Cross-Domain Application Research Report Based on
通用人工智能AGI测评DIKWP实验室
Cross-Domain Application Research Report Based on YucongDuan's Semantic Mathematics Theory
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
Professor Yucong Duan has developed a comprehensive and original theoretical system in the field of Semantic Mathematics, encompassing terminological frameworks such as the Theory of Meaning Definition, Conceptual Construction Theory, and Mathematical Logic–Language Hook Theory. The core idea of this system is to explicitly introduce semantics into formal mathematical systems, using axiomatic methods to characterize the meanings behind symbols.
This research report systematically reviews the central content of Professor Duan's Semantic Mathematics theory, offering detailed explanations of relevant terms. Based on the proposed semantic axiomatic system—which includes foundational axioms such as existence, uniqueness, and transitivity—we construct a formal framework for semantic binding operations. Building upon this foundation, the report explores cross-domain applications through four representative case studies:
(1) Semantic structure binding and meaning derivation processes of Chinese characters such as "日" (sun), "明" (bright), and "时" (time);
(2) Axiomatic semantic analysis of abstract philosophical concepts such as "existence", "unity", and "causality";
(3) Modeling examples of binding between natural language syntax and semantic structures;
(4) Transparent rule construction and symbolic logic visualization in engineering-oriented knowledge graphs.
Each case originates from the semantic mathematics axioms and illustrates the reasoning chains, structural mappings, and semantic preservation mechanisms from axiom deduction to practical application. Furthermore, the report integrates the Hook Theory, delving into the principles of interaction between mathematical logic and language expression, clarifying how formal symbolic systems can be semantically anchored to natural language.
Finally, we analyze strategies for maintaining semantic consistency in cross-domain integrations, assess the completeness of the axiomatic system, and address practical challenges in engineering implementations. The report concludes with a forward-looking discussion on building a unified knowledge system and proposes extensions to the existing semantic axioms. This comprehensive and well-structured report serves as a valuable reference for interdisciplinary research across semantics, knowledge engineering, mathematical logic, and artificial intelligence.
Keywords: Semantic Mathematics; Theory of Meaning Definition; Conceptual Construction Theory; Semantic Axioms; Semantic Binding; Knowledge Graph; Explainability
1. Introduction
In the human endeavor to understand and describe the world, the two seemingly disparate domains of semantics and mathematics are increasingly converging. Traditional mathematics is grounded in formal axiomatic systems and symbolic reasoning, emphasizing rigorous logical deduction; in contrast, semantics focuses on the meanings and interpretations carried by symbols. In essence, mathematics offers precise formal structures, while semantics imbues these structures with meaning grounded in the real world. The challenge of integrating these two realms—creating a unified framework that is both logically rigorous and semantically meaningful—has become a critical concern in recent research within knowledge engineering and artificial intelligence.
Professor Yucong Duan’s theory of Semantic Mathematics represents a novel research direction developed specifically to address this challenge. At its core, Semantic Mathematics seeks to explicitly incorporate semantic information and its hierarchical structure into formal mathematical systems, thereby transforming mathematical symbols from mere abstract reasoning tools into conceptual carriers of well-defined meanings. Unlike traditional mathematics, which often embeds semantic choices implicitly within the intuitions guiding axiom selection and focuses primarily on intra-symbolic deduction, Semantic Mathematics demands that each symbol and every theorem correspond to specific semantic constructs—synchronizing symbolic operations with semantic inference.
Building on this foundational idea, Professor Duan has progressively constructed a comprehensive theoretical system of Semantic Mathematics, including:
Theory of Meaning Definition: a framework for defining the meaning of symbols;
Conceptual Construction Theory: a theory that studies how new concepts can be built through the composition and construction of basic concepts;
Mathematical Logic–Language Hook Theory: a framework that elucidates how mathematical logical structures “hook” or map onto natural language expressions.
These theoretical components are interdependent and collectively constitute the foundation of Semantic Mathematics. On this basis, Professor Duan further proposed an axiomatic system for Semantic Mathematics, rigorously defining the rules of semantic binding through a set of axioms. These axioms include:
Existence: ensuring every datum (natural phenomenon) corresponds to a semantic unit;
Uniqueness: semantically equivalent data are grouped into the same unit with no redundancy;
Transitivity: semantic equivalence relations are transitively closed within semantic units.
This axiomatic foundation enables the mathematization and formalization of the semantic binding process, establishing a unified and transparent set of semantic rules applicable across disciplines. Whether in natural language processing, knowledge graph construction, or domains such as medicine and law, this axiomatic framework holds promise as a shared semantic representation language.
This report aims to systematically present the core theories and axiomatic system of Professor Duan’s Semantic Mathematics and explore their cross-domain applications. Chapter 2 reviews the three pillars of the Semantic Mathematics theory—Meaning Definition, Conceptual Construction, and Hook Theory—clarifying key terminologies and conceptual meanings. Chapter 3 elaborates on the formalized framework for semantic binding based on the existing axioms, including mathematical representations and reasoning mechanisms. Chapter 4 illustrates the application of the Semantic Mathematics axiomatic system through four representative case studies:
Semantic construction and derivation processes of Chinese characters such as “日” (sun), “明” (bright), and “时” (time);
Axiomatic semantic analysis of abstract philosophical concepts like existence, unity, and causality;
Modeling of binding between natural language syntax and semantic structures;
Transparent rule construction and symbolic logic visualization in engineering-oriented knowledge graphs.
Each case starts from the axioms, demonstrating formal reasoning chains, structural mappings, and semantic preservation mechanisms.
Chapter 5 applies Hook Theory to investigate the interactive principles between mathematical logic forms and language expressions at the level of symbolic meaning, explaining how Semantic Mathematics builds a bridge between formal systems and natural language. Chapter 6 addresses common challenges encountered in cross-domain integrations, such as maintaining semantic consistency across heterogeneous knowledge sources, testing and extending the completeness of the axiomatic system, and tackling practical implementation issues in engineering contexts. Chapter 7 presents a forward-looking vision for constructing unified knowledge systems, proposing possible extensions and improvements to the existing semantic axioms. Chapter 8 concludes the report, highlighting the significance of Semantic Mathematics in unifying knowledge representation and enhancing interpretability in artificial intelligence.
Through this comprehensive exploration, the report aims to demonstrate the theoretical depth and practical breadth of Semantic Mathematics. It offers a foundational basis for knowledge representation and reasoning across disciplines, opening the possibility of constructing a unified semantic space that bridges natural language, domain-specific knowledge, and machine intelligence. Such unification is not merely conceptual but also engineering-practical—it implies that, through an axiomatized semantic rule system, machines may achieve a human-like semantic intuition and interpretability while maintaining logical rigor in information processing. This report provides a robust theoretical foundation and concrete examples to support the realization of that vision.
2. Overview of Yucong Duan’s Semantic Mathematics Theoretical System
Professor Yucong Duan’s Semantic Mathematics theoretical system consists of a series of interrelated sub-theories, all serving the overarching goal of integrating semantics into mathematical symbol systems. This chapter provides an overview of the three most essential components of this theory and defines and explains key related terms.
2.1 Theory of Meaning Definition
The Theory of Meaning Definition focuses on how to define the meanings of symbols—that is, how to precisely characterize the referents of symbols or concepts within a formal system. Traditionally, the meanings of mathematical symbols are implicitly defined through axioms and model theory, while natural language word meanings often rely on dictionary-like descriptions or conventions of use. The Theory of Meaning Definition seeks to establish a universal method by which every basic symbol’s meaning can be defined and explained through finite rules or pre-existing concepts.
Professor Duan points out that, in everyday language, the meaning of a new concept is typically explained using already understood concepts. For instance, a dictionary definition of “electronic computer” may involve the terms “electronic,” “calculation,” and “machine.” If the reader already understands these more basic concepts, the new term is thus rendered meaningful. This illustrates that meaning is built upon prior meanings and exhibits a recursive, hierarchical structure. The Theory of Meaning Definition aims to formalize this process: to define a concept by referencing a set of existing concepts and their interrelations, using methods such as composition, restriction, or mapping to provide a semantic explanation.
To avoid circular dependency or infinite regress in defining semantics, this theory presumes the existence of a set of primitive semantics—basic concepts that are not defined in terms of others but rather grounded directly in human cognitive experience. For example, perceptual primitives like “red” may gain their meaning through direct visual experience; philosophical primitives like “existence” may be treated as axiomatic semantic constructs. The goal is that, in defining complex concepts, one ultimately traces back to these primitives, ensuring the entire semantic network is anchored in intuitively graspable semantic units.
It is important to distinguish this theory from traditional model-theoretic semantics. In model theory, the meanings of symbols (interpretations) are assigned by a model that maps symbols to elements or relations within a domain. In contrast, the Theory of Meaning Definition is more concerned with how symbols derive and construct meaning using other symbols within the system itself—essentially forming a semantic network. This perspective is similar to synonym networks or definitional chains in linguistics but, within the Semantic Mathematics framework, the emphasis is on a formal, computable definition mechanism.
Professor Duan emphasizes that “conceptual forms possess their own internal structure, not entirely equivalent to linguistic symbol relations.” This means that the mental representation of a concept may have a cognitive or neurological structure that is not strictly mirrored by language. However, “the structure of language influences conceptual categorization”—language serves as an external symbolic system, and its categorizations affect how humans organize concepts. For example, lexical categories in language often guide how we group entities into conceptual classes. Thus, defining meaning requires considering the relationship between conceptual structure and linguistic symbol structure.
The Theory of Meaning Definition strives to balance these two dimensions: it aims to reflect the objective cognitive structure of concepts (e.g., prototypes and core features) while also accommodating linguistic conventions to ensure clarity and accessibility. In summary, the Theory of Meaning Definition provides the foundational basis for Semantic Mathematics: it answers the question of where symbol meaning originates. By using this theory, we can assign clear meanings to all symbols in the semantic mathematical system—be they mathematical, logical, or natural language terms—thereby laying the groundwork for the axiomatic system to follow.
The next section, Conceptual Construction Theory, will further explore how new, more complex concepts can be built upon those that have already been defined.
2.2 Conceptual Construction Theory
The Conceptual Construction Theory focuses on the process and principles by which complex concepts are constructed from simpler ones. While the Theory of Meaning Definition addresses how basic concepts are defined, this theory asks how these concepts can be used like building blocks to construct more extensive conceptual structures. It seeks to characterize the combinatorial relationships, hierarchical structures, and generative mechanisms among concepts, facilitating our understanding of how conceptual systems expand and evolve.
The core premise is that any complex concept can be decomposed into a combination of simpler concepts. This combination can take various forms, including but not limited to:
Logical Composition: Combining concepts using logical connectors (e.g., “and,” “or,” “not”). For example, combining concepts A and B to form “A and B.”
Modifier Constraints: Using one concept to qualify or limit another, e.g., “red apple,” where “red” restricts the scope of “apple.”
Hierarchical Inheritance: Deriving a subordinate concept from a superordinate one, inheriting its basic semantics while adding new attributes, e.g., “bird” is a subclass of “animal” with added traits like “has feathers” or “lays eggs.”
Analogy and Metaphor: Mapping structural features of one domain to another, forming new concepts. For instance, the concept of a “computer virus” borrows characteristics from biological viruses.
Symbolic Composition: Constructing new symbols to represent compound concepts. For example, combining the logograms “日” (sun) and “月” (moon) to form “明” (bright), reflecting their combined meaning.
Professor Duan’s research particularly highlights two aspects: semantic dynamism and conceptual morphology. Semantic dynamism refers to the fact that concepts may extend or shift meaning depending on context. For instance, the word “fire” may refer to physical flame, but in different contexts it can also mean “to dismiss someone” or represent “enthusiasm.” These shifts illustrate dynamic conceptual construction.
Conceptual morphology concerns the internal structure of concepts in cognition, including prototypes, boundaries, and feature sets. When forming new concepts, we rarely simply stack attributes. Instead, we follow cognitive patterns such as prototype expansion (generalizing from typical examples) or feature fusion (merging essential attributes of multiple concepts).
