
Contextual DIKWP Semantic Mathematics
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
Building upon our previous investigations into the DataInformationKnowledgeWisdomPurpose (DIKWP) Semantic Mathematics framework proposed by Prof. Yucong Duan, this document details an updated version of the framework. The new version incorporates considerations of cognitive limitations, paradoxes such as Gödel's incompleteness theorems and Russell's paradox, and the role of the cognitive semantic space in human understanding. The updated DIKWP Semantic Mathematics aims to provide a more robust and comprehensive approach to modeling natural language semantics, knowledge representation, and cognitive processes, addressing previous limitations and integrating new insights.
1. IntroductionThe original DIKWP Semantic Mathematics framework sought to model natural language semantics through the exclusive manipulation of three fundamental semantics:
Sameness (Data)
Difference (Information)
Completeness (Knowledge)
By iteratively applying these semantics, the framework aimed to construct a comprehensive semantic space capable of representing all natural language expressions and resolving philosophical challenges related to language and meaning.
However, our previous investigations highlighted several areas requiring refinement:
Cognitive Limitations: Acknowledging that human cognition operates within an understanding space that may have inherent limits or boundaries.
Paradoxes and Mathematical Limitations: Considering Gödel's incompleteness theorems, Russell's paradox, and their implications for formal systems.
Semantic Space Construction: Addressing the paradox proposed by Prof. Duan regarding the completeness of the DIKWP Cognitive Semantic Space and the existence of explanations within it.
The new version of the DIKWP Semantic Mathematics framework integrates these considerations to enhance its theoretical foundation and practical applicability.
2. Overview of the Original DIKWP Semantic Mathematics2.1. Fundamental SemanticsSameness (Data): Recognition of shared attributes or identities between entities.
Difference (Information): Identification of distinctions or disparities between entities.
Completeness (Knowledge): Integration of all relevant attributes and relationships to form holistic concepts.
Universal Semantic Mapping: To map all natural language semantics using the fundamental semantics.
Philosophical Resolution: To address issues such as Wittgenstein's language game and Laozi's assertion on the ineffability of essence.
Cognitive Modeling: To construct a cognitive semantic space encompassing human understanding.
Our previous investigations revealed several challenges:
Cognitive Limits: Human understanding may have inherent limitations, affecting the framework's ability to model all semantics.
Formal System Limitations: Gödel's incompleteness theorems suggest that no sufficiently complex formal system can be both complete and consistent.
Paradoxes: Russell's paradox highlights the need for careful handling of selfreferential definitions.
Semantic Evolution: Language and knowledge are dynamic, requiring the framework to adapt continuously.
Acknowledging Cognitive Boundaries:
The updated framework recognizes that the cognitive semantic space may have limits, influenced by human cognitive capacities and computational resources.
Adaptive Semantic Space:
Dynamic Expansion: The semantic space is designed to evolve, accommodating new expressions and concepts as they emerge.
Iterative Refinement: Continuous updating of semantics through iterative application of fundamental principles.
Incorporating MetaSemantics:
MetaLevel Analysis: Introducing a metasemantic layer to analyze and address statements about the system itself, helping to avoid selfreferential paradoxes.
Type Theory Integration:
Hierarchical Typing: Implementing a type system to prevent paradoxes like Russell's by disallowing certain forms of selfreference.
Introduction of Additional Semantics:
While maintaining the core of Sameness, Difference, and Completeness, the framework introduces supplementary semantics to handle complex concepts:
Contextuality: Recognizing the role of context in meaning.
Temporal Dynamics: Accounting for changes in semantics over time.
Intentionality (Purpose): Incorporating the purpose behind expressions.
Revised Definitions:
Sameness: Expanded to include not just static attributes but also functional and relational similarities.
Difference: Enhanced to capture subtle distinctions, including contextual and temporal differences.
Completeness: Refined to ensure holistic integration while acknowledging potential gaps due to cognitive limits.
Mathematical Foundation:
Strengthening the mathematical underpinnings to align with formal logic and set theory while avoiding known paradoxes.
Logical Consistency:
Implementing strict logical rules and validation mechanisms to maintain consistency within the framework.
Accepting Incompleteness:
Recognizing that the framework may not be able to prove all truths within itself, in line with Gödel's theorems.
External Validation:
Allowing for the possibility of truths or explanations existing outside the current cognitive semantic space, necessitating external methods or expansions to incorporate them.
