1
From: "Human Potential & Development."
Split Justification: Development fundamentally involves both our inner landscape (**Internal World**) and our interaction with everything outside us (**External World**). (Ref: Subject-Object Distinction)..
2
From: "External World (Interaction)"
Split Justification: All external interactions fundamentally involve either other human beings (social, cultural, relational, political) or the non-human aspects of existence (physical environment, objects, technology, natural world). This dichotomy is mutually exclusive and comprehensively exhaustive.
3
From: "Interaction with the Non-Human World"
Split Justification: All human interaction with the non-human world fundamentally involves either the cognitive process of seeking knowledge, meaning, or appreciation from it (e.g., science, observation, art), or the active, practical process of physically altering, shaping, or making use of it for various purposes (e.g., technology, engineering, resource management). These two modes represent distinct primary intentions and outcomes, yet together comprehensively cover the full scope of how humans engage with the non-human realm.
4
From: "Understanding and Interpreting the Non-Human World"
Split Justification: Humans understand and interpret the non-human world either by objectively observing and analyzing its inherent structures, laws, and phenomena to gain factual knowledge, or by subjectively engaging with it to derive aesthetic value, emotional resonance, or existential meaning. These two modes represent distinct intentions and methodologies, yet together comprehensively cover all ways of understanding and interpreting the non-human world.
5
From: "Understanding Objective Realities"
Split Justification: Humans understand objective realities either through empirical investigation of the physical and biological world and its governing laws, or through the deductive exploration of abstract structures, logical rules, and mathematical principles. These two domains represent fundamentally distinct methodologies and objects of study, yet together encompass all forms of objective understanding of non-human reality.
6
From: "Understanding Formal Systems and Principles"
Split Justification: Humans understand formal systems and principles either by focusing on the abstract study of quantity, structure, space, and change (e.g., arithmetic, geometry, algebra, calculus), or by focusing on the abstract study of reasoning, inference, truth, algorithms, and information processing (e.g., formal logic, theoretical computer science). These two domains represent distinct yet exhaustive categories of formal inquiry.
7
From: "Understanding Logical and Computational Systems"
Split Justification: Humans understand logical and computational systems either by focusing on the abstract rules and structures that govern valid inference, truth, and formal argumentation, or by focusing on the abstract principles and methods that govern information processing, problem-solving procedures, and the limits of computation. These two domains represent distinct yet exhaustive categories within the study of logical and computational systems.
8
From: "Understanding Formal Logic and Deductive Reasoning"
Split Justification: Formal logic and deductive reasoning fundamentally involve two distinct yet inseparable dimensions: the abstract rules and structures governing the formation and transformation of logical expressions and arguments (syntax, proof theory), and the meaning, truth conditions, and interpretation of these expressions in relation to various models or realities (semantics, model theory). These two areas represent distinct methodologies and objects of study within logic, yet together they comprehensively cover the entire scope of understanding formal logic.
9
From: "Understanding Logical Syntax and Proof Theory"
Split Justification: The understanding of logical syntax and proof theory fundamentally divides into two distinct yet complementary areas. The first involves grasping the abstract rules that govern the correct formation and internal structure of individual logical expressions (e.g., propositions, predicates), defining what constitutes a well-formed statement within a formal language. The second involves understanding the systematic rules of inference and axiomatic frameworks that allow for the step-by-step transformation of these expressions into valid sequences of reasoning, culminating in formal proofs and arguments. One defines the building blocks of the formal language, while the other defines how to construct valid arguments from those building blocks.
10
From: "Understanding the Formal Systems of Deductive Proof"
Split Justification: Understanding the formal systems of deductive proof fundamentally involves two distinct yet complementary dimensions: the practical application of inference rules and axioms to construct specific, step-by-step derivations and arguments within a formal system, and the abstract analysis of the overarching properties and limitations of these proof systems themselves, such as their soundness, completeness, and consistency. These two areas represent distinct intentions—one focused on performing deduction, the other on analyzing its systemic characteristics—yet together they exhaustively cover the entire scope of understanding deductive proof systems.
11
From: "Understanding Proof System Metatheory"
Split Justification: Understanding proof system metatheory fundamentally involves two distinct yet complementary dimensions. The first focuses on the system's internal integrity and truth-preservation, examining whether its derivations are valid and free from contradictions (e.g., soundness, consistency). The second focuses on the system's expressive capacity and inherent boundaries, exploring what truths it can derive and where its power is limited (e.g., completeness, decidability, incompleteness theorems). Together, these two areas exhaustively cover the core meta-properties of any formal proof system.
12
From: "Understanding the Completeness and Deductive Limitations of Proof Systems"
Split Justification: ** The understanding of the completeness of proof systems focuses on whether a system can derive all valid truths within its scope. In contrast, the understanding of incompleteness and undecidability focuses on the inherent existence of truths it cannot derive or questions it cannot algorithmically resolve. These two investigations represent distinct inquiries into a proof system's overall deductive power and its fundamental boundaries, together exhaustively covering its completeness and limitations.
✓
Topic: "Understanding the Incompleteness and Undecidability of Proof Systems" (W7986)