Year 11 Cambridge Philosophy: Quick Reference Handbook of Formulas and Theorems | 剑桥哲学：公式定理速查手册

📚 Year 11 Cambridge Philosophy: Quick Reference Handbook of Formulas and Theorems | 剑桥哲学：公式定理速查手册

Philosophy is not a subject filled with ready-made formulas etched into the stone of a textbook, yet every Cambridge IGCSE or Year 11 philosopher quickly discovers that clear thinking demands its own kind of symbolic shorthand. This revision guide playfully borrows the language of mathematics to frame the recurring patterns of argument, ethics, logic, and epistemology you will meet in the Cambridge syllabus. Rather than memorising equations, you are invited to master the structures that lie beneath philosophical reasoning — your true arsenal for tackling examination questions with precision.

哲学并不是一门填满了刻在教科书石板上的现成公式的学科，但每一位学习剑桥 IGCSE 或 Year 11 哲学的同学很快就会明白，清晰的思考同样需要一套属于自己的符号速记。这本复习手册俏皮地借用了数学的语言，把你在剑桥大纲中会反复遇到的论证模式、伦理学框架、逻辑原则和认识论结构，统统整理出来。你不是在机械记忆方程式，而是要真正掌握那些藏在哲学推理底层的结构——这才是面对考试题目时最锋利的武器。

1. The Syllogistic Inference Formula | 三段论推理公式

A valid syllogism functions like a reliable algebraic identity. The most famous formula is: All M are P + All S are M → ∴ All S are P. This is the Barbara mood of categorical syllogism, and for Cambridge Philosophy, you must learn to test validity by checking whether the middle term is distributed and whether no illicit major or minor term extension occurs. Remember: If S = ‘humans’, M = ‘mortals’, P = ‘beings that die’, the conclusion follows with geometric certainty.

一个有效的三段论，就像一条可靠的代数恒等式。最著名的公式是：所有 M 是 P + 所有 S 是 M → ∴ 所有 S 是 P。这是直言三段论的 Barbara 式；学习剑桥哲学时，你必须通过检查中项是否周延、大项或小项是否非法扩张，来检验有效性。记住：如果 S = “人类”，M = “会死的”，P = “终有一死的存在”，那么结论就如同几何学一般确定无疑地推得。

All M are P
All S are M
∴ All S are P

When you encounter ethical arguments in the Cambridge exam, underline the hidden syllogisms. For example, “It is wrong to take an innocent life” (major premise) + “Abortion takes an innocent life” (minor premise) → “Therefore, abortion is wrong.” Your task is not to agree or disagree but to analyse distribution, ambiguity, and the truth of the premises — treating the argument as a formal structure to be solved.

当你在剑桥考试中遇到伦理论证时，把隐藏的三段论划出来。例如，“剥夺无辜生命是不对的”（大前提） + “堕胎剥夺了无辜生命”（小前提） → “因此，堕胎是错误的”。你的任务不是同意或不同意，而是分析周延性、语词歧义以及前提的真值——把论证当成需要拆解的形式结构来处理。

2. The Hedonic Calculus Protocol | 快乐计算协议

Jeremy Bentham’s act utilitarianism can be reduced to a decision-making formula: U = Σ (Intensity × Duration × Certainty × Propinquity) – Σ Pain factors. Although Bentham never wrote it exactly this way, Cambridge philosophers use this shorthand to illustrate the seven dimensions of pleasure — intensity, duration, certainty, propinquity, fecundity, purity, and extent. In an examination, you must show how a utilitarian would compute, not that numbers really appear.

杰里米·边沁的行为功利主义，可以归纳为一个决策公式：U = Σ (强度 × 持续时间 × 确定性 × 临近性) – Σ 痛苦因素。尽管边沁本人从未精确地这么写过，但剑桥哲学家用这套速记来阐明快乐的七个维度——强度、持续时间、确定性、临近性、繁殖性、纯度和范围。在考试中，你必须表现出功利主义者会如何“计算”，而不是真的给出数字。

U = Σ(Pᵢ× I × D × C × Pr) − Σ(Sᵢ× I × D × C × Pr)

The protocol reminds you that every pleasure and pain is to be weighed impartially. When evaluating a dilemma — say, whether to lie to protect a friend — you run the calculus: the pleasure of safety (duration moderate, intensity high) versus the pain of betrayed trust. Cambridge examiners want you to critique this by pointing out that not all pleasures are commensurable; John Stuart Mill’s distinction between higher and lower pleasures acts as a correction term in any advanced version of the ‘formula’.

