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Persistence

I operator relation keystone published
Published 2026-08-15

Type: Operator / Relation

Working Definition: Persistence is the relation under which a later stage w2 is the same concrete continuant as an earlier stage w1, despite changes in properties, parts, location, or function. A disclosed persistence basis states what must remain across that change for sameness to hold. The relation is evaluated at the two stages' own times, not at one shared snapshot.

Function in the Grammar: Persistence answers a question the encyclopedia's other entries leave open. Parthood records a whole's structure at one time. Transformation records the difference between two states. Neither settles whether the bearer of the later state is numerically the same continuant as the bearer of the earlier one. Identity addresses socially and narratively attributable persistence; Continuity addresses institutional accountability through authority chains and a Ledger. Neither currently adjudicates a repaired building, a transplanted organism, a refactored software system, or a rebuilt ship. PERSISTS_AS fills that gap without installing one universal answer to Ship-of-Theseus problems. The persistence basis discloses what matters in a given domain — matter, organization, function, causal continuity, designated essential parts, authorized succession, or a named combination — enumerated in Formal Grammar Representation below. Two analysts may disclose different bases and reach different conclusions about the same pair of stages. This grammar treats that as a disclosed disagreement, not a contradiction, because the two claims are different five-argument statements. As with Parthood's PART_OF, the five-argument PERSISTS_AS assertion generally requires reification for an OWL rendering, for example a PersistenceAssertion node carrying persistence basis, context, and modeling domain.

Formal Grammar Representation

<persists-as-statement> ::= PERSISTS_AS(
  earlier_stage: <continuant-stage>,
  later_stage: <continuant-stage>,
  persistence_basis: <persistence-basis>,
  context: <context-id>,
  modeling_domain: <modeling-domain>
)

<continuant-stage> ::= physical-object-stage | organism-stage
                     | document-stage | software-system-stage
                     | other identifiable candidate referring to
                       one thing at one time
                       (a referential label, not a commitment to
                       temporal parts; see Semantic Constraints)

<persistence-basis> ::= matter-continuity | organization-continuity
                      | function-continuity
                      | causal-spatiotemporal-continuity
                      | designated-essential-parts
                      | authorized-succession-or-provenance
                      | disclosed-combination

<modeling-domain> ::= bounded regime fixing which persistence bases
                     and continuant-stage types apply

PERSISTS_AS(w1, w2, p, c, d)

Semantic Constraints

A PERSISTS_AS statement is syntactically well-formed when both stages, a persistence basis, and the context and modeling domain are identifiable. Whether the relation actually holds is a separate question the syntax cannot settle.

t1 = STAGE_TIME(w1, c)
t2 = STAGE_TIME(w2, c)

CANDIDATE_PERSISTENCE(w1, w2, p, c, d) iff:
  t1 < t2
  PERSISTENCE_BASIS_APPLIES(p, w1, w2, c, d)
  SATISFIES_PERSISTENCE_CRITERIA(w1, w2, p, c, d, t1, t2)

PERSISTS_AS(w1, w2, p, c, d) iff:
  ADMISSIBLE(p, d)
  CANDIDATE_PERSISTENCE(w1, w2, p, c, d)

CANDIDATE_PERSISTENCE is an admissibility-independent derived relation, not a primitive: it is itself defined, by the iff above, from STAGE_TIME, PERSISTENCE_BASIS_APPLIES, and SATISFIES_PERSISTENCE_CRITERIA. What makes it admissibility-independent is only that its definition never references PERSISTS_AS or ADMISSIBLE. ADMISSIBLE is defined over CANDIDATE_PERSISTENCE, in Admissibility below, without that reference reintroducing the cycle. ADMISSIBLE(p, d) is defined from CANDIDATE_PERSISTENCE alone. PERSISTS_AS is defined last, from ADMISSIBLE and CANDIDATE_PERSISTENCE together. The dependency runs one way, so no definition here presupposes itself.