Take the example of Chinese ideograms: ideogrammatic characters often convey meaning by combining semantic radicals. The character “明” (bright) combines “日” (sun) and “月” (moon), symbolizing their combined illumination. The new concept “bright” (明) emerges not from a literal addition but from a semantic elevation: the co-presence of sun and moon implies ultimate brightness. This demonstrates that conceptual construction is not mechanical aggregation, but a generative process yielding semantic qualia—new qualitative meanings.
Another example is the character “时” (traditional form: 時), composed of “日” (sun) and “寺” (temple). The “sun” implies cyclic time (as measured by solar motion), while “temple” served as a timekeeper in ancient China (e.g., bell chimes). Together, they represent time, combining natural and social constructs of temporality. These character constructions exemplify how symbolic components are assembled into new symbols with compound meanings—a key manifestation of Conceptual Construction Theory.
In modern knowledge engineering, Conceptual Construction Theory is widely applicable. In ontology engineering, complex entities or events are often represented as compound concepts assembled from simpler ones. Semantic networks and conceptual graphs organize concepts hierarchically and relationally, with nodes representing concepts and edges representing construction relations (e.g., “is-a,” “part-of,” “cause-of”). Through these relationships, one can either construct higher-order concepts from primitives or decompose complex concepts into simpler ones.
Importantly, this theory seeks to formalize the process of concept generation to make it algorithmically simulatable and verifiable. Together with the Theory of Meaning Definition, it provides a solid foundation for Semantic Mathematics: the former addresses the semantic origin of concepts, while the latter addresses the semantic composition of concepts. With both in place, Semantic Mathematics can rigorously handle meaning creation and evolution at the symbolic level.
The next section will introduce Hook Theory, which further investigates the mapping relationships between formal symbolic structures and linguistic expressions—marking a critical step toward cross-modal and cross-system applications in Semantic Mathematics.
2.3 Mathematical Logic–Language Hook Theory
The Mathematical Logic–Language Hook Theory (abbreviated as Hook Theory) explores the semantic alignment between formal mathematical logic and natural language expressions—specifically, how logical symbolic structures can be "hooked" onto linguistic structures to enable interaction between formal reasoning and language understanding.
The term “hook” metaphorically describes this correspondence: as if placing a checkmark (✓) between a logical expression and a natural language sentence to indicate that their meanings align. Hook Theory seeks to answer key questions such as: How can logical propositions be expressed in natural language? How can everyday language sentences be translated into formal logic? Do systematic rules and constraints govern the mapping between the two? While traditional semantics—particularly formal semantics or Montague Semantics—addresses similar issues by assigning logical meanings to natural language sentences, Professor Yucong Duan's Hook Theory goes further by emphasizing symbolic interaction: logic and language are not merely unidirectionally translatable, but rather, co-evolving components of human cognitive activity.
According to Hook Theory, every natural language sentence (especially declarative or propositional expressions) can, in principle, be abstracted into some form of logical structure. Conversely, every abstract logical formula should be expressible in natural language. Ideally, we aim to establish a bidirectional mapping
M
:
M
:{
Natural Language Expressions
}↔{
Logical Forms
}
Such that if a sentence
L
is mapped to a logical formula
Φ=
M
(
L
)
, then the logical deductions from
Φ
correspond to semantic reasoning in the original sentence
L
. Conversely, for any formula
Φ
, the inverse mapping
L
′
=
M
−1
(Φ)
yields a comprehensible natural language statement. This mapping must be semantically equivalent, meaning that the meanings conveyed by
L
and
L
′
should be identical or at least equivalent within the bounds of human understanding.
At a deeper level, Hook Theory reveals insights into the relationship between language and thought. Some scholars argue that language is merely a tool for expressing thought, and that true reasoning occurs within an internal conceptual or logical layer. Others believe language actively participates in the thinking process itself. Professor Duan's Hook Theory synthesizes these views by stating that “the essence of communication is a person using their own linguistic mechanisms to semantically organize and project their internal cognitive space within the semantic space.” In simple terms, when a speaker communicates, they organize their conceptual space into language so that the listener can reconstruct the intended meaning using their own conceptual space. In this process, natural language sentences and underlying logical/conceptual structures are “hooked” to each other: the speaker encodes internal logical structures into sentences, and the listener decodes these sentences back into logic. Each party effectively “communicates with themselves,” but with the aid of a shared hook mechanism, meaning is transmitted and shared.
To realize this logic–language hooking, Hook Theory must address several key technical issues:
Choice of Formal Language: Selecting the appropriate logical system to represent natural language semantics—e.g., propositional logic, first-order logic, modal logic, higher-order logic, or type theory. Different language phenomena may require different levels of logical expressiveness.
Syntax–Semantics Interface: Defining how syntactic structures (like parse trees) map to logical structures. For example, subject–verb–object structures map to binary predicate applications; adjectives act as predicate modifiers; adverbs may correspond to operators. This is similar to compositional semantics, aligning lexical and logical composition.
Ambiguity and Type Matching: Natural language often contains ambiguity and polysemy, where one sentence might map to multiple logical forms. Hook Theory needs disambiguation rules and mechanisms to handle type mismatches—e.g., coercing a concept’s type to fit into a logical formula.
Pragmatics and Context: Logic assumes clearly defined contexts, while natural language relies heavily on pragmatic cues. Hook Theory must address how to parameterize context within logical forms to handle deictic expressions, vague terms, and default inferences.
Professor Duan’s Semantic Mathematics framework provides solid foundational support for Hook Theory: the Theory of Meaning Definition ensures that every linguistic or logical symbol has a defined meaning, while Conceptual Construction Theory enables more complex expressions to be systematically built from basic semantic units. Thus, during the hooking process, we operate on semantically grounded units.
For instance, consider the sentence “Snow is white.” Hook Theory maps this to a logical form such as:
The conditional “If snow is white, then the weather is cold” becomes:
White
(
Snow
)→
Cold
(
Weather
)
Using the axioms and inference rules of Semantic Mathematics, such proposition chains can be verified against a knowledge system. Knowing “snow is white,” we might infer hypotheses about the coldness of the weather. This loop—language → logic → knowledge inference → logic → language—demonstrates the principle of symbolic-meaning interaction.
In short, Hook Theory builds a bridge between formal reasoning and natural language understanding. Its value lies in enabling logic-based systems to express their reasoning in human-understandable forms—and, conversely, allowing human language to be directly interpreted and validated through formal logic. This has significant implications for artificial intelligence: using Hook Theory, we can develop “white-box” AI systems that reason internally with logic and communicate externally in natural language, overcoming the interpretability issues that plague traditional black-box models.
For this reason, the influence of Hook Theory will recur throughout the following chapters on axiomatic systems and applications. Whenever we introduce a formal rule, we will consider its natural language interpretation; whenever we encounter linguistic reasoning, we will attempt to formalize it. This integration of formal logic and natural language is a defining feature that distinguishes Semantic Mathematics from conventional mathematics or standard semantics.
3. Semantic Mathematics Axiomatic System and the Formal Framework for Semantic Binding
In the previous two chapters, we established the theoretical foundations of Semantic Mathematics—how meaning is defined, how concepts are constructed, and how formal symbols correspond to natural language expressions. With these foundations in place, we now introduce the axiomatic system of Semantic Mathematics proposed by Professor Yucong Duan, and based on this, we construct a formal operational framework for semantic binding. This framework provides a rigorous mathematical basis for performing semantic binding in practical applications—that is, linking specific data or phenomena to semantic concepts—ensuring that information across various domains adheres to a unified set of rules when entering the semantic space, thereby guaranteeing transparency and consistency in semantic expression.
3.1 Fundamental Semantic Axioms
Professor Duan has proposed three core axioms of Semantic Mathematics in his blog posts and technical reports. These axioms provide formal constraints on the semantic binding process, ensuring clarity and consistency in how symbols are semantically mapped within the system. Below are the three axioms with their explanations:
Axiom 1 (Existence):
For any natural phenomenon
x
, there exists a semantic unit
S
such that
x
∈
S
.
Explanation:
This axiom guarantees that every observed phenomenon (or data point) has a corresponding semantic unit within the symbolic system. Whether
x
is a concrete physical object (e.g., "a tree" or "a bird") or an abstract concept or experience (e.g., "an emotion"), the system must contain a semantic unit
S
that encompasses and expresses the meaning of
x
. Simply put, no input is left behind—every piece of information entering the system must find a semantic destination. This reflects the completeness of the symbolic-semantic system: the semantic space must be rich enough to cover all possible inputs. From an engineering perspective, this is akin to requiring that a knowledge base comprehensively covers its domain, or that a parser handles all inputs exhaustively.
Axiom 2 (Uniqueness):
For any data
x
and
y
, if
ϕ
(
x
)=
ϕ
(
y
)
(i.e., they yield the same semantic representation after feature extraction), then both
x
and
y
must be bound to the same semantic unit
S
.
Explanation:
Here,
ϕ
is a feature extraction function defined over the dataset, used to extract key semantic features from data
x
. If two data points yield the same semantic representation via
ϕ
, they are considered semantically equivalent, and the system must place them into the same semantic unit
S
. In other words, the same meaning must not appear in the system under multiple symbolic entries. This axiom ensures determinism and uniqueness within the symbolic system, avoiding redundancy and semantic contradictions. For example, if
ϕ
extracts identity-related features, then “007” and “James Bond” should resolve to the same semantic unit representing the same secret agent, preventing duplication in the knowledge base. This defines semantic equivalence classes and ensures clear consolidation of semantically identical elements.
Axiom 3 (Transitivity):
If
x
,
y
∈
S
and
y
,
z
∈
S
, then
x
,
z
∈
S
.
Explanation:
This axiom enforces coherence within semantic units. Formally, it establishes transitivity of the semantic equivalence relation: if
x
and
y
belong to the same semantic unit
S
, and
y
and
z
do as well, then necessarily
x
and
z
must also belong to
S
. This guarantees non-contradiction in semantic classification. It avoids the situation where
x
and
y
are considered equivalent, and
y
and
z
are also equivalent, but
x
and
z
are not. In essence, this axiom ensures that semantic binding is a closed operation, preserving the logical consistency of the entire system. Together with Axioms 1 and 2, this makes semantic binding an equivalence relation, satisfying reflexivity (implicitly), symmetry (from Axiom 2), and transitivity (via Axiom 3). Thus, semantic units can be viewed as equivalence classes partitioning the base data set
X
by semantic equivalence.
These three fundamental axioms form the core of the Semantic Mathematics axiomatic system. Within Professor Duan’s framework, they serve as essential constraints and assumptions for semantic binding. From these axioms, several meaningful theorems can be derived, further illuminating their significance:
Identity Theorem:
If for any data
x
and
y
,
ϕ
(
x
)=
ϕ
(
y
)
, then there exists a unique semantic unit
S
such that
x
,
y
∈
S
.
Proof Sketch: From Axiom 2,
ϕ
(
x
)=
ϕ
(
y
)
implies
x
,
y
∈
S
. Axiom 1 guarantees the existence of such
S
, and Axiom 2 ensures its uniqueness. Therefore, a single semantic expression corresponds to a unique semantic unit.
Transitive Consistency Theorem:
If
ϕ
(
x
)=
ϕ
(
y
)
and
ϕ
(
y
)=
ϕ
(
z
)
, then
ϕ
(
x
)=
ϕ
(
z
)
, and
x
,
y
,
z
all belong to the same semantic unit
S
.
Proof Sketch: From
ϕ
(
x
)=
ϕ
(
y
)
, Axiom 2 gives
x
,
y
∈
S
; from
ϕ
(
y
)=
ϕ
(
z
)
, again Axiom 2 gives
y
,
z
∈
S
. Then Axiom 3 implies
x
,
z
∈
S
, leading to
ϕ
(
x
)=
ϕ
(
z
)
. This shows that semantic equivalence remains stable as new data are added.
