📚 Year 13 OCR Philosophy: Formula & Theorem Quick Reference Handbook | Year 13 OCR 哲学：公式定理速查手册
This quick reference handbook distils the core arguments, principles and logical structures from the OCR A Level Religious Studies Philosophy of Religion and Ethics components into compact 'formulas' and 'theorems'. Each entry translates a classic philosophical move into a memorable, exam-ready skeleton. The aim is to strengthen your ability to reconstruct, evaluate and compare arguments under timed conditions.
本速查手册将 OCR A Level 宗教研究哲学与伦理学部分的核心论证、原理与逻辑结构提炼为简明的“公式”与“定理”。每个条目都把经典哲学论证转化为便于记忆、直接可用于考场的骨架，帮助你更有力地重构、评估和比较论点。
1. Anselm's Ontological Argument – The 'Greatest Conceivable' Formula | 安瑟伦本体论论证——“可设想之最大者”公式
Anselm defines God as 'that than which nothing greater can be conceived'. The argument produces a reductio ad absurdum: (1) God exists in the mind. (2) It is greater to exist in reality than in the mind alone. (3) If God exists only in the mind, we can conceive of a being that possesses all God's attributes plus existence in reality — a being strictly greater than God. (4) That contradicts the definition. (5) Therefore, God must exist in reality.
安瑟伦将上帝定义为“那无法设想有任何比之更伟大者”。论证运用归谬形式：(1) 上帝存在于心灵中。(2) 存在于现实中比仅存在于心灵中更伟大。(3) 若上帝仅存在于心灵，则我们可以设想一个拥有上帝全部属性且同时存在于现实中的存在者——该存在者严格地比上帝更伟大。(4) 这与定义相矛盾。(5) 因此，上帝必然存在于现实中。
Theorem: If God =df existence in reality is a great-making property, then God → necessarily exists in reality. ¬(God only in mind) because it generates a contradiction.
2. Gaunilo's Perfect Island – The 'Overload' Rebuttal | 高尼罗的完美海岛——过载式反驳
Gaunilo applies Anselm's logic to a 'perfect island': define the greatest conceivable island; if it does not exist in reality, it is not the greatest conceivable island; therefore, by the same reasoning, the perfect island must exist. Since this conclusion is absurd, the ontological argument form is unreliable. Gaunilo challenges the leap from concept to reality without additional evidence.
高尼罗将安瑟伦的逻辑应用于“完美海岛”：定义可设想的最伟大海岛；如果它不在现实中存在，它就不是最伟大的海岛；按同样推理，完美海岛必然存在。由于这一结论是荒谬的，本体论论证的形式不可靠。高尼罗质疑在缺乏额外证据的情况下从概念跳跃到现实的合理性。
- Gaunilo’s Counter-formula: it is not the greatest → PI must exist → absurd. ∴ Anselm’s form does not guarantee truth.
- 高尼罗反公式： 完美海岛 = 最伟大的可设想海岛。若完美海岛仅存在于心灵，则并非最伟大 → 完美海岛必须存在 → 荒谬。故安瑟伦的形式不保证真理。
3. Descartes' Ontological Argument – Existence as a Perfection | 笛卡尔本体论论证——存在是一种完满性
Descartes treats existence as a defining predicate of a supremely perfect being, just as having three angles is part of the essence of a triangle. The concept of God entails existence necessarily, because a supremely perfect being that lacks existence would be contradictory — like a mountain without a valley.
笛卡尔将存在视为至上完满存在者的一个定义性谓词，就像“三角之和等于二直角”属于三角形的本质。上帝的概念必然包含存在，因为一个欠缺存在的至上完满者将是自相矛盾的——正如没有山谷的山峰。
Formula: God = Supreme Perfect Being → Existence ∈ Essential Attributes. ∴ Necessary Existence (God cannot be conceived without existing).
4. Kant's Predicate Objection – 'Existence is not a real predicate' | 康德谓词反驳——存在不是真正的谓词
Kant argues that existence adds nothing to the concept of a thing. One hundred real thalers contain exactly the same conceptual marks as one hundred possible thalers. When we assert that God exists, we are not adding a perfection to the concept; we are positing a subject with all its predicates. Therefore, the ontological argument fails because it treats existence as a 'great-making' property when it is merely the positing of the thing.
康德指出，存在不增加事物的任何概念内容。一百个真实的银元与一百个可能的银元在概念标志上完全相同。当我们断言上帝存在时，我们并非在概念上增添一种完满性，而只是设定了一个带有全部谓词的主体。因此，本体论论证失败，因为它将存在错误地当作一种“伟大造就”的属性，而存在仅是事物之设定。
Kant’s Theorem: Existence ∉ Predicate (it is the copula of a judgement). Hence, 'God exists' does not analytically expand the concept of God.
5. Aquinas' First Cause Argument – The Unmoved Mover Theorem | 阿奎那第一因论证——不动的推动者定理
Aquinas' Second Way (First Cause) argues from the existence of causation: (1) We observe a series of efficient causes. (2) Nothing can be the cause of itself. (3) If the series of causes went back infinitely, there would be no first cause, and thus no subsequent causes. (4) We observe effects, so a first uncaused cause must exist — and this everyone calls God.
阿奎那的第二路（第一因论证）从因果关系出发：(1) 我们观察到动力因的序列。(2) 没有任何事物可以成为自身的原因。(3) 倘若因果序列可以无限回溯，则不会有第一因，继而后继的因果就不会发生。(4) 我们观察到效果存在，因此必须有一个非受动的第一因——这就是人们所说的上帝。
Formal Chain: … C₃ → C₂ → C₁ → Effect. To avoid infinite regress, ∃ First Cause (uncaused).
