Talk:Identity of indiscernibles

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This article handles the identity of indiscernibles and the indiscernibility of identicals together. The two are separate doctrines deserving separate articles. The indiscernibility of identicals, i.e., Leibniz's law, is indeed one of the two great metaphysical principles of Leibniz. The identity of indiscernibles is not one of the two great metaphysical principles of Leibniz, though Leibniz also accepted it (he thought it followed from the Principle of Sufficient Reason; he was probably wrong about that).

Moreover, it is crucial in the article to distinguish between the almost trivial version of identity of indiscernibles and the non-trivial. The almost trivial version is that if x and y have the same properties, they are identical, and this is how it is stated in the article. This version is easily shown to be true if one is liberal about what properties there are. Let P be the property of being identical with x. If x and y have the same properties, then because x has P, so does y. But then y is identical with x, since P is the property of being identical with x. To avoid such trivialization, the identity of indiscernibles needs to be restricted to purely qualitatively properties, i.e., ones that do not involve the existence of particular rigidly designated things, places, times, etc. It's hard to make this precise, but making it precise is necessary for stating the identity of indiscernibles.

I don't have the time for these revisions right now, but someone should do them. 20:23, 30 April 2007 (UTC)

Mention of duck[edit]

I'm going to delete this text:

So "if it looks like a duck, walks like a duck, and quacks like a duck, then it is a duck".

Why? Because the text is about classification, not about identity. This may be the case: If someone walks like a duck and quacks like a duck then that person is to be classified as a duck.

Controversial applications[edit]

what kind of fucking logic is this? the first 3 statements are about bill's world the conclusion is not!

we would be correct in concluding "bill believes 49/7 and the square root of 49 are two different things. And that is really how the world is!!!

Leibnitz was a genius. We have gone from an age of enlightenment to an age of darkness. We now live in a world of wikipedia half-wits RWS

I agree with you there, however you are raising a philosophical reply, some people do believe what is in the ariticle disputes Leibniz's law. Make a new section and call it replies if you want. --Aceizace 20:54, 19 February 2006 (UTC)
I think you are mistaken and the argument does not raise a valid philosophical reply. Note what you think problem with the argument is : “"bill believes 49/7 and the square root of 49 are two different things” à and therefore “And that is really how the world is!!!” This is EXACTLY the point the criticism is trying to make. The critique says that if we accept “identity of indiscernible” (Leibniz’s law) we will be led into absurd proposition that what Bill thinks makes the world that way. And since this is absurd(“what kind of fucking logic is this?” being your quote) the Leibnitz’s law is wrong.
The correct response to this attack on leibnitz’s law is to claim that what a person thinks about the object is not the property of an object.--Hq3473 04:12, 20 February 2006 (UTC)
I just stumbled across this page and I also found the argument found in this section to be, uh, weak. I'll try to express it a little more mathematically.
The claim in step 6 is that " is not identical to is absurd".
This is not absurd. They *aren't* identical. One has a 7 and a horizontal line, the other has a line with a bunch of corners. Just looking at them you can see that they are different.
To be more precise, " and mathematical expressions, and they are *different* expressions.
In some contexts, these expressions reduce to the same integer, but in others they don't. For example, if the default base is hexidecimal instead of decimal, these expressions yeild different numbers, neither of which is an integer. In other contexts, not all operators are defined, which is what is going on with poor Bill. Once you introduce some more complicated operations, Gödel's incompleteness theorems shows that even if you know how to preform all operations and have a well defined context, there exists two expressions that are equal, but that you can not prove that they are equal. (Also see the halting problem.)
It is important to distinguish between "identical" and "equivalent (under some context)".
So, on a very simplistic visual level, you can see that is not identical to , and on a much higher mathematical level you can understand that, indeed, determining if two expressions are the same can be a very hard problem. It is only fairly basic formulas that people automatically do the reductions and mentally classify them as "the same" and then make the incorrect leap to thinking they are "identical". Wrs1864 05:21, 18 November 2006 (UTC)
I changed this to a different example that does not involve evaluating math expressions, but presreves the basic problem of imperfect knowledge.--Hq3473 20:22, 18 November 2006 (UTC)
Thanks, I like your example *much* better. I think it is proably a good idea to leave the dispute tag for a little while to make sure that others agree, but as far as I'm concerned, my objections have been satisfied. Wrs1864 03:56, 19 November 2006 (UTC)
I there is no further objections i will remove the tag in a couple of days.--Hq3473 20:51, 19 November 2006 (UTC)
I am removing the tag--Hq3473 21:23, 22 November 2006 (UTC)
Good thing too, since the word problem for groups shows that there are situations where such identities are undecidable (uncomputable), and are thus, in a sense, unknowable. That is, the expressions may be equal, but it might not be knowable that they are equal. (talk) 20:14, 22 July 2016 (UTC)


I find it strange that Descartes lived and wrote Meditations before Leibniz was around, yet even the article itself says that Descartes used this reasoning. Might someone who knows more be able to include an explanation on why it is attributed to Leibniz? --Aceizace 20:54, 19 February 2006 (UTC)

