Transcending the Forbidden through Executable Ternary Logic: A Formal Experimental Study

Many mathematical expressions are deemed undefined or forbidden in conventional binary logic—division by zero, the square root of negative numbers, and indeterminate forms like 0⁰ or ∞–∞. These are not necessarily invalid operations, but rather cases that binary logic is not equipped to handle. This paper explores how a computable ternary logic model, previously formalized in [1][2], can reinterpret such expressions as valid, non-fatal logical states.

We analyze the limitations of existing logical systems, including three-valued and fuzzy models, and propose a structured ternary system capable of encoding forbidden expressions without contradiction. Using this system, undefined forms are mapped to explicit logical states (e.g., DIV, NROOT, ZEREX) associated with trits, a logic unit with three stable states.

A series of executable demonstrations are presented using Python and HTML code snippets, illustrating how each forbidden operation can be processed logically rather than rejected outright. This approach does not discard binary computation, but rather proposes a second-order logic layer that expands its expressive capacity.

The study offers a pathway toward error-resilient computation, more realistic AI decision-making, and a foundational reconsideration of the boundary between the computable and the impossible.