这个计算协议提醒你，每一种快乐和痛苦都应当不偏不倚地权衡。当评估一个两难困境——比如是否该为保护朋友而撒谎——你就运行一遍这个计算：安全感带来的快乐（持续时间中等，强度高）对比背信所带来的痛苦。剑桥考官希望你能对此提出批评，指出并非所有快乐都可通约；约翰·斯图尔特·密尔对高级快乐和低级快乐的区分，正是这套“公式”在任何高级版本里的修正项。

3. The Categorical Imperative Test Kit | 定言律令测试套件

Immanuel Kant’s moral law operates as a universalisation filter: Before acting, formulate the maxim of your action, then submit it to the test — “Can I rationally will that this maxim become a universal law?” If the maxim generates a contradiction in conception or a contradiction in will, the action is impermissible. The Cambridge syllabus wants you to treat this as a strict logical procedure, not a mere intuition pump.

伊曼努尔·康德的道德法则，正像一台普遍化过滤器：在行动之前，先把你的行为准则表述出来，然后将它送交检验——“我能否理性地意愿这条准则成为一条普遍法则？”如果这条准则导致了构想上的矛盾或意愿上的矛盾，该行动就是不被允许的。剑桥大纲希望你把它当作一套严格的逻辑程序来对待，而不是一个单纯的直觉泵。

Maxim M is permissible ⇔ M can be consistently universalised without contradiction

Consider a maxim: “I will make a false promise to secure a loan.” If universalised, the institution of promising collapses, because nobody would trust promises anymore — a contradiction in conception. Kant’s formula exposes that the very possibility of the action depends on not universalising it. In Cambridge essays, always show both the maxim and the contradiction; the “formula” is your checklist for moral permissibility.

考虑这样一条准则：“为了获得贷款，我将作出虚假承诺。”一旦普遍化，承诺这个制度就崩溃了，因为再也没有人会相信承诺——这就构成了构想上的矛盾。康德的公式揭示出，该行动之所以可能，恰恰有赖于它没有被普遍化。在剑桥的论文写作中，务必同时展示准则与矛盾；这条“公式”就是你判断道德可允许性的核查清单。

4. The Tripartite Definition of Knowledge with Gettier Shock | 三元知识定义与盖梯尔冲击

Since Plato, knowledge has been analysed as: K = J + T + B (Justified True Belief). The Cambridge course expects you to write this as a necessary-and-sufficient-conditions formula. If a subject S knows a proposition p, then: (i) p is true; (ii) S believes that p; (iii) S is justified in believing that p. This tripartite definition seemed stable until Edmund Gettier’s counter-examples showed that these three conditions are not jointly sufficient.

自柏拉图以来，知识就被分析为：K = J + T + B（经过辩护的真信念）。剑桥课程要求你把它写成一组必要且充分条件式的公式。若主体 S 知道命题 p，则：(i) p 为真；(ii) S 相信 p；(iii) S 对 p 的信念是经过辩护的。这个三元定义看起来四平八稳，直到埃德蒙·盖梯尔用反例表明，这三个条件加在一起也并非充分。

K(p) ← J(p) ∧ T(p) ∧ B(p) ? (Gettier cases reveal insufficiency)

In a classic Gettier-style case, you have a justified true belief that accidentally matches reality. For example, you see a broken clock that happens to show the correct time; your belief “It is 3:00 PM” is justified (the clock says so) and true, yet we hesitate to call it knowledge. The formula breaks down, and Cambridge examiners love to see you explain why: the justification is not appropriately connected to the truth. The search for a repaired formula — adding a “no false lemmas” condition or a causal connection — is central to epistemology questions.

