Modal DIKWP Semantic Mathematics

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Abstract

This document provides an in-depth explanation of the new version of the Data-Information-Knowledge-Wisdom-Purpose (DIKWP) Semantic Mathematics framework, as proposed by Prof. Yucong Duan. Building upon previous investigations and addressing identified limitations and paradoxes, this enhanced framework expands its capability to model and represent the full spectrum of natural language semantics and human cognition. The updated framework aims to construct a comprehensive Cognitive Semantic Space that encapsulates human expressions and provides mechanisms for resolving paradoxes and proving conjectures within its structure. This detailed explanation covers the framework's foundational principles, enhancements, formal definitions, implementation strategies, and potential applications.

The DIKWP Semantic Mathematics framework was initially developed to model and represent natural language semantics using fundamental concepts derived from the DIKWP hierarchy:

• Data: Raw facts and figures.

• Information: Data processed to be meaningful.

• Knowledge: Information applied or put into action.

• Wisdom: Insight derived from knowledge over time.

• Purpose: The overarching goals or intentions guiding actions.

The original framework focused on the explicit manipulation of three fundamental semantics:

• Sameness (Data): Recognizing shared attributes or identities between entities.

• Difference (Information): Identifying distinctions or disparities between entities.

• Completeness (Knowledge): Integrating all relevant attributes and relationships to form holistic concepts.

Despite its strengths, the initial framework faced challenges:

• Paradoxes: Issues such as Russell's Paradox revealed limitations in handling self-referential constructs.

• Incompleteness: Gödel's incompleteness theorems highlighted potential limitations in the framework's ability to prove all truths within its system.

• Cognitive Limits: The complexity of human cognition and the infinite expressiveness of natural language presented scalability challenges.

The new version addresses these issues by introducing enhancements that increase the framework's robustness, expressive power, and applicability.

The new version introduces several key enhancements:

1. Hierarchical Semantic Levels: Organizing semantics into hierarchical levels to prevent paradoxes and improve structure.

2. Integration of Type Theory: Applying type theory to enforce consistency and prevent invalid semantic constructions.

3. Expanded Fundamental Semantics: Incorporating additional fundamental semantics—Contextuality, Temporality, and Modality—to capture more nuances of natural language.

4. Formal Logical Systems: Integrating formal logic systems (e.g., Modal Logic, Temporal Logic) to enhance reasoning capabilities.

5. Mechanisms for Handling Incompleteness and Undecidability: Acknowledging and providing strategies for dealing with undecidable statements.

6. Construction of the Cognitive Semantic Space: Developing a comprehensive space that encapsulates all evolved semantics.

• Avoidance of Paradoxes: By organizing semantics into hierarchical levels, self-referential paradoxes like Russell's Paradox are prevented.

• Structured Organization: Hierarchical levels facilitate better management and understanding of complex semantic relationships.

• Level 0: Primitive Semantics

• Definition: The most basic semantic elements that cannot be broken down further.

• Examples: Existence (∃), identity (=), basic logical constants (∧, ∨, ¬).

• Level 1: Constructed Semantics

• Definition: Semantics constructed from Level 0 primitives using defined operations.

• Examples: Concepts like "cat," "tree," "red," formed by combining primitives.

• Level 2: Meta-Semantics

• Definition: Semantics about semantics; statements that describe or reference Level 1 constructs.

• Examples: "The concept of 'justice' is abstract," "Definitions of 'number' vary across contexts."

• Level n: Higher-Order Semantics

• Definition: Additional layers for more abstract or complex semantics, where n > 2.

• Examples: Discussions about the nature of meta-semantics, philosophical analyses.

Let S_n represent the set of semantics at level n.

• Level 0 (S_0): Contains primitives {p_1, p_2, ..., p_k}.

• Level 1 (S_1): Constructed using operations O on elements of S_0.

  S1={O(s0i,s0j,...)∣s0i,s0j∈S0}S_1 = \{ O(s_{0_i}, s_{0_j}, ...) \mid s_{0_i}, s_{0_j} \in S_0 \}S1​={O(s0i​​,s0j​​,...)∣s0i​​,s0j​​∈S0​}

• Level 2 (S_2): Meta-statements about S_1.

  S2={M(s1i)∣s1i∈S1}S_2 = \{ M(s_{1_i}) \mid s_{1_i} \in S_1 \}S2​={M(s1i​​)∣s1i​​∈S1​}

• Higher Levels: Similarly defined, ensuring no circular references within the same level.

• Preventing Self-Reference within the Same Level: By restricting self-reference to higher levels, paradoxes are avoided.

• Clear Separation of Semantics: Improves clarity and manageability.

• Consistency Enforcement: Types prevent invalid operations between incompatible semantics.

• Error Detection: Type mismatches highlight potential semantic errors.

• Type System: Define a set of types T = {T_1, T_2, ..., T_n}.