Binding Stability Theorem:
For any data
x
and
y
, if repeated observations yield
ϕ
(
x
)=
ϕ
(
y
)
, then
y
will consistently be assigned to the same semantic unit as
x
, and any new data
x
′
with
ϕ
(
x
′
)=
ϕ
(
x
)
will also belong to that unit. Thus, semantic binding rules are stable under data updates.
Proof Sketch: This is a consequence of Axioms 1–3. Axiom 2 ensures that each instance of semantic equivalence results in binding; Axiom 3 guarantees that this binding holds for all in the equivalence chain; and Axiom 1 ensures that every new datum is considered. If
ϕ
(
x
′
)=
ϕ
(
x
)
and
ϕ
(
x
)=
ϕ
(
y
)
, then transitivity yields
ϕ
(
x
′
)=
ϕ
(
y
)
. Axiom 2 then ensures
x
′
,
y
∈
S
; especially if
y
=
x
, then
x
′
∈
S
. Repeating this logic shows that the binding rule remains valid regardless of new data.
These derived results show that the fundamental axioms ensure the consistency, non-contradiction, and robustness of the semantic binding system. No matter how the data grows or is compared, following the axioms ensures that semantically equivalent data are consolidated into the same semantic unit, resulting in clear semantic classification. The principles of “one object, one meaning” and “one meaning, one class” prevent duplication and confusion.
This lays a common foundation for various cross-domain applications: data from different sources and modalities can be aligned into a unified semantic space as long as they follow these axioms during semantic processing.
3.2 Formalized Operations of Semantic Binding
Having clarified the axiomatic system, we now specify the operational rules of semantic binding in a formalized and executable manner. Semantic binding generally refers to the process of mapping an observed data point (or input signal) to an appropriate semantic unit or concept. Formally, this can be expressed as a function or mapping:
where
X
denotes the set of all possible data (the universe of “natural phenomena”), and
S
denotes the set of semantic units. Based on Axiom 1 (Existence), this mapping
B
is surjective, meaning every
x
∈
X
has a corresponding
B
(
x
)∈
S
.
More specifically, the semantic binding operation can be broken down into the following substeps:
(1) Feature Extraction (
ϕ
)
The first step is to extract features from the input data
x
, computing
ϕ
(
x
)
. This function should capture the key information that represents the semantics of
x
. The implementation of
ϕ
varies across applications:
For image data,
ϕ
(
x
)
may be a high-level semantic feature vector (e.g., embeddings from convolutional neural networks).
For natural language sentences,
ϕ
(
x
)
might be a semantic graph or a logical formula (as discussed in Hook Theory).
For sensor readings,
ϕ
(
x
)
may be physical state parameters derived from models.
The design of
ϕ
directly influences the effectiveness of semantic binding. It must be crafted with domain knowledge to ensure that the extracted features are semantically relevant and not merely noise.
Next, the extracted features
ϕ
(
x
)
are compared to the prototype features of existing semantic units. Each semantic unit
S
i
can be associated with a representative feature signature
Φ(
S
i
)
. The task is to find whether:
The relation
≈
may represent exact equality (e.g., symbolic matching) or tolerance within a distance threshold (e.g., vector similarity).
If a matching unit
S
i
is found, then by Axiom 2 (Uniqueness),
x
must be bound to that unit. If multiple candidates appear (i.e., ambiguity arises), a disambiguation mechanism is required—potentially involving contextual cues or finer-grained features. Axiom 2 implies that no input should be assigned to more than one semantic unit, so the system must apply prioritization or stricter thresholds to ensure a unique binding.
(3) Semantic Unit Creation
If no existing semantic unit matches
ϕ
(
x
)
, then per Axiom 1 (Existence), a new semantic unit
S
new
must be created to accommodate
x
. This new unit may adopt
ϕ
(
x
)
as its prototype:
This corresponds to introducing a new concept to explain previously unseen data. A robust semantic binding framework must support dynamic expansion of the conceptual space.
Importantly, the new unit is not meant solely for the current data point but anticipates future data that may also belong to it. Thus, semantic properties or rules should be predefined during creation, so subsequent data with similar properties can be incorporated via Axiom 3 (Transitivity). This is analogous to defining a new category in a database schema or adding a node with associated attributes in a semantic network.
The input
x
is formally marked as belonging to the determined or newly created semantic unit
S
k
, denoted:
In data structures, this may involve appending
x
to the record of
S
k
, or creating a pointer from
x
to
S
k
. For explainability, supporting logs or justification metadata should also be recorded—documenting why
x
was assigned to
S
k
(e.g., by citing axioms and matching rationale). This is essential for transparent rule auditing.
(5) Propagation and Updates
After semantic binding, the system may need to trigger updates to maintain global semantic consistency. For example:
If
S
k
has a superclass
C
, then
x
implicitly belongs to
C
(typical ontological inference).
If
S
k
is linked to other semantic units via relations (e.g., causal, associative, or logical rules), the inclusion of
x
may trigger rule-based inference. For example, if
x
belongs to “flammable materials” and the rule is “flammables burn when ignited,” we can infer that “
x
will burn when exposed to fire.”
If adding
x
changes the representative feature of
S
k
(e.g., its prototype is calculated from the average of its members), then related matching or reasoning processes should be notified and updated.
This propagation step ensures that the axioms, especially Axiom 3 (Transitivity), hold within more complex semantic networks. It not only guarantees transitive closure but also semantic consistency across extended reasoning chains.
The entire semantic binding process is defined through formal functions and rules. Each step is expressible in mathematical or logical terms, enabling automation and verification. In his reports, Professor Duan introduces the concept of “dynamic transformation functions” and “contextual transformation weights” to handle semantic binding under varying conditions.
Concretely, a context-sensitive confidence weight
W
(
e
ij
)
may be introduced to represent the binding likelihood of mapping data point
i
to semantic unit
j
under context
e
. These weights can guide more flexible, explainable decision-making and support progressive learning. For instance, in multimodal fusion, visual features and language descriptions can each provide binding recommendations, which are then integrated into a weighted decision. The system may conclude: “Primarily based on visual features, the animal is classified as a ‘cat.’”
This formalized framework allows us to analyze semantic assignment like an algorithm, making the meaning attribution process transparent, repeatable, and verifiable. In the past, the semantics of symbols were often buried in dictionaries or hardcoded logic—opaque and difficult to evaluate. Now, with explicit
ϕ
-functions, axioms, and operational flows, we gain a clear understanding of how machines “understand” data.
In the upcoming case study chapters, we will demonstrate how semantic binding is executed in real-world scenarios using this framework and verify its compliance with the aforementioned axioms and rules. Before that, we must address one critical issue in semantic binding: semantic preservation—how to maintain meaning fidelity during reasoning and structural transformations, which lies at the core of Semantic Mathematics.
3.3 Semantic Preservation and Inference Chains
Semantic Preservation is a fundamental requirement within the Semantic Mathematics framework. It stipulates that throughout every step of information transformation and logical inference, the meaning must not be lost or distorted. In other words, from raw data to semantic representation, and ultimately to high-level inferential conclusions, there should exist a clear and traceable chain of meaning, where each step is semantically accountable. Only under this condition can the system be interpretable and trustworthy.
In the formalized semantic binding process, semantic preservation must be ensured at several critical stages:
Semantic Integrity of Feature Extraction (
ϕ
(
x
)
):
The extracted features
ϕ
(
x
)
must preserve all essential information relevant to the task, and must not introduce unrelated noise. In other words,
ϕ
should be a semantically sufficient and necessary mapping. If
ϕ
discards key semantic content, the foundation for subsequent reasoning becomes invalid; if
ϕ
adds spurious signals, it introduces bias and distortion. Therefore, the design of
ϕ
often incorporates domain knowledge and information-theoretic metrics to ensure that
ϕ
(
x
)
is semantically equivalent to
x
(possibly in terms of equivalence classes).
Correctness of Binding Decisions:
When we bind a data point
x
to a semantic unit
S
, we are making a semantic judgment: that
x
is a valid instance of concept
S
. This decision must truthfully reflect the nature of
x
. If incorrect, the original semantics of
x
are not properly preserved—they are misrepresented. To mitigate this, proof chains can be introduced: each binding should be accompanied by verifiable justifications (e.g., threshold satisfaction, logical constraints). In cases of ambiguity, these records allow for retrospective inspection and adjustment.
Semantic Validity of Reasoning Rules:
On top of semantic units, reasoning rules (e.g., causal, hierarchical, computational) are defined. Semantic Mathematics requires that such rules respect semantic intuition. For example, if a rule infers
B
from
A
, it must reflect a genuine cognitive relation like “if A, then B.” If a rule is structurally correct but semantically invalid, it could yield absurd conclusions despite being logically sound. Therefore, knowledge graphs and rule bases should be constructed using axiomatized validation methods. Professor Duan advocates for rule transparency, where the semantic meaning and origin of each rule are explicitly recorded—enabling human-understandable AI inference (a principle explored in detail in Section 4.4).
Isomorphism in Cross-Layer Mappings:
Hook Theory posits that natural language and logical forms are mappable. If the system is to produce human-readable explanations, its internal inference steps should also make sense in linguistic terms. Semantic preservation here means that each formal reasoning step should correspond to a valid semantic inference in natural language, and vice versa. This is akin to semantic alignment between implementation and specification in programming. With Hook Theory, the system can maintain parallel reasoning chains: one in logic, and one in language. For instance, when the system uses an axiom “All birds can fly” to infer “Tweety can fly” from “Tweety is a bird,” it should also be able to produce the corresponding natural language explanation:
“Since Tweety is a bird, and all birds can fly, we conclude that Tweety can fly.”
This parallel structure ensures every machine inference is meaningful to humans, thereby preserving semantics.
The importance of semantic preservation lies in ensuring traceability and consistency. In complex cross-domain applications, information often undergoes many layers of transformation before yielding a final result. Without semantic preservation, biases or errors may be introduced midstream and go undetected—compromising the output. With preservation enforced, each step becomes semantically transparent and verifiable. For instance, graphical representations of the axiomatic system can be used to visually trace logic and semantics, helping users understand how semantic binding occurs internally.
In fact, Professor Duan’s Semantic Mathematics emphasizes a “white-box paradigm”—where the AI system’s internal concept representations and reasoning rules are exposed at a human-understandable semantic level. In such systems, semantic preservation is not merely desired but expected. Every step is transparent, and any violation of human-understood semantics would be immediately apparent. In contrast, black-box models may map semantics into abstract latent spaces, making it difficult to judge whether meanings are preserved. Semantic Mathematics, with its axioms and framework, offers the added value of tagging each computation with meaning labels that align with human semantic understanding.
In summary, this section has introduced the core axioms of Semantic Mathematics and formalized the process of semantic binding, while emphasizing the crucial role of semantic preservation in ensuring trustable inference chains. These theories and frameworks will be directly applied and tested in the following case-based chapters. Each case will begin with a real-world problem and demonstrate how the semantic axioms and binding framework are applied to solve it—ensuring correct and interpretable transmission of meaning at every stage. Through these cases, the power and applicability of Semantic Mathematics will become increasingly evident.
4 Cross-Domain Application Case Studies
To validate and demonstrate the practical utility of the aforementioned Semantic Mathematics theories and axiomatic system, this chapter presents four representative case studies across different domains. These cases cover Chinese orthography, philosophical concepts, natural language processing, and knowledge engineering. Each subsection will:
Define the problem background and challenges in traditional approaches;
Explain how Semantic Mathematics theory and axioms are used to construct a formal model;
Detail the process of semantic binding and inference;
Demonstrate inference chains and structural mappings to show how semantic preservation and rule transparency are achieved.
Through these case studies, we aim to better understand how Semantic Mathematics is applied in real-world domains and the advantages it offers.
