
DIKWP Semantic Mathematics: An Integrated Framework Based on Previous Investigations
Yucong Duan
International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWPSC)
World Artificial Consciousness CIC(WAC)
World Conference on Artificial Consciousness(WCAC)
(Email: duanyucong@hotmail.com)
Abstract
This document presents a detailed exposition of the new version of the DataInformationKnowledgeWisdomPurpose (DIKWP) Semantic Mathematics framework, developed by Prof. Yucong Duan. Building upon previous investigations, this updated framework incorporates insights from cognitive development, natural language semantics, philosophical challenges, and the exploration of cognitive semantic spaces. The new version aims to provide a comprehensive and robust model for understanding and representing semantics, addressing limitations identified in earlier iterations. This document outlines the foundational principles, enhancements made, and potential applications in artificial intelligence, cognitive science, and knowledge representation.
1. IntroductionThe ability to model and understand human cognition, language, and knowledge representation is a fundamental goal in cognitive science and artificial intelligence (AI). The DIKWP Semantic Mathematics framework, proposed by Prof. Yucong Duan, offers a structured approach to semantics using the core concepts of Sameness, Difference, and Completeness. Previous investigations have explored the framework's potential in modeling cognitive development, mapping natural language semantics, addressing philosophical paradoxes, and understanding the limits of human cognition.
This new version of the DIKWP Semantic Mathematics framework integrates these insights, refining its theoretical underpinnings and extending its capabilities. It addresses previous limitations, incorporates formal mathematical definitions, and proposes enhancements to better model complex semantic phenomena.
2. Overview of DIKWP Semantic Mathematics2.1. Original FrameworkThe original DIKWP Semantic Mathematics framework is based on the following components:
Data (Sameness): Recognition of shared attributes or identities among entities.
Information (Difference): Identification of distinctions or disparities between entities.
Knowledge (Completeness): Integration of all relevant attributes and relationships to form holistic concepts.
Wisdom: Application of knowledge with judgment and ethical considerations.
Purpose: Guiding principles or goals that direct the use of wisdom.
The framework operates exclusively with three fundamental semantics:
Sameness: $S$ — The set of shared attributes among concepts.
Difference: $D$ — The set of distinguishing attributes between concepts.
Completeness: $C$ — The integration of $S$ and $D$ to form a complete understanding of a concept.
Modeling Infant Development: The framework effectively models how infants recognize sameness and difference in sensory inputs, form concepts, and develop language.
Iterative Learning: Emphasizes the iterative application of fundamental semantics to refine understanding.
Universal Semantics: Proposes that all natural language expressions can be mapped using the fundamental semantics.
Resolving Ambiguity: Aims to eliminate subjectivity by providing explicit semantic representations.
Wittgenstein's Language Games: Offers a formalism to overcome the subjective nature of language meaning.
Laozi's Ineffability: Suggests that essential truths can be expressed through the integration of fundamental semantics.
Gödel's Incompleteness Theorems: Recognizes limitations in formal systems and explores strategies to address them within the framework.
Russell's Paradox: Investigates settheoretical issues and incorporates measures to prevent paradoxes.
Cognitive Semantic Space: Envisions a space constructed by the DIKWP framework that encompasses all human cognitive semantics.
Paradox: If all explanations exist within this space, then any understandable problem's explanation is inherently accessible.
The new version integrates insights from previous investigations to enhance the framework:
Acknowledging Cognitive Limits: Recognizes the inherent limitations of human cognition and formal systems.
Refining Fundamental Semantics: Expands and clarifies the definitions of Sameness, Difference, and Completeness.
Preventing Paradoxes: Incorporates mechanisms to avoid selfreferential paradoxes and inconsistencies.
To address complexities and limitations, the framework introduces additional semantics:
Contextuality ($X$): Accounts for the context in which semantics are interpreted.
Hierarchy ($H$): Recognizes levels of abstraction and organization within concepts.
Temporal Dynamics ($T$): Incorporates the temporal aspect of semantics, acknowledging that meanings can evolve over time.
Sameness ($S$):
Defined as a relation $S(a,b)$ where entities $a$ and $b$ share identical attributes within a specified context.