• Assignment Function: A function τ: S \rightarrow T assigns a type to each semantic element.

• Type Rules: Operations are permitted only if the types of operands are compatible.

  For operation O:

  If τ(si)=Ta and τ(sj)=Tb, then O(si,sj) is valid only if Ta and Tb are compatible under O.\text{If } τ(s_i) = T_a \text{ and } τ(s_j) = T_b, \text{ then } O(s_i, s_j) \text{ is valid only if } T_a \text{ and } T_b \text{ are compatible under } O.If τ(si​)=Ta​ and τ(sj​)=Tb​, then O(si​,sj​) is valid only if Ta​ and Tb​ are compatible under O.

• Types:

• T_{Entity}: Entities or objects.

• T_{Property}: Properties or attributes.

• T_{Relation}: Relationships between entities.

• T_{Statement}: Assertions or propositions.

• Valid Operations:

• Applying a T_{Property} to a T_{Entity} to create a T_{Statement}.

• Combining two T_{Statement} elements with logical connectives.

• Invalid Operation Example:

• Applying a T_{Property} directly to another T_{Property} without a valid operation defined.

• Avoids Paradoxical Constructions: By enforcing type rules, constructs that could lead to paradoxes are disallowed.

• Enhances Clarity: Types make the role and nature of each semantic element explicit.

• Importance: Context affects meaning; a word or phrase can have different meanings in different contexts.

• Implementation:

• Context Parameters: Include parameters C in semantic representations to capture context.

• Semantic Function with Context:

  $$\text{Meaning}(e,C)=s$$

  Where e is an expression, C is the context, and s is the semantic representation.

• Example:

• C_1: Financial context → Meaning("Bank", C_1) = \text{Financial Institution}

• C_2: Environmental context → Meaning("Bank", C_2) = \text{River Bank}

• Expression: "Bank"

• Contexts:

• Importance: Meanings can change over time, and statements may have temporal aspects.

• Implementation:

• G (Globally): Always in the future.

• F (Eventually): At some point in the future.

• P (Past): At some point in the past.

• Temporal Parameters: Include time t in semantic representations.

• Temporal Logic Operators:

• Example:

• Statement: "The president is addressing the nation."

• Temporal Representation:

  $$\text{At time }t,\text{ President}(t)=\text{Current President at }t$$

• Importance: Captures notions of possibility, necessity, and contingency.

• Implementation:

• ◇ (Possibility): It is possible that.

• □ (Necessity): It is necessary that.

• Modal Logic Operators:

• Example:

• Statement: "It is possible that it will rain tomorrow."

• Modal Representation:

  ◇(Rain(ttomorrow))◇(\text{Rain}(t_{\text{tomorrow}}))◇(Rain(ttomorrow​))

• Enhanced Expressiveness: Formal logics allow precise expression of complex semantics.

• Rigorous Reasoning: Enables formal proofs and deductions within the framework.

• Propositional Logic: Basic logical operators and propositions.

• Predicate Logic: Quantifiers and predicates for more detailed expressions.

• Modal Logic: Addresses necessity and possibility.

• Temporal Logic: Handles time-dependent statements.

• Deontic Logic: Deals with obligation and permission (e.g., ethics, law).

• Statement: "All humans are mortal."

• Predicate Logic Representation:

  $$\forall x(\text{Human}(x)\to \text{Mortal}(x))$$

• Deduction:

• Given Socrates is a human: \text{Human}(\text{Socrates})

• Therefore, \text{Mortal}(\text{Socrates}) follows.

• Definition: A statement is undecidable within the system if neither it nor its negation can be proven using the system's axioms and inference rules.

• Implementation:

• Tagging: Undecidable statements are tagged with a special marker (e.g., ⊥).

• Meta-Language: Use a higher-level language to discuss the undecidability.

• Example:

• Statement: "This statement is unprovable."

• Recognition: Identified as a self-referential paradox and tagged as undecidable.

• Meta-System: A higher-level system that can reason about statements in the original system.

• Approach:

• Extension: Introduce new axioms or inference rules in the meta-system.

• Consistency Checks: Ensure that extensions do not introduce contradictions.

• Example:

• Gödel's Sentence: In the original system, it is undecidable.

• Meta-Reasoning: In the meta-system, we can prove properties about the original system's limitations.

• Acknowledgment of Limits: The framework accepts that not all truths are provable within itself.

• Flexibility: Allows for growth and adaptation by incorporating new knowledge or systems.

• Comprehensive Representation: To encompass all evolved semantics derived from natural language.

• Accessibility: Provide mechanisms for discovering and retrieving explanations and proofs.

• Function: Map natural language expressions E to formal semantic representations S.

  $$f:E\to S$$

• Process:

• Parsing: Analyze the grammatical structure of the expression.

• Semantic Analysis: Assign meanings based on context, temporality, modality, etc.

• Formalization: Represent the semantics using the framework's formal language.

• Nodes: Represent semantic entities (concepts, properties, events).