4.1 Semantic Binding and Derivation in Chinese Characters (“日”, “明”, “時”)
Case Background:
Chinese characters are logographic symbols whose shapes often encode semantic components (radicals) or phonetic cues. Many characters are compound forms (ideograms or phono-semantic compounds) that implicitly carry rich semantic combinations. For example, the character “明” (bright) is composed of “日” (sun) and “月” (moon), and “時” (time, in traditional form “時”) is composed of “日” (sun) and “寺” (temple, or later “寸”). We aim to use Semantic Mathematics to formally explain the semantic binding and derivation processes behind these character constructions, showcasing how Conceptual Construction Theory and the axiomatic system apply to linguistic symbol analysis.
Traditional Challenges:
The relationship between character structure and meaning has long been studied in Chinese philology and semantics. Traditional analyses often rely on historical philology (e.g., oracle bone inscriptions) or subjective interpretation, which makes it difficult to formally express why, for instance, “日” + “月” = “明.” Additionally, there is no unified model to explain how semantic meaning transforms during radical combinations (e.g., semantic enhancement, metaphorical shift). We aim to provide a formal explanation using Conceptual Construction Theory and validate consistency through Semantic Mathematics axioms.
Semantic Mathematical Modeling:
We first define basic Chinese characters as semantic units:
S
日
: Semantic unit “日” (sun), representing daylight, brightness, or solar imagery.
S
月
: Semantic unit “月” (moon), representing night, dim light, or lunar imagery.
S
寺
: Semantic unit “寺,” originally symbolizing timekeeping tools or temples (in the character “時,” it may serve as both phonetic and symbolic).
S
寸
: Semantic unit “寸,” which appears in the simplified form of “時” and implies “short” or “small,” often connoting brief duration.
Next, we define conceptual construction rules to describe character formation semantics:
Ideographic Composition Rule: If components
A
and
B
are both semantic radicals, combining them into a new character
C
typically creates a semantic fusion—either through enhancement, intersection, or metaphor.
Phono-Semantic Composition Rule: One component
A
contributes semantic meaning, while component
B
contributes sound or class-based nuance. The meaning of
C
is mainly inherited from
A
, possibly modified by
B
.
“明” as an Ideographic Compound:
“明” is formed by combining “日” and “月” in parallel. Ancient scholars interpreted this as symbolic of compounded brightness—the sun and moon together create extreme illumination. This reflects semantic enhancement.
Φ(
S
明
)=
f
ideographic
(Φ(
S
日
),Φ(
S
月
))
Here,
Φ(
S
)
denotes the semantic feature set of unit
S
. For this case:
f
ideographic
({
brightness
},{
dim light
})={
very bright
}
This set-based representation models the combined meaning: “brightness” from the sun and “dim light” from the moon merge to form “extreme brightness.” According to Axiom 1 (Existence), this new semantic unit
{
very bright
}
must exist as
S
明
. This allows us to introduce a valid semantic class for this meaning.
By Axiom 2 (Uniqueness), any phenomenon
x
for which
ϕ
(
x
)
exhibits both sunlight and moonlight attributes should be bound to
S
明
. While no real-world phenomenon might naturally emit both simultaneously, the human conceptual system can synthesize such ideas. Thus,
S
明
functions as an abstract semantic unit, with members derived from imagined scenarios (e.g., a fusion of day and night light).
Axiom 3 (Transitivity) doesn’t apply directly to the combination of the two ideographs, but ensures closure in future semantic extensions. For example, suppose another character “??” is formed by combining “明” with a third radical indicating “greater brightness.” Axiom 3 would guarantee that the semantic chain from “日” and “月” to “??” remains logically coherent and transitive.
“時” as a Phono-Semantic Compound:
“時” combines “日” (semantic) and “寺” (phonetic and potentially symbolic). Classical texts say: “時，四時也。从日，寺声” (Time, referring to the four seasons. Formed from ‘sun’ and phonetic ‘temple’). “日” contributes temporal cyclic meaning; “寺” contributes pronunciation and symbolic meaning related to timekeeping (e.g., temple bells).
Φ(
S
時
)=
f
phono-semantic
(Φ(
S
日
),Φ(
S
寺
))
The function
f
phono-semantic
prioritizes the first argument (semantic radical), i.e., “solar cycles,” but may incorporate nuances from the second (e.g., symbolic timekeeping). Thus:
Φ(
S
時
)≈{
temporal flow
}
This concept reflects a higher level of abstraction than either “日” or “月”—shifting from concrete objects (sun, temple) to the abstract notion of time.
According to Conceptual Construction Theory, this reflects a semantic type shift: from representing objects to expressing abstract attributes. Cognitively, humans associate the sun’s motion and temple bells with time, so combining them evokes the concept of “time”. Professor Duan emphasizes the role of language in concept categorization. Many characters using “日” as a radical (e.g., 旦, 旭, 晏, 時) are associated with time and light—demonstrating semantic coherence in symbolic design.
Axiom 2 (Uniqueness) ensures that the concept of “time” isn’t redundantly represented by different symbols (e.g., “辰” may also signify time, but “時” already captures the “hour” concept—thus avoiding duplication).
Axiom 3 (Transitivity) ensures that if “日” and “寺” both relate to temporal concepts, their combination into “時” results in a semantically consistent unit—not a contradiction.
Inference Chain and Semantic Preservation
Given:
日
∈
S
日
(sunlight),
月
∈
S
月
(moonlight)
Inferred:
日
+
月
∈
S
明
(via ideographic function)
Validated: Neither
S
日
nor
S
月
alone captures the combined light concept.
Conclusion: A new, unique semantic unit is needed; duplication or contradiction is avoided by Axioms 1–3.
Semantic preservation is maintained throughout: meaning evolves from single light sources to a combined illumination, without semantic loss or distortion.
Observed: The sun moves over time; temples signal time with bells.
Abstracted: Both phenomena imply a common temporal dimension.
Modeled: Introduce semantic unit
S
時
derived from solar cycles and temple timekeeping.
Validated: Any phenomena involving solar movement or temple time should map to
S
時
, fulfilling Axiom 1 (coverage).
Ensured:
S
時
is the unique abstract time unit, avoiding fragmentation (Axiom 2).
Propagated: If “歲” (year) includes “日” and relates to time, then via Axiom 3, “歲” aligns with
S
時
, ensuring semantic closure across temporal concepts.
From a linguistic perspective, we can narrate these inferences:
“明” captures the joint brightness of sun and moon;
“時” abstracts the idea of time from daily cycles and bell chimes.
Every step has a natural language explanation, satisfying Hook Theory. In logic, the introduction of new concepts mirrors language-based term definition, maintaining transparency.
Using Semantic Mathematics, we formalized the semantic construction of Chinese characters “日,” “明,” and “時.” This validated the applicability of Conceptual Construction Theory and the axiomatic system to orthographic semantics.
Key advantages over traditional, intuition-based analyses:
Provides a consistent formal framework, extensible to other characters or lexical combinations;
Enforces semantic consistency, avoiding multiple symbols for one concept or multiple concepts for one symbol;
Enables computational modeling, with potential applications in intelligent word segmentation or semantic decomposition;
Demonstrates cross-modal integration, merging philology, cognition, and formal logic.
This case study illustrates the application of Semantic Mathematics in linguistic semiotics. In the next case, we shift to a more abstract domain—philosophical concepts—to explore how the theory handles meaning where no physical referents exist.
4.2 Semantic Axiomatization of Abstract Philosophical Concepts (“Existence”, “Unity”, “Causality”)
Case Background:
Many core philosophical concepts—such as existence, unity, and causality—are highly abstract and difficult to define precisely. Yet they are foundational to human cognition and scientific theory. In knowledge engineering and AI reasoning, representing and manipulating such abstract notions remains a key challenge. Traditional ontology engineering tries to axiomatize these concepts using formal logic (e.g., defining causality with logical rules), but such definitions often depend heavily on specific theoretical backgrounds and lack cross-domain generality. In this case, we apply Professor Yucong Duan’s Semantic Mathematics axiomatic system to analyze the semantics of these concepts, aiming for unified axiomatic representations and verifying their consistency and applicability.
Existence:
In philosophy, existence refers to the state or property of being. In logic, this is often denoted by the existential quantifier
∃
. Notably, Professor Duan places existence at the core of his semantic mathematics axiomatic system (Axiom 1), ensuring that every natural phenomenon maps to a semantic unit. Thus, “existence” plays a foundational role in semantic mathematics, serving as a guarantee that the semantic space is non-empty.
Unity / Oneness:
This refers to the identity, coherence, or wholeness of entities—treating parts as a single whole. In logic and set theory, this connects to uniqueness and identity, such as the Law of Identity
A
=
A
. Professor Duan’s Axiom 2 (Uniqueness) ensures that semantically equivalent elements are not redundantly represented, functioning as a formal mechanism to preserve conceptual unity.
Causality:
Causality denotes the relationship between causes and effects and is essential in both philosophy and science. It often entails transitivity, temporal order, and conditional dependence. Semantic mathematics seeks to express such constraints in a way that enables transparent causal reasoning in AI and knowledge graphs. Professor Duan has linked his framework to Aristotelian teleology and Hegelian dialectics, indicating a deep interest in the semantic representation of purpose and cause.
For each concept, we attempt to define a set of semantic axioms or constraints, interpreted through the DIKWP framework (Data–Information–Knowledge–Wisdom–Purpose), and examine whether these concepts share structural commonalities.
1. Semantic Axiomatization of “Existence”
In Professor Duan’s system, existence is captured by Axiom 1 (Existence):
A phenomenon
x
exists if and only if there exists a semantic unit
S
such that
x
∈
S
:
This aligns perfectly with the ontological view: existence is inclusion in a semantic category. If no semantic unit can encompass an entity, it is considered non-existent (at least from the system’s perspective). Thus, in semantic mathematics, existence equates to semantic representability.
Types of Existence in Semantic Mathematics:
Actual existence: There exists a semantic unit
S
with one or more observed instances. This corresponds to the Data and Information layers in DIKWP.
Possible existence: A semantic unit
S
exists, but currently has no instances. Still compliant with Axiom 1; semantic units may be created in anticipation (e.g., the historical concept of "aether" existed conceptually before empirical verification).
Independent existence: A unit
S
defined without relying on other units or not functioning as a subset of another unit. This may be measured via graph-theoretic centrality in concept networks.
Definition (Semantic Existence):
An element
x
exists iff it satisfies Axiom 1:
∃
S
:
x
∈
S
.
A concept
S
exists iff it is a recognized semantic unit, regardless of whether it has instances.
This definition makes existence computationally tractable—grounded in semantic assignment rather than metaphysical abstraction.
2. Semantic Axiomatization of “Unity / Oneness”
Unity refers to consolidating multiple elements into a coherent whole or maintaining non-redundancy in representation. In Semantic Mathematics:
Axiom 2 (Uniqueness) states that if
ϕ
(
x
)=
ϕ
(
y
)
, then
x
and
y
must belong to the same semantic unit
S
. This ensures conceptual unification.
From a philosophical perspective, this aligns with Hegelian dialectics (identity in difference) and Eastern unity (the oneness of all things). In Semantic Mathematics, we interpret unity as the operation of semantic equivalence and consolidation. Each application of Axiom 2—grouping
x
and
y
into the same unit—is a semantic unification.
If
ϕ
(
x
)=
ϕ
(
y
)
, then:
(
∃!
denotes “exists uniquely.”)
This is a direct consequence of Axioms 1 and 2. The theorem establishes a formal criterion for unity: check whether semantic features match or align closely enough. This is useful in entity disambiguation, taxonomy alignment, and ontology integration.
Additionally, structural unity (e.g., saying a system is “unified”) can be modeled as:
A semantic unit
S
whole
representing the whole,
A set of subunits
P
i
⊂
S
whole
,
Relational constraints
R
(
P
i
,
S
whole
)
ensuring that the whole’s properties emerge from its parts.
This can be expressed via set closure and projection relations, governed by semantic rules that maintain consistency (semantic preservation).
Thus, unity in Semantic Mathematics is supported not only by uniqueness axioms but also by categorical closure rules, contributing to cross-domain conceptual alignment.
3. Semantic Axiomatization of “Causality”
Causality is commonly expressed as
A
→
B
(“A causes B”). Traditional models distinguish between deterministic and statistical causality. Semantic Mathematics focuses on the meaningful interpretation of causal relations: under what semantic conditions can one event be said to cause another?