An undisclosed persistence basis leaves a claim incomplete rather than false. A disclosed basis whose criteria are unmet leaves the claim well-formed but not holding. A claim where t2 is not later than t1 is well-formed but can never hold; the ordering requirement is a structural condition, not a contingent one. These are three distinct states, not one undifferentiated failure.

earlier_stage and later_stage are referential labels for a candidate at one time. They are not a commitment to a perdurantist temporal-parts metaphysics over an endurantist one. An endurantist reading treats w1 and w2 as full designations of one continuant at two times. A perdurantist reading treats them as temporal parts. PERSISTS_AS is neutral between these pictures. It requires only that a persistence basis be disclosed and its criteria checked.

SATISFIES_PERSISTENCE_CRITERIA is supplied by the persistence basis, in the same way Parthood's SATISFIES_PART_CRITERIA is supplied by a unity basis. This grammar does not decide, for any basis, how much change a continuant can survive. That determination is the basis's content, disclosed by whoever asserts or evaluates the claim.

Two statements PERSISTS_AS(w1, w2, p1, c, d) and PERSISTS_AS(w1, w2, p2, c, d) with p1 != p2 may differ in whether they hold. Formally, this is never a contradiction. The two statements have different persistence-basis arguments. They are different well-formed statements, not opposite truth-values assigned to the same proposition. That formal point holds regardless of what p1 and p2 actually mean.

Substantively, whether the difference reflects a genuine disagreement worth taking seriously is a further question. That further question is only answered when p1 and p2 are truly different, each independently admissible, bases. If both are asserted to name the same persistence basis under different labels, a difference in outcome is a computational or evidentiary error, not a substantive disagreement. Resolving the formal point does not settle which basis is apt for a given decision. That remains separate.

Admissibility. PERSISTS_AS claims numerical sameness: w1 and w2 are stages of one continuant, not merely similar or continuous stages. Numerical identity is transitive and non-branching as a matter of logic, independent of this grammar's choices. If w2 is numerically identical to w1, and w3 is numerically identical to w2, then w3 is numerically identical to w1. No single w1 can be numerically identical to two distinct simultaneous later stages. ADMISSIBLE(p, d) names exactly the closure properties a basis needs to guarantee both:

ADMISSIBLE(p, d) iff, for every context c drawing on d and every
w1, w2, w3 typed for d, with t1 = STAGE_TIME(w1, c),
t2 = STAGE_TIME(w2, c), t3 = STAGE_TIME(w3, c):

  Criteria closure:
    SATISFIES_PERSISTENCE_CRITERIA(w1, w2, p, c, d, t1, t2)
      AND SATISFIES_PERSISTENCE_CRITERIA(w2, w3, p, c, d, t2, t3)
      AND t1 < t2 < t3
    =>  SATISFIES_PERSISTENCE_CRITERIA(w1, w3, p, c, d, t1, t3)

  Applicability closure:
    PERSISTENCE_BASIS_APPLIES(p, w1, w2, c, d)
      AND PERSISTENCE_BASIS_APPLIES(p, w2, w3, c, d)
      AND t1 < t2 < t3
    =>  PERSISTENCE_BASIS_APPLIES(p, w1, w3, c, d)

  Non-branching, at a shared later time:
    NOT EXISTS w2': w2' != w2 AND STAGE_TIME(w2', c) = t2
      AND CANDIDATE_PERSISTENCE(w1, w2, p, c, d)
      AND CANDIDATE_PERSISTENCE(w1, w2', p, c, d)

  Non-branching, at a shared earlier time:
    NOT EXISTS w1': w1' != w1 AND STAGE_TIME(w1', c) = t1
      AND CANDIDATE_PERSISTENCE(w1, w2, p, c, d)
      AND CANDIDATE_PERSISTENCE(w1', w2, p, c, d)

Every variable in this definition beyond p and d is bound. c, w1, w2, w3, and their stage times are universally quantified by the opening clause. w2' and w1' are existentially bound within their own non-branching clause. ADMISSIBLE(p, d) is therefore a condition on p and d alone, as its signature claims, defined entirely over CANDIDATE_PERSISTENCE, SATISFIES_PERSISTENCE_CRITERIA, and PERSISTENCE_BASIS_APPLIES. None of these reference PERSISTS_AS or ADMISSIBLE itself.

