Station Two · Axiom 0 · Distinction

Station Two · Axiom 0

Distinction - state it clean, then prove it.

Distinction Contrast → Distinction → Condition → Licence → Return → Truth Closed FOL

Station Two · Axiom 0 · Distinction

Distinction is a definition, not a refusal. To determine a thing is to set it apart from what it is not. Anything that may hold meaning, truth, or ontological standing is therefore resolved from being indistinguishable.

For anything to be true, it must not be false. Distinction carries that requirement.

Definition

A is determinate only if A is distinct from not-A. No measure. No quality. Just a clean not.

Consequence

Where items are indistinguishable, truth and meaning do not apply. Distinction is the surface on which they can appear.

What distinction is

Distinction is the clean refusal of sameness on a pair that is already in contrast. It tells apart without choosing a winner. It prepares a stable surface for rules to act.

👁A ≠ 👁B

Read: A is not B. No direction implied.

Contrast underneath

Contrast is the deeper dependency. Even a perfect tie is seen against the possibility of a difference. Distinction rides on that base.

👁A ⇄ 👁B

Tension present. No outcome.

Symmetry is not collapse

If two items coincide, that sameness is still read against the condition where difference could have existed. Symmetry is contrast at rest, not the removal of contrast.

Symmetry: tension held in balance.
Distinction: the balance breaks and a not appears.

Common confusions

Self opposition is not distinction. A thing cannot be not itself.
Opposition in both directions is unresolved. Distinction can hold while outcome does not.
Value does not live here. Truth labels appear only later at return.

How action begins

A condition can act only where contrast already holds. When it grants permission in one direction, that permission is a licence.

👁A → 👁B

A condition licences A toward B.

Opposition without decision

If both directions are licenced, there is no settled outcome.

👁A → 👁B and 👁B → 👁A

Unresolved. Distinction remains. No return.

Return and where truth enters

A return is present when one direction stands and the reverse does not. Truth is the label attached to a code of that settled read.

👁A ⇒ 👁B

No reverse return.

Diagonal ban

No self contrast. No self licence. No self return.

not (👁A ⇄ 👁A)

not (👁A → 👁A)

not (👁A ⇒ 👁A)

Axiom 0 - Contrast · Distinction · Asymmetry · Return

Signature (closed FOL)

Predicates: D(x,y) contrast (irreflexive, symmetric), DistL(x) distinction-layer relatum, C(c,x,y) condition, L(c,x,y) licence, Ret(x,y) return.

Global reading of D

∀x ¬D(x,x) (irreflexive)

∀x ∀y ( D(x,y) ↔ x ≠ y ) (we fix this extensional reading)

Interfaces - coding and value

CodeRet(p,x,y) → D(x,y)

[I0] Codes respect contrast.
True(p) := ∃x ∃y ( CodeRet(p,x,y) ∧ Ret(x,y) )

[I1] Truth attaches only at return.
False(p) := ∃x ∃y ( CodeRet(p,x,y) ∧ ¬Ret(x,y) )

[I2] False when the coded return does not hold.

Contrast and distinction

∀x ∃y D(x,y)

[C0] Base contrast.
∀x ¬D(x,x)

[C1] Irreflexive.
∀x ∀y ( D(x,y) → D(y,x) )

[C2] Symmetric.
∀x ∀y ( D(x,y) ↔ x ≠ y )

[C3] Distinction equals inequality.

Stratification on the distinction layer

∀x ∀y ( D(x,y) → DistL(x) ∧ DistL(y) )

[S0] Contrast sits on the distinction layer.

Note: with [C0] and [C3], we get ∀x DistL(x).

Contrast-respecting constraints

∀c ∀x ∀y ( L(c,x,y) → C(c,x,y) )

[LC] Licence requires condition.
∀c ∀x ∀y ( C(c,x,y) → D(x,y) )

[CD] Condition requires contrast.

Anti-parallel licensing

∀c ∀x ∀y ¬( L(c,x,y) ∧ L(c,y,x) )

[L∥] No two-way licence under the same condition. Opposition via distinct c is allowed.

Asymmetric direction and return

ADir(x,y) ≔ (∃c ( C(c,x,y) ∧ L(c,x,y) )) ∧ ¬∃c' L(c',y,x)

∀x ∀y ( Ret(x,y) ↔ ADir(x,y) )

[R≡] Return equals uncontested licence.

Opposition and lack of licence

Opp(x,y) ≔ (∃c₁ L(c₁,x,y)) ∧ (∃c₂ L(c₂,y,x))

∀x ∀y ( Opp(x,y) → D(x,y) )

∀x ∀y ( ¬∃c L(c,x,y) ∧ ¬∃c L(c,y,x) ⇒ ¬Ret(x,y) ∧ ¬Ret(y,x) )

Derivations and checks

Remark - DistL is universal

From [C0] pick y with D(x,y).
By [S0] we have DistL(x) and DistL(y).
Hence ∀x DistL(x).

Theorem - Opposition blocks return

∀x ∀y ( (∃c₁ L(c₁,x,y)) ∧ (∃c₂ L(c₂,y,x)) → D(x,y) ∧ ¬Ret(x,y) ∧ ¬Ret(y,x) )

From each L get C by [LC], then D by [CD].
The reverse licence falsifies the right conjunct of ADir(x,y).
So Ret(x,y) fails. Symmetry yields ¬Ret(y,x). Use [R≡].

Theorem - No condition, no return

∀x ∀y ( ¬∃c C(c,x,y) ∧ ¬∃c C(c,y,x) → ¬Ret(x,y) ∧ ¬Ret(y,x) )

If no condition obtains, the left conjunct of ADir fails in both directions. Apply [R≡].

Corollary - No two-way return

∀x ∀y ¬( Ret(x,y) ∧ Ret(y,x) )

Assume Ret(x,y). Then ¬∃c L(c,y,x) by ADir, so ADir(y,x) fails and hence ¬Ret(y,x).

Theorem - Return is directional

∀x ∀y ( Ret(x,y) → D(x,y) ∧ ¬Ret(y,x) )

Ret(x,y) gives ∃c ( C ∧ L ).
By [CD] we get D(x,y).
By ADir’s right conjunct we get ¬∃c' L(c',y,x), which blocks Ret(y,x) via [R≡].

Corollary - Return sits on the distinction layer

∀x ∀y ( Ret(x,y) → DistL(x) ∧ DistL(y) )

From Ret pick c with C and L. Then C implies D by [CD]. Apply [S0].

Station schema

Contrast D → Distinction (x ≠ y) → Condition and Licence → Return Ret. Opposition on a pair yields no return. Only asymmetric licensing yields Ret.

Consistency witness

Let M = { a, b } ∪ Z with Z nonempty. Equality is identity. Interpret D(x,y) as x ≠ y. Take DistL universal. Set C = L = ∅. Then C0–C3, S0, LC, CD, L∥ and R≡ hold with Ret = ∅. So contrast and distinction hold with no returns.