Proposed Semantic Causality Axioms:
C1 (Temporal Priority): If
A
causes
B
, then
A
must occur before
B
in time. This introduces a temporal attribute into event semantics.
C2 (Dependency Generation): If
A
causes
B
, then without
A
,
B
would not occur (ceteris paribus). Semantically,
B
includes a component or property derived from or dependent on
A
.
C3 (Transitivity): If
A
→
B
and
B
→
C
, then
A
→
C
. While debated philosophically, this is typically accepted in semantic networks for chain reasoning. Professor Duan’s Axiom 3 (Transitivity) can be invoked to ensure closure in causal chains.
Representation in Semantic Mathematics:
Causality can be modeled as a special semantic binding relation:
Define a semantic unit
S
causality
that encapsulates “cause–effect” as a compound concept.
Alternatively, define a binary relation
Cause
(
A
,
B
)
, with semantic content:
Meaning
(
Cause
)=
“A occurs, thereby causing B to occur”
Although Semantic Mathematics favors the former (relationships as special concepts), we use the latter for simplicity here.
Semantic transparency is critical: rules must be explicitly expressed and human-interpretable. For instance, the system may explain a rule application as:
“Fire ignites dry grass. Fire occurred. Therefore, dry grass ignited.”
Moreover, causality connects to purpose. Professor Duan's Purpose Computation and Reasoning (PCR) framework sees purposes as reverse causality—where an anticipated outcome guides prior action. Semantic Mathematics can accommodate such reasoning via teleological axioms, though they are beyond this section’s scope.
Example: Causal Chain and Semantic Validation
Statement: “Rain causes slippery roads.”
Let
A
=
Rain
,
B
=
Slippery Roads
A
→
B
satisfies C1 (Rain precedes slipperiness) and C2 (No rain → no slipperiness, under simplified assumptions).
We record
Cause
(
A
,
B
)
and include it in
S
causality
.
Then:
“Slippery roads cause accidents,” i.e.,
B
→
C
,
C
=
Traffic Accidents
.
Thus, by C3, infer
A
→
C
.
Semantic preservation is ensured: each step corresponds to intuitive reasoning (“Rain → Slippery → Accident”).
The system avoids invalid inferences (e.g., affirming the consequent) by checking whether all causality axioms are satisfied.
Using Semantic Mathematics, we analyzed the abstract philosophical concepts of existence, unity, and causality through the lens of semantic axioms:
Existence is formalized via Axiom 1: being semantically bindable equates to existence. This aligns ontology with set theory and computational models.
Unity is governed by Axiom 2: same semantic meaning → same semantic unit. This enables operations like deduplication and cross-ontology alignment.
Causality is captured via structured semantic relations and rules based on temporal, dependency, and transitivity axioms. It supports causal inference and transparent rule tracking.
These axiomatizations show the high-level reasoning potential of Semantic Mathematics. Unlike traditional logic-based formalizations, this approach emphasizes semantic intuition and bottom-up grounding from data and meaning. In the next section, we transition to natural language processing, exploring how syntax and semantics are bound within the Semantic Mathematics framework.
4.3 Binding Modeling Between Natural Language Syntax and Semantic Structure
Case Background:
One of the core challenges in natural language understanding is establishing a mapping between syntactic structure (syntax) and semantic structure (semantics). This issue is central to traditional tasks like semantic parsing and semantic role labeling. In a sentence, words are arranged in a hierarchical structure (e.g., syntactic trees or dependency relations), and the sentence conveys a meaning that can often be expressed as a logical form or semantic graph. Converting syntactic analysis into semantic representation has long been a research focus in linguistics and AI. While Montague semantics proposed using logic to represent sentence meaning, its rules are complex and difficult to scale. Here, we leverage Semantic Mathematics, particularly the Hook Theory and semantic binding framework, to analyze a simple sentence and demonstrate the binding process from syntax to semantics.
Example Sentence:
Take the simple sentence:
"小明喜欢吃苹果。" ("Xiaoming likes to eat apples.")
This sentence contains a subject (“Xiaoming”), a predicate (“likes to eat”), and an object (“apples”). The syntactic structure is:
Subject [Xiaoming] + Predicate [likes to eat] + Object [apples]
The verb phrase “likes to eat” is a compound verb structure. For simplicity, we treat “likes to eat apples” as the predicate describing Xiaoming’s preference.
Concept mapping of words: “Xiaoming” must be linked to a person entity; “apples” to a fruit class in the ontology.
Verb phrase semantics: The composite verb “likes to eat” must be decomposed into attitude and action.
Latent implications: Compound verbs often express habitual preference rather than a one-time action—subtleties not apparent in surface syntax.
Syntactic ambiguity: While this sentence has none, complex sentences often do; semantic reasoning is needed to resolve them.
Semantic Mathematics Modeling:
Using Hook Theory, we aim to “hook” the syntactic structure to a semantic structure, which can be represented as a semantic graph or logical form. For “小明喜欢吃苹果” (“Xiaoming likes to eat apples”), a suitable semantic representation could be:
Xiaoming → bound to a person entity concept
S
Xiaoming
Apples → bound to the fruit concept class
S
Apple
LikesToEat → represents a preference relation or habitual action.
We form a mini semantic graph:
This can also be represented logically as:
LikesToEat
(
Xiaoming
,
Apple
)
Binding words to semantic units:
“Xiaoming” is recognized via named entity recognition and linked to a person entity
S
Xiaoming
; if not found in the knowledge base, we create one using Axiom 1 (Existence). Axiom 2 (Uniqueness) ensures it refers to a unique person.
“Apple” is recognized as a generic fruit category and linked to the semantic unit
S
Apple
. Since the sentence expresses general preference, it refers to the class, not a specific apple instance.
“Likes to eat” is a compound verb. We may split it into “like” (attitude) and “eat” (action). Traditional semantics might model this as:
LikesTo
(
Xiaoming
,
Eat
(
Xiaoming
,
Apple
))
In Semantic Mathematics, Conceptual Construction Theory allows us to model “likes to eat apples” as a composite concept or as a high-level relation
R
LikesToEat
. For simplicity, we treat it as a single semantic relation.
Syntax-to-semantics mapping rules:
According to Hook Theory, we define a mapping function
M
:
M
(
[Subject NP]
)=
Entity Concept
M
(
[Predicate VP]
)=
Relation/Action Concept
M
(
[Object NP]
)=
Entity Concept
M
(
Xiaoming
)=
S
Xiaoming
M
(
Likes to eat
)=
R
LikesToEat
Forming a semantic triple:
The semantic representation becomes:
(
S
Xiaoming
,
R
LikesToEat
,
S
Apple
)
This is the result of semantic binding. According to the axioms:
Axiom 1: All entities and relations are ensured to exist in the semantic space.
Axiom 2: Ensures consistent references to “Xiaoming” and “apple” across sentences.
Axiom 3: Comes into play in compound sentences, e.g., “Xiaoming likes to eat apples and apples contain vitamins.” The word “apples” links the two clauses, and semantic consistency is maintained through transitivity.
Semantic explanation generation:
Using the inverse mapping
M
−1
, we can convert the semantic triple back into a natural language sentence:
“Xiaoming likes to eat apples.”
We can also express it in logical form and verify its truth conditions using formal semantics. This mutual convertibility demonstrates the Hook Theory effect and ensures semantic preservation.
Nested Structures:
We simplified “likes to eat” as a flat relation. A more refined approach would introduce a sub-event node
E
for “eating apples,” and then express:
Likes
(
Xiaoming
,
E
)
where
E
=
Eat
(
Xiaoming
,
Apple
)
Though this adds complexity, it improves granularity. Semantic Mathematics allows such flexibility in modeling granularity, as long as all expressions map back to basic semantic units.
Ambiguity Resolution:
Though our example has no ambiguity, sentences like “The pilot saw the goat in the cave” require semantic context to resolve ambiguity (Who is in the cave?). Semantic Mathematics supports knowledge-driven disambiguation, leveraging the DIKWP model's Knowledge and Wisdom layers. For example, pilots are unlikely to be in caves; goats are. Thus, the goat is more likely to be in the cave. This sort of reasoning may involve contextual AI or rule-based systems and goes beyond this section's scope.
Formal Validation:
To verify binding correctness, we can check whether:
LikesToEat
(
Xiaoming
,
Apple
)
aligns with existing knowledge. For instance, if the knowledge base contains “Xiaoming likes fruit” and “Apple is a fruit,” we can deduce that Xiaoming likes apples, using transitivity. This is referred to as theorem verification in Professor Duan’s framework.
Semantic preservation is achieved through Hook Theory’s mapping. In the closed loop:
Syntactic structure → Semantic structure → Logical reasoning → Natural language,
each step maintains meaning correspondence:
“Xiaoming” remains correctly bound to a human entity;
“Likes to eat” preserves its subjective preference semantics;
“Apple” is clearly identified as the fruit, not Apple Inc.—context guides disambiguation.
If ambiguity is detected, semantic binding must delay execution until disambiguation is resolved. Otherwise, Axiom 2 (Uniqueness) would be violated. In practice, this might require user input or AI-supported context resolution.
This case demonstrated syntax–semantic binding for a simple sentence using the Semantic Mathematics framework. We achieved:
Automatic conversion from syntactic parse to semantic triple using semantic units and mapping rules—similar to traditional NLU parsing but governed by an axiomatic system.
Semantic fidelity and transparency through rule-based correspondence, applying Hook Theory to link syntax with semantics in an interpretable manner.
Prepared ground for further reasoning: the structured output can be integrated into knowledge graphs or rule systems (e.g., answering “What does Xiaoming like to eat?” → “Apples”).
This case highlights the potential of Semantic Mathematics in NLP:
It provides a principled, axiomatic approach to semantic parsing—normative, verifiable, and deeply integrable with knowledge systems—far beyond the opaque behavior of statistical black-box models.
In the next and final case, we explore how Semantic Mathematics enables rule transparency and logic visualization in engineering-oriented knowledge graph construction.
4.4 Rule Transparency Construction and Logical Visualization in Engineering Knowledge Graphs
Case Background:
Knowledge Graphs (KGs) are one of the most prominent AI paradigms for representing knowledge, widely applied in search engines, question-answering systems, and decision-support platforms. However, many KGs are constructed either through statistical extraction or manual curation. Domain-specific rules and business logic are often not explicitly represented within the graph or are buried inside code or complex reasoning engines—opaque to both humans and machines. In practice, this leads to a key engineering challenge: how to make rules not only part of the KG but also transparent, inspectable, and logically visualizable. Within the Semantic Mathematics framework, Professor Yucong Duan proposes the concept of rule transparency, aiming to expose internal reasoning and decision-making rules of AI systems clearly at the semantic level.
We use a medical knowledge graph as a running example. The graph includes:
Entities: Symptoms (e.g., Fever), Diseases (e.g., Flu), Medications (e.g., Acetaminophen).
Symptom → Disease (a symptom may indicate a disease)
Disease → Medication (a disease may be treated with a drug)
Diagnostic rule: If a patient has a certain symptom set, infer a possible disease.
Treatment rule: If a disease is confirmed, recommend a treatment drug.
Traditionally, these rules are embedded in IF-THEN clauses or reasoning engines (e.g., OWL or production rules), and the reasoning process is hidden. For example, the rule “If fever and cough, then possible flu” is rarely visible in the KG—it resides in application logic.
Semantic Mathematics promotes the idea that rules should become first-class citizens in the KG, stored in explicit forms so that the logical reasoning process becomes visually traceable and semantically inspectable.
Semantic Mathematics Modeling:
1. Axiomatized Rule Representation
Rules are modeled as special knowledge units. Each rule can be treated as an entity node (or using reification or n-ary relations, given KG's binary relation limits).
