📚 IGCSE CIE Philosophy: Formulae & Theorems Quick Reference Handbook | IGCSE CIE 哲学：公式定理速查手册
This quick reference handbook distils the core logical structures, ethical principles, and philosophical arguments from the CIE IGCSE Philosophy syllabus into digestible ‘formulae’ and ‘theorems’. While philosophy is not mathematics, these condensed patterns help you recognise, analyse, and evaluate reasoning in an exam setting. Use them to sharpen your critical thinking and to structure your essays effectively.
本速查手册将 CIE IGCSE 哲学教学大纲中的核心逻辑结构、伦理原则以及经典哲学论证，浓缩为易于掌握的“公式”与“定理”。尽管哲学不是数学，但这些凝练的模式能帮助你在考场上快速识别、分析和评估推理过程。善用这些内容，你的批判性思维和论文结构都会更加清晰有力。
1. The Validity Formula: Deductive Argument Patterns | 有效性公式：演绎论证的基本模式
A deductive argument is valid if, when its premises are true, the conclusion must be true. The following patterns are the most common ‘formulae’ for testing validity in the IGCSE exam. Remember that validity is about logical structure, not factual truth.
如果一个演绎论证的前提为真时，其结论必定为真，则该论证是有效的。以下是最常见于 IGCSE 考试中的有效性检测“公式”。请牢记，有效性只关乎逻辑结构，与事实真假无关。
Modus Ponens: If P → Q, P, ∴ Q
Example: If it rains, the pitch is wet. It rains. Therefore, the pitch is wet. The formula is straightforward, but in exam sources you must identify whether the ‘if-then’ link is granted.
范例：如果下雨，球场就会湿。下雨了。所以球场湿了。这一公式虽然简单，但在考题素材中你需要判断该“如果-那么”关系是否被承认。
Modus Tollens: If P → Q, ¬Q, ∴ ¬P
Example: If the circuit is complete, the lamp lights. The lamp does not light. Therefore, the circuit is not complete. Modus tollens is the basis of many falsification arguments in philosophy.
范例：如果电路闭合，灯就会亮。灯没亮。所以电路没有闭合。Modus tollens 是哲学中许多证伪论证的基础。
Hypothetical Syllogism: P → Q, Q → R, ∴ P → R
This chain pattern underpins complex reasoning, such as cosmological arguments. Watch for missing links when you analyse extended prose.
这种链式结构支撑着许多复杂推理，比如宇宙论论证。在分析长篇文本时，务必留意缺失的环节。
2. The JTB Knowledge Formula: Justified True Belief | JTB 知识公式：被辩护的真信念
The tripartite definition of knowledge states that S knows that p if and only if: (i) S believes that p; (ii) p is true; (iii) S has justification for believing p. This formula — Knowledge = Belief + Truth + Justification — dominated epistemology until Gettier’s counterexamples.
知识的三元定义指出，主体 S 知道命题 p，当且仅当：(i) S 相信 p；(ii) p 为真；(iii) S 拥有相信 p 的辩护理由。这一公式——知识 = 信念 + 真相 + 辩护——在盖梯尔反例出现之前，一直主宰着认识论。
In IGCSE terms, you must be able to explain how someone can have justified true belief yet still lack knowledge (Gettier cases). A standard formula: Justified true belief + luck ≠ knowledge. For instance, a stopped clock showing the correct time accidentally.
在 IGCSE 语境中，你必须能够解释为什么一个人可以持有被辩护的真信念，却仍然缺乏知识（盖梯尔案例）。一个标准公式：被辩护的真信念 + 运气 ≠ 知识。例如，一个停止的时钟“碰巧”显示了正确的时间。
3. Cartesian Doubt Theorem: The Cogito | 笛卡尔式怀疑定理：我思故我在
Descartes’ method of doubt can be summarised as a radical sceptical formula: doubt all that can be doubted → only the doubting self remains certain. He imagined an evil demon deceiving him about everything, then concluded that the very act of being deceived proves his existence as a thinking thing.
笛卡尔的怀疑方法可以概括为一个激进的怀疑公式：怀疑一切可怀疑之物 → 唯有正在怀疑的自我确凿无疑。他设想有一个恶魔在一切事情上欺骗他，随后得出结论：被欺骗这一行为本身，恰恰证明了他作为一个思考者的存在。
Cogito ergo sum = ‘I think, therefore I am’
For the exam, you need to articulate why the cogito is held to be immune to the strongest doubt and to evaluate whether Descartes jumps too quickly from ‘I think’ to ‘I am a mental substance’.
针对考试，你需要阐明为何“我思”被认为不受最强怀疑的影响，并评估笛卡尔是否过于仓促地从“我思”跳到了“我是一个精神实体”。
4. Utilitarian Calculus: The Greatest Happiness Principle | 功利主义计算式：最大幸福原则
Classical utilitarianism, associated with Bentham and Mill, asserts a straightforward ethical formula: an action is right if and only if it produces the greatest net happiness for all affected. Happiness = pleasure minus pain, aggregated across sentient beings.
