Arxe Logic

The ArXe Theory is built upon the postulate of a profound interconnection between fundamental logic and the nature of time at its most infinitesimal scale. We’ve established that exentation (\( \neg C \))—the negation of a contradiction that resolves into a tautology (our Ex-Istence or Existence)—is directly equated with a Planck Time (\( T_p \)). This equivalence not only defines existence itself but now reveals how the universe maintains its logical coherence through temporality.

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Elementary Existence: An Unbreakable Planck Pulse

In ArXe Theory, Ex-Istence (\( E \equiv (\neg S \lor S) \)) represents what is logically valid and, by extension, physically present. The central axiom states:

\( E \equiv T_p \)

This implies that every instance of Existence manifests in a single, indivisible \( T_p \) pulse. Here lies a fundamental consequence of the Principle of Non-Contradiction (PNC) within the Planck temporal domain. If something exists within \( 1 T_p \), that same temporal pulse cannot simultaneously host contradictory states (e.g., a “beginning” and an “end,” or “being” and “non-being”). Attempting to have a single \( T_p \) contain both would imply a simultaneous contradiction, which, being logically impossible, is physically non-existent (an Istence). Therefore, simple Ex-Istence, in its \( 1 T_p \) duration, is an instantaneous, coherent unit of being, where the PNC acts as a temporal barrier against the coexistence of opposites.

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Exentia: Two Planck Times for Coherent Succession

Exentia emerges as a crucial concept for understanding the dynamics of coherence in time. We define Tensia as the contradiction of coexistence between Istence and Ex-Istence:

\( Tensia \equiv (I \land E) \equiv ((S \land \neg S) \land (\neg S \lor S)) \)

Since \( (S \land \neg S) \) is always False (Istence) and \( (\neg S \lor S) \) is always True (Ex-Istence), the conjunction \( (F \land V) \) is always False. Thus, Tensia is intrinsically a contradiction.

Exentia is the negation of this Tensia:

\( Exentia \equiv \neg Tensia \equiv \neg (I \land E) \)

Because Tensia is a contradiction (False), its negation (Exentia) will always be True (a tautology). By our axiom, Exentia correlates with a Planck Time:

\( Exentia \equiv T_p \)

However, unlike simple Ex-Istence, Exentia, being the negation of a more complex contradiction that encapsulates the tension between “non-existence” and “existence,” reveals a richer temporal structure. While the operation of Exentia as a whole is \( 1 T_p \), its complex logical nature implies the necessity of two “\( T_p \) pulses” for the coherent manifestation of contrary states.

This implication is based on how the PNC applies to Exentia. Exentia doesn’t allow the “beginning” and “end” of something to coexist simultaneously within a single \( T_p \). Instead, the nature of Exentia lies in its ability to allow two contrary states (like a beginning and an end) to exist serially, each occupying its own \( T_p \) pulse.

This means that for an entity or process to have a coherent beginning and end, two \( T_p \) pulses are required: one for the “initial” state and another for the “final” state. Exentia, as the tautology that negates the impossibility of coexistence, is the principle that allows the coherent temporal succession of opposing states without violating the PNC at any given Planck instant.

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The Axiom’s Strength: The PNC as a Temporal Driver

Here lies the true strength of the central axiom of ArXe Theory. The Principle of Non-Contradiction isn’t a static logical constraint on reality; it’s a dynamic principle that structures temporality itself.

• For simple Existence, the PNC manifests as the indivisibility of a \( T_p \) where opposites cannot coexist. It’s the “instant” of being.
• For Exentia, the PNC manifests as the guarantee that opposites (like beginning and end) do not coexist, but can succeed each other coherently in distinct \( T_p \) pulses. This mechanism transforms the prohibition of simultaneity into the permission of temporal succession.

ArXe Theory, by equating the resolution of logical incoherence with the minimum unit of time, provides a foundation for the emergence of temporality as we know it. Exentia, by “requiring” two \( T_p \) pulses to coherently manifest contrary states, becomes the basis for understanding change, evolution, and the arrow of time in the universe without compromising fundamental logic. Planck Time is not merely a measurement limit; it’s the active agent that ensures the logical coherence of reality through its intrinsic dynamics.