FuzzyLogecs · LaeLogecsLogex · LaeSpiEssentialTheorems

Truth with two channels

A claim can be true from within and false from without. Ordinary logic forces you to choose; Laegna gives the two readings independent axes and lets their algebra do the rest. Four values — I, E, O, A — and no special operators, because arithmetic on those values already behaves as inference.

The four values

I · E · O · A

I · E · O · A — the four truth values

inner channel0.75outer channel-0.40divide True by1

True is divided by 1. The values are already fuzzy — and nothing changed: the same operations, the same algebra. Fuzziness in Laegna is a magnitude on an existing channel, not a new logic.

Inner axis — I / O

Conviction and its vessel. What the system holds as true of itself: coherence, intention, the seed. The Z-side reading of a state.

Outer axis — E / A

Evidence and its refusal. What the world returns when asked: measurement, consequence, exposure. The Y-side reading of the same state.

These are the same two faces the ball has. Interior/exterior in geometry, inner/outer truth in logic, held/measured in physics: one structure, three vocabularies.

FuzzyLogecs

Already fuzzy — the fold

Laegna Logecs does not need to be extended into fuzzy logic; it needs to be restricted to stop being fuzzy. If only discrete −1 and 1 magnitudes are admitted as truth values, the system is classical binary logic. Allow any magnitude — divide True by four, say — and you have fuzzy values, using no theory that was not already there.

IEOA →(fold 1: inner)→ signed pair →(fold 2: outer)→ signed scalar →(fold 3: diagonal)→ {−1, +1} · True / 4 ⟶ fuzzy

The mapping to standard 1-dimensional true/false runs through a fractal repeat: twice through the two boolean channels, a third time through the diagonal. What survives is base-2 with unit magnitudes. Run it backwards and fuzziness is recovered for free.

Folding IEOA onto standard fuzzy logic

0 — Laegna native

Two boolean channels, four values I E O A. Inner and outer truth are independent axes; a statement can be inwardly true and outwardly false without contradiction.

Logex

Automation: arithmetic as inference

LaeLogecsLogex is the automation layer. Because truth values are numbers on the two channels, conjunction is multiplication, disjunction is the probabilistic sum, negation is sign inversion, and blends and tensions are ordinary averages and differences. An inference engine here is a numerical pipeline, not a rule interpreter — which is why it composes cleanly with LaeLane and with the ball's coordinates.

Logex — arithmetic is the inference engine

a0.75b-0.50

a · b (and): -0.375
a + b − a·b (or): 0.625
−a (not): -0.750
(a+b)/2 (blend): 0.125
a² (self-affirm): 0.563
|a−b| (tension): 1.250

No special fuzzy operators are defined. Once truth values are numbers on the Laegna channels, ordinary arithmetic already behaves as logic.

LaeSpiEssentialTheorems · LaeMath

The same claims in classical dress

Nothing on this site requires a private language to be checked. LaeSpiEssentialTheorems states the essential results in ordinary mathematical language, and LaeMath reinterprets standard mathematics through the same lens. A reader who distrusts the vocabulary can start from those two and arrive at the same ball.

Bijection

8·4^k ↔ 4^k: sphere and square agree at every octave.

Commutation

π ∘ R = R ∘ π: detail may be added before or after flattening.

Saturation

x → 2 ≡ ∞ reprojects to 0 in the next octave; no divergence survives.