surely the location of each sphere is a unique property? — QuixoticAgnostic

It's quite hard to get your head round this, because the universe is being described as if from the outside. One tends to imagine a box or a sphere with the symmetrical spheres in, and one can see that there are two, and that they are not in the same place - one on the left and the other on the right, or some such. But now imagine you are a symmetrical observer, looking at the same time from opposite sides.

Now the one on the left is also the one on the right, and the one on the right is also the one on the left. So there are still two, but they are indiscernible. Not identical, but indiscernible. How this feeds your structuralism, you'll have to tell me.

But are we not describing the universe from the outside? I don't believe indiscernible in this case means indiscernible with respect to an observer. PII only talks about properties of objects and the identity thereof. From Wiki: "For any x and y, if x and y have all the same properties, then x is identical to y."

The way I view the symmetric universe is (and this may be ignorant as I don't know the mathematics behind constructing spaces) a, say, 3-dimensional space with some point of symmetry (0, 0, 0). To maintain the symmetry of the universe, if you introduce sphere A at point (2, 4, 3), there must be sphere B at point (-2, -4, -3). Even if all other properties were equal, these clearly are discernible because they exist at different points in space. But I must be missing something because explaining like that just seems like common sense.

The way I view the symmetric universe — QuixoticAgnostic

But are we not describing the universe from the outside? — QuixoticAgnostic

You have to be careful which version of the principle it denies; not the one you quoted. I think as soon as you introduce the term 'indiscernable' the objection becomes cogent. Because to discern is to observe.In a symmetric universe, the observer is also symmetric, and cannot distinguish + and - . If you add asymmetric coordinates, the universe is not the same universe. Plus and minus is just the same as left and right, and cannot be distinguished.

I don't think i can explain it clearer, but perhaps Stanford can.

It all hinges on what is understood by 'property'. The only sense that makes PII true is the most inclusive (and deflationary): a property of a thing is anything that can be predicated of the thing, i.e. any true proposition with the thing as its subject. But that trivializes PII, because if two objects actually present themselves as numerically distinct, then you can always predicate something that would imply a distinction, if only by ostention ("A is this and B is that"). If this seems question-begging, I think that's because the PII doesn't really say anything metaphysically substantive - it is basically an explication of the concept of numerical identity.

You have to be careful which version of the principle it denies; not the one you quoted. — unenlightened

You mean between the "Identity of Indiscernibles" and the "Indiscernibility of Identicals"? I think I got the right one.

And Stanford doesn't explain it much better. However, I did find the original writing by Max Black, which (obviously) sheds some light onto his thinking. But since it's in the form of a dialogue, I'm not sure what I'm supposed to take as rational argument and what is just extraneous, such as the details of an observer passing through the mirror of the universe.

But let me propose this: you say if there is an observer, the observer is also symmetric. In fact, in your first comment you say, "But now imagine you are a symmetrical observer, looking at the same time from opposite sides." But in this case, you are a symmetrical observer, in two places at once. Is it not the case then, that this universe doesn't function as a universe where two objects exist symmetrically to each other, but that one object exists in multiple locations? In this case, it isn't the case that there exist two identical, but distinct spheres, rather there exists one sphere in two locations, given the properties of this symmetrical space we've created. That is a key distinction.

But that trivializes PII, because if two objects actually present themselves as numerically distinct, then you can always predicate something that would imply a distinction, if only by ostention ("A is this and B is that"). — SophistiCat

Did you mean numerically identical? Because if they're already numerically distinct, it wouldn't matter if you predicated another distinction. Although, if you did mean identical, I don't think it would still make sense because if we take PII to be true, it would be impossible for two things to be numerically identical in the first place, so we couldn't predicate anything to make a distinction.

I guess my confusion comes from not entirely understanding what properties are. In my own thinking, I feel I often conflate properties and relations, because I think about everything in terms of relations, but not necessarily in the philosophical sense.

I don't think I want to get involved in this conversation but I did find something you and your interlocutors may find of interest in regard to the question of the identity of indiscernibles. I found a brief discussion of this issue in Yitzhak Y. Melamed's book, Spinoza's Metaphyics, on page 32, where he says “In [Ethics Part I, Proposition 4], Spinoza presents and proves his own formulation of the identity of indiscerinbles.” At this point, there is a footnote referring the reader to Michael Dela Rocca, Representation and the Mind-Body Problem in Spinoza, UK: Oxford University Press, 1996.

He quotes P4 and goes and goes on to say:“P4 --Two or more distinct things are distinguished from one another, either by a difference in the attributes of the substances or by a difference in their affections.” The individuation principle suggested by this proposition stipulates that '(I) If x ≠ y, then there is some property (either essential or accidental) which belongs to the one but not to the other.' The proof of the proposition is relatively simple by EIpd”:

LATIN:

PROPOSITIO IV. Duae aut plures res distinctae vel inter se distinguuntur ex diversitate attributorum substantiarum, vel ex diversitate earundem affectionum.