• Edges: Represent relationships (e.g., "is a type of," "causes," "belongs to").

• Example:

• Cat is a Animal.

• Cat can be a Pet.

• Concepts: Cat, Animal, Pet.

• Relationships:

• Query System: Users can input queries in natural language or formal representation.

• Inference Engine: Uses logical reasoning to find explanations or proofs.

• Example:

• Identify relevant concepts and relationships.

• Use inference rules to construct an explanation.

• Query: "Why is a cat considered a mammal?"

• Process:

• Knowledge Discovery: Facilitates exploration of semantic relationships.

• Problem Solving: Assists in finding proofs or explanations for complex problems.

• Statement: Consider the set of all sets that do not contain themselves. Does this set contain itself?

• Hierarchical Levels: By assigning sets to levels, a set cannot contain sets of the same or higher level.

• Type Assignments: Sets are typed, and a set of type T cannot contain itself unless explicitly allowed.

• Result: The paradoxical construction is disallowed, preventing the contradiction.

• Acceptance: The framework recognizes that certain truths cannot be proven within the system.

• Meta-System Reasoning: Use higher-level systems to reason about statements undecidable in the original system.

• Continuous Expansion: The framework can be extended with new axioms or rules when justified.

• Statement: "All statements in this system are provable."

• Analysis: Recognized as problematic; the framework avoids asserting such completeness.

• Modularity: The framework is designed to be modular, allowing for incremental expansion.

• Computational Resources: Acknowledges that practical implementation depends on available computational power.

• Learning Mechanisms: Incorporates machine learning techniques to evolve and adapt the semantic space.

• Human-AI Collaboration: Facilitates cooperation between human cognition and AI to overcome individual limitations.

• Cross-Language Compatibility: The framework can map semantics across different natural languages, aiding in translation and communication.

• Interdisciplinary Integration: Bridges gaps between fields by providing a common semantic foundation.

• Natural Language Understanding (NLU): Improves AI's ability to comprehend context, nuance, and complexity in human language.

• Cognitive Computing: Supports the development of AI systems that simulate human thought processes.

• Automated Theorem Proving: Assists in generating proofs for mathematical conjectures.

• Scientific Research: Aids in hypothesis generation and validation through semantic modeling.

• Understanding Consciousness: Provides tools to model aspects of human consciousness and cognition.

• Exploring Ontological Questions: Helps in formalizing and analyzing philosophical concepts.

The new version of the DIKWP Semantic Mathematics framework represents a significant advancement in modeling natural language semantics and human cognition. By addressing previous limitations through hierarchical semantic levels, type theory integration, expanded fundamental semantics, and formal logical systems, the framework enhances its robustness and applicability.

The construction of the Cognitive Semantic Space offers a comprehensive environment for knowledge representation, discovery, and reasoning. While recognizing inherent limitations, the framework provides mechanisms to navigate challenges, making it a valuable tool for advancing artificial intelligence, facilitating universal semantic representation, and deepening our understanding of cognition and knowledge.

• Prototype Development: Building software implementations to test the framework's practical viability.

• Performance Evaluation: Assessing computational efficiency and scalability.

• Interdisciplinary Research: Engaging experts from linguistics, cognitive science, philosophy, and AI.

• User Feedback: Incorporating insights from practitioners to refine the framework.

• Responsible AI: Ensuring the framework's applications align with ethical guidelines.

• Data Privacy: Safeguarding sensitive information within the cognitive semantic space.

1. 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 not 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. ".

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3. Gödel, K. (1931). On Formally Undecidable Propositions of Principia Mathematica and Related Systems. Monatshefte für Mathematik und Physik.

4. Church, A. (1936). An Unsolvable Problem of Elementary Number Theory. American Journal of Mathematics, 58(2), 345-363.

5. Montague, R. (1974). Formal Philosophy: Selected Papers of Richard Montague. Yale University Press.

6. Barwise, J., & Etchemendy, J. (1999). Language, Proof and Logic. CSLI Publications.

7. Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Cambridge University Press.

8. Fagin, R., Halpern, J. Y., Moses, Y., & Vardi, M. Y. (2003). Reasoning About Knowledge. MIT Press.

I extend sincere gratitude to Prof. Yucong Duan for his pioneering work on the DIKWP Semantic Mathematics framework and for inspiring the development of this new version. Appreciation is also given to researchers in cognitive science, artificial intelligence, logic, and philosophy for their foundational contributions that have informed and enriched this work.

For further discussion on the new version of the DIKWP Semantic Mathematics framework and its applications, please contact [Author's Name] at [Contact Information].

Keywords: DIKWP Model, Semantic Mathematics, Cognitive Semantic Space, Hierarchical Semantics, Type Theory, Sameness, Difference, Completeness, Contextuality, Temporality, Modality, Prof. Yucong Duan, Artificial Intelligence, Knowledge Representation, Formal Logic