Technologizing Wittgenstein's Logisch-Philosophische Abhandlung with DIKWP Core Semantics

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 detailed analysis of how Ludwig Wittgenstein's Logisch-Philosophische Abhandlung (Tractatus Logico-Philosophicus) can be technologized using the core semantics of Prof. Yucong Duan's Data-Information-Knowledge-Wisdom-Purpose (DIKWP) Semantic Mathematics framework. By focusing on the mathematical formalization of the DIKWP semantics—specifically the concepts of Sameness (Data), Difference (Information), and Completeness (Knowledge)—we aim to map Wittgenstein's philosophical propositions onto a rigorous mathematical structure. This integration bridges philosophical logic with computational semantics, providing a structured approach to modeling reality, language, and thought within artificial intelligence systems.

Table of Contents

1. Introduction

• 1.1. Overview

• 1.2. Objectives

2. Background

• 2.1. Wittgenstein's Logisch-Philosophische Abhandlung

• 2.2. Core Semantics of the DIKWP Framework

3. Mathematical Foundations of DIKWP Semantics

• 3.1. Data: Sameness

• 3.2. Information: Difference

• 3.3. Knowledge: Completeness

4. Mapping Wittgenstein's Propositions to DIKWP Semantics

• 4.1. Proposition 1: The World as Data (Sameness)

• 4.2. Proposition 2: Facts and Information (Difference)

• 4.3. Proposition 3: Logical Pictures and Knowledge (Completeness)

5. Technologizing Wittgenstein's Logic Using DIKWP Mathematics

• 5.1. Formalizing Objects and States of Affairs

• 5.2. Logical Structure and Semantic Relations

• 5.3. The Limits of Language and Completeness

6. Implications for Artificial Intelligence

• 6.1. Semantic Representation in AI

• 6.2. Enhancing Logical Reasoning

• 6.3. Modeling Knowledge and Understanding

7. Challenges and Considerations

• 7.1. Complexity of Philosophical Concepts

• 7.2. Maintaining Fidelity to Original Meanings

• 7.3. Ethical Implications

8. Conclusion

9. References

1. Introduction1.1. Overview

The integration of philosophical insights into computational models is crucial for advancing artificial intelligence (AI) systems capable of deep understanding and meaningful interactions. Wittgenstein's Logisch-Philosophische Abhandlung offers a profound exploration of the relationship between language, thought, and reality through a logical framework. Prof. Yucong Duan's DIKWP Semantic Mathematics framework provides mathematical formalizations of semantics, focusing on the core concepts of Sameness (Data), Difference (Information), and Completeness (Knowledge).

1.2. Objectives

• Technologize Wittgenstein's logical propositions using the core semantics of the DIKWP framework.

• Map the structure of the Tractatus onto the mathematical formalizations of Data, Information, and Knowledge.

• Demonstrate how this integration enhances AI's semantic understanding and logical reasoning capabilities.

• Discuss the implications, challenges, and potential applications of this integration.

2. Background2.1. Wittgenstein's Logisch-Philosophische Abhandlung

The Tractatus Logico-Philosophicus is a seminal work where Wittgenstein aims to identify the relationship between language, thought, and reality. The key propositions include:

• Proposition 1: "The world is everything that is the case."

• Proposition 2: "What is the case—a fact—is the existence of states of affairs."

• Proposition 3: "A logical picture of facts is a thought."

• Proposition 4: "A thought is a proposition with a sense."

• Proposition 5: "A proposition is a truth-function of elementary propositions."

• Proposition 6: Explores the general form of a truth-function.

• Proposition 7: "Whereof one cannot speak, thereof one must be silent."

2.2. Core Semantics of the DIKWP Framework

Prof. Yucong Duan's DIKWP framework formalizes semantics as:

• Data (Sameness): Characterized by uniformity or equivalence among data elements based on shared semantic attributes. Formalized using equivalence relations in set theory.

• Information (Difference): Pertains to variability or distinctness between data elements. Quantified using distance metrics and divergence measures in metric spaces and information theory.

• Knowledge (Completeness): Refers to the extent to which knowledge encapsulates all necessary and relevant information to form a coherent and comprehensive understanding. Formalized using concepts from formal logic, specifically logical completeness and consistency.

3. Mathematical Foundations of DIKWP Semantics3.1. Data: Sameness

Definition: Sameness in data refers to the uniformity or equivalence among data elements based on shared attributes.