DEMONSTRATIO. Omnia, quae sunt, vel in se, vel in alio sunt (per axiom. 1.), hoc est (per defin. 3. et 5.) extra intellectum nihil datur praeter substantias earumque affectiones. Nihil ergo extra intellectum datur, per quod plures res distingui inter se possunt praeter substantias, sive quod idem est (per defin. 4.) earum attributa earumque affectiones. Q. E. D.

ENGLISH (Bennett):

4: Two or more things are made distinct from one another either by a difference in their attributes or by a difference in their states. Whatever exists is either •in itself or •in something else (by A1), which is to say (by D3 and D5) that outside the intellect there is nothing except •substances and •their states. So there is nothing outside the intellect through which things can be distinguished from one another except •substances (which is to say (by D4) their attributes) and •their states.

He refers above to A1, D3, D4, and D5:

A1: Whatever exists is either in itself or in something else. ·As we have already seen, a substance is in itself, a mode is in something else·.

D3: By ‘substance’ I understand: what is in itself and is conceived through itself, i.e. that whose concept doesn’t have to be formed out of the concept of something else.

D4: By ‘attribute’ I understand: what the intellect perceives of a substance as constituting its essence.

D5: By ‘mode’ I understand: a state of a substance, i.e. something that exists in and is conceived through something else.

https://www.earlymoderntexts.com/assets/pdfs/spinoza1665.pdf

I hope you find this helpful. Best wishes to you. --Stabilius

Thanks for the reference. I'd be lying if I said I fully understood the argument, but I completely, wholeheartedly, and fundamentally disagree with D3, haha. Don't mean to disregard the argument altogether, but it seems not as relevant as I'd like to my philosophy given the disagreement. I'll keep it on the shelf though in case there's anything else to gleam from it.

Thanks, just trying to help out. Tuck it away; maybe someday . . . . a bell will ring (or not). Best wishes to you, Statilius.

The thought occurs to me that there is a discernible difference between the two spheres that is to be found in the relative difference in location. Each is different from the other precisely in the fact that it is on the opposite side of the symmetry plane. For each one, the other is different from it in being "over there" rather than "here". For each one, the other is to be found outside of itself. For each one, the addition of the other makes two. This is a "difference that makes a difference", to use Bateson's phrase regarding information.

If we had some sort of absolute space with a coordinate system, we could say that the two spheres are at different locations in this space. If we draw two circles on a sheet of paper on opposite sides of a line of symmetry, we can say that each occupies a different position on the sheet. But if there is nothing analogous to that sheet in the two-sphere universe, no background relative to which the spheres can be said to be located, it would seemingly present a problem of finding a difference in locations. Not so. A relative difference is still a difference. Each sphere stands a certain distance away from the other. For each sphere, the other sphere stands at such and such a distance away, while this one does not.

Instead of thinking about a possible difference between the two spheres, we could think about the difference between there being only one sphere and there being two spheres separated by some distance. These two situations are different, no? Perhaps by comparing the two situations, we can see what is the same and what is different, and by focusing on the difference, we may discover just what it is that makes each sphere different from the other. It seems that something is revealed by considering the universe as a whole rather than just comparing the spheres within it.

First, in a one-sphere universe, there is no gap between spheres. So that separation, that distance, is something extra in the two-sphere universe. Each is far from the other. For each one, it isn't just that there is another sphere, but also that this other sphere is some distance away. So comparing the two, the first sphere is just a sphere, while the second sphere is sphere with the addition of its distance. For each sphere, the other sphere isn't just a sphere. What it is for it to be what it is is for it to be some distance away. Its "awayness" could perhaps be seen as a property it has that makes it different.

This is an interesting feature of relativistic situations. There are situations even in our universe where it is thought that things might exist from some frames of reference and not from others.

Second, there are twice as many spheres. Having a second one presents a difference, namely, that now there are two.

One possible objection though to concluding that the two spheres are therefore different is that the plane of symmetry could actually be a reflecting plane. It might be that there is only one sphere and the reflecting plane whereby there appears to be another sphere. In that case, there really is no difference between the two spheres. Their relative separation is an illusion caused by the reflecting plane. The sphere can then be seen as related to the reflecting plane or maybe even related to itself by means of that plane, but not related to another sphere.

But Black's proposed situation of there being two identical spheres on opposite sides of a plane of symmetry, and not a reflecting plane, literally defines the situation from the beginning as a case of there definitely being two distinct spheres. We start with this knowledge about this universe. This being the case, we are given to know that each one is not identical with the other, as they are numerically distinct. So an illusion of two spheres is ruled out.

