📚 Philosophy Formulae & Key Principles Quick Reference Handbook | 哲学公式定理速查手册
Welcome to your go-to revision handbook for Cambridge IGCSE Philosophy. Just as mathematics employs formulae to solve problems, philosophy relies on powerful conceptual tools—axioms, principles, and logical structures—that you can apply to arguments, essays, and critical thinking tasks. This guide presents the most essential ‘formulae’ in epistemology, ethics, metaphysics, and logic, complete with clear definitions, symbolic representations, examples, and exam-ready tips. Treat these entries as your philosophical toolbox; once memorised and understood, they will help you dissect complex ideas with confidence.
欢迎使用这本剑桥 IGCSE 哲学速查手册。正如数学依靠公式解题，哲学同样依赖强大的概念工具——公理、原则和逻辑结构——你可以将它们应用于论证、论文和批判性思维任务。本指南提炼了认识论、伦理学、形而上学和逻辑学中最重要的“公式”，并配以清晰的定义、符号表述、示例与应试技巧。请将这些条目视为你的哲学工具箱；一旦熟记并领会，你就能自信地拆解复杂的思想。
1. Cogito, Ergo Sum – The Foundational Axiom | 我思故我在：基础公理
Descartes’ seminal insight after his method of radical doubt can be treated as the first theorem of modern philosophy. The formula states that in the very act of doubting one’s own existence, a thinking subject must exist to perform that act of doubt.
笛卡尔在其激进怀疑方法之后得出的开创性洞见，可被视为现代哲学的第一条定理。该公式表明：在怀疑自身存在的那个行为中，必须存在着一个进行怀疑的思维主体。
Formula: If ‘D’ stands for ‘I doubt (or think)’, then ‘C’ (I exist) follows necessarily. D → C. In words: ‘I think, therefore I am’ (Cogito, ergo sum).
公式： 设 D 代表“我怀疑（或我思考）”，则 C（我存在）必然推出。D → C。即：“我思，故我在”（Cogito, ergo sum）。
Usage: Use this principle whenever you encounter radical scepticism about the external world. It proves that even if an evil demon deceives you about everything, your own existence as a thinking being remains indubitable. In exams, link it to foundationalism in epistemology.
应用： 每当你面对关于外部世界的彻底怀疑时，就可运用这一原则。它证明，即使有个邪恶的恶魔在一切事情上欺骗你，你作为思考者的存在仍然无可置疑。在考试中，可将其与认识论中的基础主义相联系。
2. Hume’s Fork – The Principle of Empirical Verification | 休谟之叉：经验验证原则
David Hume’s ‘fork’ is a logical formula for dividing all meaningful propositions into two exhaustive categories. It acts as a razor-sharp tool for testing whether a statement carries genuine cognitive content.
大卫·休谟的“叉子”是一个将一切有意义的命题划分为两个互斥类别的逻辑公式。它宛如一把锋利的剃刀，用来检测一个陈述是否具备真正的认知内容。
Formula: A meaningful proposition P is either (1) a ‘Relation of Ideas’ (RI) or (2) a ‘Matter of Fact’ (MF). P ∈ {RI, MF}. RI propositions are demonstrable by reason alone (e.g., mathematics), necessarily true, and their denial implies contradiction. MF propositions are known through experience, are contingent, and their denial is possible without contradiction.
公式： 一个有意义的命题 P，要么是 (1) “观念关系”（RI），要么是 (2) “事实”（MF）。P ∈ {RI, MF}。RI 命题仅凭理性就可证明（如数学），必然为真，其否定蕴含矛盾。MF 命题通过经验获知，是偶然的，否定它不会产生矛盾。
Usage: Apply Hume’s Fork to evaluate metaphysical or theological claims like ‘God exists’ or ‘the soul is immortal’. If they are neither relations of ideas nor verifiable matters of fact, Hume concludes we should ‘commit them to the flames’. This is your core revision tool for the verification principle and logical positivism.
应用： 用休谟之叉来评估形而上学或神学主张，如“上帝存在”或“灵魂不朽”。如果它们既非观念关系，也非可验证的事实，休谟的结论是应将其“付之一炬”。这是你复习证实原则和逻辑实证主义的核心利器。
3. The Is-Ought Gap – Hume’s Guillotine | 实然-应然鸿沟：休谟断头台
Hume’s guillotine is a logical principle that forbids deriving normative ‘ought’ conclusions directly from descriptive ‘is’ premises. It is a formal rule of inference that keeps moral reasoning coherent.
休谟断头台是一条逻辑原则，它禁止从描述性的“是”前提中直接推出规范性的“应当”结论。这是一条保持道德推理融贯的形式规则。
Formula: For any argument, if premises P₁, P₂ … Pₙ contain only ‘is’ statements (factual descriptions), then no conclusion C containing an ‘ought’ (moral prescription) can be validly deduced unless a bridging ‘ought’ premise is already present. ∄ (Ought-conclusion) from purely (Is-premises).
