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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. 141.161.84.89 20:23, 30 April 2007 (UTC)
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.
what kind of 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!
Leibniz 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 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)
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
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.
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.
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. -- 76.224.107.34 20:36, 10 June 2007 (UTC)
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)
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.
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)
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)
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)
Is there a reference for this "response", or is it original research?-- Hq3473 22:50, 28 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)
The example in the article concludes;
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)
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
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)
I just want to simplify that ordinary language version of the alleged apparent self- contradiction. -- Ludvikus ( talk) 21:16, 20 April 2008 (UTC)
About this, added and removed twice now;
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)
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:
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:
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)
I took out:
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)
Lestrade ( talk) 12:00, 17 August 2011 (UTC)Lestrade
I removed this:
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. 67.198.37.16 ( talk) 20:04, 22 July 2016 (UTC)
I removed this:
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)
References
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For any x and y, if x is identical to y, then x and y have all the same properties.
How about the property "has proper name ‹some name›"? For instance, in Peano arithmetic the number "two" can have "name" either or . The equality is valid, but the property "has proper name " is valid only for the first closed term. I have no questions about first-order schema of the indiscernibility of identicals: The property "has proper name " cannot be expressed by a formula of Peano arithmetic. But the second-order axiom (cited above) embraces all properties, including inexpressible by formulas.
Eugepros ( talk) 07:08, 8 April 2018 (UTC)
This looks like an individual's own rather eclectic research and reasoning, involving what seems to be a fair bit of digression. I do not understand all the concepts mentioned in this section, but reading the "symmetric universe" section is hugely less comprehensible than the rest of the article, and at the very least needs to be explained more slowly and in much greater depth. The first paragraph is ok, and I can just about follow the second one, but the discussion of how there "must" be a "quantum asymmetry" in Black's universe, the curvature of space-time and the origins of calculus leave me behind after countless readings. Indeed, in a discussion of pure logic, the involvement of so many physical experiments seems to me inexplicable.
Additionally, what seem to be bold claims are made, with a very low footnote density. The footnotes available don't all have page numbers, and one is just a google drive link. Finally, I have never seen the "--" punctuation anywhere on wikipedia before, which does not give me great confidence in the writer's familiarity with wikipedia standards and practice.
Under the "Be Bold" policy wikipedia recommends, I will delete the "symmetric universe" section in a week or so, if nobody has any other thoughts. I will leave in the first paragraph, and perhaps point the reader to the issues involved in the later discussion, if I can figure out what they are. If an article offers poorly-evidenced text that seems like gibberish in comparison with the rest of the article, it surely should not be part of any encyclopedia.
If someone can explain to me what the "symmetric universe" section is about, clearly and with sources, I suggest they do so then replace the original text with their explanation. It is possible that I am just being very thick, feel free to tell me so. — Preceding unsigned comment added by 79.66.61.34 ( talk) 16:10, 22 August 2018 (UTC)
As a mathematician, reading this article, I can't help but notice that this principle is similar in philosophy to the Yoneda lemma in category theory, which can be seen as a formalization of this principle. That lemma states (as a corollary) that two objects who can't be distinguished based on how they interact with their surroundings (i.e. X,X' satisfy Hom(X,Y)~Hom(X',Y) for all Y, naturally) have to be isomorphic (i.e. X~X', which is the mathematical way of saying "they are the same object with another name"). Maybe speaking about the link could be interesting ? A "serious" reference might be needed. 2A01:E0A:2F0:4C0:3D15:3C47:C8AE:903E ( talk) 10:40, 6 March 2020 (UTC)
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 218.219.191.130 ( talk) 00:07, 10 September 2007 (UTC)
Problem with Superman example
My second comment is about the Clarke Kent and Superman example. If I am not mistaken, one of them wears a tight costume and the other doesn't. So there is actually a difference between them in that sense. If there was absolutely no difference there could be no reason why any person would thinks the other person is able to fly and not able fly at the same time. What is more, as far as I am aware, no one can have two distinct and contradictory thoughts at the same time, so the woman that thinks one thing is not actually the same as the woman that thinks another thing - as she is now in the future and has therefore changed. All of this becomes very complicated. — Preceding unsigned comment added by 81.164.118.56 ( talk) 02:02, 18 December 2020 (UTC)
"calling the Łukaszyk–Karmowski function a metric although it isn't positive definite is a matter of naming, not a critique"
This distance function is not “a metric”, as it does not satisfy the 1st metric axiom (albeit satisfying the remaining two). So perhaps Łukaszyk–Karmowski metric should be moved to Łukaszyk–Karmowski distance (cf. [1]) and appropriately rephrased. Ł-K distance is positive definite for Dirac delta distributions. The only point here is that there exists a distance function (Ł-K distance) that does not follow the identity of indiscernibles ontological principle/1st metric axiom. That can be considered as a critique of this principle. Guswen ( talk) 11:23, 21 January 2021 (UTC)
References
"Considerations about how not to define a metric in mathematics are not relevant enough to merit their own section."