It might be best to regard the identity of indiscernibles as a principle that guides us in deciding what is the case when we are in a state of ignorance. Suppose that we are given a scenario in which there appear to be two spheres, but we don't know if there really are two. And we then use the principle of the identity of indiscernibles to try to decide. So we ask, are the two spheres different in any discernible way? If so, they are not identical. If not, there is only one sphere. But here, with this epistemological situation, we have no absolute knowledge. All differences amount to what is discernible from some vantage point. We are looking at this universe from some perspective. It is a matter of information. Where is this knower in relation to the spheres? For this knower, is one sphere further away than the other? If all that exists are the two spheres, then the knower can't be a third thing. If you are one of the spheres, if you then compare the two spheres, one is indeed at a greater distance. The two are then discernible.

Notice that for you, a knower, in order to be in a situation where one sphere is not at a greater remove, you must be separated from both spheres. You must be outside the universe, with some kind of absolute view. Or you could be at the center of this universe, perhaps with no discernible left and right sides yourself by which you can say that one sphere is on your right and the other on your left, or in front or behind or above or below. Both spheres are the same distance away. They are both the same, seemingly in every respect, no? Where then are you presented with the appearance of two spheres? It would seem that absent some means of differentiating, which, if present, would clearly distinguish the two spheres, you would be presented with only one impression, that of a sphere at some distance from you, and that's all.

I suggest that in any case where you imagine a situation of two spheres in relation to some observer, some difference between them will be revealed. For example, one sphere is in one direction, the other in another direction. But what has given you this impression of direction? Some kind of asymmetry in you must be present. We imagine looking one way and then the other, now seeing one, then seeing the other, perhaps by turning your head. A person with a turned head has a distinct left and right, which gives a means of discerning the spheres, as they can be seen as different with respect to these distinct sides of yourself.

Can you imagine a situation in which you, as an observer, could be receiving identical information from two different spheres at the same time and in which you definitely have an impression of there being two spheres? Suppose you are a head with two faces, each one facing a sphere. You are symmetrical with respect to the plane of symmetry. Each face sees a sphere in the same way, the same information arriving. Would you really have a sense of two spheres? By what means would you gain this impression? Could you actually tell the difference between a universe in which there is a two-faced head, each face seeing a different sphere, and a universe in which there is just a half-head with one face seeing a single sphere?

In order to tell the difference between two things, to have information about a difference, you must be able to compare. To know that a change has happened, for example, you must be able compare one state to another. You must be able to integrate information between the two. In the case of temporal differences, you must have memory. And this memory itself is an asymmetry in you between the two moments. In the later moment, you have more memories, more information.

If the two-faced head is to have some way of integrating the information arriving at both sets of eyes so as to yield an impression of two spheres on opposite sides of the universe, there must be a way to compare the two. And for there to be a way to compare the two, there must be a difference, an asymmetry, perhaps a difference in the brain in that head.

Suppose there is a brain in that head that is also symmetrical, and the neurons in the brain on one side of the symmetry plane are talking to the neurons on the other side. The neurons on one side send a message, saying, "I see a sphere, how about you?" These neurons also receive an identical message from the other side at the same time. Each side then concludes that yes, I see a sphere from my side and the other side also sees a sphere. But notice what's happening now. We again have a perspectival asymmetry. We are not looking at the universe from a perspective that is equidistant from the two spheres. We are looking at it from from the POV of neurons on one side, which are more distant from one sphere than the other. This is akin to being one of the spheres and regarding the other sphere, one being distant and one being close.

If what we are instead includes both halves of the brain on both sides of the symmetry plane at the same time and the symmetry is unbroken in every respect, it is hard to see how, from this point of view, we could have an impression of two spheres, or two separate impressions of one sphere. There is no way to compare, and thereby to separate, the two impressions. In order to do so, there would have to be some difference, however slight, by which we can say that brain-half A is telling me about seeing a sphere and brain-half B is also telling me about seeing a sphere. How do I have the impression of brain-halves A and B? How do I compare them and find a difference such that I am justified in concluding that I have received information from two distinct brain-halves? There must be a difference, a distinct signature from each half. It is the same problem as with the spheres.


I think that there is no conceivable situation in which an observer of any kind could have an impression of such a symmetric universe. All impressions of there being two spheres require some sort of asymmetry, either a discernible difference in the two sides of the universe, or a perspective on this universe that is asymmetric with respect to it, such that one side is more distant than the other.

If I am equidistant from the two sides, and I am also symmetric with respect to the plane of symmetry, it is impossible for me to have an impression of two distinct sides.

In imagining this situation, we are cheating by allowing ourselves this absolute perspective (non-perspective, or view from nowhere?) from outside the universe. And we are also at the same time, in our mental picture, smuggling in ways of perceiving that involve a body with asymmetries that has a location relative to the spheres. We imagine standing at some distance from the two spheres, such that one is somewhat to the left and the other is somewhat to the right. We, in our bodily experience of our world, often have just such an impression of something over to the left and something else over to the right. But here, we in fact have an asymmetric body with two distinct sides by which to compare. Our world that includes this "left-rightness" is not bisected by a symmetry plane. If it were, there would be no such left-right impression.