The principle existed LONG before Descartes, probably can be attributed to Plato. His theory of Forms had a similar concept. The law got called Leibniz law, for his formulation not for content. Therefore it is not weird that Descartes uses the principle before Leibniz formulation.--Hq3473 15:34, 23 February 2006 (UTC)

From Subjective to Objective[edit]

This principle of the identity of indiscernibles makes the claim that a subjective judgment is to be taken as correctly describing the objective world. It claims that what appears to one person has true being for everyone. Perception is reality. However, that is precisely the problem that is to be solved by almost all philosophy. Kant's whole philosphy was written in order to determinine the correctness of assuming that subjective opinions are objective. Einstein's Relativity is also about the subjective observer and his experience of objects. Berkeley, Schopenhauer, Descartes, and many others have dealt with subjectivity and its relation to objectivity. For Leibniz to proclaim the identity of indiscernibles was, itself, an attempt to assert that his own subjective observations should be considered as being truly descriptive of the objective world of experience.Lestrade 01:43, 3 June 2006 (UTC)Lestrade

Feel free to edit the article acordingly. And do not forget to site your sources!--Hq3473 18:15, 7 June 2006 (UTC)
No, it does not make that claim. It does not say "seem to have all the same properties", it says "have all the same properties". It does not presuppose being able to observe all those properties; indeed, with our knowledge of modern physics (Heisenberg's uncertainty principle) we know that one can't observe all properties at the same time. But that doesn't affect the correctness of Leibniz's claim, which is a definitional claim not a claim about human observations. greenrd 01:52, 15 February 2007 (UTC)
Sure, i agree, in my understanding identity of indiscernibles is a metaphysical principal rather then epistemological one, but if someone find sites of famous philosophers thinking otherwise, the article should reflect it. --Hq3473 03:46, 15 February 2007 (UTC)
According to greenrd, Leibniz is making a dogmatic, ontological,objective assertion about the way that the world is constituted, rather than a hypothetical, subjective statement about his own perspective of the world. Then, greenrd brings in the well–known distinction between psychology and logic. This was often used by Russell to relegate his opponents to the class of subjective, introspective psychologists, while he triumphantly stood on the firm ground of universally objective logic.Lestrade 15:48, 27 September 2007 (UTC)Lestrade


Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that 2. 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.

I know that Max Black is correct because I am in possession of a wonderful counterexample from pure mathematics--in other words, I have an elegant simple model--which proves, conclusively and persuasively, that there is at least one pair of numerically distinct objects which--nevertheless--have all their properties in common. And as soon as I have my proof published, or submitted, to a scholarly peer-reviewed philosophical journal, I look forward of the opportunity of publishing it here in this excellent Wikipedia article. Ludvikus 03:50, 2 September 2006 (UTC)

    I've transcribed here the above from the Article page - before reversion. I have written the
    comment before having become an experienced Wikipedian, understanding and following WP policy.
    Nevertheless, my observation remains true. But like Fermat? - No space to ellaborate?
    Yours truly,--Ludvikus 03:22, 14 December 2006 (UTC)

I would say that Mr. Black's critique doesn't hold water, as the two spheres he describes obviously occupy different locations in space. As location in space counts as a property, then the two spheres do not have the same properties. Anyone disagree? -Tim —Preceding unsigned comment added by (talk) 00:07, 10 September 2007 (UTC)

