PIATRA . INSTITUTE

logical morphogenesis

truth-value rhythms from self-referential sentences

based on 1993, Grim et al., Self-reference and paradox in two and three dimensions

t = 0 / 199

Phase Space

circle size = entropy

Patrick Grim's Resolution

The classical liar paradox ("This sentence is false") creates logical contradiction. Patrick Grim proposed resolving it through time semantics: evaluate each sentence at time $t$ using truth values from time $t-1$ . The liar becomes a stable oscillator: $\text{True} \to \text{False} \to \text{True} \to \ldots$

Sentence Types

Liar: "This sentence is false." Oscillates with period 2. Truth-teller: "This sentence is true." Preserves initial value (fixed point). Reference: "X is true/false" follows or negates another sentence. Conditional: $X \to \text{Self}$ or $\text{Self} \leftrightarrow X$ creates logical dependencies. Percent: Controllers that steer their sliding-window truth-average toward a target.

Emergent Dynamics

When sentences reference each other, complex patterns emerge. Mutual negation ("A says B is false, B says A is false") produces synchronized oscillation. Ring structures create waves propagating through the network. Percent-controllers generate rhythmic patterns as they balance toward their targets. These dynamics mirror regulatory networks in biology.

Cycle Detection

Since each sentence takes boolean values, the system state is a binary vector. With $n$ sentences, there are $2^n$ possible states. By the pigeonhole principle, the dynamics must eventually cycle. The attractor period tells us the fundamental rhythm of the truth-value dynamics.

The Morphogenesis Analogy

Just as gene regulatory networks produce spatial patterns in embryonic development, these sentence networks produce temporal patterns in truth-space. The timeline grid resembles a kymograph of developmental waves. This view into the "latent space" of logic reveals how structure emerges from self-reference—a morphogenesis of meaning itself.

Metrics

Mean truth: Fraction of time the sentence is true (after burn-in). Flip rate: Frequency of truth-value changes. Entropy proxy: $4 \cdot \bar{p} \cdot (1-\bar{p})$ measures variability (peaks at $\bar{p}=0.5$ ).