How do we get an impression of two spheres? You might say that we can see the gap between them. I then ask, how? How do we know there is a gap? We see the gap by seeing each sphere and comparing the two impressions and seeing that they are different in such and such a way, namely, that this sphere is on my left and that one is on my right and I am seeing them both at the same time but in different parts of my visual field. This requires a lot of information integration involving discernible asymmetries.

There likely is no absolute knowledge or objectivity of the sort that Black's proposed scenario seems to imply, where you, whatever you are, are just given to know, somehow, that there are two spheres on opposite sides of a symmetry plane.

Our real situation is perspectival, epistemological, and informational, and you can't know anything without being part of the system in question and interacting with the things in question. And here, to know distinctness, there must be a "difference that makes a difference".

Black's imagined situation is impossible. To just propose a universe in which there are two distinct spheres with all their properties in common on opposite sides of a symmetry plane might be comparable to just proposing a universe in which the rules of mathematics are called into question by there being a case of 2+2 equaling 5.

Saying that there are two things which are exactly the same in every way (having all their properties in common) while also saying that these identical things are yet being distinct is sort of like saying that there are two things which are in no way distinct, and yet are distinct. What it means to be distinct is having a difference in properties. So to say that there are two distinct spheres with all their properties in common is maybe the same as saying that there are two distinct spheres that are not distinct. A is both X and not-X at the same time. Black is here perhaps violating the law of non-contradiction. And along with this, he is invoking an absolute, objective perspective, which is arguably impossible, also implying a perfect symmetry in the knowledge or perceptual situation, while at the same time smuggling in intuitions and evoking mental pictures that only arise in an embodied, perceptually asymmetric situation, such as a human looking out at two spheres. He is asking us to bring up a mental picture that is impossible without asymmetries while telling us that this is symmetrical.

A very simple conception of the principle of the identity of indiscernibles is the following:

1. x = y [x is identical to y] if and only if x and y are indiscernible with respect to any and all attributes assignable to x and y.

2. x =/= [x is not identical to y] if and only if x and y are discernible with respect to at least one attribute assignable to x and y

3. x is a copy of y [neither 1 nor 2] if and only if x and y are indiscernible with respect to all attributes assignable to x and y except the attribute of spatial location at an instance of time.

All good so far but take a closer look at 3. If x is a copy of y then the converse, y is a copy of x, is also true. They are indiscernible except for their differing locations. If that's the case then both are original and also both are copies but a copy can't be an original - a contradiction. Ergo, it must be that, in the case of 3 also, x = y or x is identical to y.

Max Black's argument with two symmetric spheres that are not identical is covered by proposition 3 above but that leads to a contradiction: the two spheres are both original and also copies.

Spacetime seems to play a crucial role in this for at one instant a single object can't occupy two locations - Max Blacks criticism depends on this. The temporal aspect is different for a single object can occupy two different times either at the same location or at different locations. I guess my formulation of identity and nonidentity in the three ways I started off with in this post is pretty much it.

Another way of looking at the issue is to ignore spatial location altogther. Imagine Max Black's two symmetric spheres and call them A1 and A2. Hide them in a box and pick two locations in space S1 and S2. You pick the spheres at random and place one at S1 and the other at S2. Would you be able to, just on the basis of spatial location (S1 and S2), tell which sphere is A1 and which sphere is A2? No! Ergo, space or spatial location has no relevance to identity. Since the two spheres A1 and A2 are identical in all other respects, it must be that A1 is identical to A2.

:chin:

Indiscernibility of identicals: AxAy[x = y -> Fx <-> Fy] = for all x and for all y if x is identical to y then, x has property F if and only if y has property F

Identity of indiscernibles: AxAy[Fx <-> Fy -> x = y] = for all x and for all y if x has property F if and only if y has property F then, x is identical to y

Max Black's alleged "counterexample" of the symmetric spheres doesn't quite cut it because to be a counterexample that refutes the identity of indiscernibles, the symmetric spheres must be, well, indiscernible which, by virtue of their different locations in space, they're not.

The problem with denying the identity of indiscernibles is that any language rich enough to express that two entities are non-identical is rich enough to require, by virtue of that admission, that they have distinct properties: each has the property of being identical to itself, and not to the other.

So it's not clear how to make a functioning language that doesn't allow it as a principle. You'd need a primitive way to express identity that somehow couldn't be reformulated into the expression of a property.