Mathematical Formalization:

• Equivalence Relation: A relation ∼ on set $D$ satisfying reflexivity, symmetry, and transitivity.

• Equivalence Classes: Partition $D$ into disjoint subsets where each element is equivalent under ∼.

Example:

Let D={d1,d2,d3}D = \{ d_1, d_2, d_3 \}D={d1​,d2​,d3​} with attributes $f(d)$.

If d1∼d2d_1 \sim d_2d1​∼d2​ because f(d1)=f(d2)f(d_1) = f(d_2)f(d1​)=f(d2​), then [d1]={d1,d2}[d_1] = \{ d_1, d_2 \}[d1​]={d1​,d2​}.

3.2. Information: Difference

Definition: Difference in information quantifies the variability between data elements.

Mathematical Formalization:

• Distance Metric $\delta :D\times D\to \mathbb{R}$ satisfying non-negativity, identity of indiscernibles, symmetry, and triangle inequality.

• Information Set: I={δ(di,dj)∣di,dj∈D,di≠dj}I = \{ \delta(d_i, d_j) \mid d_i, d_j \in D, d_i \neq d_j \}I={δ(di​,dj​)∣di​,dj​∈D,di​=dj​}.

Example:

Using Euclidean distance:

δ(di,dj)=∑k(fk(di)−fk(dj))2\delta(d_i, d_j) = \sqrt{\sum_{k}(f_k(d_i) - f_k(d_j))^2}δ(di​,dj​)=∑k​(fk​(di​)−fk​(dj​))2​.

3.3. Knowledge: Completeness

Definition: Completeness in knowledge ensures that all necessary information is encapsulated to form a coherent understanding.

Mathematical Formalization:

• Formal System: $K=(S,⊢)$, where $S$ is a set of axioms and $⊢$ is the deduction relation.

• Logical Completeness: For every statement $\varphi$ in language $\mathbb{L}$, either $\varphi$ or $\neg \varphi$ is derivable in $K$.

4. Mapping Wittgenstein's Propositions to DIKWP Semantics4.1. Proposition 1: The World as Data (Sameness)

Wittgenstein's View:

• The world consists of facts, which are states of affairs.

• Objects constitute the substance of the world.

DIKWP Mapping:

• Data Level (Sameness): The 'objects' in Wittgenstein's world correspond to data elements $D$.

• Equivalence Relations: Objects sharing attributes form equivalence classes, reflecting the uniformity of the world's substance.

Mathematical Representation:

• D={di∣objects in the world}D = \{ d_i \mid \text{objects in the world} \}D={di​∣objects in the world}.

• Equivalence classes [di][d_i][di​] represent objects with shared properties.

4.2. Proposition 2: Facts and Information (Difference)

Wittgenstein's View:

• Facts are the existence of states of affairs.

• A state of affairs is a combination of objects.

DIKWP Mapping:

• Information Level (Difference): Facts correspond to the differences and relations between data elements.

• Distance Metrics: Quantify the distinctness between states of affairs.

Mathematical Representation:

• States of affairs s=R(d1,d2,...,dn)s = R(d_1, d_2, ..., d_n)s=R(d1​,d2​,...,dn​), where $R$ represents relations.

• Information set I={δ(di,dj)}I = \{ \delta(d_i, d_j) \}I={δ(di​,dj​)} captures the differences between objects in states of affairs.

4.3. Proposition 3: Logical Pictures and Knowledge (Completeness)

Wittgenstein's View:

• A logical picture of facts is a thought.

• Thoughts are propositions with sense.

DIKWP Mapping:

• Knowledge Level (Completeness): Thoughts represent knowledge derived from complete information.

• Formal Systems: Logical pictures are formal propositions within a complete and consistent knowledge system.

Mathematical Representation:

• Knowledge system $K=(S,⊢)$ contains propositions representing thoughts.

• Completeness ensures all logical consequences of information are included.

5. Technologizing Wittgenstein's Logic Using DIKWP Mathematics5.1. Formalizing Objects and States of Affairs

Objects as Data Elements:

• Each object did_idi​ is a data element with attributes fk(di)f_k(d_i)fk​(di​).

States of Affairs as Relations:

• A state of affairs $s$ is a combination R(di,dj,...)R(d_i, d_j, ...)R(di​,dj​,...).

Equivalence Classes:

• Objects with identical attributes form equivalence classes [di][d_i][di​].