Yeah, there is no such thing as "space", the only way to define space is in relation to other objects. So in the world with only 2 objects the only space for a spehere is defined by "distance to the other sphere" but the ther sphere will have the same prperty, so we still cannot distinguish them. See Theory of relativity.--Hq3473 02:09, 10 September 2007 (UTC)
Ok, it took me a bit of thinking to figure out what seemed wrong about your response, and here it is: First, Black says that the only two things that exist in this hypothetical universe are the two spheres. This, however, cannot be technically accurate, as the properties that we are using to describe those spheres must also exist. So what properties exist? Obviously, numerical, spatial, and physical ones exist, as the spheres exist in space and have size, shape and numerosity. Of course, their size and shape are the same. However, logical properties must also exist. And the critical flaw in Black's example is that with the very act of saying that two spheres exist, he imbues them with the logical property of not being the same object. Sphere A is sphere A. Sphere B is sphere B. Sphere A is not sphere B, and vice-versa. What made me realize this was your response to me in which you wrote "distance to the OTHER sphere." In order for "other" to have any meaning, there would have to be some property that differed between the spheres that allowed us to tell them apart - and that property was the logical one that they have been defined as two separate objects from the start. Any objections to that? - Tim —Preceding unsigned comment added by (talk) 11:48, 14 September 2007 (UTC)
No by saying there are two sphere Black does NOT give you the power to differentiate spheres. Sure if a spectator were to appear in the Black's world he would immediately identify spheres as 1 and 2. But there is no spectator. Think about it this way. Say you pick a sphere and call it Spehere 1 and the other one Spehere 2. Then you leave the world, and then come back again. WOuld you be able to tell which one is Sphere 1 and which one is spehere 2? No you would not. Because Max's world has no way to differentiate the spheres. --Hq3473 13:28, 20 September 2007 (UTC)
I'm sorry, I think I worded my comment above somewhat badly. What I meant when I wrote "there would have to be some property that differed between the spheres that allowed us to tell them apart" was not that we would be able to tell which sphere was A and which was B (after having labeled them and then re-entered Max's world). You are right; we would not be able to tell.
Let me put my argument in other words: If more than one object exists in a universe, then those objects will always be identifiable as different by means of logical properties. This is why: We know from Max's definition of his world that the Sphere A and Sphere B are separate objects. If so, then Sphere A logically *must* have the property of being "not equal to Sphere B." Likewise, Sphere B must have the property of being "not equal to Sphere A." Without these properties, we would be literally unable to conceive of Spheres A and B as being two separate objects; we would have to conclude that "Sphere A" and "Sphere B" were simply two different names for the exact same thing. In case you aren't convinced, take the example of an object lacking, say, a certain mathematical property. Let us say that this object has no numerosity. It is not a single object, nor is it many; the idea of numerosity simply does not apply. Can you imagine it? I can't. I can imagine one object and I can imagine more than one, but no matter how I try, I cannot conceive of an object without numerosity. (Nor can I talk about it! Notice how I had to use singular pronouns and verb conjugations to describe the object.) Sphere A and Sphere B are in the same boat, with reference to logical properties. We cannot conceive of their being separate objects unless each has the property of being not equal to the other.
Logical properties are so taken-for-granted that they are easy to forget. Think of a person debating whether the Law of Noncontradiction is true, not realizing that they are assuming it's true in order to have the debate. Max Black must have forgotten about logical properties, or not thoroughly understood them, when he made his argument against the law of indiscernibles.
One last thing I could say, although this argument shouldn't be necessary given the above, is that Sphere A and Sphere B *do* have different properties as per their location, despite what Hq3473 wrote before. Consider that Sphere A has the property of being 0 distance from Sphere A, while Sphere B has the property of being some non-zero distance from Sphere A. There's something else they don't have in common. -Tim
You seem to have begged the question at "Sphere A logically *must* have the property of being "not equal to Sphere B." Such thing does not follow from "Sphere A and Sphere B are separate objects." This is the whole argument that Max is trying to make -- A and B are separate objects yet Sphere A does NOT has a property of being not equal to B, in fact it IS equal to B. This is the whole point --to show that by using identity of indiscernibles we get two separate object which are nevertheless equal, and to straight up assume otherwise amounts to saying "A and B are not equal because they are not equal." In the end Max's attack works, because either you have to accept that Spheres are "separate but equal" or you eviscerate the Identity of indiscernibles by saying that all distinct objects have the property of being different from other objects and thus are different from other objects. How would such law be useful?--Hq3473 15:23, 27 September 2007 (UTC).
I agree with the grandparent, the argument doesn't hold water. Max Black must construct a space in which to embed to objects, even if this is the topological/geometric/set/etc. construct of "just two spheres." However, if we were to refer to just the simple constructivist approach of S={A,B}, where A,B are elements of the set "Sphere" then we can ask what properties they have in common (up to the Leibniz equality). However, Max Black is implicitly adding more properties: symmetry and an embedding in "the universe". Now, the description in Wikipedia is too weak to make any meaningful conjecture, but knowing the sort of reasoning logicians/philosopher's use, he's probably thinking of a closed, bounded, infinite symmetric space like unit cell of P2; see Crystallography. In that case, while P2 does not discern handedness, orientation, etc., it still provides an infinite number of exactly equivalent metric embeddings. In any of these embeddings we can determine the vector offset (for free, with no additional assumptions), uniquely between the two pairs. If you assume that no meaningful embedding occurs, then you must add in the assumption to the set construction that A does not equal B, and thus, Max Black (and the parent) are begging the question. —Preceding unsigned comment added by (talk) 16:54, 4 March 2008 (UTC)
The last time i checked there is no such thing as "absolute space". Space is only meaning-full if the reference frame is well determined (Introduction_to_special_relativity#Reference_frames_and_Lorentz_transformations:_relativity_revisited). In the world described by Black there is no well defined "space" untill you fix a frame to any one of the spheres. In any case you are welcome to present critique of Max Black's work if you find an appropriate authoritative source. --Hq3473 (talk) 17:10, 12 August 2008 (UTC)
hahaha okay I suppose I shouldn't open up this debate again, but I'm going to try and lay out the argument more clearly: this universe is the set of two items, {a,b}. Each item has the property of being a sphere of a specific size. So the universe is described as, {Sa,Sb}. These are all the properties that the objects have when taken individually, so we can't differentiate them there. Now let's think spatial. If a has the property of being one meter away from the other sphere, then b has the property of being one meter away from the other sphere, and vice versa ([Oa→Ob]&[Ob→Oa]). So giving one sphere the property of being spatially separated from the other doesn't mean that the two spheres actually have different properties. In fact, giving the property to one sphere implies that the other sphere has the exact same property. And if we say, a has the property of not being identical with b, then this implies that b has the property of not being identical with a ([a≠b]→[b≠a]), and vice versa ([b≠a]→[a≠b]). If we take these cases as exhaustive (and if space isn't absolute, which it isn't, they are exhaustive), then we have a case in which the antecedent of leibnez' second law is true Pa<-->Pb but the consequent is false a≠b, making this a counter-example to the "law." Fair enough?--Heyitspeter (talk) 05:59, 1 April 2009 (UTC)
But now of course I've thought of a counterexample to that formulation haha. We can just say, a has the property of being one meter from b. a having this property does not imply b having this property, so there we are. There is a property that one sphere has that the other doesn't implicitly have, so Black's example is finished... Has anyone read his article? That would probably help. I have no idea how he would deal with this.--Heyitspeter (talk) 06:03, 1 April 2009 (UTC)
You can't think spatially. The spheres are not spatially separated, if they where then they are discernible as separate. Consider the property P(a,b) which is true iff a and b are at different places. Then P(Sa,Sb) is true, and P(Sa,Sa) is false. So Sb would have a property that Sa lacks. Taemyr (talk) 23:20, 1 April 2009 (UTC)
Nah wouldn't work that way because P(Sa,Sb)←→P(Sb,Sa) would be true, along with ~P(Sa,Sa)←→~P(Sb,Sb) [here, because the two are logical truths], so in either case the left half of Leibniz' conditional is true but the right is false, creating a counterexample, affirming Black's hypothesis even when the sphere's are spatially separated. But still, one has the property of not being a, and the other doesn't have this property. This is the case even when the sphere's are not in different places...--Heyitspeter (talk) 06:02, 2 April 2009 (UTC)
Mmm sorry I see what you mean. P(Sa,Sb) does not imply P(Sb,Sb), so having the property of being separate from Sb is not shared. But still, we have (a≠b) not implying (b≠b), no matter where the two spheres are... --Heyitspeter (talk) 06:06, 2 April 2009 (UTC)
Yes but Identity of indiscernible defines ≠. So it's a property that you can't use to define indiscernability because if you do you create a circular argument. Taemyr (talk) 06:29, 2 April 2009 (UTC)
All I'm saying is: Suppose identity is a property. Then there is a property that one sphere has that the other does not, so we do not have a case in which the antecedent of Leibniz' theorem is true but the consequent false. This isn't begging the question. Identity of Indiscernible doesn't define identity, it's just a rule by which we can know that identity obtains. --Heyitspeter (talk) 08:27, 2 April 2009 (UTC)
My take on this is that The indiscernibility of identicals and The identity of indiscernibles together define identity. But even if you just look at it as a property that indiscernible elements have then you should not include the identity property. The identity of indiscernibles states; "For any x and y, if x and y have all the same properties, then x is identical to y." If you include the identity as a property then this can be strengthened to "For any x and y, if x and y are identical, then x is identical to y.", which is a rather trivial observation. Identity of indiscernibles is only interesting when it can lead you to conclude that x and y is identical, which means that you can't have a rule that requires you to know if they are identical in order to apply it. Taemyr (talk) 03:06, 3 April 2009 (UTC)
Mmm so Black is saying that all properties short of identity can be shared by two objects, but that those objects can nevertheless be non-identical. Leibniz' second law being wrong. Got it. Thanks!--Heyitspeter (talk) 08:55, 3 April 2009 (UTC)
Still supposing identity is a property there is no logically possible counter-example. This is a really weird situation.--Heyitspeter (talk) 08:37, 5 April 2009 (UTC)