If we had some sort of absolute space with a coordinate system, we could say that the two spheres are at different locations in this space. If we draw two circles on a sheet of paper on opposite sides of a line of symmetry, we can say that each occupies a different position on the sheet. But if there is nothing analogous to that sheet in the two-sphere universe, no background relative to which the spheres can be said to be located, it would seemingly present a problem of finding a difference in locations. — petrichor

I think Black's treatment actually disagrees with this. In the original text, his anti-PII character states:

Let me abandon the original idea of a plane of symmetry and to suppose instead that we have only a centre of symmetry. I mean that everything that happened at anyplace would be exactly duplicated at a place an equal distance on the opposite side of the centre of symmetry. — Black

So there is an absolute relative, which is the "centre of symmetry", and he even says that there is an "opposite" side, implying that despite the symmetry, one can have a notion of this side and that side. Also, sounds like by the very end of your post you seem to agree that there's something wrong with Black's thought experiment :grin:.

3. x is a copy of y [neither 1 nor 2] if and only if x and y are indiscernible with respect to all attributes assignable to x and y except the attribute of spatial location at an instance of time. — TheMadFool

I wouldn't say this is a necessary stipulation. It's strange to me how space and time are treated specially in metaphysics. Sure, they have a seemingly omnipresent role (quite literally) in our universe, but they are still just rules. They need not apply to all possible worlds. For example, Max's universe does happen to involve some notion of space because he defines spheres, but not necessarily time. Also, another unintuitive fact is that, depending on how you define Max's world, there may be objects existing in multiple places at once, something that seems impossible in our world (although it probably isn't with QM).

Also:

Max Black's alleged "counterexample" of the symmetric spheres doesn't quite cut it because to be a counterexample that refutes the identity of indiscernibles, the symmetric spheres must be, well, indiscernible which, by virtue of their different locations in space, they're not. — TheMadFool

That's exactly the point I made initially, that some have been trying to argue isn't a valid objection because the symmetry of the universe makes it such that either they aren't in different locations or that the observer can't discern the difference of location. At least I think that's what points were being made.

The problem with denying the identity of indiscernibles is that any language rich enough to express that two entities are non-identical is rich enough to require, by virtue of that admission, that they have distinct properties: each has the property of being identical to itself, and not to the other. — Snakes Alive

Are you talking about the concept of "thisness" or haecceity? I actually don't accept that as a defense of PII. Maybe it again boils down to what properties are and that I think about them in terms of relations rather than take them as fundamental.

although it probably isn't with QM — QuixoticAgnostic

QM doesn't do anything that weird, does it? You're probably referring to the concept of superposition in QM. Frankly, the idea seems not to hold up to scrutiny. They (who?) say that an electron cloud represents the the probability distribution of an electron's location in an atom, and this they claim to be the superposition for the electron - I guess this could be interpreted as an elctron being in two places at once. However, notice that the instant a measurement is made, the electron is discovered to be at a single location i.e. it's not the case that when a measurement is made the electron will occupy two different locations at the same time. So, it depends which stage of the process of pinpointing an electron's location we're at. Personally, I wouldn't take QM superposition as an instance of a single object occupying two locations at once because of what follows when a measurement is made - the localization of the electron at one place, not two or more, places.

That's exactly the point I made initially, that some have been trying to argue isn't a valid objection because the symmetry of the universe makes it such that either they aren't in different locations or that the observer can't discern the difference of location. At least I think that's what points were being made. — QuixoticAgnostic

You need to run this by me again. I don't get it. If you have the time and patience that is. Perhaps you can link me to a detailed account of Max Black's thought experiment. Thanks

I don't know, tbh. All I'm saying is, given how strange it is, and with how physicists say things of the like, I wouldn't be surprised if it was the case (though if I'm being perfectly honest with my beliefs, I expect it is).

You need to run this by me again. I don't get it. If you have the time and patience that is. Perhaps you can link me to a detailed account of Max Black's thought experiment. Thanks — TheMadFool

I'm speaking of the arguments other people have made in this thread. Specifically, unenlightened argued that even if they are in different locations, because the observer can't discern between them, PII is false. I don't think anybody actually made the argument that the two objects are actually in the same location though.

Here's the link to Black's original dialogue: http://home.sandiego.edu/~baber/analytic/blacksballs.pdf. Honestly, I think defenders of the argument might be reading too deeply into it. If you read it, there's a lot of waffling about what symmetric universe means. Like, initially the mirror of symmetry is literally a mirror, and so to be copied in the mirror means your heart is on the right side of your chest. That seems clearly ridiculous and unecessary to the argument, but shows, to me, that the argument isn't as profound as it seems to be. Of course, I'm probably just not understanding it properly.

By the way, anticipate a response in your other thread, it's basically the starting point to my metaphysics, and will probably shed some light onto my claims about space and time not being fundamental.

