Filed on 2026-10-10 from a search the owner asked for. The search found no documented case of two communicative relata that no relation differentiates, even weakly, and counted as two. Ordinary candidates fail for one reason. Identical utterances in unison are made by different speakers, and a relation such as “speaks in unison with”, holding only between distinct performers, is irreflexive and so discerns them weakly. That is Leitgeb and Ladyman’s graph with an edge, not the edgeless one (LIT-154, read in NOTE-133).
This case is constructed. What rests on a documented source is the institution’s norm: identifying marks void a ballot. The papers, the voters and the marks are stipulated. It is built to test which version of CLAIM-136 the owner holds.
The case
An election uses identical, unnumbered ballot papers. Two voters each mark a single cross for candidate C, and both papers go into one box. At the count both are valid, both are for C, and both are tallied. The result records two votes for C, and the returning officer and the candidates’ agents count them as two.
Take as the structure the relations that make a paper a vote in this election: it is an official paper of this election, deposited in this box, marked for C, valid, counted toward C. In that structure each ballot bears the same relations to everything else, the election, the box, C and the tally, and no relation to the other ballot. Formally the structure is a star. The ballots b₁ and b₂ are leaves joined only to shared nodes. They have the same atomic relations to every element, and to each other, as each has to itself, which is none. So merging them is a strong homomorphism, and every identity-free formula φ gives φ(b₁, b₂) ↔ φ(b₁, b₁). No irreflexive relation holds between them: they are not even weakly discernible. This is Leitgeb and Ladyman’s edgeless two-node graph (LIT-154), realized by an institution.
The documented anchor. The institution strips the relation that would discern the papers. UK Representation of the People Act 1983, Sch. 1, r. 47(1)(c): any ballot paper “on which anything is written or marked by which the voter can be identified except the printed number on the back … shall … be void and not counted.” The exception matters. UK papers carry a printed number, and a sealed record links it to the voter, so UK ballots are discernible and the UK is not the case. That is why the case is stipulated with unnumbered papers. Which jurisdictions use unnumbered papers was not checked.
What it can show
It forces the choice between CLAIM-136’s two versions, which the claim says the owner has not made.
- What the strict version must say. A relatum exists as an individual only in so far as its relations differentiate it. The ballots are counted as two, so the strict reading must find a relation that tells them apart. None lies inside the structure that makes a paper a vote, so it must go outside it: to the causal history of each paper (who marked it), or to its position in the box. That is CLAIM-138’s cost. Identity is then fixed by relations specified independently of the act’s constitutive structure, and r. 47 shows that the institution treats those relations as no part of the vote, since a mark that discerns the paper voids it. The alternative for the strict reading is to count the votes as one, which the tally refutes.
- What the weak version must say. The ballots are two, and nothing intrinsic grounds the plurality. That is Esfeld and Lam’s move, “one can simply accept a numerical distinction (diversity) … as primitive” (LIT-895, read in NOTE-695, p. 33), and Leitgeb and Ladyman’s edgeless graph. It concedes what CLAIM-134 says the weak version concedes, a non-relational ground of numerical identity.
So the case does not refute CLAIM-136. It makes its dilemma concrete, on an example in which the institution, not the theorist, removes the differentiating relations.
For CLAIM-138. The strict version’s only way out is the remedy CLAIM-138 says is costly: relations picked out from outside the structure. Here those relations are causal and spatial, and the institution says they are not part of what the vote is. So the case shows the cost of that remedy in a setting where the act’s own constitutive structure is fixed by rule.
What it cannot show
That a non-institutional communicative act has undifferentiated tokens. A vote is an institutional act and an expressive one. Whether it is a “communicative relatum” in CLAIM-136’s sense is arguable.
The strongest reply. A vote is a status assigned to a physical token, and the tokens are discerned physically, by position, by paper fibre, by who marked them. Secrecy hides the relation “cast by”; it does not abolish it. On this reply the strict version needs no help from outside the act, because the act includes its token. The case answers only if the structure of the vote is taken to be the rule-given relations. Which structure counts is the point at issue, and that is CLAIM-138’s question again.
That it happened. It is stipulated, and not documented as an episode. Two identical valid ballots for one candidate in one count are commonplace, but no particular count was examined.
What would close the gap. A documented count under an unnumbered-ballot system, with its rules on identifying marks, would make the anchor exact. A judgement test could ask competent officials whether two such ballots are “two votes” and whether anything about the vote, as opposed to the paper, tells them apart. That would show which structure participants take the act to have.
Sources
- Representation of the People Act 1983, Sch. 1, r. 47 (UK), https://www.legislation.gov.uk/ukpga/1983/2/schedule/1/paragraph/47. Read directly on 2026-10-10 (revised text).
- H. Leitgeb and J. Ladyman, “Criteria of Identity and Structuralist Ontology”, held as LIT-154, read in NOTE-133.
- M. Esfeld and V. Lam, “Moderate structural realism about space-time”, held as LIT-895, read in NOTE-695.
- The search’s second construction, two couples in unconnected rooms performing the same fixed exchange (Leitgeb and Ladyman’s two isomorphic components), is weaker: the participants’ biographies discern the acts unless they are excluded by fiat. It is not filed.