在一个经典的盖梯尔式案例中，你拥有一个偶然与事实相符的经过辩护的真信念。比方说，你看见一座坏掉的钟恰巧显示的是正确时间；你“现在是下午三点”的信念是经过辩护的（钟是这么显示的）并且为真，但我们却不愿称之为知识。公式就此失效，剑桥考官很乐意看到你解释原因：辩护与真相之间缺乏恰当的联结。人们尝试修补这一公式——比如增加“没有虚假引理”的条件或要求因果关联——这始终是认识论题目的核心。

5. The Divine Command Theory Equation | 神命论方程

In meta-ethics, divine command theory is often crystallised into: An act X is morally right if and only if God commands X. Conversely, X is morally wrong if and only if God forbids X. Cambridge students are asked to scrutinise this equation using the Euthyphro dilemma: “Is the pious loved by the gods because it is pious, or is it pious because it is loved?” Transposed into our formula, the dilemma asks whether the command is grounded in independent moral standards or merely in divine will.

在元伦理学中，神命论常被凝结为：一个行为 X 在道德上是正确的，当且仅当上帝命令 X。反之，X 在道德上是错误的，当且仅当上帝禁止 X。剑桥的学生被要求用欧绪弗洛两难来审视这个方程：“虔敬是因被众神所爱而是虔敬的，抑或因其是虔敬的才被众神所爱？”转换到我们的公式里，这个两难是在追问：命令到底是植根于独立的道德标准，还是仅仅植根于神的意志。

Right(X) ⇔ Commanded_by_God(X)
Wrong(X) ⇔ Forbidden_by_God(X)

If you accept the first horn — that God commands what is independently good — then morality stands above God, and the theory collapses into moral realism. If you take the second horn — that goodness is entirely constituted by God’s command — then morality appears arbitrary, for God could have commanded cruelty. This logical fork is your sharpest tool when evaluating theistic ethics in Cambridge philosophy. Always trace back the equation to the Euthyphro dilemma.

如果你接受第一个角——上帝所命令的是本来就独立为善的东西——那么道德就凌驾于上帝之上，神命论也就崩塌成道德实在论了。如果你取第二个角——善完全由上帝的命令所构成——那么道德就会显得任意武断，因为上帝本也可以命令残忍之事。这个逻辑分岔，是你在剑桥哲学中评估有神论伦理学最锋利的工具。一定要沿着方程回溯到欧绪弗洛两难。

6. The Problem of Evil Logical Deduction | 恶之问题的逻辑演绎

The logical problem of evil sets up an inconsistent tetrad: (1) God is omnipotent; (2) God is wholly benevolent; (3) God is omniscient; (4) Evil exists. Cambridge expects you to demonstrate that if any three are true, the fourth must be false — unless a theodicy breaks the entailment. This is often presented as a reductio ad absurdum: assuming an all-perfect being and the existence of evil leads to contradiction.

恶的逻辑问题搭建出一个不一致的四元组：(1) 上帝全能；(2) 上帝全善；(3) 上帝全知；(4) 恶存在。剑桥期望你证明，如果其中任意三个为真，第四个就必然为假——除非有某种神正论打破这种蕴含关系。这通常被呈现为一个归谬论证：同时假定存在一位全完美的存在者与恶，会导致逻辑矛盾。

(Omnipotence ∧ Omnibenevolence ∧ Omniscience) → ¬Evil
But Evil exists
∴ ¬(Omnipotence ∧ Omnibenevolence ∧ Omniscience)

Plantinga’s free will defence works like an error-correcting code: it introduces a logically possible proposition — “God could not create a world with free creatures who always choose good” — that breaks the entailment from the divine attributes to the non-existence of evil. In your Cambridge answers, display the deduction clearly and then evaluate whether the defence merely shows logical compatibility or whether it offers adequate moral justification.

