Deductive Reasoning

Geometry · Axiom Academy

The foundation of logical proofs — how facts, definitions, and the laws of logic turn premises into certain conclusions. Before we work with deductive reasoning, it helps to see how it differs from inductive reasoning. Watch both happen on real figures: on the left, three polygons are drawn and their angle sums measured to spot a pattern; on the right, a general fact is applied straight to one specific angle. Measures specific cases and generalizes to a conjecture. Triangle , quadrilateral , pentagon → the conjecture sum . The conclusion is probably true — until it's proven. Applies a general fact to a specific case. All right angles measure ; is a right angle ; therefore . The conclusion is certain if the premises are. (General → specific.) The Law of Detachment (modus ponens) is the most basic deductive step: if a conditional is true and its hypothesis is true, then its conclusion must be true. Watch the rule fire through a real angle — the general rule lights up, the specific angle is confirmed to measure , and the conclusion stamps directly onto the figure. If the conditional and its hypothesis hold, the conclusion is forced If an angle measures 90°, then it is a right angle. Therefore, ABC is a right angle. You cannot detach if only the conclusion is known to be true. Knowing an angle is a right angle does not, by detachment alone, tell you it measures 90° — you must know the hypothesis is true first.

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