📚 IGCSE WJEC Philosophy: Formula and Theorem Quick-Reference Handbook | IGCSE WJEC 哲学：公式定理速查手册
In philosophy, a 'formula' is not an equation of numbers but a precise pattern of reasoning. This handbook condenses the key arguments and principles of the IGCSE WJEC Philosophy course into memorable formulas, theorem-like statements and quick checks that you can recall under exam pressure.
在哲学中，'公式'不是数字方程，而是精确的推理模式。本手册将 IGCSE WJEC 哲学课程中的核心论证与原则浓缩为易记的公式、定理式陈述和可在考场上快速调用的检查清单。
Every formula below follows the same structure: a pattern, a plain-language statement of what it means, and the limitation that examiners expect you to mention. Patterns alone score low marks; patterns plus evaluation score high marks.
下文每个公式都遵循同一结构：推理模式、用日常语言说明其含义，以及考官期待你指出的局限。只背模式得分低，模式加评价才能拿高分。
1. Logic: Argument Formulas | 逻辑：论证公式
An argument is a set of premises offered as support for a conclusion. The premises are the reasons; the conclusion is what the arguer wants you to accept. In standard form, each premise is listed on its own line and the conclusion is marked with 'therefore'.
论证是一组为支持结论而给出的前提。前提是理由，结论是论证者希望你接受的东西。写成标准形式时，每条前提单独一行，结论前用 '因此' 标记。
Conditional reasoning has the form 'If P then Q' Q is the consequent. The truth of the conditional alone never tells you which of P or Q is actually true, and this single fact explains why two of the four conditional patterns are valid and two are not.
条件推理的形式是 '如果 P 那么 Q'，写作 P → Q。P 是前件，Q 是后件。条件句本身为真，绝不告诉你 P 或 Q 哪个实际为真，正是这一事实决定了四种条件推理模式中两种有效、两种无效。
Modus Ponens (affirming the antecedent):
P → Q; P; ∴ Q
Modus Tollens (denying the consequent):
P → Q; ¬Q; ∴ ¬P
Modus Ponens moves forward from the antecedent to the consequent. Modus Tollens moves backward from the falsity of the consequent to the falsity of the antecedent. Both patterns are valid in every possible world, whatever P and Q stand for.
肯定前件式由前件推向结论，否定后件式由后件的假推回前件的假。无论 P 和 Q 代表什么，这两种模式在任何可能世界中都有效。
2. Validity, Soundness and Fallacies | 有效性、可靠性与谬误
Validity is a property of the argument's form alone: an argument is valid if and only if it is impossible for all the premises to be true and the conclusion false at the same time. Validity says nothing about whether the premises are in fact true.
有效性只是论证形式的属性：当且仅当不可能出现所有前提为真而结论为假的情况时，该论证有效。有效性完全不涉及前提实际是否为真。
Soundness adds a factual requirement to the formal one. An argument is sound if and only if it is valid and all of its premises are actually true. Soundness is therefore what you want in real life; validity is what you want in logic exercises.
可靠性在形式要求之外增加了事实要求。当且仅当一个论证有效且其所有前提实际为真时，它才是可靠的。因此在现实生活中你要的是可靠性；在逻辑练习中你要的是有效性。
Valid ↔ ¬◇(all premises true ∧ conclusion false)
Sound ↔ (Valid ∧ all premises true)
irming the consequent is invalid: P → Q; Q; ∴ P. Denying the antecedent is invalid: P → Q; ¬P; ∴ ¬Q. The standard counterexample to both is: 'If it rains, the ground is wet; the ground is wet; therefore it rained' - a sprinkler would make the ground wet without any rain.
两个形式谬误必须牢记。肯定后件无效：P → Q；Q；∴ P。否定前件无效：P → Q；¬P；∴ ¬Q。二者的标准反例是：'如果下雨，地面就湿；地面湿了；所以下雨了'——洒水器也能让地面变湿而无须下雨。
Informal fallacies to name in essays include circular reasoning (assuming the conclusion), straw man (misrepresenting an opponent), ad hominem (attacking the person), slippery slope (claiming one step must lead to an extreme), false dilemma (offering only two options where more exist) and appeal to authority or popularity.