普兰丁格的自由意志辩护，就好比一段纠错编码：它引入一个在逻辑上可能的命题——“上帝不可能创造一个包含自由受造物且他们永远行善的世界”——以此切断了从神圣属性推出恶不存在的蕴含关系。在你的剑桥考卷答案里，要清晰展示这一演绎过程，然后评估这个辩护到底只是展示了逻辑相容性，还是提供了充分的道德辩护。

7. Descartes’ Cogito and the Foundationalist Formula | 笛卡尔的“我思”与基础主义公式

Descartes’ method of doubt attempts to demolish every belief that is not indubitable, searching for a fundamentum inconcussum. The outcome is the famous cogito: Cogito, ergo sum — “I think, therefore I am.” This can be expressed as a foundational belief formula: For any belief B that I hold, B is justified only if it is either self-evidently certain (like the cogito) or derived from such certainties. Cambridge epistemology requires you to assess whether this model survives the test of the Cartesian circle.

笛卡尔的怀疑方法，试图摧毁每一个并非确凿无疑的信念，以寻找那个阿基米德支点。结果就是著名的“我思故我在”（Cogito, ergo sum）。这可以用一个基础主义信念公式来表达：对我所持的任何信念 B 来说，B 能得到辩护，当且仅当它要么是自明的确定性（如“我思”），要么是从这类确定性中推导而来的。剑桥的认识论要求你评估这个模型是否能经受住笛卡尔循环的考验。

Cogito → Sum → the self exists as a thinking thing
Certainty = Indubitability

Note the subtlety: the cogito is not a syllogism but a performative insight — the very act of doubting one’s existence confirms it. However, moving from “I exist as a thinking thing” to the reliability of clear and distinct perceptions requires a non-deceiving God, which itself is justified only if clear and distinct perceptions are reliable. This circularity is often diagrammed as a feedback loop; the Cambridge candidate must decide whether it is viciously circular or whether Descartes offers a coherent foundationalist structure.

注意其中的微妙之处：我思并不是一个三段论，而是一种作用性的洞见——怀疑自身存在这个动作本身就证实了自身存在。然而，从“我作为一个思维之物存在”走到“清晰分明感知的可靠性”，中间需要一位不欺骗人的上帝作担保，而这一担保本身又只有在清晰分明感知已被假定为可靠时才能获得辩护。这个循环常被画成一个反馈回路；剑桥的考生必须判断，这到底是恶性循环，还是笛卡尔确实给出了一种融贯的基础主义结构。

8. The Teleological Design Probability Ratio | 目的论设计的概率比

William Paley’s watchmaker analogy can be reframed as a likelihood formula: P(complex ordered system | designer) >> P(complex ordered system | chance). Cambridge students use this to understand arguments from design. Paley argues that the intricate complexity of the natural world — the eye, the wing — is overwhelmingly more likely under the hypothesis of a designer than under blind natural processes. Hume’s criticisms, on the other hand, attack the probability inference as weak analogical reasoning.

威廉·佩利的钟表匠类比，可以重新表述为一个似然性公式：P(复杂有序系统 | 设计者) >> P(复杂有序系统 | 偶然)。剑桥学生用这个来理解设计论证。佩利主张，自然界的高度复杂——无论是眼睛还是翅膀——在存在设计者的假设下远比在盲目自然过程中更可能出现。而休谟的批判则攻击这一概率推理，认为它只是一种薄弱的类比推理。

L(D|H₁) ÷ L(D|H₀) → very large, but does this confirm H₁?

The post-Darwinian world introduces a third hypothesis: natural selection can generate the appearance of design without a mind. The Cambridge philosopher must calculate how the probability shifts when natural selection is acknowledged as a cumulative filter. Moreover, the problem of dysteleology — poor design features like the blind spot in the vertebrate eye — reduces the likelihood ratio. In your essays, treat the design argument as a Bayesian updating problem: prior probability of theism matters just as much as the evidence. Show that the “formula” is sensitive to background assumptions.

