Exentation in ArXe Theory: The Logical Pulse of Physical Existence

Within the ambitious framework of ArXe Theory, which seeks to bridge logic and physics, the concept of exentation emerges as a central pillar. We define it as the negation of a logical contradiction, and its postulate is key to understanding which aspects of reality can have a physical correlate in our universe.

Logical Definition of Exentation

To begin, let’s recall the classical definition of a logical contradiction. If \( S \) represents any entity or proposition, and \( \neg S \) its contradictor, then a contradiction (\( C \)), which we also call Istence (\( I \)), is expressed as the conjunction of both:

\( C \equiv I \equiv (S \land \neg S) \)

Exentation (\( E \)) is, by definition, the negation of this contradiction:

\( E \equiv \neg C \)

Applying De Morgan’s Laws (which allow us to transform the negation of a conjunction into the disjunction of the negations) and double negation (where \( \neg (\neg S) \) becomes \( S \)), we simplify exentation to a tautology (\( T \)):

\( E \equiv \neg (S \land \neg S) \equiv (\neg S \lor \neg (\neg S)) \equiv (\neg S \lor S) \equiv T \)

A tautology is a proposition that is always true, regardless of the truth value of its components. Thus, the logical sequence is unequivocal: the negation of a contradiction inevitably leads to a tautology.

The Physical Interpretation of Exentation

ArXe Theory goes beyond this abstract logical definition, providing a profound physical interpretation for these concepts:

Istence (Contradiction – \( I \equiv C \)): The logical contradiction (\( S \land \neg S \)) is called Istence. The direct physical implication is that Istence does not physically exist. In a universe presumed to be logically coherent, a contradiction cannot manifest as an observable reality. Entities that are simultaneously \( S \) and \( \neg S \) simply cannot have a presence in the physical realm. This reinforces the idea that logic is a fundamental constitutive principle of reality itself.
Exentation or Existence (Tautology – \( E \equiv T \)): The logical tautology (\( \neg S \lor S \)) is called Ex-Istence or, more simply, Existence. This is the physical correlate of the tautology. What is logically a tautology—always true—is what has physical existence in the universe. This means that everything that exists in our cosmos must be inherently non-contradictory in its fundamental constitution; it must be \( S \) or \( \neg S \), but never both simultaneously.

Exentation as Planck Time

The most audacious and central connection of ArXe Theory is that this exentation (\( \neg C \)) is directly equated with Planck Time (\( T_p \)):

\( E \equiv T_p \) or, more explicitly:

\( \neg C \equiv T_p \)

Planck Time (\( T_p \approx 5.39 \times 10^{-44} \text{ s} \)) is the tiniest temporal unit derived from the fundamental constants of physics (c, G, and ℏ). It’s the scale where quantum gravity effects become unavoidable and where our conventional notions of spacetime might break down, marking the “minimum physically meaningful time.”

This axiom implies that the very process of exentation—that is, the negation of a contradiction resulting in a tautology—is equivalent to one Planck Time. It’s not just any elementary negation that consumes \( T_p \), but specifically the fundamental act of ensuring logical coherence by moving from the “non-existence” of a contradiction to the “logical necessity of existence.”

This suggests that the most basic “pulse” of reality, its minimum unit of temporality, is the constant affirmation of non-contradiction. The universe, in its most fundamental temporal units, doesn’t just “exist,” but is the result of a continuous process where the impossibility of contradiction is resolved into the necessity of existence.

Conclusion

ArXe Theory, by directly linking exentation with Planck Time, proposes a fundamental criterion for existence. It suggests that the universe operates on principles of logical coherence at its deepest level, dictating that only what is logically non-contradictory can physically manifest. This framework invites a profound re-evaluation of the nature of time and reality, positioning logic not merely as a tool to describe it, but as its essential constituent.

Arxe Logic