We're not considering how to define (or how not to define) a metric in mathematics. We're talking about Identity of indiscernibles as a general principle. — Preceding unsigned comment added by Guswen ( talk • contribs) 00:56, 23 January 2021 (UTC)
This page is getting very long, it might be a good idea to archive it. I would use User:ClueBot_III unless there are objections. Phlsph7 ( talk) 04:59, 24 January 2021 (UTC)
"on the contrary: the proof of the ugly-duckling theorem *uses* the identity of indiscernibles"
Indeed, the proof of the ugly-duckling theorem uses the identity of indiscernibles principle to arrive at its contradiction. The proof assumes a set of 2^n objects, each having properties different than the other (no two objects in this set have all their properties in common). Each property of an object is a considered to be a Boolean-valued predicate, and thus the set form a Boolean {0, 1}^n address space, wherein each address (object) is indeed (logically) separate from the others. But the Ugly duckling theorem proves that any two addresses (objects) in this set are equally similar, as they share the same number of compound predicates, all the logical functions that can be formed from the properties of these objects, with connectives of negation, conjunction and disjunction. (*) Therefore any two addresses (objects) in this set are two things under one name. On the contrary to the statement that “to suppose two things indiscernible is to suppose the same thing under two names”.
The Ugly duckling theorem proves that no discernibility (understood as distinguishability, recognizability, identifiability, distinctability, classifiability, etc.) is possible without some sort of bias. Guswen ( talk) 12:20, 21 January 2021 (UTC)
The material in this article should be relevant to the topic and correct. Both of these points have been contested concerning the discussed addition. So far no reliable sources have been presented to dispel these concerns despite repeated requests to do so. Jochen Burghardt and I are in agreement that the material should be removed, see WP:NOTUNANIMITY. If you (Guswen) have reliable sources to present now then we can discuss them. If not then the contested material should be removed until sources are presented. Phlsph7 ( talk) 03:58, 25 January 2021 (UTC)
I think as a minimum we would need a reliable source for the relation between the UDT and indiscernibility (in the sense discussed here). Phlsph7 ( talk) 06:42, 25 January 2021 (UTC)
References
Hi there Jochen, sorry about my last edit deleting that entry from the Proof Box! When I saw it was still there after my previous edit, I assumed I had overlooked it by mistake, as often happens. I didn't actually mean to delete it twice : ]
Although I agree with you that indiscernibility of identicals does not imply reflexivity, I'm not quite following the proof summary you put in that section. The axiom of reflexivity has no predicate or relation variables, and references no predicates or relations aside from equality. This means that _all_ predicates and relations satisfy that axiom: equality does so explicitly, while everything else does so vacuously. The things that must actually satisfy it are the domain objects -- i.e., numbers -- that can stand in for x. So long as every number equals itself, relations of any arity can be defined in any manner consistent with the other axioms of logic. Even when the _domain_ is completely empty, reflexivity holds vacuously by virtue of the universal quantifier on x.
Since the other proofs in the box are both premised on reflexivity, it might make more sense if the entry that establishes reflexivity (i.e., IdIndscs → Refl) were placed up front. The titles of the other proofs ("IndscIDs ∧ Refl → Foo") would then make it clear that indiscernibility of identicals doesn't imply Refl... because, if it did, the "∧ Refl" would be superfluous. What do you think? Hobeewahn ( talk) 00:26, 8 January 2022 (UTC)
@Jochen: Thank you for this detailed explanation! At last I understand the thrust of your proof, and it makes perfect sense. To make it totally explicit what the empty relation is standing in for, I would just preface it with "When used as '='," ...
I agree that "ID" would look better than "Id" in both of those identifiers, so I will change it as per your suggestion. (There is something of a religious war going on at my workplace about whether acronyms should be uppercased within CamelCase identifiers-- but that is a long story ;) ) And thanks for fixing the oversized arrows left by my earlier edit -- they look way better now.
Hobeewahn ( talk) 21:48, 9 January 2022 (UTC)
btw: I initially made a flurry of edits to this page, not knowing how to switch between editing and previewing before publishing the final result. Now that i know how to do that, i frequently forget to comment my edits, because the comment text box disappears when you scroll down to preview them. The "Publish" button meanwhile remains visible, and lets you publish sans comment without raising any warning. Sorry for the inconvenience, and thanks for bearing with me while i get the hang of this... Hobeewahn ( talk) 22:55, 9 January 2022 (UTC)
![]() | This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
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. 141.161.84.89 20:23, 30 April 2007 (UTC)
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.
what kind of 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!
