📚 GCSE Cambridge Philosophy: Quick Reference Handbook of Formulas and Theorems | GCSE 剑桥哲学：公式定理速查手册
Philosophy may seem to resist the tidy certainty of equations, yet beneath every major argument lies a logical structure as reliable as any formula in mathematics. This handbook distils the core ‘formulas’ and ‘theorems’ of the GCSE Cambridge Philosophy syllabus – from propositional logic and ethical calculi to the analysis of free will and the defining conditions of knowledge. Each entry presents a memorable pattern you can recall quickly, apply in essays, and use to evaluate classic philosophical positions with precision.
哲学或许看起来抗拒方程式般的整洁确定性，但在每一个重要论证的背后，都隐藏着一种如同数学公式般可靠的逻辑结构。本手册提炼了 GCSE 剑桥哲学课程中的核心“公式”与“定理”——从命题逻辑、伦理学演算，到自由意志的分析以及知识的定义条件。每个条目都呈现了一种便于记忆的模式，你可以快速回顾、在论文中应用，并精准地评价经典哲学立场。
1. Propositional Logic: The Basic Conditional | 命题逻辑：基本条件式
Many philosophical arguments rest on conditional statements of the form ‘If P, then Q’ (P → Q). P is the antecedent, Q the consequent. Understanding this structure allows you to identify the premises, hidden assumptions, and valid inferences within any line of reasoning.
许多哲学论证都建立在“如果 P，那么 Q”（P → Q）形式的条件语句之上。P 是前件，Q 是后件。理解这一结构，你就能在任何推理过程中识别前提、隐含假设以及有效推论。
The truth table for the conditional is crucial: P → Q is false only when P is true and Q is false. In all other cases – P false, or Q true – the conditional is true. This counter-intuitive feature often appears in paradoxes of material implication, so use it carefully when reconstructing arguments.
条件式的真值表至关重要：只有当 P 真而 Q 假时，P → Q 才为假。其他情况下——P 假，或 Q 真——条件式皆为真。这一反直觉的特性经常出现在实质蕴涵的悖论中，因此在重建论证时需谨慎使用。
2. Valid Argument Forms: Modus Ponens & Modus Tollens | 有效论证形式：肯定前件与否定后件
The two most reliable deductive ‘formulas’ are Modus Ponens and Modus Tollens. Modus Ponens states: If P → Q, and P is true, then Q must be true. Formally: (P → Q), P, therefore Q.
最可靠的两种演绎“公式”是肯定前件和否定后件。肯定前件的表述为：若 P → Q，且 P 为真，则 Q 必定为真。形式化为：(P → Q)，P，所以 Q。
Modus Tollens works in reverse: If P → Q, and Q is false, then P must be false. Formally: (P → Q), ¬Q, therefore ¬P. This pattern underpins many falsification arguments in philosophy of religion (problem of evil) and epistemology (sceptical challenges).
否定后件则反向运作：若 P → Q，且 Q 为假，则 P 必定为假。形式化为：(P → Q)，¬Q，所以 ¬P。这一模式支撑着宗教哲学（恶的问题）和认识论（怀疑论挑战）中的许多证伪论证。
Modus Ponens: (P → Q) ∧ P ⊢ Q
Modus Tollens: (P → Q) ∧ ¬Q ⊢ ¬P
3. Logical Fallacies to Avoid | 应避免的逻辑谬误
Common fallacies can be expressed as flawed ‘formulas’ that mimic valid reasoning. Recognising these patterns sharpens your critical evaluation in the exam.
常见谬误可以表示为模仿有效推理的有缺陷“公式”。识别这些模式能让你在考试中的批判性评价更加敏锐。
Affirming the consequent: (P → Q), Q, therefore P. This is invalid because Q could be caused by something else. Denying the antecedent: (P → Q), ¬P, therefore ¬Q – similarly invalid. These mistakes frequently appear in arguments about the existence of God or the nature of reality.
肯定后件：(P → Q)，Q，所以 P。这是无效的，因为 Q 可能由其他原因导致。否定前件：(P → Q)，¬P，所以 ¬Q——同样无效。这些错误经常出现在关于上帝存在或现实本质的论证中。
Equivocation changes the meaning of a key term mid-argument, breaking the logical chain. The fallacy formula is: term X used in sense A, then sense B; therefore conclusion C, but no valid connection.
偷换概念在论证中途改变关键词的含义，破坏了逻辑链条。其谬误公式为：术语 X 先用于含义 A，后用于含义 B；因此得出 C，但并无有效关联。
4. Epistemology: The JTB Formula for Knowledge | 认识论：知识的 JTB 公式
The classic definition of knowledge is Justified True Belief (JTB): S knows that p if and only if (1) p is true, (2) S believes that p, and (3) S is justified in believing that p. Each component acts as a necessary condition; together they are supposed to be sufficient.
知识的经典定义是确证的真信念（JTB）：S 知道 p，当且仅当（1）p 为真，（2）S 相信 p，且（3）S 有理由相信 p。每个成分都是一个必要条件；合在一起则被认为是充分条件。
Gettier-style counterexamples show the formula can break down: a belief can be true and justified by luck, without genuine knowledge. The standard response is to add a fourth condition – often called the ‘no false lemmas’ condition or a reliability condition – transforming the formula into JTB+X.