以边沁和密尔为代表的古典功利主义，给出了一个直截了当的伦理公式：一个行为是正确的，当且仅当它为所有受影响者带来了最大的净幸福。幸福 = 快乐总量 – 痛苦总量，并在一切有感知的存有之间进行加总。
Action A is right ↔ Utility(A) ≥ Utility(any alternative)
In IGCSE ethics questions, you must apply the hedonic calculus: intensity, duration, certainty, propinquity, fecundity, purity, and extent. Mention Mill’s distinction between higher and lower pleasures to show deeper understanding.
在 IGCSE 伦理学考题中，你需要应用快乐计算公式中的七个维度：强度、持续时间、确定性、远近、生育力、纯度和范围。若能提及密尔对“高级”与“低级”快乐的区分，将展示更深的理解。
5. The Categorical Imperative Formula: Universal Law | 绝对命令公式：普遍法则
Kant’s deontological ethics provides a decision procedure, not a calculus. The first formulation of the categorical imperative serves as a universalisation test: act only according to that maxim whereby you can at the same time will that it should become a universal law.
康德的义务论伦理提供的是一种决策程序，而非一个计算公式。绝对命令的第一表述可以充当一项“普遍化检测”：只按照你同时能够意愿它成为一条普遍法则的准则去行动。
Permissible if: Maxim → can be universalized without contradiction
Contradictions arise either in conception (e.g., a world where everyone makes false promises destroys the practice of promising) or in the will (a rational being cannot sincerely will a maxim of neglect of talents). The second formula — treat humanity always as an end, never merely as a means — is equally central to IGCSE.
会引发矛盾的情形有两种：概念上的矛盾（例如，如果人人作虚假承诺，承诺这一实践本身就会瓦解）和意愿上的矛盾（一个理性存有不可能真诚地意愿一条漠视自身才能的准则）。第二表述——永远把人性当作目的，而绝不仅仅当作手段——在 IGCSE 中同样举足轻重。
6. Virtue Ethics Mean Formula: The Golden Mean | 美德伦理学的中道公式：黄金中道
Aristotle’s virtue ethics treats character, not acts, as primary. The doctrine of the mean provides a diagnostic formula: a moral virtue is a disposition to choose the intermediate between a vice of excess and a vice of deficiency, relative to the person and the situation.
亚里士多德的美德伦理学将品格而非行为视为首要。中道学说提供了一个诊断公式：德性就是一种选择过与不及之间恰当中道的倾向，这一中道因人和情境而异。
Virtue = mean between excess and deficiency, e.g. courage = mean between rashness and cowardice
IGCSE candidates are expected to illustrate this with examples such as temperance between licentiousness and insensibility, or generosity between prodigality and meanness. Practical wisdom (phronesis) is the ‘calculating’ faculty that discerns the mean.
IGCSE 考生需要用例证来说明：例如节制介于放纵与麻木之间，慷慨介于挥霍与吝啬之间。实践智慧（phronesis）正是那种识别中道的“运算”官能。
7. The Design Argument Formula: Analogy and Purpose | 设计论证公式：类比与目的
The teleological argument (design argument) can be captured by a structured analogy formula: the natural world exhibits order, complexity, and purpose; such properties in human artifacts are always caused by an intelligent designer; therefore, the natural world probably has an intelligent designer (God).
目的论论证（设计论证）可以用一个结构化的类比公式来捕捉：自然界展现出秩序、复杂性和目的；在人工制品中，这些特征总是源于智能设计者；因此，自然界很可能也有一位智能设计者（上帝）。
P1: Natural world resembles a watch (order & complexity) | P2: A watch implies a watchmaker | C: Nature implies a divine designer
For high marks, evaluate whether the analogy is sufficiently strong, whether natural selection offers an alternative explanation, and whether the argument proves the God of classical theism. Hume’s critiques (limited analogy, possibility of multiple designers) should be rehearsed.
要获得高分，必须评估该类比是否足够强、自然选择是否提供了另一种解释，以及该论证是否能证明经典一神论的上帝。休谟的批评（类比有限制、可能存在多个设计师）也要娓娓道来。
8. The First Cause Formula: Cosmological Reasoning | 第一因公式：宇宙论推理
Aquinas’ Second Way, and related cosmological arguments, rely on a causal chain formula: every event has a cause; the chain of causes cannot regress infinitely; therefore, there must be a first uncaused cause (God). This is often presented as a domino-sequence model.
阿奎那的第二路论证以及相关的宇宙论论证，都依赖一个因果链公式：每个事件都有原因；因果链不可能无限倒退；因此，必定存在一个第一的非受动因（上帝）。这常被展现为多米诺骨牌序列模型。
Causal series: C₁ ← C₂ ← C₃ … cannot go to infinity → First Cause (uncaused)
IGCSE responses need to engage with challenges: why must the first cause be God rather than the universe itself? Could there be an infinite regress? And does the Big Bang theory provide a scientific counter-model? These questions make for strong evaluation paragraphs.