Non-branching is restricted to stages at the same time. CANDIDATE_PERSISTENCE(w1, w2, p, c, d) and CANDIDATE_PERSISTENCE(w2, w3, p, c, d), with w2 and w3 at different times, are not branching. They are the ordinary case of a continuant with more than one later stage. The guaranteed-transitivity result below shows PERSISTS_AS(w1, w3, p, c, d) holds too in that case. An unrestricted non-branching rule would forbid that ordinary case. It would directly conflict with the transitivity this entry requires.

A candidate basis whose stated criteria would fail any of these closures is not tracking numerical sameness, whatever label it is given. It is not admissible here. It may instead name a real, useful, but weaker relation of local continuity or resemblance, which this entry does not formalize.

For an admissible p, transitivity is therefore guaranteed:

PERSISTS_AS(w1, w2, p, c, d) AND PERSISTS_AS(w2, w3, p, c, d)
  AND same (p, c, d)
  AND STAGE_TIME(w1, c) < STAGE_TIME(w2, c) < STAGE_TIME(w3, c)
  =>  PERSISTS_AS(w1, w3, p, c, d)                              [transitivity, admissible p]

This conclusion is fully licensed by the definitions above, not merely suggested by them. Both premises supply ADMISSIBLE(p, d) for the same basis and domain, and a CANDIDATE_PERSISTENCE fact for each pairwise link. Applicability closure combines those links into PERSISTENCE_BASIS_APPLIES(p, w1, w3, c, d). Criteria closure combines them into SATISFIES_PERSISTENCE_CRITERIA(w1, w3, p, c, d, t1, t3). Together with t1 < t3, these establish CANDIDATE_PERSISTENCE(w1, w3, p, c, d). ADMISSIBLE(p, d) was already established. PERSISTS_AS(w1, w3, p, c, d) follows by definition.

A basis stated in a form that would fail these closures — such as retention of a majority, by mass, of the immediately preceding stage's material, with no fixed anchor — is not admissible as stated. A basis anchored instead to a fixed original stage, such as retention of a majority of that stage's material, does not have this defect. The sorites-style erosion the Ship of Theseus puzzle turns on is evidence that a basis was stated in the non-admissible, unanchored form. It is not evidence that persistence itself fails to compose.

Formal Pattern

Direct-persistence pattern:
  t1 = STAGE_TIME(w1, c)
  t2 = STAGE_TIME(w2, c)
  t1 < t2
  PERSISTENCE_BASIS_APPLIES(p, w1, w2, c, d)
  SATISFIES_PERSISTENCE_CRITERIA(w1, w2, p, c, d, t1, t2)

  THEREFORE CANDIDATE_PERSISTENCE(w1, w2, p, c, d)

  ADMISSIBLE(p, d)

  THEREFORE PERSISTS_AS(w1, w2, p, c, d)

Composed-with-Transformation pattern:
  PERSISTS_AS(w1, w2, p, c, d)
  -- established independently, e.g. via the direct-persistence pattern

  THEREFORE it is licensed to name w1 and w2 as one subject x and write:

  Transformation(
    id: tau,
    subject: x,
    from_state: S1,
    to_state: S2,
    effective_at: STAGE_TIME(w2, c),
    governing_basis: g,
    domain: d
  )

Transformation's own entry requires that x be the same grammar-tracked subject before and after: "Simple Transformation preserves subject identity." Writing a Simple Transformation across w1 and w2 without first establishing PERSISTS_AS(w1, w2, p, c, d) presupposes the identity question this entry exists to settle. Transformation's own text directs compound, cross-subject cases to explicit Identity and Continuity rules. PERSISTS_AS supplies that rule for the physical or structural case those two entries do not cover.

CANDIDATE_PERSISTENCE is an admissibility-independent derived relation, checked directly against a basis's stated criteria without reference to ADMISSIBLE or PERSISTS_AS. Neither CANDIDATE_PERSISTENCE nor PERSISTS_AS is primitive in this grammar; both are defined from STAGE_TIME, PERSISTENCE_BASIS_APPLIES, and SATISFIES_PERSISTENCE_CRITERIA, directly or through ADMISSIBLE. PERSISTS_AS is fully defined as ADMISSIBLE(p, d) together with CANDIDATE_PERSISTENCE(w1, w2, p, c, d), with no further route to establishing it. The direct-persistence pattern is a sufficient route to CANDIDATE_PERSISTENCE, not the only one. Once CANDIDATE_PERSISTENCE and ADMISSIBLE(p, d) are both established, PERSISTS_AS follows automatically, not as an alternative or additional route.

