📚 GCSE CAIE Philosophy: Formula & Theorem Quick Reference Handbook | GCSE CAIE 哲学：公式定理速查手册
Philosophy is not a subject of formulas in the mathematical sense, but many of its most powerful arguments, theories and critical moves can be distilled into clear logical patterns. This handbook presents those key 'formulas' and 'theorems' - the structure of cosmological reasoning, the calculus of utilitarianism, the logical paradox of evil and the categorical imperative - in a revision-friendly, paired-language format. Students preparing for CAIE GCSE Philosophy can use this as a rapid recall tool to sharpen analytical writing and argument evaluation.
哲学虽然不像数学那样有公式，但其许多最有力的论证、理论和批判思路都可以提炼为清晰的逻辑模式。这本手册将那些关键的“公式”和“定理”——宇宙论推理的结构、功利主义的计算、恶的逻辑悖论、绝对命令等——以双语配对的形式呈现。备考 CAIE GCSE 哲学的同学们可以将此作为快速回顾的工具，用以加强分析写作和论证评价的能力。
1. The Cosmological Argument Formula | 宇宙论论证公式
a mover → cannot regress infinitely → therefore, an Unmoved Mover (God) exists. In short: Change → Cause → Termination of regress → First Cause.
阿奎那的第一路：运动 → 需要推动者 → 无法无限回溯 → 因此，存在一个不动的推动者（上帝）。简言之：变化 → 原因 → 回溯终止 → 第一因。
in the world → no infinite causal chain → a first efficient cause exists. Symbolically: (∀x (Cx → Cy)) ∧ ¬∞ Regress → ∃z (First Cause).
第二路：世间因果 → 没有无限的因果链条 → 存在第一个动力因。符号化为： (任一事物有因) ∧ 非无限回溯 → 存在第一因。
hing were contingent, nothing would exist now → therefore, a necessary being (God) exists.
第三路：偶然的存在者存在 → 它们依赖一个必然的存在者才能存在 → 如果一切皆偶然，则现在应无物存在 → 因此，存在一个必然的存在者（上帝）。
Kalam formulation: Everything that begins to exist has a cause; the universe began to exist; therefore, the universe has a cause (God). This is a crisp deductive formula.
卡拉姆论证表述：一切开始存在的事物都有原因；宇宙开始存在；因此，宇宙有一个原因（上帝）。这是一个干脆的演绎公式。
2. The Teleological Argument Formula | 目的论论证公式
Paley’s design argument: Complexity + Purpose → Designer. A watch found on a heath implies a watchmaker; similarly, the universe, with its complex order and purposeful parts (e.g. the eye), implies a divine Designer.
佩利的设计论证：复杂性 + 目的 → 设计者。在荒野中发现的手表意味着制表匠；同理，宇宙具有复杂秩序和目的性部件（如眼睛），意味着一位神性设计者。
Hume’s critique formula: Analogy is weak (universe ≠ watch); the effect does not dictate a single, infinite designer; the alleged design may be a result of chance operating over infinite time. Weak analogy → many possible designers → natural forces may mimic design.
休谟的批判公式：类比薄弱（宇宙≠手表）；结果并不能推断出唯一、无限的设计者；所谓的秩序可能是无限时间中偶然作用的结果。弱类比 → 多种可能的设计者 → 自然力可能模拟设计。
Darwin’s alternative: Natural selection provides a non-intentional mechanism for apparent design. Apparent design = Natural selection + Random variation − Divine intervention.
达尔文的替代方案：自然选择为表面上的设计提供了非意图的机制。表面设计 = 自然选择 + 随机变异 − 神圣干预。
3. The Ontological Argument Formula | 本体论论证公式
Anselm’s formula: God is 'that than which nothing greater can be conceived'. If God exists only in the mind, we could conceive a greater being that exists in reality as well. That would be greater. So, God must exist in reality. Perfection of concept → Necessary existence in re.
安瑟伦的公式：上帝是“可设想的无与伦比者”。如果上帝仅存在于心智中，我们便能设想一个也存在于现实中的更伟大者。那将更伟大。所以，上帝必然存在于现实中。概念上的完美 → 现实中的必然存在。
Descartes’ version: Existence is a perfection (a predicate). God, possessing all perfections, must therefore exist. Supreme Being = All perfections (including existence).
