Definition
In a deductive system like geometry, a Valid Definition is a statement that explains the meaning of a term by placing it in a known class and identifying its unique distinguishing characteristics. A definition must be reversible and use only previously accepted terms.
Why It Matters
Precise definitions are the ‘contracts’ of a logical system; without them, every argument becomes a battle over semantics, and the entire structure of deductive reasoning collapses into ambiguity and circularity.
Core Concepts
- Class and Differentia: A definition follows the pattern: Term = Class + Differentiator. For example, “A triangle (term) is a polygon (class) that has three sides (differentiator).”
- Reversibility: A good definition is a biconditional statement (). “If a figure is a triangle, it has three sides” AND “If a figure is a polygon with three sides, it is a triangle.”
- How to read: “The statement P if and only if Q.”
- Meaning: The definition works both ways — the term and its distinguishing property are logically equivalent; no extra assumptions needed.
- Use of Preceding Terms: Definitions must build upon Undefined Terms (point, line, plane) and previously defined terms to avoid circular reasoning.
- Clarity and Precision: Definitions must avoid ambiguous language and “extra” properties that can be proven as theorems (e.g., one doesn’t define a square as having four right angles and equal sides and being a parallelogram; the last part is redundant).