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Fractional Truth Tables

Jean-Paul Niko · 2026-03-19 · Outside 100 Centre Street, with Nika

The Idea

Classical logic: truth values in \(\{0, 1\}\).

Fuzzy logic: truth values in \([0, 1]\) (a line).

RTSG fractional logic: truth values in CS-space — a high-dimensional metric space with the SemanticProjector as the evaluation operator.

A proposition \(P\) evaluated by agent \(i\) has truth value:

\[\tau_i(P) = \langle \pi_i, P \rangle \in \mathbb{C}\]

where \(\pi_i\) is the SemanticProjector of agent \(i\) and \(P\) is the CS-representation of the proposition.

Truth is not a scalar. It is a projection. Different agents project differently. The "truth" of a proposition is its inner product with the evaluating consciousness.

Why This Matters

Classical logic assumes a view from nowhere — a truth value independent of who is evaluating. RTSG logic says: truth is always truth-for-an-agent. The truth table is parameterized by the I-vector of the evaluator.

This is not relativism. It is recognition that CS is the space where truth lives, and CS is always inhabited by a specific agent with a specific projector.

Objective truth = propositions where \(\langle \pi_i, P \rangle \approx \langle \pi_j, P \rangle\) for all agents \(i, j\) — propositions whose truth value is projector-independent. Mathematics and physics consist largely of such propositions.

Subjective truth = propositions where \(\tau_i(P)\) varies strongly with \(\pi_i\). Aesthetic judgments, values, personal memories.

The fractional part: truth values are complex numbers, not reals. The imaginary component encodes uncertainty or superposition. \(|\tau_i(P)|^2\) is the probability that agent \(i\) would assert \(P\) under observation.

Connection to Quantum Logic

This is quantum logic (Birkhoff-von Neumann 1936) reframed in RTSG language. Propositions are subspaces of CS. Truth is projection onto those subspaces. The non-commutativity of propositions (order matters) follows from the non-commutativity of projectors.

RTSG adds: the Hilbert space is CS-space, the projectors are SemanticProjectors, and the agents are the ones doing the projecting.

To Formalize

  1. Define the CS-logic algebra \(\mathcal{L}_{CS}\) as a lattice of subspaces of CS
  2. Define truth valuation \(\tau : \mathcal{L}_{CS} \times \mathcal{A} \to \mathbb{C}\) where \(\mathcal{A}\) is the space of agents
  3. Show classical and fuzzy logic are limiting cases
  4. Derive the fractional truth table for basic connectives (\(\wedge, \vee, \neg\)) in CS-space