Core Relations

Relation Notes
Context supplies stage time STAGE_TIME(w, c) is drawn from Context for each stage individually, not from one shared snapshot.
Persistence Basis states what must remain Continuity of matter, organization, function, causal or spatiotemporal continuity, designated essential parts, authorized succession or provenance, or a disclosed combination may serve as the basis.
Modeling Domain bounds typing Fixes which persistence bases and continuant-stage types apply.
Constitutive Rule may supply the basis When persistence turns on authorized succession or provenance, a Constitutive Rule may state or support that basis.
Parthood remains distinct, and may compose PART_OF describes a whole's organization at one snapshot. PERSISTS_AS separately adjudicates whether an earlier snapshot and a later one belong to one continuant.
Transformation requires the license PERSISTS_AS supplies A Simple Transformation presupposes that its subject is the same grammar-tracked entity before and after. PERSISTS_AS, once established, is what licenses naming w1 and w2 as that one subject.
Identity may inform, without settling Social or narrative recognition may support a persistence judgment. This encyclopedia's current Identity entry does not adjudicate physical-object persistence, the gap this entry fills.
Continuity remains distinct An institution may remain continuous through complete physical and personnel turnover. That institutional fact does not by itself establish PERSISTS_AS for any physical substrate.
Validity evaluates Whether a disclosed persistence basis's criteria actually hold is a Validity question, following the same well-formed/holds distinction as Count-As and Parthood.
Evidence supports contested application Material analysis, provenance records, or structural surveys may support or contest a claimed SATISFIES_PERSISTENCE_CRITERIA.

Typical Questions

Examples

Domain Earlier stage Later stage Persistence basis and illustrative pattern
Architecture / construction A historic building before a multi-decade restoration A candidate later stage after most original material has been replaced An anchored structural-and-causal-continuity basis's stated criterion is compatible with an unbroken repair history of this kind. An anchored matter-continuity basis's stated criterion may not be, depending on how much original material remains.
Science / biology An organism before near-total cellular turnover A candidate later stage after that turnover organization-continuity and causal-spatiotemporal-continuity bases are commonly compatible with turnover of this kind. A strict matter-continuity basis commonly is not.
Digital systems A software system before a full rewrite of its codebase A candidate later stage of the system after the rewrite A function-continuity basis's stated criterion may be compatible with a rewrite of this kind if external behavior and deployment identity are preserved. A matter-continuity basis is not typically formulated for this modeling domain.
Ordinary life / artifacts A violin before extensive restoration replacing most of its wood The instrument later sold under the same name A designated-essential-parts basis's stated criterion may be compatible with this case if a trade-recognized essential component, such as the original scroll or label, is retained. A stricter basis may not be.

Each row states a case type and the pattern a given basis's stated criterion is designed to track. None of these rows settles a specific real dispute. The actual result for any given building, organism, system, or instrument depends on the specific record and the check set out in Formal Pattern above.

Distinctions

Persistence ≠ Qualitative similarity. An exact replica need not be the same object. A perfect duplicate built alongside the original does not PERSISTS_AS it, regardless of how closely the duplicate resembles it.

Persistence ≠ Unchangedness. One object can survive substantial alteration. A repaired building may PERSISTS_AS its earlier stage under a structural-continuity basis despite significant material replacement.

Persistence ≠ Parthood. Current structure does not determine survival across structure changes. PART_OF describes organization at a single time; it does not by itself answer whether a later structure belongs to the same continuant as an earlier one.

Persistence ≠ Transformation. A Transformation describes what changed, not whether the bearer survived. Recording that a building's part-structure changed between two dates does not settle whether the later building PERSISTS_AS the earlier one.

Persistence ≠ institutional Continuity. An organization may remain legally continuous despite complete personnel and asset turnover. That institutional fact is evaluated through authority chains and a Ledger, not through any disclosed persistence basis applied to a physical substrate.

