📚 GCSE OCR Philosophy: Formula & Theorem Quick Reference Guide | GCSE OCR 哲学：公式定理速查手册
While philosophy rarely uses numbers, it contains structured arguments and ethical principles that act like logical formulas. For GCSE OCR Philosophy, mastering these ‘theorems’ helps you analyse, evaluate and build philosophical arguments with precision. This quick reference guide presents the core argument forms and moral frameworks as clear, step-by-step logical statements – your revision toolkit for the exam.
虽然哲学极少使用数字，但它蕴含着犹如逻辑公式般的结构化论证与伦理原则。对于 GCSE OCR 哲学课程，掌握这些“定理”能帮助你精确地分析、评价并建构哲学论证。本速查手册将核心论证形式和道德框架呈现为清晰的、逐步的逻辑陈述——是你备考复习的得力工具。
1. The Ontological Argument Formula | 本体论论证公式
Anselm defines God as ‘that than which nothing greater can be conceived’ – a being so great that no greater being can be imagined. The very concept forces us to accept God’s real existence.
安瑟伦将上帝定义为“不能设想有比之更伟大者”——一个如此伟大的存在，以至无法设想任何比它更伟大的存在。这个概念本身就迫使我们承认上帝的实在存在。
He argues that if this greatest conceivable being existed only in the mind, we could think of a greater being – one that also exists in reality. Since it is greater to exist in reality than merely in understanding, God must exist in reality to be the greatest conceivable being.
他论证说，如果这个可设想的最伟大存在只存在于心灵中，我们就能设想一个更伟大的存在——一个也存在于现实中的存在。因为存在于现实中比仅存在于理解中更伟大，所以上帝必须存在于现实中，才能成为可设想的最伟大存在。
God = the greatest conceivable being → existence in reality > existence in the understanding → ∴ God exists in reality.
Descartes reinforces this by treating existence as a perfection: God is a supremely perfect being; existence is a perfection; therefore God necessarily exists, just as a triangle has three angles.
笛卡尔将存在视为一种完美而强化了这一点：上帝是至上完美的存在；存在是一种完美；因此上帝必然存在，正如三角形必有三个角。
Gaunilo’s rebuttal uses the ‘perfect island’ parody: we can conceive of the greatest conceivable island; by the same logic, that island must exist, which is absurd. This shows the argument can be used to ‘prove’ imaginary entities.
高尼罗的反驳使用了“完美岛屿”的戏仿：我们可以设想一个最伟大的岛屿；依相同逻辑，那个岛屿必须存在，而这显然是荒谬的。这表明此论证可被用来“证明”虚构的事物。
Perfect Island objection: the greatest conceivable X must exist → leads to absurdity → ontological logic is flawed.
Kant’s famous criticism adds that ‘existence is not a predicate’ – it does not add to the concept of a thing. So positing existence as a perfection makes no sense; the ontological argument collapses.
康德著名的批评补充说“存在不是谓词”——它并不给事物的概念添加任何内容。因此将存在假定为一种完美是说不通的；本体论论证便不成立。
2. The Cosmological Argument Formula | 宇宙论论证公式
Aquinas’ First Way centres on motion: we observe things in motion; everything moved is moved by another; an infinite regress of movers is impossible, so there must be a First Mover unmoved, which everyone understands to be God.
阿奎那的第一路以运动为中心：我们观察到事物在运动；所有被推动者都被另一者所推动；无限回溯的推动者是不可能的，因此必定有一位不被推动的第一推动者，人人都理解这即上帝。
Motion in world → chain of movers cannot be infinite → First Mover (God).
The Second Way focuses on efficient causes: nothing can cause itself; if the chain of causes went back infinitely there would be no first cause, and thus no intermediate causes; we must arrive at a First Uncaused Cause, i.e. God.
第二路聚焦于动力因：无物能自因；如果原因链无限回溯，就没有第一因，也就没有中间原因；我们必须溯至一个不被造成的第一因，即上帝。
Every event has a cause → infinite causal chain impossible → First Cause (God).
Aquinas’ Third Way deals with contingency: everything we see is contingent (could not exist); if everything were contingent, at some time nothing would have existed, but then nothing could have begun to exist; there must be a necessary being – God – that exists of its own nature.
阿奎那的第三路处理偶性：我们所见的一切皆是偶性的（可能不存在）；若一切皆偶性，则在某个时间点可能无一物存在，但那样便无物能开始存在；因此必须有一个必然存在——上帝——其本性即存在。
The Kalam version, developed by Islamic philosophers, is simpler: everything that begins to exist has a cause; the universe began to exist (Big Bang, impossibility of an actual infinite); therefore the universe has a cause, which we call God.
卡拉姆版本更为简洁：凡开始存在之物皆有原因；宇宙开始存在（大爆炸、实际无限的不可能性）；因此宇宙有一个原因，我们称之为上帝。
Everything that begins to exist has a cause → Universe began to exist → Universe has a cause (God).
Critics like Hume and Russell argue the universe may just be a ‘brute fact’ needing no explanation, or that the argument commits the fallacy of composition – what is true of parts (needing a cause) may not be true of the whole.