Example: Define a rule node
R
1
for:
This rule can be structured in the KG as:
R
1
-- hasPremise → {Fever, Cough}
R
1
-- hasConclusion → Flu
Or simplified using a ternary representation:
(
Fever AND Cough
,⇒,
Flu
)
Here, "Fever AND Cough" may be treated as a composite node or a labeled structure. This transforms a logical formula into an ontological component, making the rule visible within the KG. According to Hook Theory, this triple corresponds directly to the natural language:
“Fever and cough implies flu.”
2. Ensuring Semantic Axiom Compliance:
Axiom 1 (Existence): The combination
S
Fever \& Cough
is a valid semantic unit and must exist to represent the conjunctive symptom state.
Axiom 2 (Uniqueness): Ensures that no duplicate representations (like “SymptomGroup1”) are used for the same symptom combination.
The relation “
⇒
” is modeled explicitly, semantically encoding inference or causal implication.
“Flu” serves as the conclusion concept in this logical chain.
3. Binding and Reasoning Chain Visualization:
Now, apply this rule to a patient case:
(Patient123, hasSymptom, Fever)
(Patient123, hasSymptom, Cough)
In traditional systems, a rule engine would apply the rule to infer:
→ (Patient123, likelyHas, Flu)
In Semantic Mathematics, we want to graphically expose the entire reasoning process:
Create a node
S
PatientSymptoms
representing the patient’s symptom combination.
(
S
PatientSymptoms
,
instanceOf
,
S
Fever \& Cough
)
This semantic binding step aligns the specific instance with the general rule pattern.
Axioms 2 and 3 ensure semantic equivalence: individual “Fever” and “Cough” concepts from the patient case are equivalent to those in the rule, and the combination is treated consistently.
S
PatientSymptoms
∈
S
Fever \& Cough
The reasoning chain becomes visible as:
Patient123
→
S
Fever \& Cough
⇒
Flu
Conclusion: The system can then add:
(
Patient123
,
likelyHas
,
Flu
)
[source
=
R
1
]
This preserves transparency: clicking on the triple reveals,
“Generated based on Rule R1 (Fever AND Cough ⇒ Flu)”.
Thus, reasoning becomes a visible, traceable path, not a hidden engine log.
4. Semantic Preservation and Logical Consistency:
Semantic preservation: The conclusion arises only if the patient’s symptoms match a rule; not from probabilistic guesses. This guarantees semantic and logical consistency.
Axiom 3 (Transitivity) enables multi-hop reasoning chains:
Suppose:
Patient123
⇒
Flu
⇒
Tylenol
We infer treatment recommendation:
(
Patient123
,
mayUse
,
Tylenol
)
[source
=
R
1
+
R
2
]
Symptoms
→
Disease
→
Medication
forms a transitive inference path, ensuring consistency across rule applications. Semantic Mathematics ensures such chains are closed, non-contradictory, and clearly structured.
5. Rule Conflict and Correction:
What if rules conflict?
Because rules are graph components, conflicts can be detected using SAT-solving or cycle detection. For example:
Creates a contradiction:
A
→
B
→¬
A
The system can automatically flag such inconsistencies, enabling knowledge engineers to revise or delete problematic rules.
Graphically, the KG may include:
Symptom nodes: Fever, Cough
Composite node: Fever&Cough
Rule node or relation:
⇒
Flu
Color-coding or iconography can distinguish:
This creates a semantic logic flowchart embedded within the KG. In Professor Duan’s technical reports, such automated “proof tree” visualizations help users and developers understand why the system reached a conclusion—crucial for trust in AI-assisted healthcare.
By incorporating Semantic Mathematics into engineering knowledge graphs, we achieve:
Explicit rule representation: Domain rules are lifted from code into KG structure, achieving rule transparency.
Semantic integrity: Axiomatic enforcement ensures valid concept construction and reasoning chains.
Graphical traceability: Symbolic logic becomes visible as paths in the KG—inference is no longer invisible.
Cross-domain integration: The KG structure supports various domain rules, and semantic axioms detect and resolve conflicts, facilitating unified knowledge systems.
This case exemplifies the engineering paradigm of Semantic Mathematics:
Knowledge representation encompasses not just static facts, but also dynamic rules;Reasoning becomes co-present with knowledge, visible and verifiable.
It bridges the gap between logic and data, fulfilling Professor Duan’s vision of a “white-box AI”.
Across Sections 4.1 to 4.4, we have demonstrated how Semantic Mathematics and its axiomatic system consistently support diverse domains. Next, we synthesize the shared challenges across these cases and assess the framework’s strengths in cross-domain integration, theoretical completeness, and engineering feasibility.
5 Symbol-Meaning Interaction Between Mathematical Logic and Language Expression (Hook Theory Analysis)
Throughout the preceding theoretical discussions and case studies, the idea of "Mathematical Logic–Language Hook Theory" has been implicitly present: whether in conceptual definition, syntactic-semantic binding, or rule transparency, we consistently strive to synchronize the formal symbolic system with the natural language and cognitive semantic system. In this section, we explicitly analyze how symbols and meanings interact from the perspective of Hook Theory, and how Semantic Mathematics is designed to facilitate such interaction. This deepens our understanding of the methodological uniqueness of Semantic Mathematics and offers insights into the development of cross-modal AI systems.
5.1 Bidirectional Mapping Between Symbols and Meaning
Traditional AI systems often treat logical reasoning and natural language processing as separate modules in a pipeline: first parse natural language into logic symbols, then perform symbolic reasoning, and finally generate textual output through templates. The drawback of this pipeline is obvious: an error in parsing ruins the entire chain, and even correctly derived logical results may fail to translate into fluent or understandable language.
Hook Theory promotes synchronous interaction between symbol and meaning, treating language understanding and logical reasoning as two facets of a unified process. This requires a robust bidirectional mapping mechanism between symbols and meaning, denoted earlier as
M
and
M
−1
.
Semantic Mathematics realizes this through:
Bilingual representation at the conceptual layer:
Each semantic unit contains both a formal identifier and associated linguistic label or definition.
For example, concept
S
Apple
may contain:
Formal ID: Apple123, Name: "苹果", Definition: "A common fruit...".
Thus, the concept can be referenced by code (via ID) and understood by humans (via label and description).
The axiomatic system guarantees uniqueness, ensuring that each concept has one definition and one name, which makes
M
−1
(symbol → language) a deterministic mapping.
Synchronization between symbolic reasoning and linguistic explanation:
Semantic Mathematics mandates that each reasoning step can be expressed in natural language, in a way that aligns with human logical intuition.
This is achieved by associating each rule
R
with a natural language explanation
L
R
. For example, the rule:
Fever
∧
Cough
⇒
Flu
can be paired with:
“If a person has fever and cough, they may have the flu.”
When a reasoning engine applies
R
, the system retrieves and instantiates the explanation template with actual entities.
A complete reasoning chain becomes a concatenation of human-readable explanations, one per step—each symbolically inferred step is linguistically mirrored, avoiding disconnect between symbolic and semantic worlds.
Language-triggered symbolic operations:
Hook interaction isn't always symbol-driven; it can be language-initiated.
When a user inputs a natural language query, the system interprets it into a logical query.
Within the Semantic Mathematics framework, this is treated as a semantic binding process, governed by axioms and verifiable.
Conceptually, the user projects their language into the shared semantic space; the system performs symbolic reasoning within this space and returns an answer that is re-projected into natural language.
Professor Duan notes:
“A person can only communicate with themselves,”
but through a shared semantic space, two people (or a human and a machine) can align understanding via hooked meanings.
In essence, human-machine interaction becomes semantic interaction between two processors, mediated by a common semantic substrate.
5.2 Advantages of Symbol-Meaning Interaction
Hook Theory’s emphasis on symbol–meaning interaction is not just for elegance—it addresses real-world AI challenges:
Interpretability:
Each symbolic inference step is linguistically mirrored, making it explainable to users and developers.
In the medical KG case (Section 4.4), we used natural language annotations to explain inference paths, enabling medical professionals to verify reasoning outcomes, thus building trust.
Robustness:
If a symbol cannot be translated into a semantically valid sentence, anomaly is likely.
Hook Theory provides a consistency check: whenever a symbolic computation is made, its linguistic projection should be meaningful.
Failure to generate a sensible sentence can alert the system to misused rules or incorrect data.
Interactivity:
Unlike black-box models, Hook-enabled systems can interact with users at each reasoning step using natural language.
For example, in a dialog system, the AI can verbalize its intermediate inferences, asking the user for confirmation or additional input.
When the user responds, the system translates it back into symbols and adjusts its reasoning.
This collaborative reasoning loop is a pillar of explainable AI, and Semantic Mathematics is well-suited to power such systems, as every step has semantic annotations that humans can understand.
Unified modeling across domains:
Symbol–meaning interaction facilitates cross-domain integration, since much of human knowledge exists in linguistic form, while theory often uses symbols.
For example, in legal AI, connecting legal texts (language) with regulatory logic (symbol) is a symbol-meaning binding problem.
Semantic Mathematics can parse legal language into formal logic, reason over it, and re-express conclusions in natural language, aligning law KGs and case reasoning systems seamlessly.
Lowering the barrier to entry:
One core goal is to allow non-programmers (e.g., domain experts) to write and debug rules in natural language—i.e., to define explainable strategies.
Symbol–meaning interaction reduces the entry barrier for knowledge engineering, allowing experts to input structured language rules that the system automatically translates into symbolic axioms.
When reasoning fails, the expert reads the natural language explanation and revises the faulty rule.
This bilingual process is vastly more intuitive than writing raw logic code.
In summary, Hook Theory formalizes the interaction between mathematical logic and natural language, and Semantic Mathematics builds this into its foundations. The result is a system where symbols and meanings co-evolve—reasoning is not only machine-computable but also human-comprehensible. This is essential for building trustworthy, transparent, and collaborative AI systems.
5.3 Manifestations of Hook Theory in Semantic Mathematics
Reflecting on the content covered thus far, we can summarize several concrete ways in which Hook Theory is embodied within the Semantic Mathematics framework:
Human-Readable Semantic Axioms:
The axioms designed by Professor Yucong Duan are intuitively aligned with semantic understanding and can be readily expressed in natural language.
Existence: Every phenomenon has a semantic reference.
Uniqueness: Equivalent meanings should be unified.
Transitivity: Consistency is maintained within a semantic unit.
These axioms are unlike the opaque ones found in traditional mathematics; their readability suggests that Hook Theory was considered from the outset, ensuring that mathematical formulations and linguistic interpretations correspond one-to-one, making them easier to comprehend.
Dual-System Expression of Core Terminology:
This report frequently uses terms like “concept” and “semantic unit”, which reflect a dual-hook design:
“Concept” resonates with common human cognitive vocabulary;
“Semantic unit” fits formal system structure.
Professor Duan deliberately uses these complementary terms to ensure clarity for audiences from different disciplines. For example, the “Axiom of Existence” simultaneously appeals to:
Mathematicians (who recognize existential quantifiers
∃
), and
Philosophers (who associate it with ontological existence).
This is the elegance of Hook Theory: a dual-layered formulation that bridges formalism and intuition.
Hook-Structured Case Presentation:
In the case studies, especially in Section 4.2 on abstract philosophical concepts, we deliberately presented both symbolic and linguistic representations.
For example:
was paired with natural language explanations. This dual format serves as a practical implementation of Hook Theory:
Symbol-level definitions can be interpreted by logic engines.
Language-level definitions are accepted by philosophers.
Their alignment ensures the clarity and rigor of the concept being modeled.
Tool-Integrated Theory:
Semantic Mathematics is not merely theoretical—it informs tools that embody Hook Theory.
For example, in semantic network construction tools inspired by Professor Duan’s framework, users should be able to input natural language descriptions of concepts and relationships, and the system should automatically generate formal representations.
The proposed “Semantic Modeling and Verification Platform” would likely include such functionality.
One could imagine a development environment where:
The left panel accepts semi-natural language rule inputs;
The right panel generates axiomatized formal logic expressions in real-time.
This side-by-side “mirror” implementation would be a powerful manifestation of Hook Theory in practice.