盖梯尔式的反例表明该公式可能失效：一个信念可以偶然为真且得到确证，却不构成真正的知识。标准回应是添加第四个条件——通常称为“无错误引理”条件或可靠性条件——将公式转变为 JTB+X。
Knowledge? = Belief + Truth + Justification + (No Gettier Luck)
5. Ethical Theories: Utilitarianism’s Happiness Calculus | 伦理学：功利主义的幸福演算
Classical Utilitarianism proposes a moral ‘formula’ for right action: an act is right if and only if it maximises overall happiness, calculated as pleasure minus pain over all affected individuals. Bentham’s felicific calculus breaks this down into seven factors: intensity, duration, certainty, propinquity, fecundity, purity, and extent.
古典功利主义为正确行动提出了一个道德“公式”：一个行为是正当的，当且仅当它能最大化总体幸福，计算方式为所有受影响个体的快乐减去痛苦。边沁的幸福演算将其分解为七个因素：强度、持续时间、确定性、邻近性、继生性、纯度和范围。
Mill refined the formula by distinguishing higher and lower pleasures, arguing that quality matters no less than quantity. In essay questions, apply the utilitarian formula to ethical dilemmas: add up the foreseeable consequences, consider all stakeholders, and judge whether net utility is maximised.
密尔通过区分高级与低级快乐完善了这一公式，主张质量的重要性不亚于数量。在论文题中，可将功利主义公式应用于伦理困境：累加可预见的后果，考虑所有利益相关方，并判断净效用是否最大化。
Right ∝ Σ (Pleasure – Pain) | Bentham: I, D, C, P, F, P, E
6. Kantian Ethics: Categorical Imperative Formulae | 康德伦理学：绝对命令诸公式
Kant’s moral law can be expressed as three interconnected formulas. The Formula of Universal Law: ‘Act only on that maxim through which you can at the same time will that it should become a universal law.’ To test a maxim, ask: can it be conceived without contradiction, and can it be willed without contradiction?
康德的道德法则可以表述为三个相互关联的公式。普遍法则公式：“只按照你同时能够意愿它成为一项普遍法则的那个准则去行动。”检验一条准则时，要问：能否不矛盾地设想它，以及能否不矛盾地意愿它？
The Formula of Humanity: ‘Act in such a way that you treat humanity, whether in your own person or in the person of another, always at the same time as an end, never merely as a means.’ This forbids using rational beings merely as instruments.
人性公式：“要以这样的方式行动，即你总是将人性——无论是你自身人格中的人性还是他人人格中的人性——同时当作目的，绝不仅仅当作手段。”这禁止将理性存在者仅仅当作工具。
The Formula of the Kingdom of Ends unites the two: act as if you were a law-making member in a systematic union of rational beings. These formulas provide a structural test for moral permissibility that exam answers can apply stepwise.
目的王国公式将前两者统一起来：仿佛你是一个理性存在者系统联合体中的立法成员那样行动。这些公式为道德可允许性提供了一种结构化的检验，考试答案中可以分步骤加以应用。
7. Metaphysics: Determinism & Free Will – Conditional Analysis | 形而上学：决定论与自由意志——条件分析
Classical compatibilism offers a formula for free will even under determinism: an agent S acts freely if S does what S wants, and if S had wanted otherwise, S would have done otherwise. This conditional analysis reduces freedom to the ability to act on one’s desires without external constraint.
古典相容论为决定论下的自由意志提供了一个公式：一个行动者 S 自由地行动，当 S 做了 S 想做的事，且如果 S 想要别的，S 就会做别的。这一条件分析将自由还原为在无外在约束的情况下按自身欲望行动的能力。
Hard determinists reject this formula, insisting that genuine freedom requires the ability to have done otherwise in the identical past circumstances – a condition they argue determinism makes impossible. Libertarians add an indeterministic component: S is free only if S’s choice is not fully determined by prior causes.
强决定论者拒绝这一公式，坚持认为真正的自由要求在同一过往情境下本可以做出其他选择的能力——他们论证决定论使这一条件不可能。自由意志论者则加入非决定论成分：S 是自由的，仅当 S 的选择并不完全由先前的原因决定。
Free Will (Compatibilist) = (Desire + Action) ∧ (If ¬Desire → ¬Action)
8. Philosophy of Mind: Identity Theory & Functionalist Formulae | 心灵哲学：同一论与功能主义公式
Mind-brain identity theory states a simple psychophysical formula: each type of mental state M is identical to some type of brain state B. For example, pain = C-fibre firing. This is a reductive claim: mental kinds are nothing over and above physical kinds.
心脑同一论陈述了一个简单的心物公式：每一类心理状态 M 都同一于某类大脑状态 B。例如，疼痛 = C-纤维激活。这是一种还原论主张：心理类型不过是物理类型的另一种说法。
Functionalism replaces the identity formula with a causal one: mental state M is defined by its functional role – the set of causal relations between sensory inputs, other mental states, and behavioural outputs. The formula becomes: M = that state which is typically caused by inputs I and leads to outputs O, given other states S.
