Philosophy Formula & Theorem Quick Reference Handbook | KS3 CCEA 哲学：公式定理速查手册

📚 Philosophy Formula & Theorem Quick Reference Handbook | KS3 CCEA 哲学：公式定理速查手册

Philosophy may not be full of equations, but it has powerful ‘formulas’ for reasoning, knowledge, and ethics. This quick-reference handbook collects key logical rules, ethical principles, and philosophical ‘theorems’ you’ll meet in KS3 CCEA Philosophy. Use these formulas to analyse arguments, evaluate moral claims, and build clear thinking.

哲学也许没有很多方程式，但它拥有用于推理、知识和伦理的强大“公式”。这本速查手册汇集了你在 KS3 CCEA 哲学中会遇到的关键逻辑规则、伦理原则和哲学“定理”。运用这些公式来分析论证、评价道德主张并建立清晰思维。

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1. Valid Reasoning: Logical Formulas | 有效推理的逻辑公式

In philosophy, a valid argument is one where if the premises are true, the conclusion must be true. The following are basic valid argument forms often treated like ‘theorems’ – if you apply them correctly, you cannot be led from true premises to a false conclusion.

在哲学中，有效论证是指如果前提为真，结论必然为真。以下是一些被视为“定理”的基本有效论证形式——如果你正确运用它们，就不可能从真前提推出假结论。

Modus Ponens (Affirming the Antecedent): If P implies Q, and P is true, then Q is true. Symbolically: P → Q, P ⊢ Q.

肯定前件 (Modus Ponens)： 如果 P 蕴含 Q，且 P 为真，则 Q 为真。符号表示为：P → Q, P ⊢ Q。

Example: If it is raining, the ground is wet. It is raining. Therefore, the ground is wet.

例子：如果下雨，地面就会湿。现在下雨了。因此，地面是湿的。

Modus Tollens (Denying the Consequent): If P implies Q, and Q is false, then P must be false. P → Q, ¬Q ⊢ ¬P.

否定后件 (Modus Tollens)： 如果 P 蕴含 Q，且 Q 为假，则 P 必为假。P → Q, ¬Q ⊢ ¬P。

Example: If the battery is charged, the torch lights up. The torch does not light up. So the battery is not charged.

例子：如果电池有电，手电筒会亮。手电筒不亮。所以，电池没有电。

Hypothetical Syllogism (Chain Argument): If P implies Q and Q implies R, then P implies R. P → Q, Q → R ⊢ P → R.

假言三段论（链式论证）： 如果 P 蕴含 Q，且 Q 蕴含 R，则 P 蕴含 R。P → Q, Q → R ⊢ P → R。

Example: If I study hard, I will pass the test. If I pass the test, I will be happy. So if I study hard, I will be happy.

例子：如果我努力学习，我会通过测验。如果我通过测验，我会很开心。所以，如果我努力学习，我会很开心。

Disjunctive Syllogism (Either/Or Removal): Either P or Q is true. P is false, so Q must be true. P ∨ Q, ¬P ⊢ Q.

选言三段论（排除选言支）： 要么 P 为真要么 Q 为真。P 为假，所以 Q 必然为真。P ∨ Q, ¬P ⊢ Q。

Example: Either the bus or the train will be on time. The bus is not on time. Therefore, the train is on time.

例子：要么公共汽车准时，要么火车准时。公共汽车不准时。所以，火车准时。

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2. Common Logical Fallacies | 常见逻辑谬误

Fallacies are errors in reasoning that look like valid arguments but do not follow logical rules. Recognising their ‘formulas’ helps you avoid being tricked by bad reasoning.

谬误是推理中的错误，它们看起来像有效论证，但实际上不遵循逻辑规则。识别它们的“公式”能帮助你避免被糟糕的推理欺骗。

Ad Hominem (Attacking the Person): Instead of addressing the argument, one attacks the character or motive of the person making it. Formula: ‘X says Y, but X is a bad person, therefore Y is false.’

诉诸人身（人身攻击）： 不去处理论点本身，而是攻击提出论点者的品格或动机。公式：“X 说 Y，但 X 是个坏人，所以 Y 是假的。”

Example: ‘You say that eating vegetables is healthy, but you are a boring person, so vegetables cannot really be that good.’

例子：“你说吃蔬菜有益健康，但你是个无趣的人，所以蔬菜不可能真的那么好。”

Straw Man (Misrepresenting an Argument): Distorting the opponent’s view into a weaker version, then attacking that distortion. Formula: ‘Person A claims X. Person B twists X into Y (an extreme or silly version), attacks Y, and claims to have refuted X.’