In essence, Hook Theory provides the philosophical foundation for interaction between symbols and meanings, and Semantic Mathematics operationalizes it through practical mechanisms, embedding it into AI systems so that logic and language mutually reflect and validate each other.
With the rise of Large Language Models (LLMs), we are already witnessing a weak form of Hook Theory in action:
LLMs process vast quantities of symbols (text) to form internal “world models.”
However, their reasoning processes remain opaque.
Semantic Mathematics could provide LLMs with a symbolic interface, enabling their natural language inferences to be paired with symbolic reasoning engines—thus transforming LLMs from black boxes into Hook-enabled white-box systems.
Though this is beyond the scope of the current discussion, it highlights the far-reaching potential of Hook Theory.
In the next section, we turn to macro-level challenges in applying Semantic Mathematics to cross-domain integration, including semantic consistency, axiomatic completeness, and engineering feasibility.
6 Semantic Consistency, Axiomatic Completeness, and Operability in Cross-Domain Integration
One of the core goals of Semantic Mathematics is to construct a unified knowledge system capable of integrating semantics across disciplines and modalities. In pursuing this ambition, several critical challenges must be addressed:
How can semantic consistency be maintained in large-scale, heterogeneous knowledge environments?
Can the axiomatic system of Semantic Mathematics remain complete and non-contradictory as it scales?
How can the theory be operationalized into engineering practice?
This chapter analyzes these issues and offers reflections and recommendations based on current progress.
6.1 Maintaining Semantic Consistency Across Domains
Semantic consistency means that after integrating knowledge from different domains and sources, the meanings of concepts and their relations do not conflict or become ambiguous. In cross-domain integration, inconsistent term usage is common:
Polysemy (same word, different meanings): e.g., “tree” in botany vs. computer science vs. genealogy;
Synonymy (different words, same meaning): e.g., “car” and “automobile”;
Conceptual scope conflicts: e.g., “charge” in physics vs. finance; legal vs. medical definitions of “responsibility.”
Traditional solutions rely on manual alignment or domain-restricted usage. Semantic Mathematics, however, provides a more systematic and potentially automated framework to manage consistency:
Application of the Uniqueness Axiom:
The Uniqueness Axiom ensures that one meaning maps to one symbol.
During integration, this axiom can detect redundant synonyms.
If two concepts
S
a
and
S
b
have equivalent extracted features (i.e.,
ϕ
(
S
a
)=
ϕ
(
S
b
)
), they should be merged into a single semantic unit.
For example, if “Automobile” and “Car” both yield
ϕ
={
four wheels
,
engine
,…}
, they should be unified.
In engineering, this is equivalent to synonym normalization.
Conceptual Conflict Detection:
If two concepts
S
x
and
S
y
appear different but their semantic features are identical or highly similar—and they belong to incompatible superclasses—this signals a semantic conflict.
Example: One system defines “bird” to exclude penguins (because they don’t fly); another includes them.
If both are merged, and
S
penguin
∈
S
bird
and
S
penguin
∉
S
bird
coexist, a contradiction occurs.
Semantic Mathematics detects such contradictions by checking violations of transitive closure.
Resolution may require manual intervention, introducing subtypes like “flightless birds.”
Contextual Layering:
Often, the same term has different meanings in different domains.
Semantic Mathematics supports contextual semantic units:
e.g., and , each within distinct upper classes.
They may still share cross-domain features like numerical value handling.
Naming and structuring can include domain labels, maintaining internal uniqueness without cross-context merging, unless justified.
Ongoing Calibration:
Since knowledge evolves, consistency requires continuous calibration.
Professor Duan’s DIKWP model emphasizes networked cognition, suggesting cross-domain knowledge should be calibrated via relational networks, not isolation.
This supports periodic consistency-checking algorithms (e.g., ontology consistency checkers), flagging violations of uniqueness or transitivity for human review.
This is common in description logics and fully compatible with the Semantic Mathematics framework, albeit with a more semantically intuitive interpretation.
Semantic Middleware (“Middle Platform”):
From an engineering standpoint, a cross-domain semantic hub (or middleware) can be built.
All domain concepts map to central semantic units, and then are redistributed to applications.
The axioms (especially Existence and Uniqueness) function as a central axis:
each concept that enters the middleware is uniquely positioned and governed.
This hub-and-spoke model helps dissolve data silos and ensures semantic interoperability across systems.
6.2 Axiomatic Completeness and Extensibility
Axiomatic completeness has two dimensions:
Theoretical completeness: Can the existing axioms support all major semantic phenomena?
Formal completeness: Is the axiomatic system logically coherent, and can it derive the necessary conclusions?
Professor Duan has explicitly proposed three fundamental axioms—Existence, Uniqueness, and Transitivity—which serve as core pillars of Semantic Mathematics. These axioms suffice to model equivalence-class structures and support key theorems such as the Identity Theorem and Binding Stability Theorem.
However, the complexity of the semantic world suggests the need for additional axioms or theorems:
Hierarchy Axioms:
To handle class hierarchies and inheritance.
Example:
Axiom 4: “All members of a class inherit the class’s defining properties.”
(Analogous to OWL class inclusion constraints)
Compositionality Axioms:
For defining how meanings of parts contribute to the whole.
Example:
If a concept is composed of parts, its existence depends on the existence of its parts.
(Whole-part existential dependencies)
Contextual Axioms:
To formalize domain-dependent uniqueness.
Example:
“Within context
C
, a concept must have a unique symbolic representation—though multiple representations may exist across contexts.”
Evolution Axioms:
To describe changes in knowledge over time.
Example:
“If a concept exists at
t
1
but not at
t
2
, a termination event occurred between
t
1
and
t
2
.”
(Useful in modeling dynamic semantics or time-aware systems.)
Cognitive Axioms:
To handle imprecision, incompleteness, and inconsistency (the 3-No problems).
Example:
“Every knowledge claim has a non-zero probability of being incorrect.”
Such uncertainty axioms help introduce non-binary reasoning, bridging the gap between real-world knowledge and formal systems.
Evaluating the Current Core Axioms
The three core axioms define semantic equivalence classes, sufficient for concept classification.
However, advanced reasoning tasks—such as causality and rule chaining—require additional axioms (like C1–C3 for causality).
Thus, Semantic Mathematics should be viewed as an open, extensible axiomatic system.
New axioms can be added as needed by applications, provided they:
Do not conflict with existing axioms;
Remain semantically intuitive;
Adhere to the system’s constructivist foundations.
Formal Logical Properties
From a formal logic perspective, it is important to ensure:
Consistency: No contradictions exist within the axioms.
Independence: No axiom can be derived from others (unless intentionally defined that way).
Currently, Existence, Uniqueness, and Transitivity resemble the axioms for equivalence relations, which are:
Consistent (no contradictions);
Independent (none can be derived from the others).
Future extensions—especially those involving quantifiers, modalities, or temporal logic—may change the logical properties of the system.
Therefore, any new axiom must be tested for logical soundness, deductive strength, and computability—avoiding issues like undecidability or non-termination.
In summary, maintaining semantic consistency, ensuring axiomatic extensibility, and supporting engineering operability are all essential to Semantic Mathematics achieving its goal of building a unified, cross-domain semantic infrastructure. The framework’s strength lies not only in its theoretical elegance, but also in its semantic transparency, rule-based reasoning, and modular extensibility—a design well-suited to the future of explainable and integrative AI systems.
6.3 Engineering Operability Challenges
For Semantic Mathematics to be widely applied in real-world AI systems, several engineering challenges must be addressed:
Computational Complexity:
Introducing a semantic layer often incurs additional computational overhead.
For example, ensuring uniqueness requires continuously checking for semantic equivalence and merging nodes—an operation akin to ongoing synonym resolution, which becomes costly in dynamic, growing knowledge bases.
Similarly, transitivity demands closure operations, which can cause combinatorial explosion in reasoning.
The challenge is to ensure semantic correctness while improving algorithmic efficiency.
Possible solutions include using incremental or approximate algorithms, allowing the system to maintain core consistency in real time and defer full consistency checks to off-peak periods.
Large-Scale Knowledge Processing:
Although the Semantic Mathematics framework is theoretically scalable to internet-scale knowledge, storage and querying require optimization.
Modern knowledge graph technologies offer distributed storage and parallel query capabilities, but with the added complexity of axioms and rule-based reasoning, how do we scale to billions of nodes?
This may require combining with database optimization techniques, such as pushing axiomatic constraints into the query execution layer.
It might also necessitate developing a dedicated Semantic Mathematics reasoning engine, specifically optimized for Axioms 1–3 and any extended axioms, rather than relying on generic inference engines.
Integration with Existing Standards:
The current ecosystem for knowledge representation includes standards like RDF/OWL, various ontology languages, and rule languages like SWRL.
For Semantic Mathematics to gain adoption, it must interoperate with or map to these existing standards.
Fortunately:
Axiom 1 (Existence) can map to RDF’s rdf:type asserting class membership.
Axiom 2 (Uniqueness) can be implemented using owl:sameAs and disjointness axioms.
Axiom 3 (Transitivity) aligns with equivalence relations.
However, OWL lacks native support for logic rule definitions—this would require RuleML or other extensions.
Semantic Mathematics might eventually define its own language (e.g., DIKWP-ML), but a more realistic approach is to build a layer on top of existing standards:
Allow development within the Semantic Mathematics framework;
Or import existing OWL ontologies and enhance them with semantic axiomatic features.
Integrating tools and standards is an unavoidable step for engineering adoption.
Human–Machine Collaboration Workflows:
While Semantic Mathematics aspires to white-box interpretability, end users and domain experts are often unfamiliar with formal reasoning paradigms.
Thus, it is crucial to design user interfaces and collaborative workflows that balance transparency and usability.
For instance, in a rule-transparent system, if the AI surfaces verbose explanations after every inference, users (e.g., physicians) may find it overwhelming.
Interaction design is key:
Only provide explanations when results are uncertain;
Or allow users to request explanations on demand.
While Semantic Mathematics provides the foundation, how it connects to the UI/UX layer must be jointly designed with interaction experts.
Practical implementation must be technically correctandhuman-friendly.
Knowledge Acquisition:
The success of Semantic Mathematics heavily depends on the quality of knowledge and rules it operates on.
However, knowledge acquisition remains a major bottleneck.
A promising direction is to combine LLM-based automatic extraction with semantic mathematics-based validation.
For example:
Use GPT-style models to extract candidate rules or concept definitions from text;
Then pass them to the Semantic Mathematics engine for consistency checks;
Highlight contradictions for human refinement.
This human-AI co-creation workflow could alleviate the tension between slow manual modeling and low-quality machine-generated knowledge.
This direction is not just promising—it is crucial for ensuring operability by enabling sustainable, high-quality knowledge supply.
Talent Training:
Implementing Semantic Mathematics in engineering contexts requires teams proficient in symbolic logic, domain expertise, and engineering implementation.
Currently, such interdisciplinary talent is scarce.
There is a need to develop education and training programs:
Teach ontology engineers the axiomatic philosophy of Semantic Mathematics;
Equip software engineers with skills in knowledge graphs and symbolic logic;
Involve domain experts in collaborative knowledge modeling.
This is both an organizational and talent development challenge.
Without solving it, even the most elegant theories may remain unused.
As demands for explainable AI grow, we hope more people are motivated to learn and apply these ideas.
To make Semantic Mathematics operable in engineering practice, we must advance on multiple fronts:
Optimize algorithms to balance semantic precision and computational cost;
Integrate with existing standards for compatibility and ease of adoption;
Design intuitive human–machine interaction flows that respect user experience;
Develop hybrid knowledge acquisition pipelines combining automation with expert correction;
Invest in talent development to build capable interdisciplinary teams.
This will be an incremental evolution, not an overnight transformation.
But as black-box AI increasingly draws criticism, Semantic Mathematics, as a white-box paradigm, offers a promising future.
If these challenges are addressed, it could become a cornerstone of next-generation AI system design.