公式： 对任何论证，若前提 P₁, P₂ … Pₙ 仅包含“是”陈述（事实描述），那么，除非已包含一座桥接性的“应当”前提，就不能有效推出一个包含“应当”（道德规范）的结论 C。∄（纯粹从“是”前提推出“应当”结论）。
Usage: Use this formula to critique naturalistic fallacies, e.g., arguments that ‘because evolution favours selfishness, we ought to be selfish’. In ethics essays, you can precisely name the gap and explain why a value premise is missing. It is essential for understanding theories like Moore’s non-naturalism and emotivism.
应用： 运用此公式批判自然主义谬误，例如“因为进化偏爱自私，我们应当自私”这类论证。在伦理学论文中，你可以准确指出这一鸿沟，并解释为何缺少价值前提。这对于理解摩尔非自然主义和情感主义等理论至关重要。
4. The Categorical Imperative – Kant’s Moral Formula | 定言令式：康德的道德公式
Immanuel Kant’s supreme principle of morality is a test for maxims, functioning like a universal moral algorithm. It has several formulations, but they are mathematically equivalent in Kant’s system.
康德的最高道德原则是对准则的检验，像一个普适的道德算法在运作。它拥有多种表述形式，但在康德体系中，它们在数学意义上是等价的。
Formula (First Formulation): Act only on that maxim M whereby you can at the same time will that M should become a universal law (UL). M → Possible UL without contradiction. If contradiction in conception or will arises, action is forbidden. Formula (Second Formulation): Treat humanity (H) always as an end (End) and never merely as a means (Means). Action A is permissible iff H is respected as End.
公式（第一表述）： 要只按照你同时愿意它成为一条普遍法则（UL）的那个准则 M 去行动。M → 可能的 UL 且无矛盾。如果构想中或意愿中产生矛盾，则该行为被禁止。公式（第二表述）： 总是将人性（H）视为目的（End），而绝不仅仅当作手段（Means）。行为 A 被允许，当且仅当 H 被尊重为目的。
Usage: Test the maxim of ‘making a lying promise to get money’. Can it be universalised? No, because universalising it would make the institution of promising collapse, yielding a contradiction in conception. This formula is your primary tool for evaluating deontological arguments, contrasting with utilitarianism.
应用： 检验“为了借钱而做假承诺”这条准则。它能被普遍化吗？不能，因为将它普遍化会使承诺制度崩溃，产生构想上的矛盾。该公式是你评估道义论论证，并与功利主义作对比的主要工具。
5. The Greatest Happiness Principle – Utility Calculus | 最大幸福原则：功利计算
Jeremy Bentham and John Stuart Mill’s ethical framework reduces moral rightness to a formula of pleasure and pain. The principle of utility states that the right action is the one that maximises net happiness for all affected.
杰里米·边沁与约翰·斯图尔特·密尔的伦理学框架将道德正当性化约为一个关于快乐与痛苦的计算公式。功利原则表明，正当的行为就是为所有受影响者带来最大净幸福的行为。
Formula: For any set of possible actions {A₁, A₂, … Aₙ}, the morally right action Aₓ is such that ∑ (Pleasureₓ – Painₓ) for all sentient beings is maximised. Bentham added hedonic calculus variables: intensity (I), duration (D), certainty (C), propinquity (P), fecundity (F), purity (Pu), and extent (E). Grand total: Value = I × D × C × P (with factors for F and Pu).
公式： 对于任意一组可能行为 {A₁, A₂, … Aₙ}，道德上正确的行为 Aₓ 是使得所有有感知者的 ∑ (快乐ₓ – 痛苦ₓ) 最大化的那个行为。边沁增添了快乐计算变量：强度 (I)、持续时间 (D)、确定性 (C)、邻近性 (P)、丰产性 (F)、纯度 (Pu) 和范围 (E)。总值 = I × D × C × P（再加上 F 与 Pu 的因子）。
Usage: Use this formula to weigh consequences in moral dilemmas, such as whether to torture a terrorist to save many lives. Mill’s refinement distinguishes higher and lower pleasures, so you must adjust the qualitative weighting. This is the core formula for act utilitarianism.
应用： 运用此公式权衡道德两难中的后果，例如是否该拷打一个恐怖分子以拯救众多生命。密尔的改进区分了高级与低级快乐，因此你必须调整质的权重。这是行为功利主义的核心公式。
6. The Veil of Ignorance – Rawls’ Justice Formula | 无知之幕：罗尔斯的正义公式
John Rawls’ theory of justice offers a procedural formula for deriving fair principles. The original position behind a veil of ignorance eliminates biases, ensuring that rational agents choose principles of justice impartially.