Leibniz 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 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)
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
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.
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.
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. -- 76.224.107.34 20:36, 10 June 2007 (UTC)
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)
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.
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)
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)
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)
Is there a reference for this "response", or is it original research?-- Hq3473 22:50, 28 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)
The example in the article concludes;
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)
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
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)
I just want to simplify that ordinary language version of the alleged apparent self- contradiction. -- Ludvikus ( talk) 21:16, 20 April 2008 (UTC)
About this, added and removed twice now;
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)
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:
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:
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)
I took out:
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)
Lestrade ( talk) 12:00, 17 August 2011 (UTC)Lestrade
I removed this:
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. 67.198.37.16 ( talk) 20:04, 22 July 2016 (UTC)
I removed this:
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)
References
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For any x and y, if x is identical to y, then x and y have all the same properties.
How about the property "has proper name ‹some name›"? For instance, in Peano arithmetic the number "two" can have "name" either or . The equality is valid, but the property "has proper name " is valid only for the first closed term. I have no questions about first-order schema of the indiscernibility of identicals: The property "has proper name " cannot be expressed by a formula of Peano arithmetic. But the second-order axiom (cited above) embraces all properties, including inexpressible by formulas.
Eugepros ( talk) 07:08, 8 April 2018 (UTC)
This looks like an individual's own rather eclectic research and reasoning, involving what seems to be a fair bit of digression. I do not understand all the concepts mentioned in this section, but reading the "symmetric universe" section is hugely less comprehensible than the rest of the article, and at the very least needs to be explained more slowly and in much greater depth. The first paragraph is ok, and I can just about follow the second one, but the discussion of how there "must" be a "quantum asymmetry" in Black's universe, the curvature of space-time and the origins of calculus leave me behind after countless readings. Indeed, in a discussion of pure logic, the involvement of so many physical experiments seems to me inexplicable.
Additionally, what seem to be bold claims are made, with a very low footnote density. The footnotes available don't all have page numbers, and one is just a google drive link. Finally, I have never seen the "--" punctuation anywhere on wikipedia before, which does not give me great confidence in the writer's familiarity with wikipedia standards and practice.
Under the "Be Bold" policy wikipedia recommends, I will delete the "symmetric universe" section in a week or so, if nobody has any other thoughts. I will leave in the first paragraph, and perhaps point the reader to the issues involved in the later discussion, if I can figure out what they are. If an article offers poorly-evidenced text that seems like gibberish in comparison with the rest of the article, it surely should not be part of any encyclopedia.
If someone can explain to me what the "symmetric universe" section is about, clearly and with sources, I suggest they do so then replace the original text with their explanation. It is possible that I am just being very thick, feel free to tell me so. — Preceding unsigned comment added by 79.66.61.34 ( talk) 16:10, 22 August 2018 (UTC)
As a mathematician, reading this article, I can't help but notice that this principle is similar in philosophy to the Yoneda lemma in category theory, which can be seen as a formalization of this principle. That lemma states (as a corollary) that two objects who can't be distinguished based on how they interact with their surroundings (i.e. X,X' satisfy Hom(X,Y)~Hom(X',Y) for all Y, naturally) have to be isomorphic (i.e. X~X', which is the mathematical way of saying "they are the same object with another name"). Maybe speaking about the link could be interesting ? A "serious" reference might be needed. 2A01:E0A:2F0:4C0:3D15:3C47:C8AE:903E ( talk) 10:40, 6 March 2020 (UTC)
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 218.219.191.130 ( talk) 00:07, 10 September 2007 (UTC)
Problem with Superman example
My second comment is about the Clarke Kent and Superman example. If I am not mistaken, one of them wears a tight costume and the other doesn't. So there is actually a difference between them in that sense. If there was absolutely no difference there could be no reason why any person would thinks the other person is able to fly and not able fly at the same time. What is more, as far as I am aware, no one can have two distinct and contradictory thoughts at the same time, so the woman that thinks one thing is not actually the same as the woman that thinks another thing - as she is now in the future and has therefore changed. All of this becomes very complicated. — Preceding unsigned comment added by 81.164.118.56 ( talk) 02:02, 18 December 2020 (UTC)
"calling the Łukaszyk–Karmowski function a metric although it isn't positive definite is a matter of naming, not a critique"
This distance function is not “a metric”, as it does not satisfy the 1st metric axiom (albeit satisfying the remaining two). So perhaps Łukaszyk–Karmowski metric should be moved to Łukaszyk–Karmowski distance (cf. [1]) and appropriately rephrased. Ł-K distance is positive definite for Dirac delta distributions. The only point here is that there exists a distance function (Ł-K distance) that does not follow the identity of indiscernibles ontological principle/1st metric axiom. That can be considered as a critique of this principle. Guswen ( talk) 11:23, 21 January 2021 (UTC)
References
"Considerations about how not to define a metric in mathematics are not relevant enough to merit their own section."