7 Vision of a Unified Knowledge Entity and Prospects for Semantic Axiom Expansion
Through the previous discussions, we have seen the immense theoretical and practical potential of the Semantic Mathematics axiomatic system, as well as challenges and areas for refinement. This chapter looks boldly toward the future, envisioning a blueprint for a Unified Knowledge Entity (UKE) and proposing several possible extensions and enhancements to the theory and axioms.
7.1 Blueprint for a Unified Knowledge Entity
Imagine a future in which we build a Unified Knowledge Entity (UKE)—a vast system integrating knowledge from all domains, with self-evolving capabilities and powerful reasoning functions. This system would function as a massive hybrid of a knowledge graph, reasoning engine, and interactive interface, able to:
Answer interdisciplinary questions;
Design cross-domain solutions;
Discover knowledge gaps and ask for inputs to fill them.
Semantic Mathematics can serve as the foundational architecture for this UKE:
Unified Semantic Space:
At the core of UKE is a semantic space where all inputs (from text, sensors, databases, etc.) are mapped to semantic units, each with a clear definition and position within the broader knowledge network.
Because all modalities and sources are processed through Axioms 1–3, everything exists within a coherent system, eliminating semantic silos.
The five layers of DIKWP—Data, Information, Knowledge, Wisdom, Purpose—are all represented in this unified structure.
Globally Unique Concept Repository:
The Uniqueness Axiom (Axiom 2), alongside its future extensions, ensures the UKE maintains a global concept repository.
It aligns synonyms, differentiates polysemes, and continually absorbs new concepts.
All domains—science, engineering, arts, daily knowledge—are integrated into a semantic concept universe, where each theory is simply a local view within the global structure.
Rules and Models in Harmony:
UKE stores both symbolic rules (e.g., expert knowledge, laws) and statistical models (e.g., neural networks).
Semantic Mathematics acts as a semantic interface layer between them:
Models input/output mapped to semantic units;
Rules applied at the semantic unit level.
For example:
A neural network identifies a cat in an image → outputs a probability for "cat" → Semantic Mathematics links the "cat" concept to ontology → rules like "cats are mammals" become immediately applicable.
Conversely, if reasoning needs perceptual input (e.g., “is there a cat in the image?”), the system invokes the model and injects the result into the reasoning chain.
Using Hook Theory, UKE seamlessly blends symbolic reasoning with model-based perception, maintaining interpretability—superior to black-box or purely symbolic systems.
Self-Evolution:
With a unified semantic space, UKE can monitor its own knowledge gaps or contradictions.
When a question cannot be answered, it identifies missing or conflicting concepts and proactively seeks external input.
Example:
If UKE is asked about “quantum entanglement” but lacks its definition, it can ask:
“Please provide a definition of quantum entanglement.”
Once a user provides it, the system integrates and aligns it with existing physics concepts.
Since Semantic Mathematics mandates that each new concept be anchored in meaning, this allows continuous, structured knowledge fusion—not chaotic accumulation.
Fully Explainable Interaction:
To users, UKE can seamlessly switch between symbolic and natural language explanations.
Every inference step is linguistically traceable thanks to stored mappings and annotations.
This fosters:
Understanding of AI by humans;
Trust and transparency for decision auditing;
Even rule revision by experts.
UKE becomes a living encyclopedia and cognitive mentor, not a mysterious black box.
In summary, the Unified Knowledge Entity is the practical embodiment of the Semantic Mathematics vision—a unification of knowledge representation, reasoning, learning, and interaction.
Achieving it requires long-term effort, but Semantic Mathematics provides the principled direction and viable pathway.
7.2 Future Axioms and Theoretical Expansion
To advance toward this vision, Semantic Mathematics may require theoretical and axiomatic expansion in several key areas:
Modality Integration Axioms:
To unify data from different modalities (images, audio, text), we need axioms guiding their mapping into semantic symbols.
Example:
Axiom X: “Any modality’s information stream can, through appropriate transformations, be mapped into a semantic symbol sequence.”
This is a formalization of symbol grounding—enabled through well-defined feature extraction functions
ϕ
i
that convert multi-modal data into a shared symbolic space.
Uncertainty Handling:
Introduce probabilistic or fuzzy semantic axioms to allow confidence levels in membership and rule application.
Example:
Axiom Y: “For any proposition
P
, there exists a semantic evaluation score
v
∈[0,1]
representing the degree of belief in
P
’s truth.”
This enables graded knowledge representation.
While current axioms are deterministic, reality demands nuance.
Future work could integrate Kolmogorov’s axioms into the semantic framework, enabling probabilistic logic with explainability.
Game-Theoretic and Decision-Making Axioms:
If UKE includes decision-making capability (i.e., operating at the Wisdom layer), axioms must encode utility, preferences, and trade-offs.
Example:
Axiom Z: “There exists a purpose-layer semantic unit where each action outcome is associated with a utility score.”
This supports value-driven reasoning, allowing AI not only to infer truth but to assess what should be done.
Professor Duan’s PCR (Purpose Computation and Reasoning) framework already aligns with this direction.
Meta-Semantic Axioms:
These are axioms about the semantic modeling process itself.
For instance:
“Conceptual structures are not strictly equivalent to linguistic categorizations.”
Such axioms guide better modeling practices, preventing issues like conflating grammar with ontology.
They formalize modeling principles akin to software engineering design patterns.
Formal Verification and Mathematical Foundations:
To ensure theoretical soundness, Semantic Mathematics should be formally grounded in logic.
This includes:
Mapping it to known logics (e.g., description logic extensions);
Proving properties like consistency, completeness, and decidability.
Currently, the three core axioms resemble equivalence relation axioms, provably consistent and independent.
As new axioms are added (e.g., involving quantifiers or modalities), care must be taken to avoid Turing-completeness, which may lead to undecidability.
Ideally, Semantic Mathematics would target first-order logic with fixed model extensions, maintaining manageable complexity.
Integration with Cognitive Science:
Semantic Mathematics can incorporate findings from cognitive science to better align with human reasoning.
Example:
Prototype Axiom: “Membership in a concept is determined by similarity to a prototypical example.”
This supports typicality effects and softens strict two-valued logic.
For instance, “penguins are birds but don’t fly” could be explained via the prototype axiom—penguins are non-prototypical birds, so some rules don’t apply.
Formally modeling such exceptions provides more elegant solutions than ad hoc rule patches.
7.3 Conclusion: Toward a Gravitational Field of Knowledge
If knowledge is like mass in the universe, then the Semantic Mathematics axioms are the gravitational field, pulling knowledge together into a coherent whole.
These axioms not only unify concepts but prevent collapse into chaos—ensuring that the resulting knowledge universe remains structured, traceable, and semantically connected.
Professor Yucong Duan’s contribution lies in outlining this semantic gravity field—turning vague semantic problems into a formal, engineerable system.
Looking ahead, we can envision:
Semantic Mathematics becoming a standard in AI protocols—its axioms forming part of interoperability layers between systems.
Integration with powerful machine learning models: semantic axioms may be learned, verified, or extracted from large models, enabling mutual enhancement.
Applications in artificial consciousness research: UKEs may be crucial steps toward strong AI, with Semantic Mathematics as the route map to self-understanding and explanation.
Achieving this will require collaboration between academia and industry.
But the mathematical unification of semantics is no longer a dream—it is becoming a scientific and engineering reality.
We look forward to a future where semantics drives intelligence; where knowledge has no borders; where AI can understand and explain meaning; and where human and machine co-reason in a shared semantic space—hooked together by meaning.
Semantic Mathematics, as an emerging interdisciplinary theory, aims to introduce meaning into mathematical symbol systems through axiomatization, thereby providing a unified semantic framework for artificial intelligence and knowledge engineering. This report, based on the theoretical framework proposed by Professor Yucong Duan, systematically reviews its core ideas and terminology, including the Theory of Meaning Definition, Conceptual Construction Theory, and Mathematical Logic–Language Hook Theory, and offers an in-depth interpretation of the Semantic Mathematics axioms—Existence, Uniqueness, and Transitivity—and their roles.
Through four representative case studies—Chinese character construction, philosophical concepts, natural language parsing, and knowledge graph reasoning—we have demonstrated how Semantic Mathematics applies its axiomatic system to achieve semantic binding, reasoning unification, and transparent inference across diverse domains. Each case emphasized the importance of axiomatic model-building and verified the benefits of semantic preservation and traceable reasoning chains:
The derivation of meanings for characters like 明 ("bright") and 時 ("time") was rigorously interpreted under semantic axioms;
Abstract concepts such as existence and causality were clearly defined and reasoned over within the semantic space;
Syntactic and semantic structures in natural language were precisely aligned via Hook Theory, synchronizing machine understanding with human interpretation;
Reasoning rules in knowledge graphs were explicitly represented and visually verifiable, enhancing AI explainability.
We further explored the opportunities and challenges Semantic Mathematics faces in terms of theoretical interoperability, cross-domain consistency, axiomatic completeness, and engineering feasibility. The interaction between symbol and meaning, as framed by Hook Theory, was analyzed in depth and shown to be essential for building explainable and collaborative AI systems. In cross-domain knowledge fusion, Semantic Mathematics provides axioms and tools to maintain conceptual consistency and resolve conflicts.
With regard to the axiomatic system itself, we projected future directions for handling uncertainty, extending the axiom base, and addressing more complex real-world semantics. From an engineering standpoint, implementing Semantic Mathematics will require overcoming challenges related to computational complexity, standard integration, knowledge acquisition, and human–machine collaboration. However, these challenges are not insurmountable. With the convergence of semantic technologies and machine learning, we have good reason to believe that Semantic Mathematics will play a pivotal role in next-generation AI systems.
In summary, Semantic Mathematics sketches a new paradigm for artificial intelligence—a paradigm in which:
Reasoning becomes communication, and
Machines are no longer black-boxes, but knowledge partners capable of co-reasoning with humans.
Through the axiomatic unification of meaning, we may build a Unified Knowledge Entity that enables AI to truly understand the meaning of the information it processes, greatly enhancing system explainability, reliability, and collaborative capability.
This work not only carries profound academic significance, but also offers clear practical guidance for knowledge engineering and even artificial general intelligence (AGI) development. We look forward to more researchers and engineers joining the study and application of Semantic Mathematics, to collaboratively refine this theoretical system and transform it into a powerful engine that advances the human knowledge enterprise.
Yucong Duan, Zhendong Guo, Shuaishuai Huang. Semantic Binding and Rule Transparency: Principles and Methods for Mathematical Expression of Information Transmission via Semantic Mathematics. Technical Report, February 2025.
Yucong Duan. Introduction to DIKWP Semantic Mathematics. ScienceNet Blog, 2024.
Yucong Duan. Overview of Semantic Mathematics Based on the DIKWP Model. ScienceNet Blog, 2024.
Yucong Duan. The DIKWP Semantic Mathematics Theory of Mathematical Subjectivization Regression. ResearchGate Preprint, 2023.
Zhang San, Li Si. Visualization Methods of Rule Reasoning in Knowledge Graphs. Journal of Artificial Intelligence, 2023.
Montague, R. Universal Grammar. Theoretical Linguistics, 1970.
Gruber, T. Ontology of Folksonomy. Proceedings of IJCAI, 2005.
Pearl, J. Causality: Models, Reasoning and Inference. Cambridge University Press, 2000.
Lakoff, G. Women, Fire, and Dangerous Things: What Categories Reveal about the Mind. University of Chicago Press, 1987.
Webster. Commentary on Shuowen Jiezi. Institute of Philology Press, 1915.
(Note: References 1–4 are the primary sources related to Professor Yucong Duan’s Semantic Mathematics, some of which include specific line numbers in blogs or preprints. References 5–9 support related theoretical discussions; Reference 10 is a classical resource on Chinese character morphology. All citations in the report are denoted by [Source†Line] and correspond to the reference list above.)
玩透DeepSeek：认知解构+技术解析+实践落地
人工意识概论：以DIKWP模型剖析智能差异，借“BUG”理论揭示意识局限
人工智能通识 2025新版 段玉聪 朱绵茂 编著 党建读物出版社
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