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: "Internal World (The Self)"
Split Justification: The Internal World involves both mental processes (**Cognitive Sphere**) and physical experiences (**Somatic Sphere**). (Ref: Mind-Body Distinction)
3
From: "Cognitive Sphere"
Split Justification: Cognition operates via deliberate, logical steps (**Analytical Processing**) and faster, intuitive pattern-matching (**Intuitive/Associative Processing**). (Ref: Dual Process Theory)
4
From: "Analytical Processing"
Split Justification: Analytical thought engages distinct symbolic systems: abstract logic and mathematics (**Quantitative/Logical Reasoning**) versus structured language (**Linguistic/Verbal Reasoning**).
5
From: "Quantitative/Logical Reasoning"
Split Justification: Logical reasoning can be strictly formal following rules of inference (**Deductive Proof**) or drawing general conclusions from specific examples (**Inductive Reasoning Case Study**). (L5 Split)
6
From: "Deductive Proof."
Split Justification: Deductive systems can be analyzed based on the relationship between whole statements (**Propositional Logic**) or the properties of objects and their relations (**Predicate Logic**). (L6 Split)
7
From: "Predicate Logic"
Split Justification: Predicate logic extends reasoning to include variables and quantities (**Understanding Quantifiers**) and applying these to sets of objects (**Basic Set Theory Proof**).
8
From: "Basic Set Theory Proof"
Split Justification: This dichotomy distinguishes between two fundamental methodologies for constructing basic set theory proofs: element-wise proofs, which analyze the membership of individual elements using predicate logic definitions of set operations, and algebraic proofs, which manipulate set expressions using established set identities and laws. These two approaches represent distinct, yet comprehensive, methods for proving set theoretic statements.
9
From: "Algebraic Set Theory Proof"
Split Justification: This dichotomy distinguishes between proofs that primarily rely on the manipulation of symbols and application of axioms within a formal algebraic system (e.g., Boolean algebra) and proofs that leverage the interpretation of those algebraic expressions in terms of set-theoretic models, often involving element-level reasoning or the specific properties of sets as the underlying structure. Together, these methods comprehensively cover the approaches to constructing algebraic set theory proofs.
10
From: "Semantic Algebraic Proofs"
Split Justification: This split categorizes semantic algebraic proofs based on the scope of truth being established: universal truth across all possible interpretations (validity) versus truth within at least one specific interpretation (satisfiability). This fundamental dichotomy addresses the two primary aims of semantic analysis in algebraic contexts.
11
From: "Proofs of Semantic Satisfiability of Algebraic Expressions"
Split Justification: This dichotomy distinguishes between two fundamental approaches to proving the semantic satisfiability of an algebraic expression. Constructive proofs demonstrate satisfiability by explicitly providing a model or assignment of values that makes the expression true. Non-constructive proofs establish the existence of such a satisfying model or assignment without necessarily furnishing it, often through indirect methods like proof by contradiction or by demonstrating that a known satisfiable condition must hold. These two categories are mutually exclusive and together comprehensively cover all methods for proving semantic satisfiability.
12
From: "Non-Constructive Proofs of Semantic Satisfiability of Algebraic Expressions"
Split Justification: Non-constructive proofs fundamentally establish the existence of a satisfying assignment without providing it. This can be achieved either indirectly, by showing that its non-existence leads to a logical contradiction (proof by contradiction), or directly, by invoking theorems, principles, or cardinality arguments that guarantee existence without explicit construction. This dichotomy covers all forms of non-constructive proofs.
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Topic: "Non-Constructive Proofs by Direct Existence Arguments" (W8159)