📚 IGCSE Cambridge Philosophy: Quick-Reference Handbook of Formulas & Theorems | IGCSE 剑桥哲学：公式定理速查手册
This handbook distils the essential logical forms, argument structures, and ethical principles you need for Cambridge IGCSE Philosophy. Instead of memorising long paragraphs, treat these patterns as 'formulas' that unlock rigorous analysis. Whether you are evaluating a syllogism, testing the validity of a moral argument, or deconstructing a fallacy, this quick-reference tool will sharpen your reasoning.
本手册提炼了剑桥 IGCSE 哲学所需的关键逻辑形式、论证结构和伦理原则。你不必死记硬背长篇大论，可以把这些模式视作“公式”来解锁严谨分析。无论你是在评估三段论、检验道德论证的有效性，还是在拆解逻辑谬误，这份速查工具都将提升你的推理能力。
1. Propositional Connectives and Truth Tables | 命题逻辑联结词与真值表
Every philosophical argument relies on the basic building blocks of propositional logic. These five connectives determine the truth conditions of compound statements. The table below shows their symbols, names, and truth-functional definitions. Understanding how these work is vital for assessing validity.
每个哲学论证都依赖命题逻辑的基本构建块。以下五种联结词决定了复合命题的真值条件。下表展示了它们的符号、名称及真值函数定义。理解它们的原理是评估有效性的关键。
| Symbol | English Name | Truth Table Pattern |
|---|---|---|
| ¬ P | Negation | If P is true, ¬P is false; if P is false, ¬P is true. |
| P ∧ Q | Conjunction | True only when both P and Q are true. |
| P ∨ Q | Disjunction | True when at least one of P, Q is true. |
| P → Q | Conditional | False only when P is true and Q is false. |
| P ↔ Q | Biconditional | True when P and Q share the same truth value. |
记住，条件式 P → Q 并不要求 P 和 Q 之间存在因果关联，它只关乎真值分配。当你在哲学文本中遇到“如果……那么……”的主张时，立即用真值条件加以检验，这能暴露许多隐藏的假设。
2. Valid Rules of Inference | 有效推理规则
Inference rules are the 'theorems' of natural deduction. If you accept the premises, the conclusion necessarily follows. Here are the most frequently used forms in philosophical argumentation. Each is given as a formula; memorising them allows you to spot valid moves instantly.
推理规则是自然演绎中的“定理”。只要你接受前提，结论必然随之而来。以下是哲学论证中最常用的形式，每个都以公式呈现。记住它们能让你瞬间识别有效推导。
Modus Ponens: P → Q, P ∴ Q
If the tree is an oak, it sheds leaves in autumn. The tree is an oak. Therefore, it sheds leaves in autumn. The pattern affirms the antecedent.
若这棵树是橡树，它会在秋天落叶。这棵树是橡树。所以，它会在秋天落叶。该模式肯定了前件。
Modus Tollens: P → Q, ¬Q ∴ ¬P
If it is raining, the pavement is wet. The pavement is not wet. Therefore, it is not raining. This denies the consequent.
如果下雨，人行道会湿。人行道不湿。因此，没有下雨。这否定了后件。
Hypothetical Syllogism: P → Q, Q → R ∴ P → R
If we study, we pass the exam. If we pass the exam, we graduate. Therefore, if we study, we graduate.
假如我们学习，就会通过考试。假如通过考试，就能毕业。因此，假如我们学习，就能毕业。
Disjunctive Syllogism: P ∨ Q, ¬P ∴ Q
Either the butler did it or the maid did it. The butler did not do it. Therefore, the maid did it.
要么是管家干的，要么是女仆干的。不是管家干的。因此，是女仆干的。
3. Valid Syllogistic Forms | 有效三段论式
Categorical syllogisms, studied since Aristotle, remain a core test of deductive reasoning. The four standard categorical propositions are A (All S are P), E (No S are P), I (Some S are P), and O (Some S are not P). The mnemonic names encode mood and figure; the ones below are unconditionally valid.
直言三段论自亚里士多德以来一直是演绎推理的核心检验。四个标准直言命题是 A（所有 S 是 P）、E（没有 S 是 P）、I（有些 S 是 P）和 O（有些 S 不是 P）。助记名称编码了式与格，下列几式是无条件有效的。
Barbara: AAA-1
All M are P. All S are M. Therefore, all S are P.
所有 M 是 P。所有 S 是 M。所以，所有 S 是 P。
Celarent: EAE-1
No M are P. All S are M. Therefore, no S are P.
没有 M 是 P。所有 S 是 M。所以，没有 S 是 P。
Darii: AII-1
All M are P. Some S are M. Therefore, some S are P.
所有 M 是 P。有些 S 是 M。所以，有些 S 是 P。
Ferio: EIO-1
No M are P. Some S are M. Therefore, some S are not P.
