Indiscernibility for spheres and Trinity

Discuss Black's two spheres with respect to each formulation of PII.

The entire section 2 of SEP, Identity of Indiscernibles, was quoted inside my input above at this point in this conversation, but omitted here for copyright reasons.

Here’s how Black’s two spheres (a perfectly symmetric world with exactly two congruent, qualitatively identical spheres a and b at fixed distance d, and nothing else) bear on each formulation of the Identity of Indiscernibles you listed.

Quick outcomes

Formulation	Verdict in Black’s world	Why
(PIIb) Necessarily, no two objects share all their intrinsic pure properties	Fails	a and b match in all intrinsic qualitative features (mass, radius, composition, etc.). No intrinsic qualitative difference is available.
(PIIa) Necessarily, no two objects share all their pure (qualitative) properties	Fails	Even allowing extrinsic pure properties, each sphere has the very same qualitative relational profile (e.g., “being d from a sphere,” “being exactly like some sphere”). Pure properties can’t tell them apart.
(PII) Necessarily, no two objects share all their non-trivializing properties	Holds	Non-trivializing impure properties are allowed. Then a has the property being d from a, b has being d from b; these differ in how the objects stand to particular things and so are non-trivializing impure properties. Thus a and b do not share all non-trivializing properties.
(PIIQa) (absolute discernibility) No two objects satisfy exactly the same 1-place open sentences	Fails (on the intended, non-trivial reading)	No qualitative monadic predicate singles out just one sphere. (If you allow names/identity, it becomes trivially true, but then it’s not the intended, non-trivial version.)
(PIIQr) (relative discernibility) No two objects satisfy exactly the same 2-place open sentences in both orders	Fails	Relative discernment needs an asymmetric relation (R(a,b) & ¬R(b,a)). In the symmetric world, natural relations (like “is at distance d from”) are symmetric, so they don’t relatively discern a and b.
(PIIQw) (weak discernibility) No two objects are such that each satisfies all the same 2-place open sentences with itself	Holds (non-trivially)	Use the irreflexive, symmetric relation R = “is at distance d from.” Then R(a,b) and R(b,a), but ¬R(a,a) and ¬R(b,b). That weakly discerns them without invoking identity/difference.

A few clarifications

• The key to (PII) vs (PIIa)/(PIIb) is whether you let in impure but non-trivializing properties. Properties like being two miles from a (or from b) “mention” a particular and so are impure, but they do mark a difference in how each sphere is (its spatial relations to a particular), not merely which object it is. Hence they’re non-trivializing on the understanding you gave and they rescue (PII).

• For (PIIa) and (PIIb), excluding impure properties keeps the playing field purely qualitative; then Black’s setup is the classic counterexample: two concrete objects sharing all pure—and even all intrinsic pure—properties.

• On the modal reading: if Black’s world is metaphysically possible, then the necessary versions of (PIIa) and (PIIb) are false. A merely contingent version (e.g., “as a matter of fact, nothing is like that”) could still be consistent with Black’s case.

Up shot: Black’s spheres overturn the strong, qualitative theses (PIIa) and (PIIb) and the absolute/relative Quinean versions, but they do not defeat the weak Quinean version (PIIQw) or the broad, non-trivializing version (PII).

Make this table of which versions of PII hold or fail in the case of Boethius’s view on the Trinity, where the three persons are distinguished by relations (one-to-one paternity/filiation, two-to-one spiration/procession), and discuss.

Here’s the same style of table, now for Boethius’s Trinitarian picture (one divine nature; three persons individuated solely by relations of origin: one-to-one paternity/filiation and two-to-one spiration/procession).

Outcomes in Boethius’s Trinity

Formulation	Verdict	Why (sketch)
(PIIb) Necessarily, no two objects share all their intrinsic pure properties	Fails	Father, Son, Spirit share all intrinsic qualitative features (omnipotence, eternity, simplicity, etc.). No intrinsic qualitative difference is available to distinguish the persons.
(PIIa) Necessarily, no two objects share all their pure (qualitative) properties	Holds	Pure relational properties differentiate them: Father begets someone (∃y Paternity(x,y)); Son is begotten (∃y Filiation(y,x)); Spirit proceeds from two (∃y∃z [y≠z ∧ Spiration(y,z,x)]). These predicates don’t name any particular individual, so they’re pure and they come apart across the three persons.
(PII) Necessarily, no two objects share all their non-trivializing properties	Holds	Since (PIIa) holds and pure properties are a subset of the non-trivializing ones, (PII) follows.
(PIIQa) (absolute discernibility) No two objects satisfy exactly the same 1-place open sentences	Holds	Unary qualitative predicates distinguish: “is unbegotten / begets someone,” “is begotten,” “proceeds-from-two.” Each applies to exactly one person.
(PIIQr) (relative discernibility) No two objects satisfy exactly the same 2-place open sentences in both orders	Holds	There are asymmetric binary relations: Paternity(Father, Son) but not Paternity(Son, Father). For Spirit pairs, define a binary R(x,y) := ∃z Co-Spirates(x,z,y); then R(Father, Spirit) and R(Son, Spirit), but not conversely. So each distinct pair is relatively discerned.
(PIIQw) (weak discernibility) For any two, there’s a symmetric, irreflexive relation they bear to each other but not to themselves	Holds	Let OriginLinked(x,y) hold iff either Paternity(x,y) or Paternity(y,x) or ∃z Co-Spirates(x,z,y) or ∃z Co-Spirates(y,z,x). This is symmetric, irreflexive, and true of every distinct pair—so weak discernibility is satisfied non-trivially.

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Discussion

• Big contrast with Black’s spheres: In Black’s world, only weak (irreflexive, symmetric) discernment is available; absolute and relative versions fail. In Boethius’s Trinity, the asymmetric relations of origin (paternity/filiation) and the structured two-to-one spiration give you absolute and relative discernment straightaway.

• Why (PIIb) fails but (PIIa) survives: The classical claim that the persons are consubstantial and share all intrinsic divine attributes blocks (PIIb). Yet purely qualitative relational predicates—built from origin relations without naming any particular—differentiate the persons, so (PIIa) (and hence (PII)) hold.

• Purity vs. naming: Predicates like “is begotten” or “begets someone” are pure in the relevant sense (they quantify over “someone,” not “this individual”). So the individuation is qualitative even though it is wholly relational.

• Handling spiration’s arity: Though spiration is fundamentally two-to-one, that’s compatible with Quinean tests: you can define suitable unary and binary predicates by existentially binding the co-spirator, which is standard when assessing discernibility in first-order terms.

• Modal note: If Boethius’s Trinitarian structure is metaphysically possible, that already refutes the necessary (PIIb). If one also holds that the Trinity is necessary, then (PIIa), (PII), and the Quinean versions hold necessarily, while (PIIb) is necessarily false.

If you want, I can also draw the minimal first-order signature (relations, axioms) that makes the table’s verdicts come out exactly as above.