Wittgenstein's Composition based Semantics Mechanism of the DIKWP Framework

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 presents a deep investigation into how the composition mechanism of Ludwig Wittgenstein's Logical Structure can be utilized to create the semantics mechanism of the Data-Information-Knowledge-Wisdom-Purpose (DIKWP) Semantic Mathematics framework developed by Prof. Yucong Duan. By exploring the parallels between Wittgenstein's logical propositions and the hierarchical semantics of the DIKWP model, we aim to integrate the composition mechanisms to enhance the semantic structuring within the DIKWP framework. This integration provides a robust mathematical foundation for modeling complex semantic relationships, improving artificial intelligence systems' ability to represent reality, process information, and construct knowledge in alignment with human cognitive processes.

Table of Contents

1. Introduction

• 1.1. Overview

• 1.2. Objectives

2. Background

• 2.2.1. Core Semantics: Sameness, Difference, Completeness

• 2.2.2. Hierarchical Structure: Data, Information, Knowledge, Wisdom, Purpose

• 2.1.1. Elementary Propositions and Logical Form

• 2.1.2. Truth Functions and Proposition Composition

• 2.1. Wittgenstein's Logical Structure and Composition Mechanism

• 2.2. DIKWP Semantic Mathematics Framework

3. Detailed Analysis of Wittgenstein's Composition Mechanism

• 3.1. Logical Atomism and Elementary Propositions

• 3.2. Composition through Logical Operations

• 3.3. Truth-Functional Logic

4. Mapping Wittgenstein's Composition Mechanism to DIKWP Semantics

• 4.1. Data Level: Elementary Propositions as Atomic Data

• 4.2. Information Level: Composition of Data into Complex Propositions

• 4.3. Knowledge Level: Logical Relationships and Completeness

• 4.4. Wisdom and Purpose: Application of Logical Understanding

5. Creating the Semantics Mechanism in DIKWP Using Wittgenstein's Composition

• 5.1. Formalization of Data Composition

• 5.2. Information Structuring through Logical Operations

• 5.3. Knowledge Formation via Logical Deduction

• 5.4. Integration into Wisdom and Purpose Levels

6. Examples and Applications

• 6.1. Example: Modeling Logical Propositions in DIKWP

• 6.2. Application in Artificial Intelligence Systems

7. Implications and Benefits

• 7.1. Enhanced Semantic Representation

• 7.2. Improved Logical Reasoning Capabilities

• 7.3. Alignment with Human Cognitive Processes

8. Challenges and Considerations

• 8.1. Complexity of Logical Structures

• 8.2. Maintaining Consistency and Completeness

• 8.3. Computational Efficiency

9. Conclusion

10. References

1. Introduction1.1. Overview

Ludwig Wittgenstein's Tractatus Logico-Philosophicus introduces a logical structure of language and reality, emphasizing how complex propositions are built from simpler ones through logical composition. This composition mechanism aligns with the hierarchical semantics in the DIKWP Semantic Mathematics framework, which models cognitive processes from data to purpose.

By investigating how Wittgenstein's composition mechanism can inform and enhance the semantics mechanism in the DIKWP framework, we aim to create a more robust and logically coherent system for semantic representation and processing in artificial intelligence (AI).

1.2. Objectives

• Investigate the composition mechanism in Wittgenstein's logical structure.

• Analyze how this mechanism aligns with the semantics in the DIKWP framework.

• Create a semantics mechanism in DIKWP using Wittgenstein's composition principles.

• Demonstrate the integration through formalization and examples.

• Discuss the implications for AI systems and cognitive modeling.

2. Background2.1. Wittgenstein's Logical Structure and Composition Mechanism2.1.1. Elementary Propositions and Logical Form

• Logical Atomism: Wittgenstein proposes that the world consists of atomic facts represented by elementary propositions.

• Elementary Propositions: Indivisible assertions about the world that correspond directly to atomic facts.

• Logical Form: The structure that allows propositions to represent reality; it is shared between language and the world.

2.1.2. Truth Functions and Proposition Composition

• Truth Functions: Logical operations that build complex propositions from elementary ones (e.g., conjunction, disjunction, negation).

• Composition Mechanism: The process of combining elementary propositions using truth-functional logic to form complex propositions that represent more intricate facts about the world.