It seems Max Black's argument is more nuanced than I thought. Nothing unusual here - I always think simple. Will get back to you after I read the dialog in the link. :up:

Are you talking about the concept of "thisness" or haecceity? I actually don't accept that as a defense of PII. Maybe it again boils down to what properties are and that I think about them in terms of relations rather than take them as fundamental. — QuixoticAgnostic

It's not a matter of accepting or not.

If you say:

a =/= b

Then I can write you a property that a has but b doesn't, namely:

LAMBDA x[x = a]

The only way to deny the conclusion is to stop me, somehow, from being able to express this property. But you have to do it using a language that admits there is something about a that b doesn't have, viz. being a. Admit it, and then I can isolate the property a has but b doesn't – refuse to admit it, and you can't claim to have a counterexample.

"But that's not a property!" etc. Sure it is. I just wrote it. It's well defined, and meets any reasonable formal definition of a property you'd care to list. OK, so maybe you want to arbitrarily rewrite the rules so that this one doesn't count. Alright, but now you're just playing with words, so what's the point? We can gerrymander our definitions so that anything doesn't 'count' as a property anyway, so who cares? It's just a word game, then.

If you say:

a =/= b

Then I can write you a property that a has but b doesn't, namely:

LAMBDA x[x = a]

The only way to deny the conclusion is to stop me, somehow, from being able to express this property. — Snakes Alive

What formalism is this? I'm somewhat familiar with basic logics (propositional, first-order, second-order) but this doesn't quite look like any of those. Is it lambda calculus? How does the statement function then?

But I'm pretty sure your argument is circular; in trying to prove PII, you're assuming PII. You argue "If $a \neq b$, then there exists a property $a$ has that $b$ doesn't", but that is exactly what you're trying to prove! Your argument is of the form, $\forall x \forall y \left [ x \neq y \rightarrow \exists F \left ( \neg \left ( Fx \leftrightarrow Fy \right ) \right ) \right ]$,
which is just the contrapositive of PII, $\forall x \forall y \left [ \forall F \left ( Fx \leftrightarrow Fy \right ) \rightarrow x = y \right ]$. So your premise is equivalent to the conclusion, and your reasoning is circular.

Furthermore, if we grant you that the existing relation is that of identity, then the premise becomes, $\forall x \forall y \left [ x \neq y \rightarrow ( \neg \left ( x = x \leftrightarrow y = x) \right ) \right ]$ and logically reduces to:
$\forall x \forall y \left [ x \neq y \rightarrow \left ( \left ( x = x \wedge y \neq x \right ) \vee \left ( x \neq x \wedge y = x \right ) \right ) \right ]$,
$a \neq b \rightarrow \left ( \left ( a = a \wedge b \neq a \right ) \vee \left ( a \neq a \wedge b = a \right ) \right )$,
$a \neq b \rightarrow \left ( \left ( \mathrm T \wedge b \neq a \right ) \vee \left ( \mathrm F \wedge b = a \right ) \right )$,
$a \neq b \rightarrow \left ( b \neq a \right ) \vee \mathrm F$,
$a \neq b \rightarrow a \neq b$, which is a tautology.

I'm probably mistaken in my logic somewhere but I spent far too much time thinking about this to throw it away. So if I am wrong, please let me know, would be a good learning experience.

Another way of looking at the issue is to ignore spatial location altogther. Imagine Max Black's two symmetric spheres and call them A1 and A2. Hide them in a box and pick two locations in space S1 and S2. You pick the spheres at random and place one at S1 and the other at S2. Would you be able to, just on the basis of spatial location (S1 and S2), tell which sphere is A1 and which sphere is A2? No! Ergo, space or spatial location has no relevance to identity. Since the two spheres A1 and A2 are identical in all other respects, it must be that A1 is identical to A2. — TheMadFool

I believe this leaves out the fact that they ARE in different locations. i'm not sure you can leave out that fact to claim they are identical but not the same. I could just as easily say "well that one is different because it's over there, and that one is different because it's over there. It's not important to know which one is S1 or S2 only that they are different from each other at that instance in time.
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Also for there to be no ability to discern who's a copy and whose the original wouldn't this :

3. x is a copy of y [neither 1 nor 2] if and only if x and y are indiscernible with respect to all attributes assignable to x and y except the attribute of spatial location at an instance of time. — TheMadFool

have to be this :

"x is a copy of y [neither 1 nor 2] if and only if x and y are indiscernible with respect to all attributes assignable to x and y except the attribute of spatial location at ANY instance of time."

It may be nitpicking, in which case, I apologize. But it seems without it we can claim all sorts of difference as any other time.

In anycase, I really enjoyed the 3 premises! It was very clear and insightful.

Pick any formalism you like. Let's say we have a first-order logic with identity. Now let's assume a non-identity relation between a and b:

~[a = b]

Now suppose I define by a meaning postulate a predicate F as follows:

Ax[Fx <-> x = a]

It follows that Fa, but ~Fb.

