Arxe Logic

Building upon ArXe Theory, which interlaces fundamental logic and the physical realm at its most basic scale, we’ve established that exentation (\( \neg C \)), defined as the negation of a contradiction, gives rise to a tautology (\( E \equiv T \)). This process, in turn, is directly equated with Planck Time (\( T_p \)), representing a foundational pulse of physical existence. This principle not only delineates what can exist but also establishes an essential framework of logical coherence for physical reality.

We now delve deeper into an implication of the theory that reinforces its internal consistency: the exentation of the coexistence between non-existence and existence.

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Recapping the Second Exentation

To understand the Third Exentation, let’s recall its key components:

• Tensia (Te): This represents the contradiction arising from the purported coexistence of Istence (\( I \), that which does not exist) and Ex-Istence (\( E \), that which does exist).

Given that \( I \) is False (F) and \( E \) is True (V), \( Tensia \equiv (F \land V) \equiv F \). Tensia is a logical contradiction, inherently false.

Exentia (Xt): This is the negation of Tensia and, therefore, a tautology.

As Tensia is false, \( Exentia \equiv \neg F \equiv V \). Exentia is a logical tautology, inherently true. Physically, it implies the possibility of coherent succession in time, requiring two units of \( T_p \).

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The Third-Order Contradiction: Perience

We now form a new conjunction using Tensia and Exentia:

\( (Te \land Xt) \)

Substituting the known truth values:

\( (Te \land Xt) \equiv (F \land V) \)

The result of this conjunction is, once again, False:

\( (F \land V) \equiv F \)

This expression \( (Te \land Xt) \) is, consequently, a logical contradiction. It represents the impossibility for the contradiction of coexistence (Tensia) and the affirmation of coherent succession (Exentia) to exist simultaneously.

This third-order contradiction is termed Perience:

\( Periencia \equiv (Te \land Xt) \)

Perience denotes a more complex logical impossibility, a deeper level of intrinsic incoherence. Its name suggests an entity “at the periphery” of possibility, inherently excluded from reality.

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The Third Exentation: Ex-Perience, the Coherence of Change Processes

As with all contradictions in ArXe Theory, Perience must be “exentated” for reality to maintain its consistency. We apply negation to Perience:

\( \neg Periencia \equiv \neg (Te \land Xt) \)

Since Perience is False, its negation \( \neg Periencia \) will always be True. Applying De Morgan’s Laws:

\( \neg (Te \land Xt) \equiv (\neg Te \lor \neg Xt) \)

As \( Te \) is False, \( \neg Te \) is True. As \( Xt \) is True, \( \neg Xt \) is False.

Substituting these values:

\( (\neg Te \lor \neg Xt) \equiv (V \lor F) \)

The disjunction of a True value and a False value is always True:

\( (V \lor F) \equiv V \)

Therefore, the negation of Perience is a tautology. This tautology, resulting from the Third Exentation, is termed Ex-Perience:

\( Ex-Periencia \equiv \neg Periencia \equiv (\neg Te \lor \neg Xt) \)

The name Ex-Perience is apt, signifying what lies “beyond” or “outside” the impossibility of Perience, representing the affirmation of coherence at this higher level.

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Physical Implication: Planck Time and the Stability of Temporal Processes

Consistent with the central axiom of ArXe Theory (\( \neg C \equiv T_p \)), this Ex-Perience, as an exentation, must also be correlated with Planck Time:

\( Ex-Periencia \equiv T_p \)

However, the implication is richer. Ex-Perience is the negation of a contradiction that encapsulates the tension between the impossibility of coexistence (Tensia) and the coherence of succession (Exentia). This suggests that Ex-Perience guarantees the stability and validity of the process of change and succession itself.

If Existence implies \( 1 T_p \) (a “pulse of being”), and Exentia implies \( 2 T_p \) (a “pulse of coherent change” allowing for beginning and end), Ex-Perience, as the tautology that negates the contradiction between change itself and its impossibility, could imply a superior temporal duration or complexity. Alternatively, it could signify the Planck-scale coherence that stabilizes sequences of events, ensuring their non-contradictory unfolding throughout time.

Perience represents a level of logical incoherence where even the dynamics of existence and non-existence (Tensia and Exentia) conflict. Ex-Perience, by negating this, affirms that reality is coherent not only in what exists and how it changes but also in the coherence of the rules of change themselves. This “third layer” of Planck Time ensures that the flow of events, as permitted by Exentia, is fundamentally free of contradictions at a higher level.