Set Theory and Logic

2504 Submissions

[4] viXra:2504.0191 [pdf] submitted on 2025-04-29 00:32:12

Resolution Matrix Semantics for Modal Logic: Philosophical Implications of Indeterminacy and Poly-Logic Thinking

Authors: Andrey Kuznetsov
Comments: 9 Pages.

This paper explores the philosophical implications of Resolution Matrix Semantics (RMS) as an alternative foundation for modal logic. Unlike traditional Kripkean models, which interpret modality through relations between multiple possible worlds governed by classical logic, RMS treats indeterminate truth values as fundamental, operating within a single world. RMS introduces "blinking" truth assignments and sub-interpretations to resolve uncertainty, capturing the inherently poly-logical nature of human thought. Drawing a parallel to quantum physics, we argue that Kripke models resemble Everett’s Many-Worlds interpretation, while RMS aligns with the Copenhagen interpretation’s emphasis on intrinsic uncertainty. RMS offers a new view of modal reasoning—not as a proliferation of worlds, but as a diversification of perspectives within one world. The framework’s philosophical significance is examined through connections to poly-logic thinking, quantum cognitive models, and potential applications in artificial intelligence and parallel computing. RMS ultimately provides a dynamic, pluralistic model for rationality that better reflects the complexity of human cognition and decision-making under uncertainty.
Category: Set Theory and Logic

[3] viXra:2504.0181 [pdf] submitted on 2025-04-28 12:41:34

Some Formulas and an Equivalence Between Two Sets of Axioms in Axiomatic Propositional Logic

Authors: Harry Willow
Comments: 6 Pages.

We derive some formulas in axiomatic propositional logic and we show that two sets of different axioms are equivalent. The outputs were manually derived through logical deductions and subsequently typeset in LaTeX, without the use of automated computational tools.
Category: Set Theory and Logic

[2] viXra:2504.0147 [pdf] submitted on 2025-04-23 19:56:06

A Formalization of the Structural Dependency Principle

Authors: Theophilus Agama
Comments: 4 Pages.

We introduce the notion of a emph{foundation} of a structure $B$ as a minimal free (or reflective) substructure through which $B$ is generated from $A$. After defining "subobject," "ascending chain condition," and "free object," we prove the emph{Principle of Structural Dependency}: $$ A ;overset{iota}{hookrightarrow}; mathbb{F}(B);overset{jmath}{hookrightarrow}; BquadLongrightarrowquadmathcal{P}(A)subseteqmathcal{P}(B).$$Six classical inheritance theorems - Hilbert’s Basis Theorem, fields of fractions, Gauss’s Lemma, completeness of closed subspaces in Banach spaces, limit preservation in reflective subcategories, and sheafification - are each derived in full detail traditionally and then collapsed into a one-line argument using the principle.
Category: Set Theory and Logic

[1] viXra:2504.0051 [pdf] submitted on 2025-04-07 15:46:04

On the Nature of Logic and the P vs NP Problem

Authors: Natalia Tanyatia
Comments: 12 Pages. https://github.com/NataliaTanyatia/Logical-Manifesto.git (Note by viXra Admin: For the last time, please submit article written with AI assistance to ai.viXra.org)

We prove P = NP by demonstrating that NP-completeness arises from an avoidable computational overhead: the exponential cost of constructing higher-order logical (HOL) frameworks from first-order primitives. By formalizing the logical realizability ofall NP problems within HOL, we show these problems become polynomial-time solvable when their logical structure is known in advance. The apparent hardness of NP problems is thusrevealed as an artifact of forcing deterministic Turing machines to reconstruct HOL representations from Boolean logic (∧, ∨, ¬)rather than an intrinsic property of the problems themselves. Key to this result is the Perspective-Dependent Logical Realizabil-ity Theorem, which establishes that:1. Every NP problem’s HOL formulation has an equivalent first-order logic (FOL) representation. 2. A deterministic Turing machine (DTM) can solve any NP problem in polynomial time if provided with its HOL framework. 3. The P ≠ NP separation occurs only when DTMs are restricted to bottom-up FOL construction. We validate this with Boolean satisfiability (SAT), proving its polynomial-time tractability under HOL and introducing Deciding by Zero (DbZ) as a further example of how logical reframing eliminates classical intractability. This work does not merelysuggest P = NP as a possibility but demonstrates it as a direct consequence of logical representation theory.
Category: Set Theory and Logic