笛卡尔的版本：存在是完美（谓词）。上帝拥有所有完美，故必然存在。至上存在者 = 所有完美（包含存在）。
Kant’s refutation: Existence is not a real predicate; it adds nothing to the concept. One hundred real thalers contain no more conceptually than one hundred possible thalers. Existence ≠ Predicate → Ontological argument fails.
康德的驳斥：存在不是真实的谓词；它对概念无所增添。一百个真的银币在概念上并不多于一百个可能的银币。存在 ≠ 谓词 → 本体论论证不成立。
4. The Problem of Evil Logical Formula | 邪恶问题的逻辑公式
The inconsistent triad (Mackie): If God is omnipotent, He can prevent evil. If God is wholly good, He would want to prevent evil. Yet evil exists. Therefore, such a God does not exist. Omnipotence + Omnibenevolence + Evil → Contradiction.
不相容的三位一体（麦凯）：如果上帝全能，祂便能阻止邪恶；如果上帝全善，祂便愿意阻止邪恶。然而邪恶存在。因此，这样的上帝不存在。全能 + 全善 + 邪恶 → 矛盾。
Evidential problem: The sheer amount and distribution of gratuitous evil makes God’s existence improbable. Gratuitous evil → likely ~God.
证据性问题：大量且无端分布的恶使得上帝的存在显得不可能。无端的恶 → 可能 ~上帝。
Logical flow: (P1) If God exists, evil does not exist. (P2) Evil exists. (C) Therefore, God does not exist. (Modus tollens)
逻辑流程：(前提1) 如果上帝存在，邪恶便不存在。(前提2) 邪恶存在。(结论) 因此，上帝不存在。（否定后件推理）
5. The Free Will Defence Formula | 自由意志辩护公式
Plantinga’s defence: A world with morally significant free creatures is of greater value than a world of automatons. God cannot create a world with free creatures who always choose good, because freedom entails the possibility of evil. Free will → Possible evil → Greater good (love, moral growth) → Justifies God.
普兰丁格的辩护：具有道德意义上自由受造物的世界，比自动机械的世界更有价值。上帝无法创造一个自由受造物却总选择良善的世界，因为自由必然包含恶的可能性。自由意志 → 可能的恶 → 更大的善（爱、道德成长） → 为上帝辩护。
Natural evil as a consequence: Some natural evil may result from the free choices of non-human moral agents (fallen angels) or be necessary for soul-making. Natural evil = fallen free agents + environment for character building.
自然恶作为后果：某些自然恶可能源自非人的道德能动者的自由选择（堕落天使），或为塑造灵魂所必需。自然恶 = 堕落的自由能动者 + 塑造品格的环境。
Crucial distinction: The defence does not claim this is the actual reason; it only shows God and evil are logically compatible. Logical possibility → No contradiction → Defence succeeds.
关键区分：该辩护并不主张这就是实际理由；它只是表明上帝与邪恶在逻辑上可相容。逻辑可能性 → 无矛盾 → 辩护成功。
6. Religious Experience Argument Formula | 宗教经验论证公式
an experience of God → unless there are defeaters, S is justified in believing God exists. Perceptual model: Experience of X → prima facie justification for belief in X.
基本论证：主体S拥有对上帝的经验 → 除非存在否决因子，否则S有理由相信上帝存在。感知模式：对X的经验 → 对X之相信的初步证成。
William James’ formula: Religious experiences have four marks (ineffability, noetic quality, transiency, passivity). Their fruits (moral and spiritual transformation) confirm their divine origin. Experience + Fruits → Warrant.
威廉·詹姆士的公式：宗教经验有四个特征（不可言说、知性品质、瞬时性、被动性）。其果实（道德和灵性转化）证实其神圣来源。经验 + 果实 → 保证。
Swinburne’s principle of credulity: In the absence of special considerations, it is rational to accept that things are as they seem to be. Seems → Is (unless undercut).
