📚 Philosophy Formula & Theorem Quick Reference for Year 11 Edexcel | 爱德思 Year 11 哲学公式定理速查手册
This article presents essential philosophical ideas as compact “formulas” and “theorems” – structured principles and argument patterns you will encounter in the Edexcel Year 11 Philosophy course. Think of each entry as a ready-made reasoning tool, helping you to break down complex ethical theories, proofs for God’s existence, or dilemmas about life after death into clear, repeatable steps. Use these as a memory aid and a checkpoint when structuring essays or revising key topics.
本文将关键的哲学理念浓缩为简洁的“公式”和“定理”——即在爱德思 Year 11 哲学课程中会遇到的结构化原则与论证模式。你可以把每一个条目看作现成的推理工具，帮助你将被复杂的伦理理论、上帝存在的证明或死后生命的困境分解为清晰、可重复的步骤。请将它们用作记忆辅助和检查点，在构建论文或复习关键主题时发挥作用。
1. The Greatest Happiness Formula | 最大幸福公式
Action is right ⇌ it produces the greatest total pleasure minus pain for all affected.
e pleasure (P) and pain (Pn) that result from an action, then choose the option that maximises the net balance: Total Utility = Σ(P) – ts over mere physical satisfaction. When analysing an ethical dilemma, apply this formula step by step: identify all beings affected, estimate the intensity, duration, certainty, and remoteness of the pleasures and pains for each, and finally calculate which action yields the highest overall utility.
在功利主义中，一个行为的道德价值完全由其后果决定。边沁的快乐计算法要求我们将行为带来的快乐（P）与痛苦（Pn）相加，再选择能够最大化净值的选项：总功利 = Σ(P) – Σ(Pn)。约翰·斯图尔特·密尔通过区分高级（H）和低级（L）快乐对这一公式进行了细化，给予智力和道德的享受比单纯的肉体满足更大的权重。在分析伦理困境时，请逐步运用以下公式：识别所有受影响的生灵，估计每种快乐和痛苦的强度、持续时间、确定性和远近程度，最后计算出哪一个行为能产生最高的总体功利。
2. The Categorical Imperative Formula of Universal Law | 定言命令普遍法则公式
Act only according to that maxim whereby you can, at the same time, will that it should become a universal law.
Immanuel Kant’s ethical formula tests the logical coherence of a proposed action. A maxim (M) is a personal principle like “I will break a promise when it suits me.” The test asks: if M becomes a universal law (U) that everyone must follow, does the action still make sense? Two outcomes are possible: a contradiction in conception (C1), where the very idea of the action becomes impossible in a world where everyone does it (e.g., promise-breaking would destroy the institution of promising), or a contradiction in will (C2), where you cannot rationally will such a world, even if it is logically possible. If either contradiction occurs, the action is forbidden. This formula removes emotion and consequences from moral reasoning, focusing purely on rational duty.
康德的伦理公式检验一个拟议行为在逻辑上是否自洽。一个准则（M）是一条个人原则，如“当对我有利时，我会背弃承诺”。该检验问：如果 M 成为人人都必须遵守的普遍法则（U），这一行为是否仍然合理？可能出现两种结果：构想上的矛盾（C1），即在一个人人都如此行事的世界中，该行为本身变得不可能（例如，背弃承诺会破坏承诺这一制度）；或者意志上的矛盾（C2），即即使逻辑上可能，你也无法理性地意愿这样一个世界。如果出现任何一种矛盾，该行为就是被禁止的。该公式将情感和后果从道德推理中移除，专注于理性义务。
3. The First Cause Theorem (Cosmological Argument) | 第一因定理（宇宙论论证）
Everything that exists has a cause; an infinite regress of causes is impossible; therefore, there must be a first uncaused cause (God).
n of reasoning: (1) We observe that things in the world are in a state of change or motion – . (3) If we trace back the chain of causes, it cannot go on to infinity, because without a first cause there would be no subsequent causes and thus nothing would exist now. (4) Therefore, we must arrive at a first cause, which everyone calls God. The theorem can be written as: for any existing entity E, there is a cause C. The series C₁ ← C₂ ← C₃ … cannot be infinite (¬∞). Therefore ∃ the necessary being or whether the first cause must have the attributes of a theistic God.