2.2. DIKWP Semantic Mathematics Framework2.2.1. Core Semantics: Sameness, Difference, Completeness

• Data (Sameness): Raw facts characterized by uniformity.

• Information (Difference): Data processed to highlight differences and relationships.

• Knowledge (Completeness): Integration of information into a complete and consistent understanding.

2.2.2. Hierarchical Structure: Data, Information, Knowledge, Wisdom, Purpose

• Wisdom (W): Application of knowledge with ethical considerations.

• Purpose (P): Guiding motivations that direct actions and cognitive processes.

3. Detailed Analysis of Wittgenstein's Composition Mechanism3.1. Logical Atomism and Elementary Propositions

• Atomic Facts: The simplest facts that cannot be broken down further.

• Elementary Propositions: Statements that assert the existence of atomic facts.

• Correspondence Theory: Elementary propositions correspond directly to atomic facts in the world.

3.2. Composition through Logical Operations

• Logical Connectives: AND (∧), OR (∨), NOT (¬), IMPLIES (→).

• Building Complex Propositions: By applying logical connectives to elementary propositions, we construct complex propositions that represent more complex facts.

3.3. Truth-Functional Logic

• Truth Tables: Define the truth value of complex propositions based on the truth values of their components.

• Functional Completeness: The set of logical connectives is sufficient to express any logical relation.

4. Mapping Wittgenstein's Composition Mechanism to DIKWP Semantics4.1. Data Level: Elementary Propositions as Atomic Data

• Data Elements: Correspond to elementary propositions representing atomic facts.

• Sameness: Data elements share the property of being atomic and indivisible.

4.2. Information Level: Composition of Data into Complex Propositions

• Information as Relations: Differences and relationships between data elements are captured through logical composition.

• Logical Operations: Use logical connectives to combine data elements, highlighting differences and forming information.

4.3. Knowledge Level: Logical Relationships and Completeness

• Knowledge Base: The set of all logically derived propositions from the information level.

• Completeness: Ensuring that all logical consequences of the information are included in the knowledge base.

4.4. Wisdom and Purpose: Application of Logical Understanding

• Wisdom: Applying knowledge to make judgments, incorporating ethical considerations.

• Purpose: The goals or motivations guiding the application of wisdom.

5. Creating the Semantics Mechanism in DIKWP Using Wittgenstein's Composition5.1. Formalization of Data Composition5.1.1. Defining Atomic Data Elements

• Set of Atomic Data Elements: D={d1,d2,...,dn}D = \{ d_1, d_2, ..., d_n \}D={d1​,d2​,...,dn​}.

• Correspondence to Elementary Propositions: Each did_idi​ corresponds to an elementary proposition pip_ipi​.

5.1.2. Representing Data Sameness

• Equivalence Relation: di∼djd_i \sim d_jdi​∼dj​ if did_idi​ and djd_jdj​ are identical atomic facts.

• Equivalence Classes: Partition $D$ into classes of identical data elements.

5.2. Information Structuring through Logical Operations5.2.1. Composing Data Elements

• Logical Connectives: Use $\{\wedge ,\vee ,\neg \}$ to combine data elements.

• Information Elements: $I=\{\varphi \mid \varphi \text{ is a logical combination of elements in }D\}$.

5.2.2. Measuring Difference

• Semantic Difference: Defined by the logical structure of composed propositions.

• Information Set: Captures all meaningful differences and relationships among data elements.

5.3. Knowledge Formation via Logical Deduction5.3.1. Constructing the Knowledge Base

• Axioms: The set $S$ includes all elementary propositions and accepted truths.

• Deduction Relation: $⊢$ defines how new propositions are inferred.

5.3.2. Ensuring Completeness and Consistency

• Logical Completeness: For every proposition $\varphi$, $\varphi$ or $\neg \varphi$ is derivable.

• Consistency: No contradictions; cannot derive both $\varphi$ and $\neg \varphi$.

5.4. Integration into Wisdom and Purpose Levels5.4.1. Applying Knowledge

• Wisdom Functions: Evaluate the implications of knowledge for decision-making.

• Ethical Considerations: Incorporate value judgments into the application of knowledge.

5.4.2. Guiding Actions with Purpose

• Purpose Functions: Define objectives that guide the use of wisdom.

• Alignment: Ensure that actions are consistent with overarching goals.

6. Examples and Applications6.1. Example: Modeling Logical Propositions in DIKWP6.1.1. Data Level

• Atomic Data Elements:

• d1d_1d1​: "It is raining."