斯温伯恩的轻信原则：若无特别考虑，理性上应接受事物如其所呈现的那样。看起来如此 → 即如此（除非被削弱）。
Challenge: Religious experiences are diverse and contradictory; they depend on cultural background. Diversity → Naturalistic explanation → Undercutting defeater.
挑战：宗教经验多样且相矛盾；它们取决于文化背景。多样性 → 自然主义解释 → 削弱性否决因子。
7. Pascal’s Wager Formula | 帕斯卡赌注公式
ures). If you do not believe and God exists → infinite loss (hell). If you do not believe and God does not exist → finite gain. Rational calculation favours belief. Expected value of belief >> expected value of unbelief.
决策矩阵：若你信上帝且上帝存在 → 无限收益（天堂）。若你信但上帝不存在 → 有限损失（某些世俗快乐）。若你不信而上帝存在 → 无限损失（地狱）。若你不信且上帝不存在 → 有限收益。理性计算倾向于信仰。信仰的预期价值 >> 不信的预期价值。
Criticism: Many gods objection – which god? We cannot simply choose to believe. Multiple possible gods → wager requires probability assignments that are impossible.
批判：多神反驳——信哪一个神？我们无法轻易选择相信。多个可能的神 → 赌注需要不可能的概率分配。
Pascal’s response: The wager is an appeal for the agnostic to start practising faith, not a proof of God. Wager = decision under uncertainty → embark on a religious way of life.
帕斯卡的回应：这赌注是对不可知论者的呼吁，让其开始实践信仰，而非对上帝的证明。赌注 = 不确定下的决策 → 踏上宗教生活之路。
8. Utilitarianism Formula | 功利主义公式
Bentham’s principle of utility: An act is right if it tends to produce the greatest happiness for the greatest number. Rightness = Maximise (Total pleasure − Total pain).
边沁的功利原则：一个行为若倾向于为最大多数人产生最大幸福，便是正确的。正确性 = 最大化（总快乐 − 总痛苦）。
Hedonic calculus: Seven factors – intensity, duration, certainty, propinquity, fecundity, purity, extent. Utility = Σ (I×D×C×P×F×Pu×E). Simply, add up all the pleasure units, subtract pain units, for all affected.
快乐计算：七因素——强度、持续时间、确定性、邻近性、丰产性、纯度、范围。效用 = Σ (I×D×C×P×F×Pu×E)。简言之，加总所有快乐单位，减去痛苦单位，涵盖所有受影响者。
Mill’s higher pleasures: Quality matters, not just quantity. 'It is better to be a human being dissatisfied than a pig satisfied.' Higher pleasures (intellectual) > lower pleasures.
密尔的高级快乐：品质要紧，不止数量。“做一个不满足的人，胜于做一头满足的猪。”高级快乐（智性） > 低级快乐。
Act vs rule utilitarianism: Act: compute for each act. Rule: follow rules that, if generally adopted, maximise utility. Rule U: Adopt rule R such that everyone obeying R → max utility.
行为功利主义与规则功利主义：行为：对每个行为计算。规则：遵循那些若被普遍采纳则将最大化效用的规则。规则功利主义：采纳规则R，使得所有人遵守R → 最大效用。
9. Kant’s Categorical Imperative Formula | 康德绝对命令公式
First formulation: 'Act only according to that maxim whereby you can at the same time will that it should become a universal law.' Maxim → Universal law → Contradiction test → Pass/Fail.
第一表述：“只依据你能同时意愿其成为普遍法则的准则去行动。”准则 → 普遍法则 → 矛盾检验 → 通过/失败。
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.' Humanity = End-in-itself → Never mere means.
人性公式：“行动时，无论对自己还是对他人，始终将人性作为目的，而绝不只当作手段。”人性 = 目的本身 → 绝不只当作手段。
Duty vs inclination: An act has moral worth only if done from duty, not from inclination or expected consequences. Moral worth = Acting from duty (good will).
义务与禀好：只有出于义务而非禀好或预期后果的行为才具有道德价值。道德价值 = 出于义务而行动（善良意志）。
Kingdom of ends: A systematic union of all rational beings through common laws. Rational agents → Self-legislating → Kingdom of ends.