Difference ($D$):
Defined as a relation $D(a,b)$ where entities $a$ and $b$ have distinct attributes that differentiate them within a context.
Completeness ($C$):
Defined as $C(a)=S(a,b)∪D(a,b)$, representing the full set of attributes that define entity $a$ in relation to others.
Contextuality ($X$):
A function $X(a)$ that maps an entity $a$ to its contextual parameters.
Hierarchy ($H$):
A mapping $H(a)$ that assigns entity $a$ to a level within a hierarchical structure.
Temporal Dynamics ($T$):
A function $T(a,t)$ that describes the attributes of entity $a$ at time $t$.
Scalability: The framework incorporates computational models to handle complex semantics at scale.
Adaptive Learning: Emphasizes iterative refinement and learning mechanisms that mimic human cognitive development.
Type Theory Integration: Utilizes type hierarchies to prevent selfreferential paradoxes.
Axiomatic Constraints: Adopts axioms from formal logic and set theory to ensure consistency.
The new version establishes a formal mathematical foundation:
Set Theory Foundations: Based on axiomatic set theory (e.g., ZermeloFraenkel with Choice) to avoid paradoxes.
Logic and Proof Systems: Incorporates formal logic systems to enable rigorous proofs and reasoning.
Semantic Algebra: Develops an algebraic structure for semantics, allowing operations such as union, intersection, and complement.
Simulating Cognitive Development: Models how humans learn and develop concepts over time.
Understanding Cognitive Limits: Explores the boundaries of human cognition and strategies to extend them.
Natural Language Processing (NLP): Enhances AI's ability to understand and generate human language with greater semantic depth.
Explainable AI: Provides transparent reasoning mechanisms, improving trust and interpretability.
Ontologies and Semantic Networks: Constructs rich semantic representations for complex domains.
Interoperability: Facilitates sharing and integration of knowledge across systems.
Philosophical Problems: Addresses issues like Wittgenstein's language games and Laozi's ineffability.
Mathematical Challenges: Offers new perspectives on problems like Gödel's incompleteness theorems and Russell's paradox.
Example Sentence: "Time flies like an arrow."
Semantic Analysis:
Sameness: Recognize entities (time, flies, arrow) and shared attributes.
Difference: Differentiate between literal and metaphorical meanings.
Completeness: Integrate semantics to capture the full meaning in context.
Contextuality: Consider the idiomatic usage in English.
Hierarchy: Identify the sentence structure and grammatical roles.
Temporal Dynamics: Acknowledge that interpretations may change over time.
Type Hierarchies: Assign sets to different types, preventing a set from containing itself.
Formal Definition: Redefine set membership to avoid selfreferential definitions.
The new version of the DIKWP Semantic Mathematics framework represents a significant advancement in modeling semantics and human cognition. By integrating insights from previous investigations, expanding fundamental semantics, and establishing a formal mathematical foundation, the framework addresses earlier limitations and enhances its applicability.
This comprehensive approach holds promise for advancing AI, cognitive science, and knowledge representation, providing tools to better understand and model complex semantic phenomena.
ReferencesDuan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".
Russell, B. (1908). Mathematical Logic as Based on the Theory of Types. American Journal of Mathematics, 30(3), 222262.
Gödel, K. (1931). On Formally Undecidable Propositions of Principia Mathematica and Related Systems. Monatshefte für Mathematik und Physik.
Chomsky, N. (1957). Syntactic Structures. Mouton.
Piaget, J. (1952). The Origins of Intelligence in Children. International Universities Press.
I extend sincere gratitude to Prof. Yucong Duan for his pioneering work on DIKWP Semantic Mathematics and for inspiring the development of this new version. Appreciation is also given to researchers in cognitive science, artificial intelligence, and philosophy whose insights have contributed to refining this framework.
Author InformationFor further discussion on the new version of the DIKWP Semantic Mathematics framework, please contact [Author's Name] at [Contact Information].
Keywords: DIKWP Model, Semantic Mathematics, Cognitive Semantic Space, Sameness, Difference, Completeness, Prof. Yucong Duan, Cognitive Limits, Artificial Intelligence, Knowledge Representation, Formal Semantics
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