后达尔文的世界引入了第三个假说：自然选择无需心智就可以产生设计的表象。剑桥的哲学学生必须计算，当认识到自然选择是一个累积式过滤器之后，概率会发生怎样的变化。此外，反面目的论的问题——比如脊椎动物眼睛的盲点这类糟糕的设计特征——会降低似然比。在你的论文中，把设计论证当成一次贝叶斯更新来处理：有神论的先验概率与证据同样重要。要表现出这条“公式”对背景假设的敏感度。

9. The Emotivist Meaning Equation | 情感主义意义方程

A. J. Ayer’s verification principle consigns ethical statements to non-cognitive expression. His formula: A statement S is literally meaningful only if it is either analytically true (tautology) or empirically verifiable. Ethical utterances fail this test, so “Murder is wrong” does not assert a factual proposition but expresses the speaker’s feeling of disapproval. Cambridge meta-ethics asks you to unpack this with symbolic precision: Ethical language = emotive expression + imperative command.

艾耶尔的证实原则把伦理陈述归入了非认知表达的范畴。他的公式是：一个陈述 S 具有字面意义，当且仅当它要么是分析的真理（同义反复），要么是经验上可证实的。伦理话语通不过这个检验，因此“谋杀是错误的”并不是在断言一个事实命题，而是在表达说话者不赞同的情感。剑桥的元伦理学要求你用符号般的精确性来拆解这一点：伦理语言 = 情感表达 + 祈使命令。

“X is wrong” → “Boo to X!” + “Do not do X.”

The boo-hoorah theory, however, faces the Frege-Geach problem: how can emotive statements feature in logical inferences? If “If lying is wrong, then getting your brother to lie is wrong” is a moral argument, the antecedent “lying is wrong” must have a consistent meaning across both asserted and unasserted contexts. Cambridge candidates need to show whether the expressivist can provide a logic of attitudes without collapsing back into propositional truth conditions — a deep test of any meta-ethical “formula”.

然而，“呸—好哇”理论面临着弗雷格—吉奇问题：情感性语句如何能参与逻辑推理？如果“如果撒谎是错误的，那么让你弟弟去撒谎也是错误的”是一个道德论证，那么作为前件的“撒谎是错误的”就必须在断定和未断定的语境中拥有恒常的意义。剑桥考生需要说明，表达主义者能否在不退回到命题真值条件的前提下，提供一套态度逻辑——这是对任何元伦理“公式”的深层检验。

10. The Relativistic Consistency Check | 相对主义一致性检验

Cultural relativism often appears as: Truth of moral claim C relative to culture X. But the universal claim “All moral truths are relative to a culture” seems to be itself a non-relative assertion. Cambridge expects you to spot this paradox: If relativism is true, then it is only relatively true; if it is absolutely true, then at least one moral truth is absolute. This self-referential inconsistency is the internal “bug” in the relativist formula.

文化相对主义常表现为：道德主张 C 的真假是相对于文化 X 的。然而，那个普遍的宣称“所有道德真理都相对于文化”本身，却似乎是一个非相对性的断言。剑桥期待你发现这个悖论：如果相对主义为真，那它本身也只是相对为真；如果它是绝对为真，那么至少有一个道德真理是绝对的。这种自我指涉的不一致性，就是相对主义公式内在的“漏洞”。

∀p (MorallyTrue(p) → ∃c (Culture(c) ∧ TrueRelativeTo(p,c)))
Self-application: Is this truth relative?

The Cambridge discussion moves beyond simple self-refutation: some relativists respond that their theory is not a first-order moral claim but a second-order meta-ethical thesis about the nature of moral truth, and thus immune. Your task is to evaluate whether this move succeeds or merely postpones the problem. Outline the regress: if all statements are only relatively true, then the statement “all statements are only relatively true” requires its own cultural frame — leading to an infinite complexity or an arbitrary stopping point.

