Deductive Reasoning
Deductive reasoning draws certain conclusions from general principles, moving from accepted truths to specific results with logical necessity.
Definition
Deductive reasoning means using known rules to reach guaranteed conclusions; if the starting facts are true and the logic is valid, the conclusion must be true. It proceeds from general premises to specific conclusions using logical rules, and mathematical proofs are entirely deductive, deriving specific results with certainty from axioms and theorems; a deductive argument can be valid (correct form) even with false premises, but the conclusion is only guaranteed true if the premises are also true. Formal deductive logic uses inference rules (modus ponens, modus tollens, etc.) applied to well-formed formulas; a deduction is a finite sequence of formulas each an axiom or following from earlier ones by an inference rule, and soundness/completeness connect what is deducible to what is true in all models (Godel's Completeness Theorem for first-order logic).
Example
"All squares have $4$ equal sides" (general rule); "this shape is a square"; therefore "this shape has $4$ equal sides," a guaranteed conclusion. More formally: "All differentiable functions are continuous," "$f(x) = x^2$ is differentiable," therefore "$f(x) = x^2$ is continuous," a valid and certain argument. Modus ponens (from $P$ and $P \to Q$, deduce $Q$) and modus tollens (from $\neg Q$ and $P \to Q$, deduce $\neg P$) are the atomic inference rules underlying all mathematical proof.
Key Insight
Deductive reasoning is the gold standard of mathematical proof: unlike guessing, a valid deduction cannot be wrong if the premises are right. The formalization of deductive reasoning by Frege, Russell, and Hilbert in the late 19th and early 20th centuries led directly to the invention of computers, since logical circuits implement deductive inference mechanically.