稻草人谬误（歪曲论点）： 把对方的观点扭曲成较弱的版本，然后攻击那个扭曲版。公式：“甲主张 X。乙把 X 歪曲成 Y（一个极端或愚蠢的说法），攻击 Y，并声称推翻了 X。”

Example: A: ‘We should spend more on schools.’ B: ‘So you think we should give all our money to schools and ignore hospitals?’

例子：甲：“我们应该给学校多拨一些款。” 乙：“所以你认为我们该把所有钱给学校，不管医院了？”

False Dilemma (Either/Or Fallacy): Presenting only two options when more exist. Formula: ‘You must either accept A or accept B. Therefore, since B is unacceptable, A must be true.’

错误两难（非黑即白）： 只给出两个选项，而事实上还有更多可选。公式：“你要么接受 A，要么接受 B。既然 B 不可接受，那 A 必定为真。”

Example: ‘Either you support this leader completely, or you are against your country.’

例子：“你要么完全支持这位领袖，要么就是反对自己的国家。”

Slippery Slope: Assuming that one small step will inevitably lead to a chain of extreme events without sufficient evidence. Formula: ‘If A happens, then B will happen, then C, and eventually disaster Z; therefore we must not allow A.’

滑坡谬误： 在没有充分证据的情况下，假定一小步必然引发一连串极端事件。公式：“如果 A 发生，那么 B 会发生，接着 C，最终导致灾难 Z；因此我们决不能允许 A。”

Example: ‘If we let students re-take one test, next they will want to re-take every assignment, and soon no one will study at all.’

例子：“如果我们允许学生补考一次，接下来他们会要求重做所有作业，很快就没有人会学习了。”

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3. The JTB Formula for Knowledge | 知识的 JTB 公式

For centuries, philosophers defined knowledge using the ‘JTB’ theorem: a person knows something if it is a justified true belief. This classical formula is written as: S knows that P if and only if (i) P is true, (ii) S believes that P, and (iii) S is justified in believing that P.

几个世纪以来，哲学家使用“JTB”定理来定义知识：一个人知道某事，当且仅当它是被证实的真信念。这个经典公式写作：S 知道 P，当且仅当 (i) P 为真，(ii) S 相信 P，并且 (iii) S 相信 P 是有理由的。

Think of JTB as a three-part checklist: without truth, it is not knowledge but a mistake; without belief, you cannot claim to know it; without justification, it is just a lucky guess.

把 JTB 想象成一个三步核查表：没有真，那就不是知识而是错误；没有信念，你就不能声称知道它；没有确证，它只是一个幸运的猜测。

Example: Priya sees a clock showing 3:00. Unbeknownst to her, the clock is broken, but by coincidence it really is exactly 3:00. Priya believes it is 3:00, her belief is true, but she lacks justification because the clock is unreliable. So she does not have knowledge.

例子：普丽娅看到一个钟显示 3:00。她不知道那个钟坏了，但碰巧此刻确实正好是 3:00。普丽娅相信现在是 3:00，她的信念是真的，但她缺乏确证，因为那只钟不可靠。所以她并不拥有知识。

This has led to refinements, but the JTB formula remains the starting point for all theories of knowledge.

这导致了后来的修正，但 JTB 公式仍然是所有知识理论的出发点。

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4. Descartes’ Cogito Formula | 笛卡尔的“我思”公式

René Descartes, searching for one absolutely certain truth, arrived at the formula: ‘Cogito, ergo sum’ – ‘I think, therefore I am.’ Even if you doubt everything, the very act of doubting proves that a thinking ‘I’ exists.

勒内·笛卡尔在寻找一个绝对确定的真理时，得出了这个公式：“Cogito, ergo sum”——“我思故我在。” 即使你怀疑一切，怀疑这个行为本身就证明了一个思考着的“我”存在。

We can write this as a logical short-cut: The statement ‘I do not exist’ cannot be consistently thought without a thinker. Thus, someone is doing the thinking. Symbolically: (Doubt → Thinker) → Existence of thinker.

我们可以将它写成一个逻辑捷径：“我不存在”这个陈述，没有一个思考者便不可能被思考。因此，必定有人在思考。用符号表示：（怀疑 → 思考者）→ 思考者存在。

Descartes’ theorem becomes the foundation for building further knowledge: whatever is as clear and distinct as the cogito must also be true.