IGCSE 的回答需直面质疑：为什么第一因必须是上帝，而不能是宇宙本身？是否可能存在无限倒退？大爆炸理论是否提供了一个科学的替代模型？这些问题将撑起高质量的评估段落。
9. The Problem of Evil Logical Formula: Inconsistent Triad | 恶的问题的逻辑公式：不一致三元组
The logical problem of evil forms a powerful inconsistent triad: if God is all-powerful, he could prevent evil; if God is all-loving, he would want to prevent evil; yet evil exists. Therefore, an all-powerful, all-loving God does not exist (or the attributes must be redefined).
恶的逻辑问题形成了一个强有力的不一致三元组：如果上帝全能，他就能阻止恶；如果上帝全善，他就愿意阻止恶；然而恶确实存在。因此，一位全能全善的上帝并不存在（或者说，这些属性必须被重新定义）。
Omnipotence + Omnibenevolence + Evil → Contradiction
The free will defence proposes a ‘theorem’ of greater goods: God allows moral evil because free will is a greater good that logically requires the possibility of wrong choices. Candidates should also explain natural evil and the soul-making theodicy to show range.
自由意志辩护提出了一条关于“更大善”的“定理”：上帝容许道德之恶，是因为自由意志是一种更大的善，而自由意志在逻辑上要求有做出错误选择的可能性。考生还应解释自然之恶及灵魂塑造神义论，以展现知识广度。
10. Mind-Body Position Formulae: Dualism vs. Physicalism | 心身问题立场公式：二元论与物理主义
Philosophy of mind offers competing ‘formulae’ for the relation between mental and physical states. Substance dualism states: mind and body are two distinct substances; mind = res cogitans (thinking thing), body = res extensa (extended thing).
心灵哲学为心理状态与物理状态的关系提供了几个相互竞争的“公式”。实体二元论主张：心灵与身体是两种截然不同的实体；心灵 = 思之物，身体 = 广延之物。
Dualism: Mental substance ≠ Physical substance | Physicalism: Mind = Brain process
The identity theory (a reductive physicalism) claims that each type of mental state is identical to a type of brain state, e.g., pain = C-fibre firing. Functionalism rejects the identity formula, proposing instead that mental states are defined by their causal roles: mind is to brain as software is to hardware.
同一论（一种还原物理主义）声称，每一类心理状态都等同于某一类大脑状态，例如，疼痛 = C神经纤维激发。功能主义拒绝这个同一公式，转而提出心理状态由其因果角色来定义：心灵与大脑的关系，就如同软件与硬件的关系。
11. Informal Fallacy Spotting Table | 非形式谬误速查表
Recognising fallacies is crucial for evaluating arguments in IGCSE papers. These recurring patterns of flawed reasoning should be memorised as ‘error formulae’.
识别谬误对于评估IGCSE试卷中的论证至关重要。以下这些反复出现的错误推理模式，应当作为“错误公式”牢记于心。
| Fallacy (English) | Pattern (Formula) | 中文简释 |
|---|---|---|
| Ad hominem | Attacks the person, not the argument | 人身攻击，而非反驳论证 |
| Straw man | Misrepresents an opponent’s view to refute it easily | 歪曲对方观点以便轻松驳倒 |
| False dilemma | Presents only two alternatives when more exist | 假两难，无视其他可能选择 |
| Begging the question | Assumes the conclusion in the premises | 循环论证，前提已预设结论 |
| Slippery slope | Claims a small step leads inevitably to an extreme outcome without evidence | 无证据断言一小步必然导致极端后果 |
| Appeal to authority | Cites an unqualified authority as decisive support | 引用不合格的权威作为定论支撑 |
12. Critical Thinking Toolkit: Argument Reconstruction | 批判性思维工具箱：论证重构
In the 12-mark and essay questions, you are often asked to analyse and evaluate philosophical passages. A reliable ‘algorithm’ can be followed: (1) Identify the main conclusion; (2) Extract and number the premises; (3) Check for hidden assumptions; (4) Test validity and soundness; (5) Apply relevant counterarguments from your formula bank.
在12分题及论文题中，你常常需要分析和评估哲学文本。可以遵循一套可靠的“算法”：(1) 找出主结论；(2) 提取前提并编号；(3) 检查隐含假设；(4) 检验有效性和可靠性；(5) 从你的公式库中调用相关反论证。
A sound argument = valid form + true premises. Always distinguish between ‘I find the argument unconvincing because the premise is false’ and ‘the argument is invalid because the conclusion does not follow’. This precision impresses examiners.
一个可靠的论证 = 有效形式 + 真实的前提。务必辨明“我觉得这个论证没有说服力，因为某个前提为假”和“这个论证无效，因为结论推不出”这两种批评之间的区别。这种精确性能赢得阅卷人的好感。
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