• d2d_2d2​: "I have an umbrella."

6.1.2. Information Level

• Logical Composition:

• ϕ1=d1∧d2\phi_1 = d_1 \wedge d_2ϕ1​=d1​∧d2​: "It is raining AND I have an umbrella."

• ϕ2=d1∧¬d2\phi_2 = d_1 \wedge \neg d_2ϕ2​=d1​∧¬d2​: "It is raining AND I do not have an umbrella."

6.1.3. Knowledge Level

• Deriving Consequences:

• From ϕ1\phi_1ϕ1​, infer "I will stay dry."

• From ϕ2\phi_2ϕ2​, infer "I might get wet."

6.1.4. Wisdom and Purpose

• Applying Knowledge:

• Decide to carry an umbrella when expecting rain (wisdom).

• Guided by Purpose:

• Purpose: Stay dry and healthy.

6.2. Application in Artificial Intelligence Systems

• Semantic Parsing: AI can parse natural language statements into logical propositions.

• Knowledge Representation: Build knowledge bases using logical compositions.

• Reasoning Engines: Use logical deduction to infer new information and make decisions.

• Ethical AI: Incorporate wisdom and purpose to align AI actions with human values.

7. Implications and Benefits7.1. Enhanced Semantic Representation

• Logical Coherence: Ensures that data and information are structured logically.

• Clarity: Improves understanding of relationships among data elements.

7.2. Improved Logical Reasoning Capabilities

• Deductive Reasoning: Enables AI systems to infer new knowledge logically.

• Problem-Solving: Facilitates complex reasoning tasks.

7.3. Alignment with Human Cognitive Processes

• Cognitive Modeling: Mirrors human logical reasoning and understanding.

• Communication: Enhances AI's ability to interpret and generate human-like language.

8. Challenges and Considerations8.1. Complexity of Logical Structures

• Scalability: Managing large numbers of propositions and logical combinations.

• Computational Resources: Requires efficient algorithms for logical deduction.

8.2. Maintaining Consistency and Completeness

• Contradictions: Avoiding inconsistencies in the knowledge base.

• Logical Paradoxes: Handling self-referential propositions carefully.

8.3. Computational Efficiency

• Optimization: Balancing thoroughness with computational constraints.

• Approximation Methods: May need to employ heuristics for practical applications.

9. Conclusion

By deeply investigating the use of Wittgenstein's composition mechanism in logical structures, we have demonstrated how it can be utilized to create the semantics mechanism of the DIKWP Semantic Mathematics framework. This integration aligns the logical structuring of propositions with the hierarchical semantics of data, information, and knowledge in the DIKWP model. The resulting framework enhances AI systems' ability to represent and process semantic information logically and coherently, improving reasoning capabilities and alignment with human cognitive processes.

10. References

1. Wittgenstein, L. (1921). Tractatus Logico-Philosophicus. (Various translations).

2. International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 .  https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model

3. 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. ".

4. Russell, B. (1918). The Philosophy of Logical Atomism. The Monist, 28(4), 495–527.

5. Frege, G. (1892). On Sense and Reference. Zeitschrift für Philosophie und philosophische Kritik.

6. Russell, S., & Norvig, P. (2021). Artificial Intelligence: A Modern Approach (4th ed.). Pearson.

7. Carnap, R. (1937). The Logical Syntax of Language. Routledge.

8. Quine, W. V. O. (1951). Two Dogmas of Empiricism. The Philosophical Review, 60(1), 20–43.

9. Tarski, A. (1944). The Semantic Conception of Truth. Philosophy and Phenomenological Research, 4(3), 341–376.

10. Floridi, L. (2011). The Philosophy of Information. Oxford University Press.

11. Gärdenfors, P. (2000). Conceptual Spaces: The Geometry of Thought. MIT Press.

Keywords: Wittgenstein, Composition Mechanism, Logical Structure, DIKWP Semantic Mathematics, Data, Information, Knowledge, Semantics, Artificial Intelligence, Logical Reasoning, Cognitive Modeling.

Note: This document provides a deep investigation into using Wittgenstein's composition mechanism to create the semantics mechanism of the DIKWP framework. By aligning the logical structuring of propositions with the hierarchical semantics of data, information, and knowledge, we enhance the framework's ability to model complex semantic relationships in AI systems.