没有 M 是 P。有些 S 是 M。所以，有些 S 不是 P。
4. Common Logical Fallacies Quick Check | 常见逻辑谬误速查
Spotting fallacies is as important as knowing valid forms. A fallacy mimic a valid pattern but contains a structural flaw. The list below gives the formula and a brief example. When you suspect a fallacy, map the argument onto these forms.
识别谬误与掌握有效形式同样重要。谬误模仿有效模式，却含有结构性缺陷。下列清单给出了公式与简例。当你怀疑有谬误时，尝试将论证套入这些模式。
Affirming the Consequent: P → Q, Q ∴ P (无效)
错误示例：如果下雨，地面湿。地面湿，所以下雨了。 (地面湿可能是因为洒水车)
Denying the Antecedent: P → Q, ¬P ∴ ¬Q (无效)
错误示例：如果是猫，就是哺乳动物。这不是猫，所以不是哺乳动物。 (狗也是哺乳动物)
Begging the Question (petitio principii): 结论隐藏在前提中。
范例：允许言论自由是有益的，因为它对社会有好处。为何有好处？因为人们可以自由表达。循环论证。
False Dilemma: P ∨ Q 是虚假的选言，忽略第三可能。
要么你完全支持我，要么你就是敌人。忽略了中立或善意的批评。
5. Knowledge as Justified True Belief and Gettier Cases | 知识作为确证的真信念与盖梯尔案例
The traditional 'formula' B: 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. This tripartite definition collapses under Gettier-style counterexamples where justification and truth are merely accidentally connected.
命题性知识的传统“公式”为 K = JTB：S 知道 p，当且仅当 (i) p 为真，(ii) S 相信 p，且 (iii) S 确证地相信 p。这一三元定义在盖梯尔式反例面前崩塌，因为确证与真值只是偶然关联。
Consider a stopped clock that happens to show the correct time. If a subject looks at it and forms the belief that it is 3pm, the belief is both true and justified (the clock usually works), yet intuitively they do not know it is 3pm. Therefore, the JTB formula is insufficient.
设想一座停摆的钟偶然显示着正确时间。某人看到该钟并形成“现在是下午3点”的信念，这一信念既为真又得到确证（钟通常正常运行），但直觉上他并不知道现在是3点。因此，JTB 公式并不充分。
6. The Ontological Argument Logical Structure | 本体论论证的逻辑结构
Anselm's ontological argument can be condensed into a logical deduction. It attempts to prove God's existence a priori by analysing the concept of a maximally great being. Below is a simplified formulation often used in IGCSE analysis.
安瑟伦的本体论论证可浓缩为一个逻辑推演。它试图通过分析最伟大存有的概念来先天地证明上帝存在。以下是 IGCSE 分析中常用的简化表述。
Premise 1: God is defined as that than which nothing greater can be conceived.
前提1：上帝被定义为“无法设想比之更伟大之物”。
Premise 2: Existence in reality is greater than existence in the understanding alone.
前提2：存在于现实中，比仅仅存在于理解中更伟大。
Conclusion: If God existed only in the understanding, we could conceive a greater being that also exists in reality. Therefore, God must exist in reality.
结论：若上帝只存在于理解中，我们就能设想一个更伟大的、也存在于现实中的存有。因此，上帝必然存在于现实中。
Gaunilo's parody uses the same form with a 'perfect island', challenging the move from concept to reality.
高尼罗的仿拟用“完美的岛屿”套用相同形式，质疑从概念到实在的跳跃。
7. Cosmological and Teleological Argument Schemas | 宇宙论与目的论论证图式
Cosmological arguments typically follow a chain of causation or contingency. Thomas Aquinas' Third Way can be seen as a modal formula: If everything is contingent, at some time nothing existed. If nothing existed, nothing would exist now. Since things do exist now, there must be at least one necessary being.
宇宙论论证通常遵循因果或偶然性链条。托马斯·阿奎那的第三路可视为一个模态公式：假如万物皆偶存，则曾有一刻无物存在。若曾无物存在，则现在也无物存在。既然现在确有事物存在，则必须至少有一个必然存在者。
teleological argument (design argument) points to regularities and apparent purpose. Paley' gy can be formulated as: Complex, purpose-ordered arrangement → intelligent designer. Hume's criticisms challenge the strength of the analogy and the uniqueness of the designer.
目的论论证（设计论证）指向规律性和显见的目的性。佩利的钟表类比可表述为：复杂、有目的指向的排列 → 智慧设计者。休谟的批评质疑这一类比的有效性及设计者的唯一性。
8. The Logical Problem of Evil | 恶的逻辑问题
The classic deductive argument from evil attempts to show an inconsistency between the existence of an omnipotent, omnibenevolent God and the existence of evil. The logical form is often presented as a set of inconsistent statements. If two of them are true, the third must be false.
