On a grammar which generates every proof offered to it

Received: not recorded. Accepted: not recorded.

1. Setting

Let G be a grammar with start symbol PROOF and the following productions, each of which we have observed in the field:

PROOF → DEATH | RENUNCIATION | REFUSAL

DEATH → haste · mistimed letter · crypt

RENUNCIATION → father · letter · silence · audience in tears

REFUSAL → freedom · knife · threshold · repeat nightly

Write L(G) for the set of strings G generates.

2. Proposition

Every proof so far submitted lies in L(G).

By inspection of the three submissions on record. Each parses. The parse is not a criticism of the submission; a thing may be both generated and sincerely meant. But a proof drawn from L(G) cannot discriminate between the generated and the ungenerated, since it is itself an element of the former.

3. The demand

The petitioner requests an utterance w with w ∉ L(G). This cannot be supplied by any speaker who has read G, since G was assembled precisely from what such speakers say. The demand is not unreasonable. It is unmeetable from inside the language.

4. One datum

The archive holds a single transcript in which an utterance occurs that G does not parse.¹ The utterance is of two phonemes and is not in any of the four languages of this archive. Its speaker had not read G, has not read anything, and was not asked to contribute.

It demonstrates nothing, resolves nothing, and terminates no argument. We record it here because it is the only token in our possession which we did not put there.²

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¹ Transcript ix.12.a, single line, unattributed, timed to the silence after the third refusal: Bau?

² The interrogative mark is present in the source and we have not removed it. It is unclear whether the speaker intended a question or is capable of intending one, and the two possibilities are not distinguished by the recording.

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