休谟和罗素等批评者认为宇宙可能只是一个“粗鲁事实”而不需解释，或该论证犯了组合谬误——部分之物需有原因，未必适用于整体。
3. The Teleological (Design) Argument Formula | 目的论（设计）论证公式
William Paley’s watchmaker analogy: if you found a watch on a heath, its intricate parts working together for a purpose would compel you to infer a watchmaker. The universe displays far greater complexity and purpose; thus it demands a divine designer.
威廉·佩里的钟表匠类比：若在荒野上发现一只表，其精密零件协同运作以达目的将必然使你推导出制造者。宇宙展示出更巨大的复杂性与目的；因此它要求一位神圣设计者。
Complex, purposeful arrangement → intelligent designer → Universe has such arrangement → God the designer.
The aesthetic argument adds that nature’s beauty and order, such as the Fibonacci sequence in sunflowers, strengthens the inference to design.
美学论证补充说，自然之美与秩序，如向日葵中的斐波那契数列，加强了设计者推论。
David Hume’s criticisms predate Paley: the world might be more like a vegetable than a machine; even if design is shown, we cannot infer the designer is God – it could be a team of gods or an infant deity; plus, the existence of evil and imperfections suggests a flawed designer.
休谟的批评早于佩里：世界可能更像植物而非机器；即使显示了设计，我们不能推论设计者就是上帝——可能是一队神祇或幼年神祇；此外，恶与不完美的存在暗示着有缺陷的设计者。
Darwin’s theory of evolution by natural selection offers a non-theistic explanation for apparent design: random variation plus natural selection can produce complex adaptations without a designer. The design argument is thereby severely weakened.
达尔文的自然选择进化论为表面设计提供了非神论解释：随机变异加自然选择无需设计者便能产生复杂适应。设计论证因此被严重削弱。
Apparent design ← natural selection (evolution) → no need for God as designer.
4. The Moral Argument Formula | 道德论证公式
Kant observed that human beings experience an objective moral law within themselves – the sense of duty. This moral law demands that we attain the summum bonum (highest good), a state where virtue and happiness coincide. To make this achievable, we must postulate God to ensure that moral efforts are rewarded.
康德观察到，人类内心体验着客观道德律——义务感。这道德律要求我们达成至善，即德性与幸福合一的状态。为使这成为可能，我们必须设定上帝以保障道德努力获得回报。
Objective moral law → need for summum bonum → God as moral guarantor.
Modern versions are simpler: if objective moral duties exist, they require a transcendent source; the best explanation is a moral lawgiver, God. Without God, morality would be merely human convention or social contract.
现代版本更简洁：若客观道德义务存在，它们需要一个超越性的来源；最佳解释是一位道德立法者，上帝。没有上帝，道德将只是人类约定或社会契约。
Critics such as Freud argue that conscience is a product of socialisation and the superego, not a divine voice. Also, the Euthyphro dilemma challenges: is something good because God commands it, or does God command it because it is good? Either horn poses problems.
弗洛伊德等批评者认为良心是社会化和超我的产物，绝非神圣之声。此外，尤西弗罗困境提出挑战：某事物是因上帝命令而善，还是因为它是善的上帝才命令之？无论哪种选择都带来问题。
Moral law → either independent of God (then God unnecessary) or dependent on God’s will (then arbitrary) → Euthyphro dilemma weakens the argument.
5. The Problem of Evil ‘Disproof’ Formula | 恶的问题“反证”公式
The logical problem of evil presents a direct contradiction: if God is omnipotent, He can prevent evil; if omnibenevolent, He would want to prevent evil; yet evil exists. Therefore such a God does not exist.
逻辑恶问题提出一个直接矛盾：若上帝全能，祂能够防止恶；若祂全善，祂想要防止恶；然而恶却存在。因此这样的上帝不存在。
God is omnipotent ∧ omnibenevolent ∧ omniscient → no evil should exist. Evil exists → ¬ (omnipotent ∧ omnibenevolent ∧ omniscient) exists.
This formulation, famously posed by Epicurus and later Mackie, challenges the coherence of classical theism.
这一由伊壁鸠鲁提出、麦基后来明确阐述的表述，挑战了古典有神论的融贯性。
The free-will defence counters that a world with free morally significant beings is better than a world of robots; morally significant free will requires the possibility of evil. So God allows evil for the sake of a greater good.
自由意志辩护回应说，拥有具有道德意义的自由存在物的世界优于一个傀儡世界；具有道德意义的自由意志要求存在作恶的可能性。因此上帝为了更大的善而容许恶。
Free will is a great good → its exercise may produce evil → God is justified in allowing evil for greater good.
The evidential problem points to the sheer amount and distribution of suffering – particularly innocent suffering – which seems disproportionate. This casts doubt on God’s goodness rather than producing a logical contradiction.
证据恶问题指向苦难的庞大数量与分布——特别是无辜者受难——这看起来不成比例。它并非产生逻辑矛盾，而是引发对上帝的善的怀疑。
Responses include soul-making theodicies (Irenaeus/Hick) and the claim that we cannot see the full cosmic picture; but the problem remains a major challenge to theism.