功能主义用一个因果公式取代了同一公式：心理状态 M 由其功能角色定义——即感官输入、其他心理状态和行为输出之间的因果关系的集合。公式变为：M = 在给定其他状态 S 的情况下，通常由输入 I 引起并导致输出 O 的状态。
This shift explains multiple realizability: the same mental state can be implemented by different physical substrates (brain, silicon), just as the same software can run on different hardware.
这一转变解释了多元可实现性：同一种心理状态可以由不同的物理基础（大脑、硅芯片）实现，就像同一个软件可以在不同的硬件上运行。
9. Philosophy of Religion: The Ontological Argument Formula | 宗教哲学：本体论论证公式
Anselm’s ontological argument can be rendered as a logical proof seeking to derive God’s existence from the very concept of God. The ‘formula’ runs: God is by definition the greatest conceivable being (GCB). A GCB that exists in reality is greater than one that exists only in the mind. Therefore, if God existed only in the mind, we could conceive of a greater being (one that also exists in reality). Thus God must exist in reality.
安瑟伦的本体论论证可以呈现为一个逻辑证明，试图从上帝的概念本身推导出上帝的存在。其“公式”为：上帝根据定义是可设想的最伟大存在者（Greatest Conceivable Being, GCB）。存在于现实中的 GCB 比仅存在于心灵中的更伟大。因此，如果上帝仅存在于心灵中，我们就能设想一个更伟大的存在者（同时存在于现实中）。所以，上帝必定存在于现实中。
If GCB = df. that than which nothing greater can be conceived → GCB must exist in re
Gaunilo’s parody using a perfect island and Kant’s objection that existence is not a predicate challenge the formula’s validity. In essays, reconstruct the logical steps and assess whether the definitional move is question-begging.
高尼罗用完美岛屿进行的模仿推理，以及康德“存在不是谓词”的反对意见，都对这一公式的有效性构成了挑战。在论文中，应重建逻辑步骤，并评估这一定义性操作是否属乞题。
10. Political Philosophy: Social Contract Formula | 政治哲学：社会契约公式
Social contract theories provide a formula for legitimate political authority: a state is legitimate if and only if all rational individuals would consent to it under specified conditions. Hobbes’ version: in a state of nature life is ‘solitary, poor, nasty, brutish, and short’, so rational agents consent to an absolute sovereign to secure peace.
社会契约理论为政治权力的合法性提供了一个公式：一个国家是合法的，当且仅当所有理性个体在特定条件下都会同意它。霍布斯的版本：自然状态下生活“孤独、贫困、污秽、野蛮而短促”，因此理性行动者同意让渡权利给一个绝对主权者以保障和平。
Locke’s formula differs: individuals consent to government only to protect natural rights – life, liberty, property – and retain the right to revolution if the contract is broken. Rawls modernises the formula with the original position and veil of ignorance, producing two principles of justice that would be chosen by self-interested but rational persons unaware of their place in society.
洛克的公式则不同：个体同意政府只是为了保护自然权利——生命、自由和财产——并保留革命权，如果契约被破坏。罗尔斯用原初状态和无知之幕的公式使其现代化，产生了自利但理性的个体在不知道自己在社会中位置的情况下会选择的两个正义原则。
Legitimacy = Hypothetical Consent under Conditions C (no coercion, full information)
11. Exam Application: Structuring Philosophical Arguments | 考试应用：构建哲学论证
To apply these formulas in the GCSE Cambridge examination, follow a clear structure for each evaluative essay: (1) State the thesis clearly. (2) Deconstruct the relevant philosophical formula step by step. (3) Present an objection, using a counterexample or a logical challenge to one of the formula’s components. (4) Evaluate the strength of the objection and offer a balanced conclusion.
要在 GCSE 剑桥考试中应用这些公式，每篇评价性论文可以遵循清晰的结构：（1）明确陈述论点。（2）逐步解构相关的哲学公式。（3）提出一个反对意见，使用反例或对公式某一成分的逻辑挑战。（4）评价反对意见的力度，并给出平衡的结论。
For instance, when discussing the JTB formula of knowledge, you could present a Gettier-style counterexample, explain why the traditional formula fails, and then assess whether adding a ‘no false lemmas’ condition satisfactorily solves the problem. Always connect your reasoning back to the specific formula being tested.
例如，在讨论知识的 JTB 公式时，你可以呈现一个盖梯尔式的反例，解释传统公式为何失效，然后评价添加“无错误引理”条件是否令人满意地解决了问题。始终要将你的推理与所检验的具体公式连接起来。
Practice translating each philosophical position you study into a concise formula. This habit will help you memorise key arguments, spot fallacies quickly, and produce structured, high-mark answers under timed conditions.
练习将你学到的每一个哲学立场转化为简洁的公式。这一习惯将帮助你记忆关键论证，快速识别谬误，并在限时条件下写出结构清晰、得分高的答案。
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