论文中应当点名的非形式谬误包括：循环论证（预设结论）、稻草人（曲解对手）、人身攻击、滑坡（声称一步必然导致极端后果）、虚假两难（在存在更多选项时只给两个）以及诉诸权威或诉诸大众。
3. Epistemology: The Knowledge Formula (JTB) | 知识论：知识公式（JTB）
Propositional knowledge is knowledge that something is the case, as opposed to knowledge how to do something or knowledge by acquaintance. The traditional analysis of propositional knowledge is Justified True Belief, usually abbreviated JTB.
命题性知识是'知道某事如此'的知识，区别于'知道如何做'的技能知识和'亲知'的知识。命题性知识的传统分析是'得到辩护的真信念'，通常缩写为 JTB。
S knows that p ↔ (p is true ∧ S believes that p ∧ S is justified in believing that p)
Each of the three conditions is claimed to be individually necessary and jointly sufficient. Truth is necessary because you cannot know what is false, although you can believe it. Belief is necessary because you cannot know what you do not believe. Justification is necessary because a lucky guess is not knowledge.
这三个条件被认为各自必要、合起来充分。真为必要，因为你不能知道假的东西，尽管你可以相信它。信念为必要，因为你不可能知道你不相信的东西。辩护为必要，因为碰巧猜中不叫知识。
In an exam, define justification carefully: S is justified in believing p when S has adequate evidence, a reliable source, or a sound reason for p. Then add that the classical account is attacked by Gettier cases, which is the bridge to the next section.
考试时要谨慎定义辩护：当 S 拥有充分证据、可靠来源或可靠理由相信 p 时，S 对 p 的信念才是有辩护的。接着补充：这一经典解释受到盖蒂尔案例的攻击，这就自然过渡到下一节。
4. Gettier and the No-Luck Condition | 盖蒂尔问题与无运气条件
Edmund Gettier published counterexamples in 1963 in which a person has a justified true belief but still does not know. The cases work by separating the justification from the truth: the belief is true, but only by luck.
埃德蒙·盖蒂尔于 1963 年发表了一些反例，其中一个人拥有得到辩护的真信念，却依然不构成知识。这些案例的机制是让辩护与真相互脱节：信念为真，但只是靠运气。
The classic stopped clock case runs as follows. You look at a clock that happens to be stopped at exactly the correct time, and you form the belief 'It is 3 o'clock'. The belief is true, you believe it, and your evidence (reading the clock) normally justifies the belief. Yet you do not know the time, because the truth of your belief is accidental.
经典的'停摆的钟'案例是这样的：你看了一眼恰好停在正确时刻的钟，形成信念 '现在是三点'。该信念为真，你相信它，而你的证据（看钟）通常也能为它辩护。但你并不知道时间，因为你信念之真纯属偶然。
Knowledge = JTB + no essential luck
No false lemmas: JTB + every inferential step is true
Safety: if S believes p, then p could not easily have been false
Standard repairs include the no-false-lemmas account, which forbids reasoning through a false premise; reliabilism, which requires the belief to be produced by a reliable cognitive process; the safety condition, which requires that in nearby possible worlds the belief would not be false; and virtue epistemology, which requires the belief to be an achievement of the knower's ability rather than an accident.
常见的修补方案包括：无假前提说，禁止推理经过任何假前提；可靠主义，要求信念由可靠的认知过程产生；安全性条件，要求在最接近的可能世界中该信念不会为假；以及德性知识论，要求信念是认知者能力的成就而非偶然。
5. Rationalism and Empiricism: Sources of Knowledge | 理性主义与经验主义：知识的来源
Rationalists hold that reason alone can be an independent source of substantive knowledge about reality. Descartes argues that clear and distinct perceptions cannot be doubted, that the cogito 'I think, therefore I exist' survives radical doubt, and that some ideas such as infinity and perfection are innate.
理性主义者认为，理性本身可以独立提供关于实在的实质性知识。笛卡尔主张：清楚分明的知觉不可怀疑；'我思，故我在'这一命题能经受彻底怀疑；无限、完美等观念是天赋的。
Empiricists hold that all knowledge and all concepts derive from experience. Locke attacks innate ideas and compares the mind at birth to a blank slate; simple ideas come from sensation and reflection, and complex ideas are built from them. Berkeley reduces physical objects to collections of ideas; Hume reduces all ideas to prior impressions.