We're not considering how to define (or how not to define) a metric in mathematics. We're talking about Identity of indiscernibles as a general principle. — Preceding unsigned comment added by Guswen ( talk • contribs) 00:56, 23 January 2021 (UTC)
This page is getting very long, it might be a good idea to archive it. I would use User:ClueBot_III unless there are objections. Phlsph7 ( talk) 04:59, 24 January 2021 (UTC)
"on the contrary: the proof of the ugly-duckling theorem *uses* the identity of indiscernibles"
Indeed, the proof of the ugly-duckling theorem uses the identity of indiscernibles principle to arrive at its contradiction. The proof assumes a set of 2^n objects, each having properties different than the other (no two objects in this set have all their properties in common). Each property of an object is a considered to be a Boolean-valued predicate, and thus the set form a Boolean {0, 1}^n address space, wherein each address (object) is indeed (logically) separate from the others. But the Ugly duckling theorem proves that any two addresses (objects) in this set are equally similar, as they share the same number of compound predicates, all the logical functions that can be formed from the properties of these objects, with connectives of negation, conjunction and disjunction. (*) Therefore any two addresses (objects) in this set are two things under one name. On the contrary to the statement that “to suppose two things indiscernible is to suppose the same thing under two names”.
The Ugly duckling theorem proves that no discernibility (understood as distinguishability, recognizability, identifiability, distinctability, classifiability, etc.) is possible without some sort of bias. Guswen ( talk) 12:20, 21 January 2021 (UTC)
The material in this article should be relevant to the topic and correct. Both of these points have been contested concerning the discussed addition. So far no reliable sources have been presented to dispel these concerns despite repeated requests to do so. Jochen Burghardt and I are in agreement that the material should be removed, see WP:NOTUNANIMITY. If you (Guswen) have reliable sources to present now then we can discuss them. If not then the contested material should be removed until sources are presented. Phlsph7 ( talk) 03:58, 25 January 2021 (UTC)
I think as a minimum we would need a reliable source for the relation between the UDT and indiscernibility (in the sense discussed here). Phlsph7 ( talk) 06:42, 25 January 2021 (UTC)
References
Hi there Jochen, sorry about my last edit deleting that entry from the Proof Box! When I saw it was still there after my previous edit, I assumed I had overlooked it by mistake, as often happens. I didn't actually mean to delete it twice : ]
Although I agree with you that indiscernibility of identicals does not imply reflexivity, I'm not quite following the proof summary you put in that section. The axiom of reflexivity has no predicate or relation variables, and references no predicates or relations aside from equality. This means that _all_ predicates and relations satisfy that axiom: equality does so explicitly, while everything else does so vacuously. The things that must actually satisfy it are the domain objects -- i.e., numbers -- that can stand in for x. So long as every number equals itself, relations of any arity can be defined in any manner consistent with the other axioms of logic. Even when the _domain_ is completely empty, reflexivity holds vacuously by virtue of the universal quantifier on x.
Since the other proofs in the box are both premised on reflexivity, it might make more sense if the entry that establishes reflexivity (i.e., IdIndscs → Refl) were placed up front. The titles of the other proofs ("IndscIDs ∧ Refl → Foo") would then make it clear that indiscernibility of identicals doesn't imply Refl... because, if it did, the "∧ Refl" would be superfluous. What do you think? Hobeewahn ( talk) 00:26, 8 January 2022 (UTC)
@Jochen: Thank you for this detailed explanation! At last I understand the thrust of your proof, and it makes perfect sense. To make it totally explicit what the empty relation is standing in for, I would just preface it with "When used as '='," ...
I agree that "ID" would look better than "Id" in both of those identifiers, so I will change it as per your suggestion. (There is something of a religious war going on at my workplace about whether acronyms should be uppercased within CamelCase identifiers-- but that is a long story ;) ) And thanks for fixing the oversized arrows left by my earlier edit -- they look way better now.
Hobeewahn ( talk) 21:48, 9 January 2022 (UTC)
btw: I initially made a flurry of edits to this page, not knowing how to switch between editing and previewing before publishing the final result. Now that i know how to do that, i frequently forget to comment my edits, because the comment text box disappears when you scroll down to preview them. The "Publish" button meanwhile remains visible, and lets you publish sans comment without raising any warning. Sorry for the inconvenience, and thanks for bearing with me while i get the hang of this... Hobeewahn ( talk) 22:55, 9 January 2022 (UTC)