Formal Statements

Section 10.1 Formal Statements

Table 10.1.1. Types of Formal Statements

Term	Requires Proof?	Empirical?	Context	Role
Definition	❌	❌	Mathematics / Logic	Introduces precise meaning of a term
Axiom / Postulate	❌	Sometimes	Foundations	Undemonstrated starting assumption
Theorem	✅	❌	Mathematics	Major derived result requiring rigorous proof
Lemma	✅	❌	Foundations	Intermediate helper theorem used to prove larger results
Corollary	Derived Directly	❌	Research	Necessary consequence following immediately from a theorem
Proposition	✅	❌	Research	Mid-level proven result of moderate significance
Conjecture	Unproven	Sometimes	Research	Tentative proposition believed true based on patterns/intuition, awaiting proof
Hypothesis	Testable (Not Proven)	✅	Science / Physics	Definition D.1.1
Claim	Unverified	Sometimes	General Research	Asserted statement requiring evidence or argument to be accepted
Principle	❌	Sometimes	Physics / Math	Fundamental rule guiding theory construction (often heuristic)
Law	❌ (Mathematically Derived)	✅	Science	Descriptive generalization of empirical regularities
Model	Validated	✅	Science	Representational system designed to explain/predict phenomena