Epistemological Version[edit]

The articles gives the above rather than an ontological principle:

  The identity of indiscernibles is an ontological principle that states that if there is no way of
  telling two entities apart then they are one and the same entity. That is, entities x and y are
  identical if and only if any predicate possessed by x is also possessed by y and vice versa.
Yours truly,--Ludvikus 03:53, 14 December 2006 (UTC)

Ontological principle[edit]

I've modified/corrected the opening sentence from the above, to the following:

    The identity of indiscernibles is an ontological principle; i.e., that if (two
    or more) object(s), or entity/ies have all thier/its property/ies in
    common then they (it) are identical (are one and the same entity). That is, entities x and
    y are identical if and only if any predicate possessed by x is also possessed by y and
    vice versa.
Yours truly,--Ludvikus 04:05, 14 December 2006 (UTC)

So you're the one I should kick in the balls for making it needlessly illegible. Great. I'll change it back to English now. -- 20:36, 10 June 2007 (UTC)

Criticism Counterexample[edit]

The proposed criticism is: "Opponents of this counterexample would claim that a contradiction can be found between proposition (2) and (3) (i.e. Lois Lane cannot have opposite thoughts about the same object, regardless of the name)." To me this objection seems like begging the question. Lane think that the person can and can't fly at the same time because she does not know that it is the same person. So she DOES have opposite thoughts, and denying it begs the question: I.E. it is arguing for "Identity of indiscernibles" like this: "I know that Identity of indiscernibles is true, and therefore your counterexample(no matter what it is) cannot work". Thus i propose deleting this weak objection. --Hq3473 23:24, 1 March 2007 (UTC)

Quine's Variation[edit]

The most well spoken version of the identification of indiscernibles I have encountered is found in Quine's "Identity, Ostension, and Hypostasis," as follows: "Objects indistinguishable from one another within the terms of a given discourse should be construed as identical for that discourse." This gets us away from descriptions about properties and the like, which of course invite the confusion of supposing that the creation of two objects with identical sets of properties might disprove the proposition (Liebniz would argue that, for this to be the case, you would have to find a way to have two identical objects occupying the same spatio-temporal location as well, which makes a refutation of this kind rather hard to manage, unless you can imagine two individual objects occupying the same space), or suggesting that a single object, seen, say, from two different perspectives, would also disprove the proposition. Of course, the Quinean version is not ontological in the sense of defining specificity to real objects in the physical universe. It is a deliberately broad definition, intended to deal with another set of representational philosophical problems that are only partly related to what Liebniz was interested in demonstrating. Nevertheless, it would seem to me a worthy candidate for admission in this article, for some plucky chap willing to add it in.

