Logical Morphogenesis: Paradoxes as Rhythms

Abstract

"This sentence is false." Classical logic chokes on it: call it true and it is false, call it false and it is true. This playground takes a different stance, borrowed from the revision theory of truth: instead of demanding a single answer, let the truth value update in time. The Liar then stops being a contradiction and becomes an oscillation. Networks of self-referential sentences become dynamical systems with cycles and attractors, and the dynamics underneath are exact, which the calibration pins.

Truth in time

Each sentence has a truth value that updates from the previous step according to what it asserts:

• A constant is always true or always false.
• The Liar ("this sentence is false") returns the negation of its own previous value, so it flips every step.
• A truth-teller ("this sentence is true") keeps its value.
• Assertions copy or negate another sentence's previous value.
• A biconditional is true exactly when it and its target agreed last step.

All sentences update synchronously from the previous global state, the same discrete-time convention as a cellular automaton. The calibration checks the single-step logic of these rules directly (the biconditional, the negating assertion).

Paradox becomes period

Run the Liar and it produces ...true, false, true, false..., a cycle of period 2. The calibration confirms it. The truth-teller, by contrast, is a fixed point, period 1. This is the heart of the revision-theoretic reading: a paradox is not a broken statement but a statement with no stable value, whose revision sequence never settles, instead it cycles.

Because the global state is a finite vector of bits, the trajectory must eventually repeat (pigeonhole), so every run ends in a cycle. The first repeated state marks where the attractor begins, and the distance to it is the period. The calibration verifies the detector on a constructed two-state alternation (period 2). This eventual periodicity is a theorem, not a hope.

Networks and "morphogenesis"

Wire many sentences together, mutual negation, reference rings, mixed assertions, and the cycles get richer: longer periods, transients before the attractor, sensitivity to the initial assignment. Calling the resulting stable temporal patterns "morphogenesis" is a metaphor: just as morphogenesis is the emergence of biological form, here logical form (rhythm, cycle, fixed point) emerges from local self-referential rules. It is a lens on self-reference, not a model of development, and the assumptions panel says so.

One sentence with memory

Most rules look back one step, but a "percent controller" looks at a moving average of its own recent history and tries to hold its truth rate near a target. That adds memory and changes the achievable cycles. It is an honest extension; the calibration deliberately targets the memoryless rules, where the periods are unambiguous.

Where this sits among theories of paradox

The dynamical reading is one of several responses to the Liar. Truth-value-gap theories say the Liar is neither true nor false; glut (paraconsistent) theories let it be both; the revision theory used here says it has no stable value and models the unstable revision process itself. The playground commits to the revision/dynamical view because it is the one that turns into something you can watch and measure.

References

• Gupta, A., and Belnap, N. (1993). The Revision Theory of Truth.
• Kripke, S. (1975). Outline of a theory of truth.
• Standard references on finite-state dynamics and eventual periodicity.