托马斯·阿奎那将此表述为一个推理链条：(1) 我们观察到世间万物处于变化或运动之中——它们是由他物引起的存在。(2) 没有东西能是自身的原因，因为那样它就必须先于自身存在，这是不可能的。(3) 如果我们沿着因果链追溯，它不可能无限倒推，因为没有第一因就不会有后续的因，也就不会有现存的事物。(4) 因此，我们必须达到一个第一因，也就是众人所称的上帝。该定理可写为：对任何存在物 E，都有一个原因 C。序列 C₁ ← C₂ ← C₃ … 不能为无穷 (¬∞)。因此 ∃ C₀（不动的推动者）。批评者们质疑宇宙本身是否可能就是必然存在物，或者第一因是否必须具有有神论上帝的属性。
4. The Design Theorem (Teleological Argument) | 设计定理（目的论论证）
Complex order with a purpose implies a designer; the universe exhibits such order; therefore, the universe has a designer (God).
o serve the purpose of telling time, you rightly infer a watchmaker. The natural world (N) – eyes, wings, planetary orbits – ion. David Hume offered a pre-emptive criticism: even if we accept the analogy, the theorem cannot prove the God of classical theism; it could just as well suggest a team of lesser gods, or an imperfect apprentice deity who got things wrong. Modern science also advances natural selection as an alternative mechanism that mimics design without a designer. Nonetheless, for revision, remember the syllogistic form: Order → Designer, Universe has order, ∴ Designer.
威廉·佩利的钟表匠类比建立在这个公式之上。如果你在荒野中发现一块手表（W），其精密零件共同作用以显示时间为目的，你自然会推断出存在一位钟表匠。自然界（N）——眼睛、翅膀、行星轨道——展现出更为复杂且具有明显目的的设计。通过类比，存在一位设计智能（D）是最合理的解释。大卫·休谟提出了一项先发制人的批评：即使我们接受这一类比，该定理也无法证明古典有神论的上帝；它同样可能暗示一个由多个较低级神明组成的团队，或者一个把事情弄糟的不完美学徒神。现代科学也提出了自然选择作为一种无需设计者的替代机制，却能模拟出设计。无论如何，为了考试复习，请记住其三段论形式：秩序 → 设计者，宇宙有秩序，∴ 存在设计者。
5. The Problem of Evil Logical Formula | 恶之问题逻辑公式
If an omnipotent, omnibenevolent God exists, gratuitous evil would not exist. Gratuitous evil does exist. Therefore, such a God does not exist.
ly: (1) God is omnipotent (O). (2) God is wholly good (G). (3) Evil exists (E). Add the hidden premise that a wholly good being always prevents evil as far as it can, and an omnipotent being can prevent all evil. Then O ∧ G → ¬ lly possible (free-will defence), or some evils are necessary for a greater good (Irenaean theodicy), or evil is an absence of good, not a positive reality (Augustine). For your exam, translate the problem into the formula ¬(O ∧ G ∧ explain how each theodicy aims to show that O, G, and E can coexist without contradiction.
恶的逻辑问题（J.L.麦基的不一致三元组）可以表达为三个不能同时为真的命题：(1) 上帝是全能的 (O)。(2) 上帝是全善的 (G)。(3) 恶存在 (E)。加上隐含前提：全善的存在总会在能力范围内阻止恶，全能的存在能够阻止一切恶。那么 O ∧ G → ¬E。既然观察到 E 存在，我们就面临一个矛盾。有神论者必须否认至少一个前提：也许上帝的全能只意味着做逻辑上可能的事（自由意志辩护），或者某些恶是为了更大的善所必需（爱任纽神义论），或者恶是善的缺乏而非一种积极的实在（奥古斯丁）。在考试中，请将这一问题转化为公式 ¬(O ∧ G ∧ E)，并准备好解释每一种神义论如何旨在表明 O、G 和 E 可以无矛盾地共存。
6. The Dualist Survival Theorem | 二元论幸存定理
The mind is a non-physical substance distinct from the body; therefore, the death of the body does not entail the death of the person.