Feel free to write about Quine's version, its advantages and shortcomings, etc. Make sure to source to Quine.--Hq3473 03:46, 19 April 2007 (UTC)

Descartes' argument[edit]

I don't think that Descartes' argument should be described as an application of the identity of indiscernables. Note that the conclusion, that the body and the mind are different, states that two things are not identical. If anything, this would be an instance of principle 1, the indiscernibility of identicals. Zarquon 03:48, 19 April 2007 (UTC)

The definition in the first paragraph says: "The identity of indiscernibles is an ontological principle: that if and only if. Not that the "if and only if" part makes the Identity of indiscernibles work both ways. --Hq3473 04:59, 19 April 2007 (UTC)

"Controversial Applications" not true[edit]

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.

The conclusion "Since in proposition 6 we come to a contradiction with proposition 2, we conclude that at least one of the premises is wrong." has been obtained ridiculously. To show that this is an invalid argument, firstly we consider the statement "Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly.". Simply put a property of an object must be inherent to itself and not based on some observers view. It is also possible that we cannot confirm that an object has a certain property or not, in which case be contradictory by saying that an electron is a wave and not a particle or vice versa, then when observed we "think" it is a wave or particle, thus appearing contradictory based on the identity of indiscernibles. In that case we cannot say whether the electron is identical to itself and cannot make any conclusions.

Nicholaslyz 10:30, 9 July 2007 (UTC)

Note that you say that " Simply put a property of an object must be inherent to itself and not based on some observers view", this is the point of the "proof" precisely. Descartes tried to rely on a human belief about an object as a property of an object, this exact line of reasoning the "proof" aims to debunk.--Hq3473 20:24, 9 July 2007 (UTC)

Contradictory and incorrect(?) definitions; Proposed article split[edit]

The lead defines identity of indiscernables as being: two objects are equal if and only if they have all properties in common. However, further down, identity of indiscernables is distinguished from indiscernability of identicals: the two halves of the if-and-only-if. But it can't be half of itself...

Moreover, many authors use Leibniz's Law to mean only indiscernability of identicals, and the first comment on this very talk page says that identity of indiscernables is not one of Leibniz's great metaphysical principles, although he accepted it.

I think it would make sense to split this page into two separate articles: identity of indiscernables and indiscernability of identicals. I mean, Black's objection is directed at the identity of indiscernables, and the Superman confusion relates to the indiscernability of identicals. Vaccillation between covering the two principles makes for a confusing article.

Let me ask the question: Is there any evidence that any reliable source apart from Wikipedia has treated these two principles together - or that the value of doing so outweighs any confusion created?—greenrd 01:45, 27 October 2007 (UTC)

Oppose split. The introduction can be altered from "if and only if" to the correct statement. I can see where the confusion lies in the Superman example. As for splitting it, I think it would be better to just rename the article to something more appropriate, and have a distinct separation within the article itself. Necessary and sufficient accomplishes this. — metaprimer (talk) 13:18, 27 October 2007 (UTC)
First, thanks for removing the self-contradiction. I think there is an important difference between this article and necessary and sufficient - this article could probably benefit from more expansion (e.g. where they have been applied to try and prove various statements, other controversies about them, etc.); and if it is likely to become a long article containing two sub-articles without much overlap between them (and with potential for confusion!) it makes sense to split it up into two articles.
My point about the Superman section - which I didn't make very clearly, I admit - was that both Descartes' argument and the Superman "paradox" are applications of the indiscernibility of identicals (contrary to what the section currently says).
Also, what would be a good new name, if we kept this article as one article? "Identity of indiscernibles and indiscernibility of identicals" is too long and awkward, in my opinion.—greenrd 13:42, 27 October 2007 (UTC)

Response to Black's critique[edit]

Is there a reference for this "response", or is it original research?--Hq3473 22:50, 28 October 2007 (UTC)

It's original research. I'm aware of WP:NOR, but I added it in the hope that no-one would object, in the spirit of WP:IAR. Feel free to remove it.—greenrd 08:19, 29 October 2007 (UTC)
I will remove, because it addition to being OR it does not seem a particularly strong response to Max Black. Sure the universe can be looped, but this just goes to show that Identity of indiscernibles will lead to weird counter-intuitive results. --Hq3473 13:26, 29 October 2007 (UTC)

Black is rather obviously wrong in that he first defines a universe model that contains two distinct objects (say, two parts containing "identical spheres", because that is what reflection symmetry suggests) only to then claim the spheres in both objects are one and the same. To then go on to "refute" that by constructing yet another bilaterally symmetrical universe wherein you place two objects, and also that you have no way to spatially tell them apart when you've just defined them as being spatially distinct, doesn't really help people see the point. I seem to recall Hacking exposed that rather more elegantly and elaborately than the article now suggests. JeR (talk) 19:54, 31 March 2010 (UTC)

Hahah I hadn't thought of that critique. Do you think you could clarify Hacking's argument on the main article?--Heyitspeter (talk) 20:05, 31 March 2010 (UTC)

Secret identity.[edit]

The example in the article concludes;

Leibniz's law is wrong; or else
A person's knowledge about x is not a predicate of x, thus undermining Descartes' argument.

