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Proof by contradiction, or reductio ad absurdum, is a form of indirect proof. It establishes the truth of a proposition by showing that assuming the proposition to be false leads to a logical contradiction. To prove a proposition [latex]p[/latex], one assumes its negation, [latex]\neg p[/latex], and deduces a contradiction, such as [latex]q \land \neg q[/latex], thereby concluding that [latex]p[/latex] しなければならない be true.
背理法の論理的基礎は、命題が真かつ偽であることはあり得ないという無矛盾律と、命題は真か偽のどちらかでなければならないという排中律である。この方法は、証明したいことの反対を仮定することから始まる。例えば、 2の平方根 は 非合理的, one starts by assuming it is rational. If [latex]\sqrt{2}[/latex] is rational, it can be expressed as a fraction [latex]a/b[/latex] in lowest terms, where a and b are integers. This leads to [latex]2 = a^2/b^2[/latex], or [latex]a^2 = 2b^2[/latex]. This implies [latex]a^2[/latex] is even, which means [latex]a[/latex] must also be even. So, [latex]a = 2k[/latex] for some integer k. Substituting this back gives [latex](2k)^2 = 2b^2[/latex], or [latex]4k^2 = 2b^2[/latex], which simplifies to [latex]2k^2 = b^2[/latex]. This means [latex]b^2[/latex] is even, and therefore [latex]b[/latex] is also even. If both a and b are even, the fraction [latex]a/b[/latex] was not in lowest terms, which contradicts the initial assumption. This contradiction forces the conclusion that the initial assumption—that [latex]\sqrt{2}[/latex] is rational—must be false. This method is powerful but can be non-constructive, as it proves a statement is true without providing a direct example or construction.
矛盾による証明 (不条理の還元)
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