can clearly and distinctly conceive the mind (res cogitans) existing without the body (res extensa). (2) Whatever is clearly and distinctly conceivable is logically possible. (3) If A can exist without B, A is not identical to B. (4) Therefore, the mind is a separate substance and can survive bodily annihilation. Descartes’ formula: Mind (M) ≠ Body (B), and survival = M persists after B ceases. Challenges come from materialism: if mental states are brain states (identity theory), then the destruction of the brain eliminates the mind; there is no surviving subject. Also, the problem of interaction: how does an immaterial substance cause physical effects? t ? For revision, know that Plato’s argument from recollection and the cyclical argument in the Phaedo also serve as “proofs” of the soul’s pre-existence and immortality, following a similar essential structure.
实体二元论，与柏拉图和笛卡尔密切相关，为相信死后生命提供了形而上学基础。推理如下：(1) 我能够清晰而明确地设想心灵（思维实体）脱离肉体（广延实体）而存在。(2) 凡是清晰明确可设想的事物在逻辑上就是可能的。(3) 如果 A 可以脱离 B 而存在，那么 A 与 B 不是同一的。(4) 因此，心灵是一种独立的实体，能够在肉体毁灭后存活。笛卡尔的公式：心灵 (M) ≠ 肉体 (B)，幸存 = M 在 B 终止后持续存在。挑战来自唯物主义：如果心理状态就是大脑状态（同一论），那么大脑的毁坏就会消灭心灵；不存在幸存的自我主体。此外，还有互动难题：非物质的实体如何引起物理效应？复习时请记住，柏拉图在《斐多篇》中的回忆论证和循环论证也作为证明灵魂预先存在和不朽的“证据”，遵循类似的基本结构。
7. The Hickean Replica Theorem | 希克复制人定理
A person can survive death if God creates an exact replica of that individual in a resurrected world, preserving psychological continuity and bodily form.
John Hick’s “replica” n P₀ ceases to exist in this space-time. However, in an instant, God creates an exact replica P₁ in a different space (the resurrection world). This P₁ is psychologically continuous with P₀ (shares memories, character, dispositions) and bodily indistinguishable from P₀ as it was at its peak. The formula: Survival = (Psychological continuity) ∧ (Bodily similarity) ∧ we would consider them the same person; resurrection is analogous on a cosmic scale. Critics raise the duplication problem: what if God creates multiple replicas? Which, if any, is the original you? e survival, or is it just a perfect copy that replaces you while the original remains dead? This highlights the difference between numerical identity and exact similarity.
约翰·希克的“复制品”理论不依赖于非物质的灵魂，旨在处理死后的人格同一性问题。其序列如下：在死亡的刹那，人 P₀ 在这个时空中不复存在。然而，在一瞬间，上帝在另一个空间（复活世界）中创造出一个精确的复制品 P₁。这个 P₁ 在心理上与 P₀ 连续（享有相同的记忆、性格、倾向），在身体上与 P₀ 鼎盛时期的模样难以区分。公式：幸存 = (心理连续性) ∧ (身体相似性) ∧ (神圣记忆库保证)。希克认为，如果我们能够设想一个人在伦敦消失瞬间又在纽约重现，我们会将他们视为同一个人；复活在宇宙尺度上与此类似。批评者提出复制难题：如果上帝创造了多个复制品怎么办？哪一个是真正的你，或者一个都不是？此外，一个单纯的复制品是否真正构成幸存，还是说它只是一个完美副本，取代了你而你本人依然死去？这凸显了数量同一性与高度相似性之间的区别。
8. The Euthyphro Dilemma Formula | 尤西弗罗困境公式
Is it right because God commands it, or does God command it because it is right?
ng from Plato’s dialogue, can be expressed as two horns. Horn 1: Right = Divine command (R ⇌ God commands it, then morality is arbitrary; God could command cruelty and it would become “good.” ltimate source of morality. The formula reveals a challenge for divine command theory: either morality loses its objectivity and becomes based on divine whim, or God becomes redundant for moral knowledge. A theist might try to escape the dilemma by arguing that God’s nature is goodness itself, making the two horns merge: God commands in accordance with his necessarily good character. For essays, always outline both horns and the attempted third way, then evaluate whether the dilemma ultimately undermines a religious foundation for ethics.
