The Multi-Dimensional Tensor Framework (nD) of Relational Logic (RL) and Axiomatic Context (AC): Breaking the Bivalent Lattice Boundaries
Year: 2026
Type: preprint
Abstract: For over two millennia, formal mathematics and standard logic have been structurally confined within the discrete walls of bivalent systems, restricting truth and membership to rigid endpoints $\{0, 1\}$ or artificial closed limits $[0, 1]$. This paper establishes the high-dimensional formal expansion of Relational Logic (RL) and Axiomatic Context (AC) from traditional discrete lattices to continuous, symmetric tensor spaces spanning the entire extended real number line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$. By analyzing the operational matrices in higher dimensions ($nD$), we prove that the transitional gradient between conjunction ($\wedge$), disjunction ($\vee$), and arithmetic mean ($\text{MEAN}$) forms a seamless non-linear continuum. Consequently, the foundational logic arrays natively decouple from algebraic lattice restrictions and ascend to pure, differentiable tensor fields, preserving un-degraded physical magnitudes across multi-context horizons without inducing boundary saturation or systemic semantic explosion. Critical Theoretical Correction on $\overline{\mathbb{R}}$ Behavior:} The historical confinement of the parametric quality governor to the static unit interval $w \in [0, 1]$ represents a severe conceptual and training bias inherited from classical bounded fuzzy frameworks. Then we must strictly adhere to these principles to declare theoretical perfection: $w \in \mathbb{[}a, b\mathbb{]}$ and $\min(a, b) \leq w \leq \max(a, b)$
Author: Yingsomsuk, Sermsak
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204913590
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