经典的恶之演绎论证试图展示全能、全善的上帝与恶的存在之间具有不一致性。其逻辑形式通常表现为一组矛盾的陈述：若其中二者为真，则第三者必假。
1. God is all-powerful. 2. God is all-good. 3. Evil exists. A being that is all-powerful can eliminate all evil; an all-good being would want to eliminate all evil. Yet evil exists. Therefore, one of the premises is rejected.
1. 上帝全能。2. 上帝全善。3. 恶存在。全能者可以消除一切恶，全善者会愿意消除一切恶。然而恶仍然存在。因此，上述前提之一须被否定。
The free-will defence adds a fourth element: God cannot eliminate evil without eliminating free will, and free will is a greater good. This transforms the logical case but does not answer the evidential problem of gratuitous suffering.
自由意志辩护增添了第四要素：上帝无法在不取消自由意志的情况下消除恶，而自由意志是更大的善。这改变了逻辑论证的格局，但并未回应无端苦难的证据性问题。
9. The Principle of Utility (Bentham's Hedonic Calculus) | 功利原则（边沁的幸福计算）
Classical utilitarianism provides a decision-making formula: the right action is the one that produces the greatest happiness for the greatest number. Bentham's hedonistic calculus attempts to quantify pleasure and pain using seven dimensions.
古典功利主义提供了一种决策公式：正确的行为就是为最大多数人带来最大幸福的行为。边沁的享乐计算法试图用七个维度来量化快乐与痛苦。
The dimensions form a 'utility formula': Intensity, Duration, Certainty, Propinquity (nearness in time), Fecundity (chance of leading to more pleasure), Purity (chance of not being followed by pain), and Extent (number of people affected). Summing the net utility for each stakeholder identifies the moral course.
这些维度构成了一个“效用公式”：强度、持续时间、确定性、临近性（时间上的远近）、多产性（产生更多快乐的机会）、纯粹性（不被痛苦跟随的机会）和范围（受影响的人数）。对每一利益相关者的净效用求和，即可确定道德行动方案。
10. Kant's Categorical Imperative Formulations | 康德绝对命令的表述
Kantian ethics rejects consequences and looks to the form of the maxim. The first formulation of the categorical imperative acts as a universalisability test. The logical structure is: Act only according to that maxim whereby you can at the same time will that it should become a universal law.
康德伦理学拒绝后果，而着眼于准则的形式。绝对命令的第一表述作为一项可普遍化检验而运作。其逻辑结构为：只依据那些你同时能够意愿其成为普遍法则的准则去行动。
If universalising a maxim leads to a contradiction in conception (e.g. making a false promise would destroy the practice of promising) or a contradiction in the will, the act is impermissible. The second formula, the Formula of Humanity, commands: Act so that you treat humanity, whether in your own person or in that of another, always as an end and never merely as a means.
如果把一条准则普遍化会导致概念上的矛盾（例如，虚假许诺会摧毁许诺本身这一实践）或意志上的矛盾，那么该行为是不被允许的。第二表述，即人性公式，命令道：你要这样行动，永远都把你自身中的人性和他人中的人性，不仅仅当做手段，而同时当做目的。
11. Natural Law Primary Precepts | 自然法首要诫命
Aquinas' Natural Law theory can be seen as a teleological formula. Human beings, by nature, are oriented toward certain goods. The primary precepts are the fundamental principles of practical reason: preserve life, reproduce and educate offspring, know truth about God, live in an ordered society, and avoid ignorance and offending others.
阿奎那的自然法理论可被视为一个目的论公式。人类在本性上朝向某些善。首要诫命是实践理性的基本原理：保存生命、繁衍并教育后代、认识上帝真理、生活于有秩序的社会、避免无知及不冒犯他人。
This can be expressed as: Good Act + Right Intention + Suitable Circumstances → Morally Good Action.
一个道德上善的行为必须通过三重检验：行为本身必须善、意图必须善、处境必须适当。这可以表示为：善的行为 + 正当意图 + 合宜处境 → 道德上善的行动。
12. Descartes' Cogito and Method of Doubt | 笛卡尔的我思与怀疑方法
Descartes' foundational project uses radical doubt as a method. The famous formula 'Cogito, ergo sum' (I think, therefore I am) is reached after imagining an evil genius who deceives him about everything. Even if he is deceived, he must exist to be deceived. The thinking self is thus indubitable.
笛卡尔的基础主义方案将彻底怀疑作为一种方法。著名的公式“我思故我在”（Cogito, ergo sum）是在设想一个恶神可能欺骗他一切之后得出的。即使他被欺骗，为了被欺骗他也必须存在。因此思维着的自我是无可怀疑的。
The logical core can be written as: For any X, if X doubts, then X exists. I doubt. Therefore, I exist. This argument is immune to the most extreme sceptical hypotheses and serves as the first certain truth in his epistemology.
其逻辑核心可写为：对任意 X，若 X 怀疑，则 X 存在。我怀疑。所以，我存在。该论证免受极端怀疑论假设的攻击，并成为他认识论中第一个确定无疑的真理。
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