约翰·罗尔斯的正义理论提供了一个推导公平原则的程序性公式。在无知之幕背后的原初状态消除了偏见，确保理性行动者不偏不倚地选择正义原则。
Formula: Let OP be the original position with parties behind a Veil of Ignorance (VI), unaware of their own class, talents, or conception of the good. Under VI, principles P are chosen iff P would be agreed upon by rational, self-interested agents under conditions of fairness. Outcome: Two Principles — (1) Equal basic liberties (L) for all, and (2) social/economic inequalities (Ine) arranged so that they are (a) to the greatest benefit of the least advantaged (Difference Principle), and (b) attached to positions open to all under fair equality of opportunity (FEO).
公式： 设 OP 为原初状态，其中各方处在无知之幕 (VI) 之后，不知晓自身的阶级、天赋或善观念。在 VI 下，原则 P 被选中，当且仅当 P 会得到处于公平条件下的理性自利行动者同意。结果：两条原则——(1) 平等的基本自由 (L) 属于所有人；(2) 社会与经济不平等 (Ine) 的安排应 (a) 最有利于最不利者（差别原则），且 (b) 在公平的机会平等 (FEO) 下，职位向所有人开放。
Usage: Apply the veil of ignorance as a test for proposed policies or social arrangements. Ask: would a person unaware of their own position choose this rule? It helps you construct principled arguments about the welfare state, taxation, and discrimination. In exams, contrast it with Nozick’s entitlement theory.
应用： 将无知之幕作为对提议政策或社会安排的检验。问：一个不知晓自身处境的人会选择这条规则吗？这有助于你建构关于福利国家、税收和歧视的原则性论证。考试中，可与诺齐克的资格理论进行对比。
7. Occam’s Razor – The Parsimony Formula | 奥卡姆剃刀：简约公式
Occam’s Razor, or the principle of parsimony, is a methodological rule for theory choice. It instructs us not to multiply entities beyond necessity, favouring simpler explanations that still account for the evidence.
奥卡姆剃刀，即简约原则，是一条关于理论选择的方法论规则。它教导我们：如无必要，勿增实体，倾向于仍能解释证据的更简单说明。
Formula: Given two competing hypotheses H₁ and H₂ that equally explain the data D, if H₁ posits fewer additional assumptions or kinds of entities than H₂ (|Assumptions(H₁)| < |Assumptions(H₂)|), then H₁ is to be preferred. In symbols: Prefer H_minEntity. The razor slices off superfluous metaphysical parts.
公式： 给定两个竞争假设 H₁ 和 H₂，它们同样地解释了数据 D，如果 H₁ 提出的额外假设或实体种类比 H₂ 少（|假设(H₁)| < |假设(H₂)|），则应优先选择 H₁。符号化：选择 H_最少实体。这把剃刀削去了多余的形而上学部件。
Usage: Use Occam’s Razor to evaluate dualism vs. materialism in the philosophy of mind. Materialism posits only physical substance, while dualism adds mental substance. If both explain experience, the razor favours materialism unless dualism provides better explanatory power. It’s also key in arguments about the existence of God.
应用： 用奥卡姆剃刀评估心灵哲学中的二元论与唯物主义。唯物主义只设定物理实体，而二元论增添了心灵实体。如果两者都能解释经验，剃刀就偏向唯物主义，除非二元论提供了更好的解释力。它在关于上帝存在的论证中也很关键。
8. The JTB Formula – Tripartite Definition of Knowledge | JTB 公式：知识的三元定义
Since Plato’s Theaetetus, the standard analysis of knowledge has been captured by the formula: Knowledge is Justified True Belief. This provides the necessary and sufficient conditions for someone S to know a proposition P.
自柏拉图的《泰阿泰德篇》以来，对知识的标准分析就凝聚为这样的公式：知识是得到证成的真信念。它为某人 S 知道一个命题 P 提供了必要且充分的条件。
Formula: S knows that P iff (1) P is true (T), (2) S believes that P (B), and (3) S’s belief that P is justified (J). K = J ∧ T ∧ B. The justification condition is the tricky element; Gettier cases show that this formula might be incomplete without a fourth condition to defeat luck.
公式： S 知道 P，当且仅当 (1) P 为真 (T)，(2) S 相信 P (B)，且 (3) S 对 P 的信念得到证成 (J)。K = J ∧ T ∧ B。证成条件是个棘手元素；葛梯尔案例表明，若缺少第四个排除运气的条件，该公式可能并不完整。
Usage: This is the central formula in epistemology. In exam questions about the definition of knowledge, you should first set out the JTB analysis. Then introduce a classic Gettier case (e.g., the stopped clock showing the correct time by chance) to demonstrate that JTB allows accidental true belief, prompting upgrades like reliabilism or the ‘no false lemmas’ condition.