经验主义者认为一切知识与概念都源于经验。洛克反对天赋观念，把心灵比作白板；简单观念来自感觉与反省，复杂观念由此组合而成。贝克莱把物体化约为观念的集合；休谟则把一切观念化约为先前的印象。
Rationalism: reason → knowledge (innate ideas, a priori insight)
mpiricism: experience → all ideas (Nihil est in intellectu quod non prius fuerit in sensu)
Hume's fork and his analysis of causation are the strongest weapons for the empiricist. Causation is never perceived as a necessary connection; we only observe constant conjunction, and habit makes us expect the effect. Therefore induction cannot be justified by reason, since any attempt to justify it uses induction itself.
休谟的'分叉'及其因果分析是经验主义者最强的武器。因果关系从来不能被感知为必然联系；我们只观察到恒常会合，习惯让我们期待结果。因此归纳无法由理性证明，因为任何证明都会再次使用归纳本身。
6. Philosophy of Religion: Arguments for God | 宗教哲学：上帝存在的论证
The cosmological argument reasons from the existence of the universe to a cause of the universe. In its Kalam form: everything that begins to exist has a cause; the universe began to exist; therefore the universe has a cause. Aquinas argues that chains of motion, causation and contingency require a first member that is not itself moved, caused or contingent.
宇宙论论证从宇宙的存在推出宇宙的原因。其凯拉姆形式为：凡开始存在的事物都有原因；宇宙开始存在；因此宇宙有原因。阿奎那主张，运动、因果与偶然性之链都需要一个自身不被推动、不被引起、非偶然的第一环。
Kalam: ∀x (Begins(x) → HasCause(x)); Begins(universe); ∴ HasCause(universe)
The teleological or design argument reasons from apparent order and purpose to a designer. Paley's watch analogy is the standard statement: if you found a watch on a heath you would infer a watchmaker, so the far greater complexity of living things implies a designer. Hume replies that the analogy between artefacts and nature is weak, that order can arise without design, and that the argument supports only a limited and possibly multiple designer.
目的论或设计论证从表面上的秩序与目的推出设计者。佩利的手表类比是标准表述：若你在荒地上捡到一块手表，你会推断有制表者，因此生命体远为复杂的结构也意味着设计者。休谟回应说，人工制品与自然的类比很弱，秩序可以无设计地产生，且该论证最多只能支持一个有限、甚至可能是多位的设计者。
The ontological argument tries to prove God from the concept of God alone. Anselm defines God as 'that than which nothing greater can be conceived'. If God existed only in the understanding, something greater - the same being existing in reality - could be conceived, which is contradictory; therefore God exists in reality. Gaunilo objects with a perfect island, and Kant adds that existence is not a real predicate: 'exists' adds no property to a concept.
本体论论证试图仅从上帝的概念证明上帝存在。安瑟伦把上帝定义为 '无法设想比之更大者'。若上帝只存在于理解之中，则可设想一个更大者即同一存在者也在现实中存在，这就自相矛盾；故上帝现实中存在。高尼罗以'完美的岛屿'反驳；康德补充说，存在不是实在的谓词：'存在'并不为概念增加任何属性。
7. The Problem of Evil and Theodicy | 恶的问题与神义论
The logical problem of evil states that the existence of evil is logically incompatible with a God who is omnipotent, omniscient and omnibenevolent. An omnipotent God could prevent evil, an omniscient God would know about it, and an omnibenevolent God would want to remove it; yet evil exists.
恶的逻辑问题主张，恶的存在与全知、全能、全善的上帝在逻辑上不相容。全能的上帝能够阻止恶，全知的上帝知道恶，全善的上帝愿意消除恶；然而恶存在。
Omnipotence ∧ Omniscience ∧ Omnibenevolence ∧ Evil exists → contradiction
Escape route: there exists a morally sufficient reason for God to permit evil
J. L. Mackie presented the problem as an inconsistent triad: God is all-powerful and wholly good; evil exists; and a good being always eliminates evil as far as it can, provided it knows how and is able to do so. Remove any one of the three and the contradiction dissolves; the theist must show why a good God might allow evil.