Now, the only way to stop the conclusion is to prevent me from defining such a property. But then, it's just a matter of stipulations on the language. But in the typical first-order logic, you can't do that, for any predicate symbol is definable, so long as it's well-formed, by showing to which individuals it is true of and which it is false of. Well, we let F be true of a and false of b. It is not possible for you to prevent this move, since you have admitted that a and b are distinct individuals in the domain. Et viola.

If you want to block this move, you must invent a logic in which a predicate like F is not expressible. The first order logic is not such a logic.

Did you mean numerically identical? Because if they're already numerically distinct, it wouldn't matter if you predicated another distinction. Although, if you did mean identical, I don't think it would still make sense because if we take PII to be true, it would be impossible for two things to be numerically identical in the first place, so we couldn't predicate anything to make a distinction. — QuixoticAgnostic

No, I meant numerically distinct, as in the example of two spheres. If you are allowed to predicate anything by way of specifying a property, up to and including haecceity, then you will find a way to distinguish two apparently distinct objects, no matter how qualitatively alike they are.

My point is similar to that of @Snakes Alive, except that he too readily trivializes the discussion by reinterpreting it in the formal logic context. Informal language is richer than simple logics in that you can block moves like counting haecceity as a property, without violating any rules. But then again, even if you find yourself in a situation in which, playing by the rules of the game, you cannot express something with words that is otherwise apparent, all that this signifies is that the game has its limitations.

If there is a wider lesson here, it is that the traditional discourse of properties with its atomistic character, in which objects subsist without any external context, is inadequate.

Informal language is richer than simple logics in that you can block moves like counting haecceity as a property, without violating any rules. — SophistiCat

I actually don't think this is right. You can't block haecceity in natural language, either, which is why you need to come up with an artificial language that blocks it.

In English, for example, you'd have to say: "suppose there are two distinct spheres, but they're the same in every way."

The response is: what do you mean? You just said they're distinct. Surely the one is not the other, then – but I've just predicated, in the natural language, a property of one that the other doesn't have, viz. the property of being the one as opposed to the other.

I'm hopelessly confused. Let's try to start over from the beginning, leaving aside the logic of the statement for now.

First of all, to be clear: you are for PII, but only because the mere idea that two things are different can be predicated as thing 1 is thing 1 and thing 2 is thing 2. Is this a sort of haecceity or "thisness" as I eluded to before? That there is some substance underlying the properties that gives things their identity?

Secondly, and this is a rudimentary question, but what do you understand properties, predicates, and relations to mean, and how are they related or unrelated to PII? Are relations different from properties such that PII doesn't consider them as relevant to whether things are identical?

^ The above was written before Sophisticat's response.

No, I meant numerically distinct, as in the example of two spheres. If you are allowed to predicate anything by way of specifying a property, up to and including haecceity, then you will find a way to distinguish two apparently distinct objects, no matter how qualitatively alike they are. — SophistiCat

Which is why I don't think there's such a thing as haecceity, I suppose. If everything can be said the same about two things, except that one is this and one is that, then you're assuming PII is false, because if PII is true, then there aren't actually two things. You can't say one is this and one is that because there being two objects is just an illusion.

If there is a wider lesson here, it is that the traditional discourse of properties with its atomistic character, in which objects subsist without any external context, is inadequate. — SophistiCat

If this is what property is taken to mean, that they are intrinsic to objects and don't require external context, then I would do away with the notion of properties entirely. This is where my structuralist perspective comes in, because I don't believe objects have intrinsic qualities, rather that they are defined by their places in some structure, where the structure is just the composite of relations between things. However, property could still reasonably be defined of a thing as any relationship it has derived from the structure. This is why PII seems so important to me: if it were the case that two things related to everything else in exactly the same way, and those things were not actually the same thing, it would just shatter my worldview.

I actually don't think this is right. You can't block haecceity in natural language, either, which is why you need to come up with an artificial language that blocks it.

In English, for example, you'd have to say: "suppose there are two distinct spheres, but they're the same in every way." — Snakes Alive

You might instead want to say "they are alike in every way," making it clear that you mean "internal", or "qualitative" properties.

The response is: what do you mean? You just said they're distinct. Surely the one is not the other, then – but I've just predicated, in the natural language, a property of one that the other doesn't have, viz. the property of being the one as opposed to the other. — Snakes Alive

Ah, but unless you point at one or the other sphere as you say "one" and "the other," this will not get you out of the bind. Because without ostention, the sentence "the property of being the one as opposed to the other" is equally applicable to both spheres; the "property" therefore is exactly the same. It's frustrating, but there you are. If you are forbidden to point, you may find that you lack the words to distinguish between the two spheres in the absence of some additional structure in the world, such as an asymmetrically positioned observer. But this only says something about the limitations of expression.

making it clear that you mean "internal", or "qualitative" properties. — SophistiCat

Natural language draws no such distinction, AFAIK. So you're already inventing technical language precisely suited to denying the identity of indiscernibles.