Persistence ≠ the existing social-narrative Identity. Recognition and narrative can support some persistence judgments without settling physical-object identity. A community may narrate a rebuilt hall as "the same hall" while a stricter material basis would not.

Persistence ≠ Succession. A successor may inherit functions, rights, or provenance without being numerically identical to its predecessor. A newly incorporated company that acquires a defunct firm's brand and contracts need not PERSISTS_AS that firm.

Persistence ≠ Resurrection or reconstruction. Rebuilding from preserved plans does not automatically re-create the original object. A modern reconstruction built to a lost building's original drawings is a new later stage. Its PERSISTS_AS claim against the destroyed original depends on the disclosed basis. That claim commonly fails under a causal-spatiotemporal-continuity basis. The same claim can succeed under a designated-essential-parts or authorized-succession basis.

Common Failure Modes

Mode Description
Basis undisclosed or basis-shopped A PERSISTS_AS claim is asserted without naming a persistence basis, or a basis is chosen after the fact to fit a preferred conclusion. Where to look: whether the basis was disclosed before the criteria were checked, and whether an alternative basis would have been rejected by the same claimant.
Qualitative similarity substituted for persistence A replica or near-duplicate is treated as the same continuant because it closely resembles the original. Where to look: whether any disclosed basis's criteria were actually checked, or only visual or functional resemblance.
Formal non-contradiction and substantive disagreement conflated, in either direction A genuine difference in disclosed basis is reported as a factual contradiction, or a difference between two claims that name the same basis under different labels is excused as legitimate disagreement instead of flagged as an error. Where to look: whether the two claims' persistence-basis arguments are genuinely distinct and each independently admissible; a real difference in basis is never a contradiction, and a same-basis discrepancy is never a permitted disagreement.
Succession or resurrection smuggled in as persistence A successor entity or a reconstruction is treated as numerically identical to its predecessor without checking a disclosed basis's criteria against it. Where to look: whether provenance, inherited rights, or preserved plans are being substituted for causal or material continuity.
Institutional Continuity conflated with physical Persistence An organization's legal continuity through personnel and asset turnover is treated as evidence that some physical substrate persisted. Where to look: whether the claim is really about authority chains and a Ledger, which is Continuity's domain, rather than about a disclosed persistence basis.
Inadmissible basis used as if numerically sound A persistence basis is stated in a pairwise, unanchored form — for example, majority retention relative to the immediately preceding stage rather than a fixed original — that fails transitivity or non-branching across a longer span. Where to look: whether the basis's criterion is anchored to a fixed stage or drifts stage-to-stage; an unanchored basis is not admissible for PERSISTS_AS even where each adjacent pair looks satisfied.
Ordering violation treated as a mere failure to satisfy A claim where the later stage's time is not after the earlier stage's time is treated as a basis-dependent failure rather than a structural one. Where to look: whether t1 < t2 was checked before evaluating PERSISTENCE_BASIS_APPLIES or SATISFIES_PERSISTENCE_CRITERIA at all.

Minimum Viable Test Case

w1 = Building_A_stage1
w2 = Building_A_stage2
p1 = StructuralAndCausalContinuityBasis
p2 = MaterialContinuityBasis
c  = Context(Restoration_Record_7)
d  = Architecture

t1 = STAGE_TIME(w1, c)
t2 = STAGE_TIME(w2, c)
t1 < t2

p1 states: the persistence criterion is an unbroken causal-repair
chain plus retention of the load-bearing structural organization,
regardless of which material components are replaced. This
criterion is anchored to the unbroken chain itself, not to any
pairwise comparison against an immediately preceding stage.

p2 states: the persistence criterion is retention of a majority,
by mass, of the original constituent material, where "original"
names the designated founding stage of this continuant's history,
fixed once and not re-anchored at each later stage.

ASSUME ADMISSIBLE(p1, Architecture)
  -- stipulated for this worked case. Establishing ADMISSIBLE in
     general requires checking criteria closure, applicability
     closure, and non-branching over every w1, w2, w3, c in the
     Architecture domain, which this single case does not attempt.
     p1's anchored, unbroken-chain criterion is the kind of
     criterion those closures are designed to hold for; the check
     below is a spot check against this stipulation, not a proof
     of the general claim.

