# For those that believe in it: Why is Leibniz' Law of Indiscernables true to you?

There has been a criticism of it wich goes like this:

https://en.wikipedia.org/wiki/Identity_of_indiscernibles#Critique

Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it is sufficient that one provide a model in which there are two distinct (numerically nonidentical) things that have all the same properties. He claimed that in a symmetric universe wherein only two symmetrical spheres exist, the two spheres are two distinct objects even though they have all their properties in common.[15]

Black argues that even relational properties (properties specifying distances between objects in space-time) fail to distinguish two identical objects in a symmetrical universe. Per his argument, two objects are, and will remain, equidistant from the universe's plane of symmetry and each other. Even bringing in an external observer to label the two spheres distinctly does not solve the problem, because it violates the symmetry of the universe.

For those that believe if All the properties and attributes of X and Y are identical,they are the one entity/object/selfsame/absolutely one,how do you respond to this?

• Kant already criticized LL (Leibniz law) similarly as your quoted example using a pair of gloves. The thing is you’re implicitly holding absolute space view as Kant, so you can identify left from right from outside of your imagined closed world. But for Leibniz holding different views your ideal logical case is like Buridan’ ass, yes, you cannot identify your symmetric shadow from yourself, but metaphysically this never applies to LL, same reason as Buridan’s ass “paradox”… Apr 30, 2022 at 21:25
• On the deeper note, as Leibniz himself explained in some personal correspondence that this absolute space view is in contrast with PSR (principle of sufficient reason) as contingent truths, and he believes PSR is the correct one. If this is true, and there seems no logical contradiction between PSR and LL expressed in 2nd-order logic, thus logically you cannot really determine you're now in the left or right world even in this ideal thought experiment, thus effectively they're identical intuitively and phenomenologically. Think about this for a moment, actually it's not that surprise... May 1, 2022 at 0:08
• They have separate position, which is an attribute like anything else, and so are different objects. If they shared the same position(in space and time), then they are certainly the same object. (The universe you propose lacks any orientation, so we cannot really assign them positions, but they have them, and different ones too, nonetheless.) May 1, 2022 at 19:30
• @causative Max Black is well aware of your defense of LL from the usual absolute space view, he criticized this kind of quick response from external observer as quoted in the last sentence above in Wikipedia Even bringing in an external observer to label the two spheres distinctly does not solve the problem, because it violates the symmetry of the universe. And Leibniz is famous for not holding this kind of external observer view about space contrary to most other people including Newton. Whatever coordinate you choose, you're still viewing the other sphere (world) as an external observer. May 2, 2022 at 2:22
• @DoubleKnot When you say "absolute left or right," I cannot read this as anything other than talking about an absolute coordinate system in which there is an absolute "left" (say, the negative direction of the x-axis) and an absolute "right" (the positive direction). In physics there is no absolute right or left, but there are coordinate systems. What is right and what is left depends on your coordinate system, which is relative. So these words, "absolute space view," especially as you clarify it as meaning an "absolute left or right," are not descriptive of the use of coordinates in physics. May 2, 2022 at 2:41

Although Black speaks of properties, the problem is a more general labeling problem. Speaking of properties relates it to the logicism influencing the period in which he worked.

Both Kant and Wittgenstein criticized the logical import of the principle using geometric language. Importantly, the criticized it differently. Kant invoked numerical difference. Wittgenstein, by treating names as geometric points, invoked numerical identity. Black's model follows the Wittgensteinian criticism.

With regard to symbolic algebra, de Morgan asserted that the sign of equality cannot be purely formal because substitutivity requires warranting. This would be compatible with the Kantian criticism.

The development of logical calculi foreshadowed problems with the use of the sign of equality. Responding to both Frege's identity puzzles and Bradley's regress, Russell wrote his paper "On denoting." There is a specific passage in that document addressing the notion of "difference."

This speaks directly to the nature of truth in so far as naive use of language seems to use a correspondence theory. So, if one cannot denote what appears to be two objects, one cannot speak meaningfully of there being a difference.

There is, in this problem, an essential circularity that had been expressed quite clearly by Strawson in his book, "Individuals." If one insists upon using a geometric account for numerical identity, then one is confronted with the problem that parts of space separate points of space and that points of space differentiate parts of space. And, as ought to be expected, Strawson is arriving at this analysis after having used a visual illustration to explain why qualitative identity cannot serve as numerical identity.

It is well known that the received view of the first-order paradigm is based upon the folklore of arithmetization to ground the mathematical subject of real analysis. This has been problematic and one need not accept this view.

So, one might presume, for the sake of argument, that the use of Venn and Euler diagrams in the pedagogy of logic need not be divorced from geometric intuition. With that said, the formal sentences in bullet item 4 of the example section of the n-catlab entry,

https://ncatlab.org/nlab/show/gauge+space#examples

should be somewhat interesting.

There is no "singular" attached to the denial of distinctness in these sentences. Of course, this seems somewhat opposite to what Black is trying to explain. These formal sentences are using denoting symbols.

What I am looking at, however, are the propositional connectives.

The unary negation can be eliminated in favor of the denied conditional. Now, both connectives occur as loci in the free Boolean lattice on two generators.

This is a finite system. Where logicism focuses attention on Boolean polynomials because the great advance of symbolic algebra is the development of logical algebra, compositionality need have nothing whatsoever to do with polynomials.

One can use any definite system of truth tables with named third columns to identify each locus of the Boolean lattice with a 16 x 16 Cayley table. And, these Cayley tables are a grand expression of "identity" relative to the totality of relations. It is only possible because of finiteness.

These 16 tables are a precise construction of the infinite regress which foundationalism tries to avoid.

The physicists ought to appreciate he next step.

Who needs 'T' and 'F'?

Replace those two symbols arbitrarily with symbols expressing opposed orientation (one horizontal line and one vertical line). Eliminate the names used to construct the Cayley tables.

This will leave 16 4-vectors composed of horizontal and vertical lines.

If you really want to understand the significance of Black's paper (or, for that matter, the nature of counterfactuality associated with Wittgenstein's states of affairs), start with these 16 4-vectors and tell me your criterion for labeling your two abstract symbols in binary opposition as 'T' and 'F'.

If you think I am an asshole, compare the essential circularity observed by Strawson with what Kant says about mathematics and logic through his proxies sensibility and intelligibility.