Ah, but unless you point at one or the other sphere as you say "one" and "the other," this will not get you out of the bind. Because without ostention, the sentence "the property of being the one as opposed to the other" is equally applicable to both spheres; the "property" therefore is exactly the same. — SophistiCat

Sure, but you can just say 'call one A, the other B.' Problem solved.

I want to home in on the problem here: if you have no way to refer to them separately, you can't even coherently frame the scenario. If you do, then you have a way to distinguish their properties. You cannot have it both ways, where you have the vocab to frame the scenario, but not to distinguish between the two.

First of all, to be clear: you are for PII, — QuixoticAgnostic

I'm not 'for' it, I doubt it's even a substantive question. I just think it's a matter of what kind of language you want to use, and using a language that doesn't admit it is pointlessly complicated and serves no purpose.

That there is some substance underlying the properties that gives things their identity? — QuixoticAgnostic

You don't need any commitments to substance. You just say one, call it A, has the property of being A, while the other doesn't, because it's not A. You cannot say: but in virtue of what is it not the other? Well, you tell me – you imagined the scenario! If you insist there is nothing, not even one's being A and the other not being, then you cannot even coherently stipulate the very scenario you're trying to prove your point with. As soon as you admit one is not the other, I have the property distinguishing them.

Secondly, and this is a rudimentary question, but what do you understand properties, predicates, and relations to mean, and how are they related or unrelated to PII? — QuixoticAgnostic

That's just a question of what language you want to engineer. In the regular first order logic, a propetty is the denotation of any predicate symbol, and any predictable symbol is definable in terms of which individuals it applies to. So as soon as you admit a domain with two distinct individuals A and B, I can define a property that holds of A and not of B – indeed this is all defining a property consists in, specifying to which individuals it applies or doesn't.

If this is what property is taken to mean, that they are intrinsic to objects and don't require external context, then I would do away with the notion of properties entirely. This is where my structuralist perspective comes in, because I don't believe objects have intrinsic qualities, rather that they are defined by their places in some structure, where the structure is just the composite of relations between things. However, property could still reasonably be defined of a thing as any relationship it has derived from the structure. This is why PII seems so important to me: if it were the case that two things related to everything else in exactly the same way, and those things were not actually the same thing, it would just shatter my worldview. — QuixoticAgnostic

Then you are in trouble, because in a symmetric structure, such as the one with the two spheres, you still can't individuate an isolated part, even using relational properties, because symmetrical parts will have identical relational properties. But I don't think this undermines the structuralist view. The structuralist view is holistic, and when describing the structure of the system as a whole you don't run into any such difficulties. That's because you are not obligated to produce an exhaustive specification of an isolated part of the system; you always have the entire system as a background, so there is no danger of collapsing distinct objects into one, simply because you can't individuate them separately from the rest of the system. There is a difference between a structure with two spheres and a structure with one - so what does it matter that you cannot identify one of these spheres by its properties?

Traditional ontological ideas come under even more pressure in modern physics, where the notion of a bounded material object does not sit well with such "things" as fields. And event if we grudgingly grant objecthood to subatomic particles, we then have to deal with bosons (integer-spin particles, such as photons) that can share all of their properties, including their position. So that while we may know that there is more than one particle, it is not possible, even in principle, and even with the loosest definition of a property, to individuate any of these particles. All we can say is that there are n particles in the system sharing the same state (and even that n may not be precisely known, being subject to Heisenberg uncertainty).

Natural language draws no such distinction, AFAIK. So you're already inventing technical language precisely suited to denying the identity of indiscernibles. — Snakes Alive

I disagree, I think there is a difference in usage between "the same" and "alike." In any case, technical or not, this is more than a stripped-down language of basic logical systems.

Sure, but you can just say 'call one A, the other B.' Problem solved. — Snakes Alive

Before you can call one A, the other B, you have to indicate which is A and which is B - otherwise A and B will be interchangeable and you are back where you started. So it's haecceity or bust, it seems.

I want to home in on the problem here: if you have no way to refer to them separately, you can't even coherently frame the scenario. If you do, then you have a way to distinguish their properties. You cannot have it both ways, where you have the vocab to frame the scenario, but not to distinguish between the two. — Snakes Alive

I agree with the spirit of what you are saying, but I still maintain that individual properties are not always the way to go. They are very convenient, but only up to a point. When we start pushing against edge cases, we should be prepared to give them up. We don't always need to refer to individual parts by their properties in order to know that they are there - we can infer their existence from the system as a whole.

You can describe the system as a whole so as to imply that there are two distinct objects in it. That should satisfy your reasonable demand that we should be account for what we know. But further steps, such as introducing pseudo-properties, are superfluous.