ASSUME ADMISSIBLE(p2, Architecture)
  -- likewise stipulated. p2's anchor to the fixed founding stage,
     rather than to whichever stage happens to precede a given
     comparison, is the kind of criterion those closures are
     designed to hold for

NOT EXISTS w2': w2' != w2 AND STAGE_TIME(w2', c) = t2
  AND CANDIDATE_PERSISTENCE(w1, w2', p1, c, d)
  -- confirms this instance is consistent with the ADMISSIBLE(p1, ...)
     stipulation: Building_A_stage2 is the only candidate stage
     occupying this site at t2 under p1

NOT EXISTS w2': w2' != w2 AND STAGE_TIME(w2', c) = t2
  AND CANDIDATE_PERSISTENCE(w1, w2', p2, c, d)
  -- the same instance-level check for p2

PERSISTENCE_BASIS_APPLIES(p1, w1, w2, c, d)
SATISFIES_PERSISTENCE_CRITERIA(w1, w2, p1, c, d, t1, t2)
  -- the repair history is unbroken and the load-bearing structure
     is unchanged; p1's criterion holds

PERSISTS_AS(w1, w2, p1, c, d)

PERSISTENCE_BASIS_APPLIES(p2, w1, w2, c, d)
NOT SATISFIES_PERSISTENCE_CRITERIA(w1, w2, p2, c, d, t1, t2)
  -- fewer than half of the founding stage's original material
     remains by mass; p2's criterion fails

NOT PERSISTS_AS(w1, w2, p2, c, d)

Result:
  Both claims are well-formed and both bases are disclosed. They
  reach different conclusions about the same pair of stages. This
  is a disagreement between disclosed bases, not a contradiction:
  PERSISTS_AS(w1, w2, p1, c, d) and PERSISTS_AS(w1, w2, p2, c, d)
  are different statements with different third arguments.

Incompleteness case, undisclosed basis:
  PERSISTS_AS(w1, w2, unstated, c, d)

  This statement is incomplete rather than false. It cannot be
  evaluated until a persistence basis is supplied.

Ordering-violation case:
  w3 = Building_A_stage0   -- STAGE_TIME(w3, c) < t1

  PERSISTS_AS(w1, w3, p1, c, d)

  STAGE_TIME(w1, c) < STAGE_TIME(w3, c) does not hold, since w3 is
  the earlier stage. This claim is well-formed but can never hold,
  regardless of which basis is supplied. The failure is structural,
  not a matter of unmet criteria.

Intellectual Lineage

The Ship of Theseus puzzle this entry addresses is recorded in Plutarch, Life of Theseus (circa 75 CE), where it is already reported as disputed among earlier philosophers, some holding the repaired ship remained the same and others that it did not. Thomas Hobbes, De Corpore (1655), Part II, Chapter 11, extends the puzzle with a second ship reassembled from the original's discarded planks, sharpening it into a two-candidate dilemma. That extension anticipates the fission problem this entry's non-branching requirement is built to foreclose: at most one of two distinct candidate successors can be numerically identical to the same earlier stage. Derek Parfit, Reasons and Persons (1984), analyzes fission cases in depth and treats branching as the sharpest challenge to identity-based persistence; this entry's admissibility requirement follows the standard response in that literature, deriving non-branching from the transitivity and symmetry of numerical identity itself rather than stipulating it as an independent rule. David Wiggins, Sameness and Substance (1980; revised as Sameness and Substance Renewed, 2001), argues that persistence and identity criteria are relative to a sortal concept — what counts as the same ship, building, or organism depends on which kind is in question. This entry's disclosed persistence basis adapts that sortal-relativity into an explicit grammar argument, rather than leaving the relevant sortal implicit. The modern endurantism/perdurantism debate over how objects persist through time, given its canonical statement in Ted Sider, Four-Dimensionalism (2001), motivates this entry's explicit neutrality: earlier_stage and later_stage are referential labels compatible with either metaphysical picture, and this entry does not adjudicate between them.

Cross-References

Required

Consequential

Evidential

Cross-References

This entry is AI-assisted. Reviewed by the human author before publication.