这个源于柏拉图对话录的困境可以表述为两个角。第一角：正当 = 神圣命令 (R ⇌ D)。如果 X 是正当的仅仅因为上帝命令如此，那么道德就是任意的；上帝可以命令残忍之事，而这就会变成“善”。第二角：上帝命令 X 是因为它已经是正当的 (G → R)。那样的话，上帝之外就存在一个独立的善的标准，上帝便不再是道德的终极源头。这一公式揭示了神命论面临的挑战：要么道德失去其客观性，变得基于神圣的心血来潮，要么上帝在道德知识中变得多余。有神论者可能会试图逃逸困境，主张上帝的本性就是善本身，从而使两角合二为一：上帝依据其必然善良的品格发出命令。在论文中，请务必概述两角以及尝试的第三条路径，然后评估该困境是否最终削弱了宗教作为伦理基础的可能。
9. The Act–Omission Distinction Formula | 作为–不作为区分公式
Harming by acting is morally worse than allowing harm by omitting to act, even if the consequences are identical.
of a trolley, killing them to save five; (B) you see a runaway trolley headed for five people, but do nothing to divert it, letting the five die. Many hold that A is murder while B is merely a failure to assist. The formula: Moral blameworthiness (M) = [Intent (I) × Action type (At)] ÷ Justification (J). When the act directly causes death, the At factor is high; when the death results from an omission, At is low. However, utilitarians reject the distinction as morally irrelevant: both acts produce the same net happiness. The Edexcel course often requires you to apply this to issues like euthanasia, where passive steps (withdrawing treatment) are often legally distinguished from active steps (administering lethal injection). Be ready to criticise the formula by asking: is there a real difference between intending to kill and merely foreseeing death as a side-effect, or is that line also blurry?
这一区分在医学伦理和生命神圣性的许多辩论中居于核心。考虑两种情景：(A) 你将陌生人推向电车轨道，致其死亡以拯救五人；(B) 你看到失控电车冲向五人，却无所作为任其死去。许多人认为 A 是谋杀，而 B 仅仅是未能援救。公式：道德可责性 (M) = [意图 (I) × 行为类型 (At)] ÷ 正当理由 (J)。当行为直接导致死亡时，At 因子高；当死亡源于不作为时，At 因子低。然而，功利主义者认为这种区分在道德上无关紧要：两种行为产生相同的净幸福。爱德思课程通常要求你将此应用于安乐死等问题，其中被动措施（撤除治疗）在法律上常与主动措施（注射致死药物）区分开来。请准备好批判这一公式，询问：意图杀人跟仅仅预见死亡作为副作用之间真的存在差异，还是那条界线也模糊不清？
10. The Secular Moral Authority Formula | 世俗道德权威公式
If God does not exist, objective moral values and duties would not exist; objective moral values and duties do exist; therefore, God exists.
of God as required for the highest good. The syllogism: (1) Without a transcendent lawgiver, morality would be a product of biological evolution or social convention, and thus relative and subjective. (2) However, we experience morality as objectively binding – er is the existence of a perfectly good God who commands it. The formula: Objective moral duty (O) → God (G). ¬G → ¬ n be defended without theism. Edexcel philosophy students should be ready to compare this theorem with the Euthyphro dilemma and with utilitarian or Kantian attempts to ground ethics in non-religious foundations.
这是一则关于上帝存在的道德论证的现代形式，常与威廉·莱恩·克雷格联系在一起，但其根源可追溯到康德将上帝设定为最高善所必需的公设。三段论如下：(1) 没有超越性的立法者，道德就只是生物演化或社会习俗的产物，因而是相对且主观的。(2) 然而，我们体验到的道德是具有客观约束力的——有些事情（如以折磨儿童取乐）是真正错误的，与人的意见无关。(3) 对这一客观道德秩序的最佳解释就是存在一位完美良善的上帝，是祂命令了它。公式：客观道德义务 (O) → 上帝 (G)。¬G → ¬O。O, ∴ G。反对者挑战前提 1 和前提 2：也许客观道德可以在无神的宇宙中作为原初事实存在，或者道德实在论无需有神论亦可捍卫。爱德思哲学学生应准备好将此定理与尤西弗罗困境以及功利主义或康德主义的非宗教伦理基础建构进行比较。
11. The Free Will Defence Theorem | 自由意志辩护定理
A world containing free creatures who sometimes choose evil is more valuable than a world of automata that always choose good; therefore, God may permit evil to preserve free will.
