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,
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.