However it seems to me that it might just as well be the claim that superman is equal to clark kent that is wrong. Ie. the claim that they are the same person is weaker than the claim that they are equal.

An example that does not involve other peoples believes would be the Supreme Governor of the Church of England and the Paramount Chief of Fiji. The first having the right to formally appoint high-ranking members of the church of England. Taemyr (talk) 17:56, 6 April 2008 (UTC)

I don't like this. Supreme Governor of the Church of England and the Paramount Chief of Fiji COULD be different in, for example, an unlikely event that Fiji succeeds. Clark Kent and Superman or any other real or fictional person with a secret identity are the same people no matter what.--Hq3473 (talk) 16:53, 12 August 2008 (UTC)
Try Supreme governor of England in the year 2000 and the Paramount Chief of Fiji in the same year. Or for an even clearer example, two pointers that point to the same variable. The label is different from the thing. Unless this counter argument is sourced I will remove it as OR. Taemyr (talk) 05:32, 11 September 2008 (UTC)
Secret identity is a common counterexample to desecrates argument. I do not remember which particular article i was quoting at the time. But i believe this article is a sufficient source(although this one take batman as an example): " Alter Egos and Their Names, David Pitt, The Journal of Philosophy, Vol. 98, No. 10 (Oct., 2001), pp. 531-552, page 550", you can find the artcile in full at [1].--Hq3473 (talk) 13:41, 11 September 2008 (UTC)
It seems that this source argues my point though;

Moreover, I do not share Saul’s puzzlement about these cases; for it seems to me that the most straightforward explanation of the substitution failures – namely, that ‘Superman’ and ‘Clark Kent’, ‘Bruce Wayne’ and ‘Batman’ are not coreferential – is correct.

— David Pitt, Alter Egos and Their Names
Although presumably Saul takes an other view. Taemyr (talk) 00:28, 12 September 2008 (UTC)
The point is that the article lists the "alter-ego" as an example of a counter argument to a certain use of Identity of indiscernibles. The article also criticizes this counter-example. So i feel this example should stay, as opposed to making up a different example. As for the criticism, feel free to add it and reference the same source.--Hq3473 (talk) 13:33, 12 September 2008 (UTC)

The Principle "states that two or more objects...are identical..."?![edit]

Surely the principle doesn't state, as the article now says it does, that "two or more objects or entities are identical if...." If it really does state that, then it's clearly absurd; for how can two objects be identical? Isokrates (talk) 20:56, 19 April 2008 (UTC)

It is usual in formal arguments to interpret "two objects" as "two objects that might be instances of the same object." When you require them to be two seperate object this usually needs to be stated. So a relation ≤, is antisymmetric if for any two objects a≤b and b≤a implies a=b. Compare to ... for any two objects a≤b and b≤a is a contradiction. Taemyr (talk) 19:36, 20 April 2008 (UTC)
  • Your example is not as helpful as it appears. Every two objects are instances of a<b. And every one object is an instance of a=a. No object(s) is/are instances of both. --Ludvikus (talk) 21:22, 20 April 2008 (UTC)
  • I'm saying that you really cannot instantiate the relation you give above - although you do conform by it to standard practice (it's as if 2 contradictions are wiping each other out). --Ludvikus (talk) 21:30, 20 April 2008 (UTC)
  • What a strange relation: "something is greater than or equal to something else"! --Ludvikus (talk) 21:32, 20 April 2008 (UTC)
I should perhaps not have used the symbol ≤. Remember that we are defining our relation. So don't pressupose arithmetic "less than or equal", arithmetic "less than or equal" is simply an instance of an antisymetric relation. And no, every two objects need not be instances of where a and b is different. You usually has to specify it explicitly when you want to say that you are reasoning about pairs of un-equal elements. Taemyr (talk) 07:28, 21 April 2008 (UTC)

Two objects are one if such and such is the case.[edit]

I just want to simplify that ordinary language version of the alleged apparent self-contradiction. --Ludvikus (talk) 21:16, 20 April 2008 (UTC)

How about Hepsherus and Phosporus both being Venus? Hesperus#"Hesperus is Phosphorus".--Hq3473 (talk) 01:08, 21 April 2008 (UTC)
"Two objects are one", I think this is the heart of the reason why the above poster sees a contradiction. His view is that two objects are never one. So getting around the percieved contradiction while retaining formal correctness would require something like "Two differing objects never share every property." Taemyr (talk) 07:33, 21 April 2008 (UTC)
I personally like the definition from Quine given earlier on this talk page. "Objects indistinguishable from one another within the terms of a given discourse should be construed as identical for that discourse." Because this definition includes which properties are of intererest or not. Taemyr (talk) 07:39, 21 April 2008 (UTC)