目的王国：所有理性存在者通过共同法则形成的系统联合。理性能动者 → 自我立法 → 目的王国。
10. The Naturalistic Fallacy Formula | 自然主义谬误公式
Hume’s is–ought gap: From descriptive premises (what is the case), one cannot logically deduce a prescriptive conclusion (what ought to be). 'Is' → 'Ought' invalid without bridging principle.
休谟的“是-应当”裂隙：从描述性前提（是什么）无法逻辑地推导出规范性结论（应当是什么）。“是” → “应当” 在缺少桥接原则时无效。
Moore’s open question argument: Goodness cannot be defined in natural terms (e.g. pleasure), because we can always ask 'But is pleasure good?' This remains an open question. Natural property P ≠ good → Open question test.
摩尔的开放问题论证：善不能用自然属性来定义（如快乐），因为我们总可以问“但快乐是善吗？”这仍是一个开放问题。自然属性P ≠ 善 → 开放问题检验。
Application: 'Human beings naturally seek pleasure, therefore pleasure is good' commits the fallacy. Naturalistic fallacy = confusing natural with normative.
应用：“人自然追求快乐，因此快乐是善的”犯了此谬误。自然主义谬误 = 混淆自然的与规范的。
Ethical non-naturalism: Moral properties are sui generis, known through intuition, not reducible to science. Moral property → Irreducible → Intuition.
伦理非自然主义：道德属性是独特的，通过直觉认识，不可还原为科学。道德属性 → 不可还原 → 直觉。
11. Existentialism: Existence Precedes Essence Formula | 存在主义：存在先于本质公式
Sartre’s core formula: For humans, existence comes first; there is no pre-defined human nature. You are not a manufactured object with a purpose. Existence → Freedom → Self-definition → Essence.
萨特的核心公式：对于人，存在在先；没有预先确定的人性。你不是一个有预定目的的制造物。存在 → 自由 → 自我界定 → 本质。
Radical freedom: You are condemned to be free; you are responsible for all your actions. Freedom = Condemnation → Total responsibility.
激进的自由：你被判定为自由；你要对自己的全部行为负责。自由 = 宣判 → 完全责任。
Creating meaning: Life has no intrinsic meaning; we must create it through projects and commitments. Meaning = Personal project + Authentic choice.
创造意义：生命没有内在意义；我们必须通过筹划和投身来创造。意义 = 个人筹划 + 本真的选择。
Bad faith: Denying your freedom by pretending you are determined (like a waiter acting only as a waiter). Bad faith = Self-deception about freedom.
自欺：通过假装自己是被决定的来否认自由（如一个只作为侍者表演的侍者）。自欺 = 关于自由的自欺。
12. Science and Falsifiability Formula | 科学与可证伪性公式
Popper’s demarcation: A theory is scientific if it can be falsified; it must make risky predictions. Scientific = Falsifiable (in principle). Astrology and Freudian psychoanalysis are unfalsifiable.
波普尔的划界：一个理论若能证伪便属科学；它必须做出风险的预测。科学 = 可证伪（原则上）。占星学和弗洛伊德精神分析不可证伪。
Religious claims: Many philosophers argue that 'God exists' is not a scientific hypothesis; it is a foundational belief or 'blik'. God-exists = Non-falsifiable → Not scientific (but possibly meaningful).
宗教主张：许多哲学家认为“上帝存在”并非科学假说；它是基础信念或“布利克”。上帝存在 = 不可证伪 → 非科学（但可能有意义）。
Falsification applied to evil: The believer may allow nothing to count against God’s existence, making the belief unfalsifiable. Flew’s challenge: 'What would have to occur to constitute for you a disproof of God’s love?' No possible counter-evidence → Death by a thousand qualifications.
证伪原则应用于恶：信徒可能不让任何事证伪上帝存在，使该信念不可证伪。弗卢的挑战：“对于你来说，有什么事情的发生会构成上帝之爱的反驳？”没有可能的反证 → 死于千般限定。
Faith as interpretation: Some see religious beliefs as interpretive frameworks, not hypotheses. World viewed through a religious lens → Not directly falsifiable → Meaning, not fact.
信仰作为诠释：有人视宗教信仰为诠释框架，而非假说。通过宗教透镜看世界 → 不可直接证伪 → 意义，而非事实。
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