potent God could have created a world with free creatures who always freely choose good. The theorem states: It is logically impossible for God to cause or determine a person to freely choose only good, because freedom is incompatible with determinism. The value of a free world (Wₓ) is greater than the value of a deterministically good world (Wg). Wₓ inevitably carries the risk of moral evil (Eₘ). Therefore, Eₘ does not contradict God’s existence; it is a logically necessary side effect of a higher good. The formula: God’s attributes → desire for free moral agents. Free will → possibility of sin. ∴ Evil is possible even with an all-powerful, all-loving God. Note that this defence deals with moral evil (caused by human choices) but does not directly explain natural evil (earthquakes, disease). To cover natural evil, some argue that it is a consequence of the fallen state of creation, or necessary for soul-making (Irenaeus/Hick).
自由意志辩护，最为知名的表述来自阿尔文·普兰丁格，它通过挑战“全能的上帝能够创造出一个拥有自由受造物且他们总是自由地选择善的世界”这一前提，来回应恶的逻辑问题。该定理表明：上帝在逻辑上不可能致使或决定一个人只自由地选择善，因为自由与决定论不相容。一个自由世界 (Wₓ) 的价值大于一个决定论式的至善世界 (Wg)。Wₓ 不可避免地带有道德之恶 (Eₘ) 的风险。因此，Eₘ 并不与上帝的存在相矛盾；它是一项更大善在逻辑上必然的附带品。公式：上帝属性 → 渴望拥有自由的道德行为者。自由意志 → 罪的可能性。∴ 即使有一位全能全爱的上帝，恶也是可能的。注意，此辩护处理的是道德之恶（由人的选择引起），但并未直接解释自然之恶（地震、疾病）。为涵盖自然之恶，有人认为它是受造界堕落状态的后果，或是塑造灵魂所必需（爱任纽/希克）。
12. The Verificationist Meaning Theorem | 证实主义意义定理
A statement is meaningful only if it is either analytically true or empirically verifiable.
The logical positivists’ verification principle can be formulated as: For any statement S, S is cognitively meaningful iff S is a tautology (true by definition) or S can in principle be verified by sense experience. This was used to challenge religious language. Theological statements like “God loves us” (T) cannot be verified by any possible observation; they are not tautologies. Therefore, applying the theorem, T is cognitively meaningless – not false, but literally nonsensical. Critics note that the theorem itself cannot pass its own test: “A statement is meaningful only if it is verifiable” is neither a tautology nor empirically verifiable, so it is self-refuting. A.J. Ayer proposed weak verification (direct or indirect observational support), but the demarcation remains a challenge. In the Edexcel syllabus, this intersects with arguments about the meaningfulness of religious language and responses like R.M. Hare’s “blik” (non-cognitive but still significant) or Wittgenstein’s language games.
逻辑实证主义的证实原则可以表述为：对于任何陈述 S，S 在认知上有意义，当且仅当 S 是一个同义反复（依定义为真）或 S 在原则上能被感官经验所证实。这一原则被用来挑战宗教语言。神学陈述如“上帝爱我们”(T) 无法被任何可能的观察所证实；它们也不是同义反复。因此，应用该定理，T 在认知上是无意义的——并非错误，而是字面上无意义。批评者指出该定理本身无法通过其自身的检验：“一个陈述有意义仅当它能被证实”本身既非同义反复，也无法用经验证实，因此是自我反驳的。艾耶尔提出了弱证实（直接或间接的观察支持），但划界仍是一个难题。在爱德思教学大纲中，这与关于宗教语言有意义性的论证以及 R.M. 黑尔的“布利克”（非认知但仍有重要意义）或维特根斯坦的语言游戏等回应交叉在一起。
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