About this, added and removed twice now;

Saul Kripke, in Naming and Necessity, argued that Superman and Clark Kent are rigid designators that both refer to the same person. Whatever Lois Lane believes about Superman, she necessarily believes about Clark Kent, though she is not aware of that fact. In other words, she does believe that Clark Kent can fly, because when she is forming beliefs about Superman, she is in fact referring to Clark Kent. They are the same person. Under this construction, Lois Lane has a pair of conflicting beliefs, but there is no property that Superman has that Clark Kent lacks, and vice versa

This is largely irrelevant. If Lois Lane is capable of holding conflicting beliefs about the properties of Clark Kent due to her beliefs about Superman. Then Descartes is capable of holding conflicting beliefs about the entity that is his body and the entity that is Descartes. Taemyr (talk) 21:43, 20 December 2008 (UTC)

Pitt, in the article that the argument is sourced on does however raise a valid objection. Since it questions whether the fact that Clark Kent is the same person as Superman is sufficient for Clark Kent and Superman to be corefferential. Using it to source this would be OR though, since the argument is not treated directly by Pitt.Taemyr (talk) 21:47, 20 December 2008 (UTC)
I agree perhaps it is not the best reference. But the argument i quite common in modern philosophy. For example: [2]. I also think this might mentioned here: Steinhart, E. (2002) Indiscernible persons. Metaphilosophy 33 (3), 300 - 320. Will find better references later. --Hq3473 (talk) 19:45, 29 January 2009 (UTC)
Have any sources drawn comparisons with the Masked man fallacy? Taemyr (talk) 13:26, 30 January 2009 (UTC)

Suggestions for Rewrite[edit]

There is a profound evolution of thought surrounding the principle of Identity of Indiscernibles spanning more than 2500 years in the West, and successive formulations range from trivial, tautologically true constructions to the metaphysically-laden statement invented by Leibniz. As an axiomatic law of thought, any given instance of this principle can be understood and analyzed only in context, within the given metaphysical framework for that instance. That is, no meaningful discussion of this principle can take place outside of the historical philosophical traditions in which the various instances of this principle have been conceived.

This article fails in this regard, and by implication more or less equates Leibniz' Rule with its own statement of the principle of Identity of Indiscernibles, which is quite different. By declaring merely that "a form" of this rule also was presented by Leibniz, while failing to identify any difference in Leibniz' statement, it is likely readers will wrongly conclude that any nuance adopted by Leibniz is of little import. The actual statement of Leibniz' Rule is as follows:

For any individuals, x and y, if for any intrinsic, non relational property f, x has f if and only if y has f, then x is identical with y.

Consequently for Leibniz, if x and y are distinct they must differ in terms of some intrinsic, non relational property. If the editors had included such detail then the article would not have invited to no avail such sophomoric (at times puerile) banter and facile epistemological refutation.

A useful article on the Identity of Indiscernibles should enumerate and order its most important formulations, and for each provide some metaphysical context for its motivation and limits of application. Thus, in the section on Leibniz' Rule, a minimal outline of his metaphysics, giving special attention to his ontology (real entities, well-founded phenomena, actual existents, i.e. monads), as well as to his meaning of relational and non relational properties, is essential to understanding his formulation of the principle. For instance, Leibniz sought to avoid commitment to space as an independent (ontological) entity, relying instead upon the notion of relational properties between material objects. As such, this whole discussion of Black's thought experiment, utterly divorced as it is from any pertinent context, is absurd. This is because the axioms of Black's imaginary "universe", at least as far as these have been presented in this article, are incomplete, and his assertion is undecidable, as we have not sufficient ground for comparing the intrinsic, non relational properties, whatever these are, of the two hypothetical spheres. I suspect that a first hand reading (not a wiki) of Black would reveal a far deeper and nuanced position than that presented by the editors thus far. Someone should check this. Next, properties such as "x believes N about y" are extrinsic and relational and thus cannot be used in the formulation of so-called thought experiments designed to refute Leibniz' Rule, which precludes such arguments out of hand. Black's impact on other formulations of the principle could be examined.

The article begins as follows:

The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical (are one and the same entity), if they have all their properties in common.

This statement thus tells us that whenever two entities share all properties in common then they are the same entity, and from this we can derive the contrapositive assertion that if the entities are not identical then they must differ with respect to some property; however, the statement does not say what conclusion can be drawn from the converse, that is, when the objects differ with respect to some property. What then? Since the structure of Descartes' reasoning as it applies here conforms to the unstated converse of the principle given in the article (e.g. If some property is not shared between two objects then they are not identical), the later statement from the article, "one famous application of the indiscernibility of identicals was by René Descartes in his Meditations on First Philosophy," is not supported. Again, if anything meaningful is to obtain from that allusion then one would need to precisely articulate the particular law of thought Descartes was relying upon and then directly compare this with the appropriate formulation and its embedding metaphysical context by now included in the enumeration of formulations of the principle. Nor do we know what constitutes "entities" or "properties" within the sparse construct given in this article. Beyond this, we are not given any epistemological context, which then opens the floodgates to all the tired controversies between rationalists, empiricists, foundationalists, pragmatists, ... ad infinitum. In sum, the present form of this article is poorly conceived, and this whole tangent involving modal logic and intensional contexts is misplaced.

