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Station One · Axiom −1

Contrast before truth. Axiom −1 sets the ordering so nothing is smuggled in.

Proto truth Contrast to Distinction to Condition to Licence to Return to Truth Closed FOL layer

Axiom −1 · Contrast before truth

If you try to start from truth, you already assumed a split between true and false. That split is distinction. Distinction rests on contrast. Contrast rests on the first NOT. Axiom −1 exists to keep the stack honest.

The core claim

For anything to be true, it cannot be false. That requires difference. That requires a pair. A pair requires a NOT.

Truth is a late label. The early work is done by contrast and the rule that resolves it.

Coin flip intuition

Heads and tails are two potential outcomes. Before the flip, the contrast is there, but no outcome is resolved. The words heads and tails give you a distinction, but still no result.

The condition is the floor that sorts one result from the other. Once the condition is met, you get a return: the one outcome that holds. Only then does a truth label make sense.

What fails if true and false coincide

If true and false were indistinguishable, no statement could resolve into a final outcome. You would still have raw contrast, but no stable distinction, no working condition, and no return to bind.

Flow chart of the argument

  1. 1
    NOT is the first mark of difference.
  2. 2
    Contrast holds: something against not something.
  3. 3
    Distinction reads the contrast as A is not B.
  4. 4
    Condition acts: it tests and sorts.
  5. 5
    Licence allows one directed read.
  6. 6
    Return is the uncontested direction that holds.
  7. 7
    Truth is the label we give to that standing return.

Interactive cycle

Buttons or keyboard. Left for Back. Right or Space for Next. Escape for Reset.

Start the builder
Begin. Place the first eye. See why self distinction fails.
Press Space to begin
Contrast diagram 👁 A 👁 B 👁 C
Distinction A,B: off Contrast: primal Return: none Truth bound: none

Presets

Duad uses neutral arrows for raw contrast. Licences sit on their own lane. Returns ride the edge and stay thicker. Collapse overlaps A and B and removes distinction on that pair. Contrast still holds as the deeper dependency.

Formal layer - full axiom set

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, S(p) statement, Truth(p) true, False(p) false.
  • Interfaces: CodeD(p,x,y) codes D(x,y); CodeRet(p,x,y) codes Ret(x,y).
  • Constant: N the null-eye.
  • Shorthand: Val(p) := Truth(p) ∨ False(p).

Value axioms

  1. ∀p ( Truth(p) → S(p) ) ∧ ∀p ( False(p) → S(p) )

    [V1] Truth and falsity apply only to statements.

  2. ∀p ( Truth(p) → ¬False(p) ) ∧ ∀p ( False(p) → ¬Truth(p) )

    [V2] A statement carries at most one value.

Minimal properties of contrast

  1. ∀x ¬D(x,x)

    [C1] Irreflexive.

  2. ∀x ∀y ( D(x,y) → D(y,x) )

    [C2] Symmetric.

Contrast respecting constraints

  1. ∀c ∀x ∀y ( L(c,x,y) → C(c,x,y) )

    [LC] Every licence rests on a condition.

  2. ∀c ∀x ∀y ( C(c,x,y) → D(x,y) )

    [CD] Conditions apply only on contrasted pairs.

Return as uncontested licence

  1. ∀x ∀y ( Ret(x,y) ↔ ( (∃c ( C(c,x,y) ∧ L(c,x,y) )) ∧ ¬∃d L(d,y,x) ) )

    [R≡] Return equals an uncontested one way licence.

  2. ∀x ∀y ¬( Ret(x,y) ∧ Ret(y,x) )

    [RA] No opposing returns.

Derivations and checks

How the pieces fit

  • Start with contrast.
  • Conditions apply only where contrast holds [CD].
  • A licence is a condition granting a one way read [LC].
  • A return is that one way licence with no counter licence [R≡].
  • A statement is true when it codes a return (see the code layer in your Volume I text).

Theorem - Contrast before truth

∀p ( Truth(p) → ∃x ∃y D(x,y) )

Meaning: any truth claim presupposes at least one contrasted pair on the layer.

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