笛卡尔的定理成了建立更多知识的基础：任何像“我思”一样清楚分明的东西，也必定为真。

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5. Utilitarian Formula: The Greatest Happiness Principle | 功利主义公式：最大幸福原则

Jeremy Bentham and John Stuart Mill proposed a moral ‘calclus’: an action is right insofar as it tends to promote happiness and wrong insofar as it tends to produce the reverse of happiness. Joy can be measured in units they called ‘utils’. We can summarise it as a utility function:

杰里米·边沁和约翰·斯图尔特·密尔提出了一种道德“计算法”：一个行为正确与否，取决于它倾向于增进幸福还是带来不幸。快乐可以用他们称之为“效用”的单位衡量。我们可以总结为效用函数：

Net Utility = (Total Pleasure) – (Total Pain)

净效用 = (总快乐) – (总痛苦)

According to this formula, when facing a choice, you should select the action that produces the greatest net happiness for everyone affected. Every person counts equally – ‘each to count for one, and none for more than one.’

根据这个公式，当面临选择时，你应当选择能为所有受影响者带来最大净幸福的行为。每个人都被平等计数——“每个人都算作一个，没有人可以多算一个。”

Example: You have £10 to share between two charities. Charity A creates 100 utils, Charity B creates 70 utils. Ceteris paribus, giving to A produces the greater net utility. A utilitarian would choose A.

例子：你有 10 英镑可以捐给两个慈善机构。慈善 A 产生 100 效用，慈善 B 产生 70 效用。在其他条件相同下，捐给 A 产生更大的净效用。功利主义者会选择 A。

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6. Kant’s Categorical Imperative Formula | 康德的绝对命令公式

Immanuel Kant argued that morality is not about consequences but about duty and reason. He gave us a test-formula called the Categorical Imperative. The key version: ‘Act only according to that maxim whereby you can at the same time will that it should become a universal law.’

伊曼努尔·康德认为，道德无关后果，而是关乎义务和理性。他给了我们一种称为绝对命令的检验公式。其主要版本是：“只按照你同时愿意它成为一条普遍法则的准则去行动。”

To use the formula: (1) Identify the maxim (rule) behind your action, e.g., ‘I will lie to get out of trouble.’ (2) Imagine a world where everyone follows this maxim. (3) Ask yourself: could this still work, and can I rationally will such a world? If lying were universal, trust would collapse and lying would become pointless. You cannot will the maxim without contradiction.

运用该公式：(1) 找出你行动背后的准则（规则），例如：“我会为了摆脱麻烦而说谎。” (2) 想象一个人人遵守此准则的世界。(3) 问自己：这还能行得通吗？我能理性地意愿这样的世界吗？如果说谎成了普遍法则，信任会崩溃，说谎变得毫无意义。你无法毫无矛盾地意愿这条准则。

Thus, the formula acts as a universalisation checker. If a maxim passes the test, it is morally permissible; if it fails, it is wrong.

因此，这个公式起着一套普遍化检验器的作用。如果一条准则通过了检验，它就是道德上允许的；如果通不过，它就是错的。

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7. Pascal’s Wager: Expected Value Calculation | 帕斯卡的赌注：期望值计算

Blaise Pascal offered a decision-theoretic ‘formula’ for believing in God. The wager is often presented as a decision matrix where you calculate expected value under uncertainty. The possible payoffs can be shown as follows:

布莱兹·帕斯卡为相信上帝提供了一个决策论“公式”。这个赌注常以决策矩阵展示，你可以在不确定性下计算期望值。可能的收益如下所示：

Your choice	God exists	God does not exist
Believe in God	Infinite happiness (Heaven)	Finite earthly loss or gain
Do not believe	Infinite suffering (Hell)	Finite earthly loss or gain

English version of table: If you believe and God exists, you gain infinite reward. If you do not believe and God exists, you face infinite punishment. If God does not exist, the gains and losses are limited either way. Pascal argued that the ‘expected value’ of believing is infinite and therefore rational to choose belief.

表格中文版：如果你相信而上帝存在，你将获得无限奖赏。如果你不信而上帝存在，你将面临无限惩罚。如果上帝不存在，无论信还是不信，得失都是有限的。帕斯卡认为，相信的“期望值”是无限的，因此选择相信是理性的。

This formula treats religious faith like a bet, but many critics point out that we cannot simply decide to believe something for pragmatic reasons.

这个公式将宗教信仰当作一场赌博，但许多批评者指出，我们不能仅仅出于实用理由就决定相信某事。

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8. Ockham’s Razor: The Principle of Parsimony | 奥卡姆剃刀：简约原则

William of Ockham gave us a powerful thinking tool: ‘Do not multiply entities beyond necessity.’ In modern terms: when you have competing explanations, the one with fewer assumptions should be preferred — all else being equal. The formula is sometimes written as:

奥卡姆的威廉给了我们一个强大的思维工具：“如无必要，勿增实体。”用现代话说：当你面对若干个互相竞争的解释时，

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