Dynamic and OpenEnded Space:
The cognitive semantic space is not static but evolves with new inputs and discoveries.
Layered Structure:
Base Layer: Fundamental semantics (Sameness, Difference, Completeness).
Meta Layer: Semantics about semantics, allowing selfreflection and analysis.
Semantic Mapping:
Decomposition: Breaking down expressions into fundamental semantic units.
Mapping: Associating units with appropriate semantics.
Integration: Reconstructing meaning through synthesis of mapped semantics.
Semantic Evolution:
Feedback Mechanisms: Incorporating feedback to refine semantics based on new information.
Learning Algorithms: Employing machine learning techniques to adapt and expand the semantic space.
Type Theory Application:
Preventing SelfReference: Using type hierarchies to avoid paradoxes like Russell's.
Consistency Checks:
Validation Procedures: Regular checks for contradictions within the semantic space.
Acceptance of Undecidability:
Recognizing that some statements may be undecidable within the framework, in accordance with Gödel's incompleteness.
Enhanced Semantic Representation:
More nuanced modeling of language, capturing context, intent, and temporal aspects.
Ambiguity Resolution:
Improved mechanisms for resolving ambiguities through expanded semantics.
Cognitive Modeling:
Providing a structured approach for AI systems to model humanlike understanding.
Knowledge Representation:
Enabling AI systems to represent knowledge in a way that is both rigorous and adaptable.
Addressing Philosophical Challenges:
Offering a framework that acknowledges limitations while striving for comprehensive understanding.
Contributions to Mathematics:
Providing new perspectives on handling paradoxes and formal system limitations.
Example: The Liar Paradox  "This statement is false."
Traditional Challenge:
The statement cannot be consistently labeled as true or false without contradiction.
Framework Application:
Type Assignment: Classify the statement at a metalevel to prevent selfreference at the same level.
Semantic Decomposition:
Sameness: The structure of the statement.
Difference: The selfreferential aspect.
Completeness: Integration reveals the paradox.
Resolution:
Acknowledge the undecidability within the system.
Use metasemantic analysis to explain the paradox without contradiction.
Example: Goldbach's Conjecture
Framework Application:
Sameness: Identifying properties of even numbers and primes.
Difference: Distinguishing between numbers that can and cannot be expressed as the sum of two primes.
Completeness: Integrating known data to form a holistic understanding.
Outcome:
The framework aids in organizing knowledge about the conjecture but recognizes that a proof may be beyond its current capacity, accepting potential incompleteness.
The updated DIKWP Semantic Mathematics framework builds upon the original by:
Acknowledging Cognitive and Formal Limitations: Integrating an understanding of inherent limits in human cognition and formal systems.
Enhancing Semantic Tools: Introducing additional semantics and refining definitions to handle complexity.
Strengthening Mathematical Foundations: Ensuring logical consistency and robustness against paradoxes.
Adopting a Dynamic Approach: Emphasizing evolution and adaptability in the cognitive semantic space.
Future Directions:
Research and Development: Further exploration into practical implementations and applications.
Interdisciplinary Collaboration: Working with experts in cognitive science, linguistics, mathematics, and artificial intelligence to refine the framework.
Empirical Validation: Testing the framework's effectiveness in realworld scenarios.
Duan, 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. ".
Gödel, K. (1931). On Formally Undecidable Propositions of Principia Mathematica and Related Systems. Monatshefte für Mathematik und Physik.
Russell, B. (1908). Mathematical Logic as Based on the Theory of Types. American Journal of Mathematics, 30(3), 222262.
Tarski, A. (1933). The Concept of Truth in Formalized Languages. Studia Philosophica.
Chalmers, D. J. (1995). Facing Up to the Problem of Consciousness. Journal of Consciousness Studies, 2(3), 200219.
I extend sincere gratitude to Prof. Yucong Duan for his pioneering work on DIKWP Semantic Mathematics and for inspiring this comprehensive update to the framework. Appreciation is also given to researchers in cognitive science, mathematics, and artificial intelligence whose insights have contributed to refining the model.
11. Author InformationFor further discussion on the updated DIKWP Semantic Mathematics framework, its applications, or collaborations, please contact [Author's Name] at [Contact Information].
Keywords: DIKWP Model, Semantic Mathematics, Cognitive Semantic Space, Paradox Resolution, Sameness, Difference, Completeness, Prof. Yucong Duan, Cognitive Limits, Formal Systems, Artificial Intelligence
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