应用： 这是认识论中的核心公式。在关于知识定义的考题中，你应首先列出 JTB 分析。接着引入一个经典的葛梯尔案例（如停了的钟偶然显示正确时间），来展示 JTB 容许了偶然的真信念，从而促使升级，如可靠主义或无错误引理条件。
9. The Problem of Evil – Logical Inconsistency Formula | 恶的问题：逻辑不一致公式
The problem of evil presents a challenge to traditional theism, framed as a contradiction within the set of divine attributes. J.L. Mackie formalised it as a logical inconsistency between the existence of an omnipotent, omnibenevolent God and the presence of evil.
恶的问题对传统有神论提出了挑战，其形式为神性属性集合内部的一个矛盾。J.L. 麦基将其形式化为全能、全善的上帝存在与恶的存在之间的逻辑不一致。
Formula: The inconsistent triad: ① God is omnipotent (all-powerful, ∀ situations S, God can prevent S). ② God is omnibenevolent (perfectly good, ∀ evil E, God desires to prevent E). ③ Evil exists (∃ E such that E occurs). Mackie argues (① ∧ ② ∧ ③) → Contradiction. If ① and ② are true, ③ must be false; if ③ is true, at least one of ① or ② must be adjusted. There is no possible world where all three hold simultaneously.
公式： 不一致三元组：① 上帝全能（全能的，∀ 状况 S，上帝能阻止 S）。② 上帝全善（完美的善，∀ 恶 E，上帝欲阻止 E）。③ 恶存在（∃ E 使得 E 发生）。麦基论证 (① ∧ ② ∧ ③) → 矛盾。如果 ① 和 ② 为真，③ 必为假；如果 ③ 为真，① 与 ② 中至少有一个须调整。不存在所有三项同时成立的可能性界。
Usage: This formula is a deductive argument against the classical theistic God. In essays, you must portray the theist’s responses: the Free Will Defence (evil is a consequence of a greater good, free will) or soul-making theodicy. These replies deny that an omnibenevolent being must always prevent evil if a higher-order good is only achievable through it. Use the formula to structure your analysis of the problem.
应用： 该公式是一个反对古典有神论上帝的演绎论证。在论文中，你必须描绘有神论者的回应：自由意志辩护（恶是更高善——自由意志的后果）或灵魂塑造神义论。这些回应否认全善的存在者必须总是阻止恶，如果更高阶的善唯有通过恶才能达成。运用此公式来构建你对这一问题的分析。
10. Plato’s Theory of Forms – The One Over Many Formula | 柏拉图的理型论：多上之一公式
Plato’s metaphysics explains the existence of properties by positing a separate realm of perfect, immutable Forms. The basic formula is that for any set of many particular things sharing a common name, there is a single corresponding Form that causes them to be what they are.
柏拉图的形而上学通过设定一个独立、完美的、不变的理型领域来解释属性的存在。其基本公式是：对于任何一组共享共同名称的众多具体事物，都存在着一个与之对应的单一理型，是它使得这些事物成为它们所是的样子。
Formula: Let F be a property (e.g., beauty, justice, largeness). For all particular things x₁, x₂ … xₙ that are F, there exists a Form Φ (F-itself) such that (i) Φ is perfectly and purely F, (ii) Φ is eternal and non-spatial, and (iii) each particular xᵢ participates in or imitates Φ. xᵢ is F in virtue of its relation to Φ. Φ ≠ xᵢ and Φ occupies a higher ontological level.
公式： 设 F 为一种属性（如美、正义、大）。对于所有是 F 的具体事物 x₁, x₂ … xₙ，存在着一个理型 Φ（F-本身），满足：(i) Φ 是完美的、纯粹的 F；(ii) Φ 是永恒的且非空间的；(iii) 每一具体事物 xᵢ 分有或模仿 Φ。xᵢ 是 F 乃凭借其与 Φ 的关系。Φ ≠ xᵢ，且 Φ 占据更高的本体论层次。
Usage: Apply this formula to explain Plato’s argument from the Phaedo about the Form of Equality: we never see perfectly equal sticks, yet we possess the concept of Equality itself. This recollection argument suggests innate knowledge of Forms. In exams, use it to contrast Plato’s rationalism with empiricism, or to introduce the Third Man argument as an objection (infinite regress).
应用： 运用此公式来解释柏拉图在《斐多篇》中关于“相等”理型的论证：我们从未见过完全相等的木棍，却拥有“相等本身”的概念。这个回忆论证暗示了关于理型的天赋知识。考试中，可用它来对比柏拉图的理性主义与经验主义，或引入“第三人论证”作为反对意见（无限回溯）。
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