Finally, the opening discussion of First Order logical representations, which may or may not apply to any given formulation of the principle, lacks motivation, is somewhat misleading, and should come later, probably under a heading such as 'applications' or 'mathematical representations' or the like. Including this discussion at the very beginning without qualification suggests that any given formulation of the Identity of Indiscernibles principle, including Leibniz' Rule, is essentially an axiom or theorem of First Order Logic, which is not the case. Moreover, the discussion of tautological identity without sufficient exposition suggests that Leibniz' Rule permits this, which it does not. I suggest a complete rewrite of this article. I should mention that I am not calling for original research here but rather what one at minimum would expect from an accessible, peer-reviewed exposition: namely, an informed discussion of the principle, its history, metaphysical context, applications, and current status as a viable precept.

G.W. Leibniz, 'On the Principle of Indiscernibles', in Leibniz: Philosophical Writings, ed. and tr. G.H.R. Parkinson and M. Morris (London 1973). C.D. Broad, Leibniz: An Introduction (Cambridge 1975). Oxford Companion to Philosophy, ed. Honderich, Ted, (Oxford 1995).

--Devala1 (talk) 23:01, 24 May 2010 (UTC)

Be bold! As long as you cite your sources it should be received positively.--Heyitspeter (talk) 23:46, 24 May 2010 (UTC)

Rm Kant: why[edit]

I took out:

In his Critique of Pure Reason, Immanuel Kant argues that it is necessary to distinguish between the thing in itself and its appearance.[1] Even if two objects have completely the same properties, if they are at two different places at the same time, they are numerically different (see: identity)

which had the tag "A citation linking this argument explicitly to identity of indiscernables is required here.", though I'd taken it out before reading that. Firstly, "it is necessary to distinguish between the thing in itself and its appearance" is irrelevant, and must presumably have been a confusion on the part of someone. Secondly, "...they are numerically different" is wrong, in that the use of "numerically" is wrong / meaningless. Third, it isn't at all clear that Kant makes the second argument "Even if two objects have completely the same properties, if they are at two different places at the same time", and I hope he doesn't, because clearly location is a property. And fourth, that point is made in the following section William M. Connolley (talk) 11:17, 17 August 2011 (UTC)

Kant claimed that Leibniz mistakenly considered two perceived objects as though they were mere concepts. Two similar concepts can be thought to be the same or indiscernible. However, if the two similar perceived objects are correctly considered to be phenomena, then they would be known as being in different places in space. In this way, they would not be considered to be the same or indiscernible.
In Critique of Pure Reason, A 264, Kant wrote: "Leibniz took phenomena to be things by themselves, intelligibilia, that is, objects of the pure understanding … and from that point of view his principle of their indiscernibility (principium identitas indiscernibilium) could not be contested. As, however, they are objects of sensibility, and the use of the understanding with regard to them is not pure, but only empirical, their plurality and numerical diversity are indicated by space itself, as the condition of external phenomena. For one part of space, though it may be perfectly similar and equal to another, is still outside it …."

Lestrade (talk) 12:00, 17 August 2011 (UTC)Lestrade

  1. ^ Critique of pure reason, Immanuel Kant (1781/1787), transl. Norman Kemp Smith, Macmillan Press, 1929, pp 278–79

Rm Quantum[edit]

I removed this:

Unfortunately this is arguing from the existence of macroscopic objects with possibly hidden variables which are in fact different. And indeed, this just proves the impossibility of there being only 2 distinct objects in a universe. Indeed in quantum mechanics, the relation is between an object (say the object's electric field) and the superposition of all possible identical objects relative to it. So e.g. one could argue that an isolated charged particle has a total field through the spherical surface on its center which is invariant (independent of radius of the sphere). And this is the content of the integral equation version of the divergence of the electric field (Maxwell's 1st law discovered before him).[original research?]

Whoever wrote that appears to not understand what hidden variables are, what quantum mechanics is, or Maxwell's electrodynamics. Seems to be some original research. (talk) 20:04, 22 July 2016 (UTC)

Revising the Descartes section[edit]

I removed this:

A response may be that the argument in the Meditations on First Philosophy isn't that Descartes cannot doubt the existence of his mind, but rather that it is beyond doubt, such that no being with understanding could doubt it. This much stronger claim doesn't resort to relational properties, but rather presents monadic properties, as the foundation for the use of Leibniz's law. One could expound an infinite list of relational properties that may appear to undermine Leibniz's law (e.g., Lois Lane loves Clark Kent, but not Superman. etc.) but nonetheless any approach focused on monadic properties will always produce accurate results in support of Descartes' claim.[1]

The argument is incorrect, as doubt is not a monadic property; not surprisingly, it is also not the argument made by Carriero. Ben Standeven (talk) 15:35, 27 January 2017 (UTC)

  1. ^ Carriero, John Peter (2008). Between Two Worlds: A Reading of Descartes's Meditations. Princeton University Press.