麦凯把该问题表述为不一致的三元组：上帝全能且全善；恶存在；善的存在者只要知道并有能力，就会尽可能消除恶。移除任一条矛盾即解，因此有神论者必须说明良善的上帝为何允许恶。
Theodicies answer this demand. The free will defence (Plantinga) holds that it is possible that God created free creatures, and that free creatures may choose evil; a world with free beings who sometimes sin may be better than a world of automata. The soul-making theodicy (Irenaeus, Hick) holds that suffering is a necessary instrument for building character and achieving virtue. Augustinian theodicy treats evil as a privation, an absence of good rather than a positive substance, and traces it to the misuse of free will.
神义论回应这一要求。自由意志辩护（普兰丁格）主张：上帝创造自由存在者、而自由存在者可能选择恶，这是可能的；有自由而偶会犯错的世界可能优于全是自动机的世界。灵魂塑造神义论（伊里奈乌斯、希克）主张，苦难是塑造品格、成就德性的必要工具。奥古斯丁式神义论把恶视为匮乏，是善的缺失而非实体，并把它追溯到自由意志的误用。
By contrast, the evidential problem of evil (Rowe) claims not that evil makes God's existence impossible, but that the amount and distribution of apparently pointless suffering makes God's existence unlikely. It attacks degrees of probability rather than logical consistency, so the defences above answer it only partly.
与之相对，恶的证据问题（罗伊）主张的不是恶使上帝存在变得不可能，而是大量且分布不合理的、看似无意义的苦难使上帝存在变得不大可能。它攻击的是概率高低而非逻辑融贯，因此上述辩护只能部分回应它。
8. Ethics: The Utility Formula | 伦理学：功利公式
Utilitarianism is a consequentialist theory: the rightness of an action depends entirely on its consequences, not on the motive of the agent or on absolute rules. Bentham's principle of utility is stated as the greatest happiness of the greatest number, where happiness means pleasure and the absence of pain.
功利主义是后果论理论：行为的对错完全取决于后果，而不取决于行为者的动机或绝对规则。边沁的功利原则表述为'最大多数人的最大幸福'，其中幸福指快乐与痛苦的免除。
An action is right ↔ it produces the greatest total net happiness for the greatest number
Net happiness = Σ (pleasure − pain) over all affected beings, each counted equally
Bentham's hedonic calculus measures pleasure by seven criteria: intensity, duration, certainty, propinquity (nearness in time), fecundity (chance of producing further pleasure), purity (chance of not producing pain) and extent (the number of people affected). In exam answers, list these criteria and then immediately criticise them as difficult and unreliable to measure.
边沁的快乐计算法用七项标准度量快乐：强度、持久性、确定性、远近（时间上的接近）、丰度（引发更多快乐的可能）、纯度（不引发痛苦的可能）和广度（受影响的人数）。答题时应列出这些标准，随即批评其难以测量、并不可靠。
Mill modifies Bentham by distinguishing higher and lower pleasures, arguing that intellectual and moral pleasures are qualitatively superior, and that competent judges who have experienced both prefer the higher. He also offers the harm principle as a rule for limiting interference with liberty.
密尔修正边沁，区分高级快乐与低级快乐，主张理智与道德的快乐在质上更优，而同时体验过两者的'合格评判者'会偏好高级快乐。他还提出伤害原则，作为限制干预他人自由的规则。
Main objections to memorise are: it can permit injustice to individuals if the majority gains; pleasures and pains cannot be measured on one scale; act utilitarianism is implausibly demanding; and Nozick's experience machine suggests we value reality and autonomy, not just pleasant experience. Rule utilitarianism replies by testing rules rather than single acts.
必须记住的主要反驳有：若多数人获益，它可以允许对个人的不公；快乐与痛苦无法在单一尺度上度量；行为功利主义的要求高得不合理；诺齐克的'体验机器'表明我们看重真实与自主，而不只是愉悦体验。规则功利主义则改为检验规则而非单个行为。
9. Kant: The Categorical Imperative | 康德：绝对命令
Kant's ethics is deontological: the moral value of an action lies in the maxim on which it is done, not in its consequences. Only a good will is good without qualification, and a good will acts from duty rather than from inclination.
康德伦理学是义务论的：行为的道德价值在于其依据的准则，而不在后果
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