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 Semantics and Model Theory"
Split Justification: Understanding Logical Semantics and Model Theory fundamentally involves two distinct yet complementary aspects: first, establishing the basic mechanisms for assigning meaning to formal language elements and determining the truth of formulas within specific mathematical structures (models); and second, investigating the overarching properties of these models, the relationships between them, and their connections to formal theories. These two areas represent the foundational definitional layer and the subsequent theoretical exploration, together exhaustively covering the discipline.
10
From: "Understanding Interpretations, Valuations, and Truth in Models"
Split Justification: ** This dichotomy fundamentally separates the act of defining the semantic context—which encompasses specifying the domain of discourse, interpreting non-logical symbols (constants, functions, predicates), and assigning values to variables—from the subsequent formal process of recursively defining how complex formulas acquire a truth value (satisfaction) within that established context. These two areas are distinct yet together comprehensively cover the full scope of understanding interpretations, valuations, and truth in models.
11
From: "Recursive Definition of Formula Satisfaction and Truth"
Split Justification: The recursive definition of formula satisfaction and truth fundamentally consists of two distinct parts: the base cases, which establish satisfaction for atomic formulas directly based on the model's interpretation, and the inductive steps, which define satisfaction for complex formulas based on the satisfaction of their subformulas and the rules of logical connectives and quantifiers. These two components are mutually exclusive (a formula is either atomic or compound) and together comprehensively cover the entire recursive definition.
12
From: "Satisfaction and Truth for Compound Formulas"
Split Justification: Compound formulas, by definition, are constructed in one of two fundamental ways: either by combining simpler formulas using propositional connectives (like negation, conjunction, disjunction, implication) or by applying quantifiers (universal or existential) over variables within a formula. These two construction methods dictate distinct recursive clauses in the definition of satisfaction and truth, are mutually exclusive in form, and together comprehensively cover all possible types of compound formulas.
✓
Topic: "Satisfaction and Truth for Formulas with Logical Connectives" (W5810)