剑桥的讨论并不仅仅停留在简单的自驳上：一些相对主义者回应说，他们的理论不是一个一阶的道德主张，而是一个关于道德真理本性的二阶元伦理学论题，因此是对这种悖论免疫的。你的任务是评估这一步是否成功，或者只是在推迟问题而已。勾勒出这个无穷倒退：如果所有陈述都只是相对为真，那么“所有陈述都只是相对为真”本身也需要另一个文化框架——最终走向无限复杂性，或者在一个随意的地方停下。

11. The Consequentialist vs. Deontological Decision Matrix | 后果论与义务论的决策矩阵

Cambridge ethical dilemmas often expect you to compare two rival “algorithms”. Act consequentialism computes: Choose the action A such that A maximises expected good outcomes. Deontology filters: Choose the action A such that A passes the universalisability test and does not use anyone merely as a means. In many scenarios, the outputs differ; your job is to show how the two systems treat the same case — for instance, lying to a murderer at the door. Display this with a simple decision table.

剑桥的伦理困境题常要求你比较两种对立的“算法”。行为后果论的计算：选择行动 A，使得 A 能最大化被预期的好结果。义务论的过滤：选择行动 A，使得 A 通过普遍化检验，并且不把任何人纯粹当作手段来利用。在很多情境中，两者给出的结果是不同的；你的任务是展现出这两个系统如何处理同一个案例——比如，面对门口杀手的谎言。用一张简单的决策表来呈现这一点。

Case: Lying to murderer at the door	Consequentialist output	Deontological output
Expected outcome	Life saved → Good	Lying cannot be universalised
Moral verdict	Permissible/Obligatory	Impermissible

The Cambridge examiner will then ask you to criticise each “formula”: the consequentialist might struggle to guarantee justice, for it could sanction punishing an innocent if it maximises utility; the deontologist might be accused of caring more about rule-purity than about the life of an innocent friend. Your essay should treat these as competing formalisms, each with strict derivation rules and problematic edge cases.

剑桥的考官接下来就会要求你批评每一种“公式”：后果论者可能难以保障正义，因为如果惩罚无辜者能使效用最大化，它就可能被允许；义务论者则可能被指责，比起一个无辜朋友的生命，他更在意规则的纯洁性。你的论文应该将这两者视为相互竞争的形式系统，各自都拥有严格的推导规则和成问题的极端案例。

12. The Philosopher’s Toolkit: Validity, Soundness, and Fallacy Detectors | 哲学工具箱：有效性、健全性与谬误探测器

No quick-reference handbook is complete without the fundamental evaluative formulas that apply to every argument on the Cambridge paper. Validity is a structural formula: An argument is valid if it is impossible for the premises to be true and the conclusion false. Soundness adds a factual condition: An argument is sound if it is valid and all its premises are actually true. A fallacy occurs when this logical mechanism breaks — and Cambridge prizes the ability to name the fallacy, from ad hominem to post hoc ergo propter hoc.

没有哪一本速查手册能缺少那条适用于剑桥试卷上每一个论证的根本评估公式。有效性是一个结构性的公式：一个论证是有效的，当且仅当不可能前提为真而结论为假。健全性则引入事实条件：一个论证是健全的，当且仅当它是有效的且所有前提实际为真。谬误就发生在这一逻辑机制失灵之时——而剑桥很看重你给谬误命名的能力，从人身攻击到以先后为因果。

Validity: If (P₁ ∧ P₂ ∧ … ∧ Pₙ) → C necessarily
Soundness: Validity + True(P₁) ∧ True(P₂) ∧ … ∧ True(Pₙ)

When you read a passage in the exam, mentally run these checks. Is the inference valid? Could the conclusion be false even if every premise were granted? Then identify the unstated assumptions — those hidden “Pᵢ” terms needed to make the argument valid. The gap between evidence and conclusion is exactly the space where philosophical genius (and good marks) resides. Always write with the precision of a logician, even when discussing the deepest mysteries of existence.

当你考试时读到一段材料，就在心里运行这些检测。推理有效吗？即便每个前提都为真，结论仍可能为假吗？然后识别那些未说出口的假定——那些为了让论证有效而必须添上的隐匿“Pᵢ”项。证据与结论之间的缝隙，恰是哲学才华（以及高分）藏身的地方。即使在探讨最深邃的存在之谜，也永远要带着逻辑学家的精确去下笔。

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Year 11 Cambridge Philosophy: Quick Reference Handbook of Formulas and Theorems | 剑桥哲学：公式定理速查手册