Arxe Logic

The ArXe Theory is an ambitious conceptual framework that seeks to unify the General Relativity (GR) and Quantum Field Theory (QFT). Its originality lies in establishing a bridge between logic and the fundamental properties of the universe, such as time and physical dimensions, through a single, unique axiom.

Axiom of Identity

The central axiom of ArXe Theory is:

\(\neg = Tp\)

or its extended form:

\(\neg() = Tp\)

This axiom postulates that each instance of logical negation (outside of parentheses, according to De Morgan’s Laws) is equivalent to a Planck Time (Tp), defined as the minimum possible physical time.

Fundamental Logical Definitions

To construct its structure, ArXe Theory introduces the following logical definitions:

• Contradictories: A pair of propositions is defined as contradictory if they are \(S\) and \(\neg S\).
• Exentation (Exent): This is the negation of a logical contradiction. It is defined as:
  \(Exent = \neg (S \land \neg S)\).
  Applying De Morgan’s Laws, this simplifies to:
  \(Exent = (\neg S \lor \neg \neg S) = (\neg S \lor S) = (S \lor \neg S)\).
  Since \((S \lor \neg S)\) is a tautology, an Exentation is always True. By the Axiom of Identity, an Exentation \((S \lor \neg S)\) is equivalent to a Planck Time (\(Tp\)).

Successive Exentations

The theory constructs a sequence of logical concepts derived from the fundamental definitions:

• First Exentation (\(e1\)): Represents the transition from Istence (\(Is\)) to Ex-Istence (\(ExIs\)). This transition is a necessary or tautological truth (\(Is \rightarrow ExIs\)).
  • Istence (Is): Represents a logical contradiction, defined as \(Is = (S \land \neg S)\). By nature, an Istence is always False.
  • Ex-Istence (ExIs): Is defined as the exentated Istence. That is, it is the tautology resulting from the exentation: \(ExIs = (S \lor \neg S)\).
• Second Exentation (\(e2\)): Represents the transition from Citance (\(Ci\)) to Ex-Citance (\(ExCi\)). This transition is a necessary or tautological truth (\(Ci \rightarrow ExCi\)).
  • Citance (Ci): Is defined as the conjunction of Istence and Ex-Istence: \(Ci = (Is \land ExIs)\). Since \(Is\) is False and \(ExIs\) is True, their conjunction \(F \land V\) is always False.
  • Ex-Citance (ExCi): Is the exentation of Citance, i.e., \(ExCi = \neg (Is \land ExIs)\). Applying the axiom of identity and De Morgan’s laws, this resolves to: \(ExCi = (\neg Is \lor \neg ExIs)\). Since \(Is\) is False and \(ExIs\) is True, this translates to \((\neg F \lor \neg V) = (V \lor F) = V\). Therefore, Ex-Citance is always True. It is described as a type of contingent truth (depending on the original values of \(S\)), although its final result is always true.
• Third Exentation (\(e3\)): Represents the transition from Perience (\(Pe\)) to Ex-Perience (\(ExPe\)). This transition is a necessary or tautological truth (\(Pe \rightarrow ExPe\)).
  • Perience (Pe): Is defined as the conjunction of Citance and Ex-Citance: \(Pe = (Ci \land ExCi)\).
  • Ex-Perience (ExPe): Is defined as the disjunction of Citance and Ex-Citance: \(ExPe = (Ci \lor ExCi)\).

Generalization of Exentations

The construction pattern is generalized as follows:

• An Entation of order \(n\) is defined as the conjunction of the Entation of order \(n-1\) and the Exentation of order \(n-1\):

\(Ent_n = (Ent_{n-1} \land ExEnt_{n-1})\)

An Exentation (\(ne\)) of order \(n\) is defined as the disjunction of the Entation of order \(n-1\) and the Exentation of order \(n-1\):

\(ExEnt_n = (Ent_{n-1} \lor ExEnt_{n-1})\)

Function of the Temporal Exponent in ArXe Theory

ArXe Theory establishes a fundamental connection between the exentation number (\(n\)) and the fundamental temporal dimension (T). This relationship is defined through the following temporal exponent function:

\(f_{exponent}(n) = \begin{cases} 0 & \text{if } n=1 \\ (-1)^n \cdot \lfloor n/2 \rfloor & \text{if } n>1 \end{cases}\)

This formula allows for transforming the exentation number \(n\) into an exponent for the temporal dimension \(T\), resulting in \(T^{f(n)}\). In other words, each level of exentation (\(n\)) determines the value to which the basic temporal dimension \(T\) is raised, thus integrating the logical aspects of the theory with the physical properties of time.