回应包括灵魂塑造神正论（爱任纽/希克）以及宣称我们无法看到全宇宙图景的说法；但对有神论而言这仍是一大挑战。
6. Utilitarianism: The Happiness Calculus | 功利主义：幸福计算公式
Jeremy Bentham’s principle of utility states: an action is right if it produces the greatest happiness for the greatest number. Happiness equals pleasure minus pain.
边沁的功利原则指出：若一个行动能为最大多数人带来最大幸福，则它是正确的。幸福等于快乐减去痛苦。
Action right ↔ Σ(pleasure − pain) is maximised for all affected.
Bentham devised the hedonic calculus to measure pleasures and pains using seven factors: intensity, duration, certainty, propinquity, fecundity, purity and extent. The act with the highest net score is the moral choice.
边沁设计了快乐计算法，用七个因子衡量快乐与痛苦：强度、持续期、确定性、邻近度、孕育力、纯洁度和范围。净得分最高的行动即为道德选择。
John Stuart Mill improved this by distinguishing higher (intellectual) and lower (bodily) pleasures, insisting ‘it is better to be a human being dissatisfied than a pig satisfied’. He thus introduced a qualitative dimension to the calculus.
密尔改进了这一点，区分了高级（智识）快乐与低级（身体）快乐，坚称“宁愿做一个不满足的人，也不做一只满足的猪”。他因此为计算引入了质的维度。
Act utilitarianism judges each individual act by its consequences, while rule utilitarianism applies the principle to general rules: follow the rule that, if generally obeyed, maximises utility. Rule utilitarianism avoids some absurd counterexamples (e.g. punishing an innocent person in a mob).
行为功利主义根据后果逐一评判每一行为，而规则功利主义则把原则应用于一般规则：遵循那若被普遍遵守便将功利最大化的规则。规则功利主义避免了某些荒谬的反例（如暴民中惩罚无辜者）。
Hedonic Calculus inputs → net utility > 0 and > alternatives → right action.
Criticisms include the difficulty of measuring pleasure, the tyranny of the majority, and the demand of impartiality ignoring personal relationships. Yet the calculus remains a powerful tool in ethical reasoning.
批评包括衡量快乐的困难、多数暴政以及忽略个人关系的要求。然而此计算法仍是道德推理中的有力工具。
7. Kantian Ethics: The Categorical Imperative Formula | 康德伦理学：绝对命令公式
For Kant, morality is grounded in reason, not consequences. The supreme principle is the categorical imperative – an unconditional command. Its first formulation is the formula of universal law.
对康德来说，道德基于理性而非后果。最高原则是绝对命令——无条件的律令。其第一表述为普遍法则公式。
Act only according to that maxim whereby you can at the same time will that it should become a universal law.
You must test the maxim (your personal rule) by imagining everyone following it. If it leads to a contradiction in conception (e.g. stealing – the concept of property would break down) or a contradiction in will (you cannot rationally will a world where no one helps others), the act is forbidden.
你必须通过想象人人都遵循此准则来检验它。如果它导致构想矛盾（如偷窃——财产的概念将会瓦解）或意志矛盾（你无法以理性意愿一个无人互帮的世界），则该行为是被禁止的。
Maxim → universalise → contradiction? Yes → impermissible; No → permissible.
The second formulation, the formula of humanity, requires that you treat humanity, in yourself and others, always as an end in itself, never merely as a means. This forbids exploitation and upholds human dignity.
第二表述，即人性公式，要求你将人性，无论在自己还是他人身上，始终视为目的本身，决不仅仅用作手段。这禁止了剥削并维护了人的尊严。
Treat humanity as an end, never merely as a means.
Kant’s system is deontological: the rightness of an act depends on its adherence to duty, not on outcomes. A good will is the only unqualifiedly good thing. Critics argue duties can clash, and the theory is too rigid (e.g. lying is always wrong even to save lives).
康德的体系是义务论的：行为的对错取决于是否恪守义务，而非结果。善良意志是唯一无条件善的事物。批评者指出义务可能冲突，且该理论太僵化（如即使为救命说谎也总是错的）。
8. Natural Law Theory: The Primary Precepts Formula | 自然法理论：首要戒律公式
Aquinas’ Natural Law is grounded in the idea that God designed the world with a purpose (telos). By using reason, we can discover the natural moral law written into creation. The fundamental rule is ‘do good and avoid evil’.
阿奎那的自然法基于上帝设计世界有目的的观念。通过运用理性，我们可以发现写进受造界中的自然道德律。基本规则是“行善避恶”。
Reason → purpose (telos) → natural inclinations → primary precepts → human flourishing (eudaimonia).
From our natural inclinations, Aquinas derives five primary precepts: preserve life, reproduce, educate the young, live in an ordered society, and worship God. These are absolute and universal.
阿奎那从自然倾向推导出五项首要戒律：保全生命、繁衍、教育幼小、生活在有序社会中、敬拜上帝。这些是绝对且普遍的。
Secondary precepts are derived from the primary ones using human reason; for example, from ‘preserve life’ we get ‘do not kill
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