上一片我们把三个连接词接进了电路。可符号一多就容易数漏：“A ∧ B ∨ ¬C”到底有几种情况？宝箱管理员立下规矩——别猜，把所有可能性一行行列出来。这张表，就叫真值表。Last leaf we wired the three connectives into circuits. But with several symbols, cases are easy to miss: how many situations does “A ∧ B ∨ ¬C” have? The chest keeper's rule: never guess — list every possibility, one per row. That sheet is a truth table.
第 1 步：每种情况占一行Step 1: One row per case
先看最简单的：一个命题，一张小表就够。Start simple: one proposition needs only a tiny table.
每个命题只有两种可能，表格里用 T（true，真）和 F（false，假）表示。机器人拿起铅笔：每一行写一种组合，最后一列写整句话的结果。一行一种情况、一种也不漏——这就是填表法。Every proposition has just two possibilities, written T (true) and F (false). The robot picks up its pencil: each row takes one combination, and the last column gives the result of the whole sentence. One case per row, none missed — that is the truth-table method.
那要是有两个、三个命题呢？And if there are two or three propositions?
第 2 步：n 个命题，2ⁿ 行Step 2: n propositions, 2ⁿ rows
一个命题 2 行，两个命题 2×2=4 行，三个命题 2×2×2=8 行——每多一个命题，情况翻一倍，公式就是 2ⁿ。写表有秘诀：第一列“隔一行换一次”（T F T F…），第二列“两行一组换”（T T F F…），跟数二进制一个道理。看 A ∧ B 的表：只有第一行结果是 T——两真才真。One proposition takes 2 rows, two take 2×2=4, three take 2×2×2=8 — each extra proposition doubles the cases, giving 2ⁿ. Writing tip: let the first column alternate every row (T F T F…), the second switch every two rows (T T F F…), just like counting in binary. In the A ∧ B table only the first row's result is T — both true is required.
表列全了，它能当“裁判”用吗？With every case listed, can the table act as a judge?
第 3 步：当裁判——等价与重言式Step 3: Judge duty — equivalence and tautology
能！想验证两个公式是不是一回事，就把它们的表并排填完：结果列一模一样，就说明等价。还有一种公式，不管 A 填什么都为真，比如 A ∨ ¬A（要么下雨 ∨ 要么不下雨）——结果列全是 T，它叫重言式（tautology）；永远为假的叫矛盾式。表一填完，谁真谁假一目了然。It can! To check whether two formulas are the same, fill their tables side by side: identical result columns prove equivalence. Some formulas are true no matter what — A ∨ ¬A (either it rains ∨ it does not) has a result column that is all T — that is a tautology; one that is always false is a contradiction. Fill the table, and the truth is plain to see.
🎮 补全真值表（30 秒）🎮 Complete the truth tables (30 seconds)
点空格在 ? → T → F 之间切换，4 张表全部填对就撒花；最后一张藏着重言式的秘密。Tap a blank to cycle ? → T → F. Fill all 4 tables correctly to finish; the last one hides the secret of tautology.
一句话记住它：真值表 = 把所有真假组合一行行排出来的清单；n 个命题 2ⁿ 行，填完表，一切一目了然。Remember it in one line: a truth table lists every true/false combination, one per row; n propositions take 2ⁿ rows, and once filled, everything is plain.
每行一种组合，T 真 F 假，一行不漏One case per row, T for true and F for false, none missedn 个命题 → 2ⁿ 行：1 个 2 行，2 个 4 行，3 个 8 行n propositions → 2ⁿ rows: 1 → 2, 2 → 4, 3 → 8结果列全是 T 的公式叫重言式（如 A ∨ ¬A）A formula whose result column is all T is a tautology (e.g. A ∨ ¬A)
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
Back to tree