Example:

Let D={d1,d2}D = \{ d_1, d_2 \}D={d1​,d2​} where f(d1)=f(d2)f(d_1) = f(d_2)f(d1​)=f(d2​), so d1∼d2d_1 \sim d_2d1​∼d2​ and [d1]={d1,d2}[d_1] = \{ d_1, d_2 \}[d1​]={d1​,d2​}.

5.2. Logical Structure and Semantic Relations

Information as Differences Between Objects:

• Use distance metrics to quantify differences: δ(di,dj)\delta(d_i, d_j)δ(di​,dj​).

Building Information Sets:

• Information set $I$ includes all δ(di,dj)\delta(d_i, d_j)δ(di​,dj​) representing states of affairs.

Constructing Knowledge Systems:

• From $I$, derive axioms $S$ for knowledge system $K$.

• Ensure logical completeness: all propositions about the world are derivable.

Example:

Given I={δ(d1,d2)}I = \{ \delta(d_1, d_2) \}I={δ(d1​,d2​)}, construct $K$ with propositions like "Object d1d_1d1​ is related to d2d_2d2​ by $R$."

5.3. The Limits of Language and Completeness

Wittgenstein's Proposition 7:

• Recognizes that some aspects cannot be expressed in language.

DIKWP Perspective:

• Completeness in knowledge accounts for all derivable propositions but acknowledges the limits defined by the formal system.

Mathematical Implication:

• There exist statements $\varphi$ not in $\mathbb{L}$, and thus not in $K$.

6. Implications for Artificial Intelligence6.1. Semantic Representation in AI

Data Modeling:

• Objects in the world are represented as data elements with attributes in AI systems.

Information Processing:

• Differences between data elements are quantified, enabling AI to recognize patterns and states of affairs.

Knowledge Representation:

• AI constructs knowledge bases $K$ that are logically complete and consistent.

6.2. Enhancing Logical Reasoning

Formal Logic in AI:

• Employ formal systems $K$ to enable AI to perform deductions and reason about the world.

Completeness and Consistency:

• Ensuring knowledge bases are complete and consistent improves AI's reasoning capabilities.

6.3. Modeling Knowledge and Understanding

Cognitive Modeling:

• By mapping Wittgenstein's logic to DIKWP semantics, AI can model human-like understanding.

Handling Limits of Language:

• AI systems recognize the boundaries of expressible knowledge, aligning with Wittgenstein's insights.

7. Challenges and Considerations7.1. Complexity of Philosophical Concepts

• Translating complex philosophical ideas into mathematical models requires careful interpretation.

7.2. Maintaining Fidelity to Original Meanings

• Ensure that the technological mapping preserves the essence of Wittgenstein's propositions.

7.3. Ethical Implications

• AI systems must handle the limits of knowledge responsibly, avoiding overextension beyond what can be meaningfully expressed.

8. Conclusion

By starting from the core semantics of the DIKWP Semantic Mathematics framework—specifically the mathematical formalizations of Sameness (Data), Difference (Information), and Completeness (Knowledge)—we have technologized Wittgenstein's Logisch-Philosophische Abhandlung. This mapping provides a rigorous mathematical foundation for integrating philosophical logic into AI systems, enhancing their ability to model reality, process information, and construct knowledge in a manner aligned with human cognition.

9. References

1. Wittgenstein, L. (1921). Logisch-Philosophische Abhandlung (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. (1903). The Principles of Mathematics. Cambridge University Press.

5. Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423.

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

7. Hinton, G., & Salakhutdinov, R. R. (2006). Reducing the Dimensionality of Data with Neural Networks. Science, 313(5786), 504-507.

8. Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann.

9. Frege, G. (1892). On Sense and Reference.

10. Hilbert, D., & Ackermann, P. (1928). Principles of Mathematical Logic. Chelsea Publishing.

Keywords: Wittgenstein, Logisch-Philosophische Abhandlung, DIKWP Semantic Mathematics, Sameness, Difference, Completeness, Data, Information, Knowledge, Artificial Intelligence, Semantic Representation, Logical Reasoning.

Note: This document focuses on starting from the core semantics of the DIKWP framework, as per the provided reference, to technologize Wittgenstein's Logisch-Philosophische Abhandlung. By leveraging the mathematical formalizations of Sameness, Difference, and Completeness, we establish a rigorous mapping between Wittgenstein's propositions and the DIKWP semantics, facilitating their integration into computational models for AI.