not sure what it is. But he’s right – what I’m presenting is indisputably Ind of Id, and not Id of Ind. For what its worth, I discussed this formulation over lunch with two colleagues – both PhDs and expert in logic and philosophy of language, and both were clear that this is a formulation of Ind of Id.

    Neither the “Nec” nor the “only” are useless. The whole thing ought to be a necessary truth, not a contingent one. And as Anthony said, the only is part of “only if” – that is, the material conditional.

  14. villanovanus says:

    Dale,

    All I can say is that it is very embarrassing that no less than “two colleagues – both PhDs and expert in logic and philosophy of language” get so clumsily fooled by the way two logically … identical formulations of the same principle are phrased. Not content with the blunder, you all embarrassingly invoke some imaginary “modality” of the “pristine formulation”, whereas the initial “Necessarily” is, once again, purely redundant and rhetorical, and the original “only if for any P, x has P if and only if y has P” is a clumsy formulation of the following premise, containing an IFF clause: “if, for every property P, object x has P if and only if object y has P”.

    If you still are unconvinced (it is quite normal for scholars to rely on authority more than on evidence …), I invite you and friends to check SEP > “The Identity of Indiscernibles” (http://plato.stanford.edu/entries/identity-indiscernible), where there are also indisputable straightforward formulations in terms of symbolic logic.

    MdS

  15. villanovanus says:

    I had overlooked this “note” …

    [Anthony Bigg, #4, February 13, 2013 at 9:14 am] Note that [A only if B] is equivalent to [if A then B].

    … that, I believe, clinches it all.

    Maybe Anthony, Dale, his “two colleagues – both PhDs and expert in logic and philosophy of language” (and perhaps also Colin McGinn) can try the alleged “equivalence” with these “items”:

    A: Colin can put his shoes on
    B: Colin has already put his socks on

    (hint: the “A only if B”, although, maybe, not logically cogent, suggests that Colin is a very tidy and proper person; the “if A then B” is hardly a logical necessity, especially in the summer …; OTOH, “if B then A”, conveys very much the same idea as “A only if B”, albeit with less rhetoric emphasis on Colin’s tidiness …)

    MdS

  16. Dale says:

    Hi Anthony,

    I see your point. I left a qualification out of the simplified formula: simultaneously *or timelessly*. It seems just as evident that if something exists timelessly, it couldn’t differ from itself. So if God had never created, still at that point, so to speak, he couldn’t have differed from himself.

    I do think it is possible that there should be no time. That is, it’s not a necessary truth that time is real. But as to time’s being impossible – so that necessarily, there is no time – I think that is a counterpossible. If time is possible, it is necessarily possible. So it is necessarily false that time is impossible. So if we ask, what if time were impossible… that’s not merely a counterfactual, but also a counterpossible. I’m inclined to think that these are neither T nor F, though the majority view is that they are all trivially true.

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