归谬法，或称反证法，是一种间接证明的形式。它通过证明假设命题为假会导致逻辑矛盾，从而确立该命题的真理性。 要证明命题 [latex]p[/latex]，需假设其否定命题 [latex]\neg p[/latex]，并由此推导出矛盾，例如 [latex]q \land \neg q[/latex]，从而得出结论：[latex]p[/latex] 必然为真。.
反证法（归谬法）
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The logical foundation for proof by contradiction is the law of non-contradiction, which states that a proposition cannot be both true and false, and the law of the excluded middle, which states that a proposition must be either true or false. The method begins by assuming the opposite of what one wants to prove. For example, to prove that the square root of 2 is irrational, 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.
UNESCO Nomenclature: 1201
逻辑
类型
抽象系统
中断
基础
用法
广泛使用
前体
- 苏格拉底式辩论法（交叉质询）
- 埃利亚学派哲学（例如，芝诺悖论）
- 亚里士多德对形式逻辑的发展
应用程序
- 欧几里得证明素数无穷大
- 证明 2 的平方根是无理性的
- 康托尔的对角线论证证明了实数的不可数性
- 在计算机科学中证明停机问题是不可判定的
专利：
NA
潜在创新理念
Related to: contradiction, reductio ad absurdum, indirect proof, irrational numbers, logic, law of non-contradiction, negation, assumption, Cantor, halting problem.
历史背景
反证法（归谬法）
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