In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions. In order to directly prove a conditional statement of the form "If p, then q", it suffices to consider the situations in which the statement p is true. Logical deduction is employed to reason from assumptions to conclusion. The type of logic employed is almost invariably first-order logic, employing the quantifiers for all and there exists. Common proof rules used are modus ponens and universal instantiation.
In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. For example instead of showing directly p ⇒ q, one proves its contrapositive ~q ⇒ ~p (one assumes ~q and shows that it leads to ~p). Since p ⇒ q and ~q ⇒ ~p are equivalent by the principle of transposition (see law of excluded middle), p ⇒ q is indirectly proved. Proof methods that are not direct include proof by contradiction, including proof by infinite descent. Direct proof methods include proof by exhaustion and proof by induction.
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“In Europe life is histrionic and dramatized, and ... in America, except when it is trying to be European, it is direct and sincere.”
—William Dean Howells (18371920)
“The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
—Andrew Michael Ramsay (16861743)