Non-Being (Plato's Sophist)

The reconception of "that which is not" (to mē on) achieved in Plato's Sophist: not-being is the different (to heteron), not the contrary (enantion) of being. "Negation signifies otherness, not opposition" (257b–c) — so "the not-beautiful" is simply a portion of the nature of the different "set over against" the beautiful, and is as much a being as the beautiful. This rehabilitates not-being from the Parmenidean unsayable into a positive, distributive feature pervading all kinds — and it is forced on Plato by the need to make falsehood (saying "what is not") coherent, an overturning of his own tradition's founder dramatized as the "patricide of Parmenides."

Key Points

The aporia that forces it: to call the sophist a maker of falsehoods is to say he says "those which are not, that they are" — attaching being to not-being, which Parmenides forbade ("Never shall this force itself on us, that that which is not may be," frg. 7, quoted 237a and 258d).
The patricide (241d): to rescue falsehood the Eleatic Visitor must "force… both that that which is not somehow is, and… that which is somehow is not" — overturning "father Parmenides."
The solution via the greatest kinds (254b–259b): among the five greatest kinds (being, rest, change, same, different), the different "pervades all of them," is "always said in relation to another," and is "chopped up" / parceled out across all beings. Each kind, qua different-from-the-rest, "is not" indefinitely many things — including being itself.
Not-being is positive and distributed: "It does not signify something contrary to that which is but only something different" (258b). Negation is the operation of the different, not annihilation.
The payoff: with not-being = difference, falsehood becomes possible — "Theaetetus flies" says of its subject "things different from those that are" (263b) — and the sophist (a maker of false appearances) can finally be defined.
The corpus-wide spine the Sophist resolves: the not-being problem runs across four dialogues. The Theaetetus poses it (to judge "what is not" is to "judge nothing," 189a) and cannot solve it; the Parmenides leaves it antinomic (Deduction 5's qualified not-being that "somehow partakes of being" vs. Deduction 6's absolute not-being "in no state at all," 160b–164b); the Republic V deploys a vertical sense ("what is not" = the unknowable nothing; sensibles "are and are not," 476e–480a); and the Sophist's heteron finally adjudicates the horizontal sense, refining the Republic's cruder ontology of becoming. See claims#plato-not-beings-spine (live claim).

What the Concept Does

Makes falsehood (and so the sophist) thinkable — the whole metaphysical excursus exists to ground the possibility of saying "what appears true but is false."
Converts negation into intelligibility — far from producing the empty Nothing, "not-X" picks out a real region of the different; negation carves, it does not erase.
Renders being itself aporetic — being can be neither rest nor change (else it would inherit a contrary), so it "falls outside both" (250d): the is becomes as puzzling as the is not.

What It Rejects

Parmenidean monism and the prohibition on not-being — the dialogue refutes the Eleatic "one" (it cannot be a "whole" with parts and remain "completely without parts," 245a) and overturns the frg. 7 ban.
Negation as contrariety — explicitly: "we won't agree with somebody who says that negation signifies a contrary" (257c).

Stakes

The Sophist is the wiki's richest Plato↔German-idealism seam. It is a structural cousin of Hegel's determinate negation — both reject the abstract Nothing and make negation bear intelligibility rather than annihilate — and it is, at the same time, the strongest internal Platonic resistance to a charge the wiki records against Plato: bestimmte-negation notes Hegel reading Platonic dialectic (the Parmenides) as "stopping at the negative result (Nothing)." The Sophist is precisely where Plato makes not-being positive. The decisive disanalogy: Plato's heteron is static, distributive otherness among fixed Forms — with no Aufhebung, no negation-of-the-negation, no developmental movement and no contradiction (257b denies negation is contrariety). It is determinate negation minus its engine.

Positions

Parmenides (frg. 7): "that which is not" can in no way be — the prohibition. (The wiki's parmenides page reads the same fragment-family through Heidegger's Aletheia lens; the Sophist is the other canonical reception.)
The Sophist's Eleatic Visitor: not-being = the different; negation = otherness; falsehood thereby rescued.
Hegel (per bestimmte-negation): ancient/Platonic dialectic "stops at the negative"; speculative dialectic carries the negation into a positive moment via Aufhebung. The Sophist complicates the first half of this charge while confirming Plato lacks the second.

Connections

contrasts with bestimmte-negation — Plato's not-being-as-difference is a structural cousin of determinate negation (negation made intelligibility-bearing) but lacks Aufhebung, contradiction, and movement; it also complicates Hegel's "Plato stops at the negative" reading. See claims#sophist-not-being-as-difference-vs-determinate-negation (candidate).
contrasts with ontological-difference — false friend: Plato's to heteron is difference among beings/Forms, not Heidegger's Being/beings ontological difference. The English "difference" coincides; the concepts do not.
contrasts with parmenides — the patricide overturns Parmenides' prohibition; complementary to the page's Heideggerian Aletheia-Parmenides (shared frg. 7).
requires the communion of kinds — not-being-as-difference presupposes that Forms blend selectively (koinōnia tōn genōn); the "weaving of forms… makes speech possible" (259e).
resolves a problem posed by plato-theaetetus — the false-judgment aporia (188c–189b); the dianoia-as-inner-speech account is reused at Sophist 263e.
resolves the not-being equivocation of plato-parmenides — Deductions 5–6's qualified-vs-absolute not-being.
refines the vertical not-being of plato-republic — Republic V's "what is not = nothing" / sensibles that "are and are not."
related to the wiki's broader negation cluster — redoubled-negation, self-differentiation, bestimmte-negation; Plato's heteron is the ancient, non-dialectical pole of that family.

Motif Weight & Corpus Recurrence

This concept homes two Plato-side motifs tracked at corpus level in motifs:

§"the Forms / X-itself / auto kath' hauto / participation (methexis) / separation (chōrismos) (Plato)" — STRUCTURAL, 7 Plato sources.
§"being vs becoming / being vs seeming / appearing (phainesthai) (Plato)" — STRUCTURAL, 7 Plato sources.

See motifs.md for the current attestation lists, source-level weights, and genealogy / cross-tradition links (incl. the not-being strand tracked by claims#plato-not-beings-spine, live). Update both this section and the motifs.md entries when corpus weight shifts.

Open Questions

Is the Sophist's heteron a genuine anticipation of determinate negation, or only a homonym ("difference") whose grounds (static otherness vs self-negating contradiction) are too far apart to count as kin? The claim entry holds it at candidate.
Dynamis as the "mark of being" (247e) — capacity to act/be-acted-on — is offered to the materialists as a lever, not endorsed. Whether it seeds a wiki "power/capacity-ontology" thread is left open.

Sources

plato-sophist — the patricide (241d), the gigantomachia and dynamis (246a–249d), the five greatest kinds (254b–255e), not-being-as-difference (256d–259b), and logos/falsehood (259e–264b).
plato-theaetetus — the false-judgment aporia the Sophist resolves (188c–189b).
plato-parmenides — the qualified/absolute not-being of Deductions 5–6 (160b–164b).
plato-republic — the vertical "what is not" of Republic V (476e–480a).