6. The Kalam Cosmological Argument – A Temporal Beginning | 卡拉姆宇宙论论证——时间性开端
The Kalam formulation (Al-Ghazali, later Craig) states: (1) Everything that begins to exist has a cause. (2) The universe began to exist. (3) Therefore, the universe has a cause. This cause is argued to be a personal, transcendent Creator. Support often appeals to the impossibility of an actual infinite in the real world and to Big Bang cosmology.
卡拉姆版本（安萨里，后经克雷格发展）陈述如下：(1) 凡开始存在者皆有原因。(2) 宇宙已开始存在。(3) 故宇宙有一原因。该原因被论证为有位格、超越的创造者。支持证据常指出现实世界中不可能存在实无限，以及大爆炸宇宙学。
Kalam Syllogism: ∀x (Begin(x) → Cause(x)); Begin(universe) ⊢ Cause(universe).
7. Paley's Design Argument – The Watchmaker Formula | 佩利的设计论证——钟表匠公式
Paley' . Likewise, the universe exhibits far greater complexity and apparent design; therefore, by analogy, the universe has a divine Designer. Purposeful order (A → B for a function) signals mind.
佩利的类比论证：如果你在荒野发现一块手表，其复杂性与目的性安排必然使人推断出一位智慧的钟表匠。同理，宇宙展现出远为复杂的表面设计；依据类比，宇宙有一位神圣设计者。目的性秩序（为某种功能而安排的 A → B 关系）标志着心智的存在。
Analogical Theorem: Watch : Watchmaker :: Universe : Universe-Maker (Design inferred from ordered complexity toward ends).
8. Hume's Sceptical Toolkit against Design | 休谟对设计论证的怀疑工具箱
Hume, through Philo, deploys multiple undermining moves: the analogy is weak — the universe might be more like a vegetable or a spider's web; we have no experience of world-making to ground the inference; even if a designer were conceded, the marks of imperfection in nature would point to an amateur or a committee of gods, not the all-perfect God of theism. Design does not entail a single, omniscient, omnibenevolent deity.
休谟借斐罗之口展开多重削弱：类比过于薄弱——宇宙可能更像一株蔬菜或一张蛛网；我们没有任何创造世界的经验可为推理奠基；即便承认设计者存在，自然中的种种缺憾反而指向一个学徒或一队众神，而非有神论的全善上帝。设计并不必然推出单一、全知、全善的神。
| Hume's Counter-principle | 休谟反制原则 |
| Like effects prove like causes — but only proportionally. | 相似结果证明相似原因——但仅成比例。 |
| A finite, imperfect world suggests a finite, imperfect cause. | 有限且不完美的世界意味着有限且不完美的原因。 |
9. Logical Problem of Evil – The Inconsistent Triad | 恶的逻辑问题——不一致三元组
The logical problem charges that the statements (1) God is omnipotent, (2) God is wholly good and (3) evil exists are logically incompatible. If God is willing and able to prevent evil, evil should not exist. Since evil does exist, the theistic conception of God appears inconsistent — effectively a deductive disproof.
逻辑恶问题指责以下三个命题在逻辑上不能同时成立：(1) 上帝全能，(2) 上帝全善，(3) 恶存在。若上帝愿意且能够防止恶，恶便不应存在。既然恶确实存在，有神论的上帝概念便显得不一致——实质上构成一个演绎上的反证。
Inconsistent Triad: (Omnipotence ∧ Omnibenevolence) → ¬Evil. But Evil is true; ∴ ¬(Omnipotence ∧ Omnibenevolence).
10. Plantinga's Free Will Defence – A Possible Worlds Theorem | 普兰丁格自由意志辩护——可能世界定理
Plantinga argues that it is logically possible that God could not create a world containing moral good without also creating free beings who sometimes choose evil. Even an omnipotent being cannot actualise a logically impossible state of affairs, such as a world where agents are significantly free yet never go wrong. Hence, the existence of moral evil does not entail a logical contradiction with God's attributes.
普兰丁格论证，在逻辑上可能存在着这样的状况：上帝无法创造一个包含道德善的世界，而不同时创造有时选择恶的自由被造物。即使是全能者也不能实现逻辑上不可能的事态，例如一个受造物拥有重大自由却永不出错的世界。因此，道德恶的存在并不必然与上帝属性构成逻辑矛盾。
Modal Defence: It is possible that (God actualises morally valuable freedom ↔ some evil exists). Logical compatibility is preserved.
11. The Verification Principle – Ayer's Meaning Formula | 证实原则——艾耶尔的意义公式
A.J. Ayer's logical positivism asserts that a factual statement is meaningful only if it is either true by definition (analytic) or verifiable in principle by sense-experience (synthetic). Religious statements like 'God loves us' fail both tests: they are not analytic truths, nor can any empirical observation confirm or disconfirm them. Hence, they are cognitively meaningless, expressing emotion but no factual claim.
艾耶尔的逻辑实证主义主张，一个事实性陈述只有满足以下条件之一才具有意义：或者是因定义而为真（分析命题），或者原则上可被感官经验所证实（综合命题）。宗教陈述如“上帝爱我们”未能通过这两项检验：它们并非分析真理，也没有任何经验观察能够证实或否证它们。因此，它们在认知上无意义，仅表达情感而非事实声称。
Meaning criterion: Meaningful = Analytic ∨ Empirically Verifiable. 'God-talk' ∉ both sets → non-cognitive.
12. Bentham's Hedonic Calculus – The Utility Theorem | 边沁的快乐计算——功利定理
Bentham's act utilitarianism evaluates actions by the quantity of pleasure produced minus pain. The hedonic calculus weighs seven factors: intensity, duration, certainty, propinquity (nearness), fecundity (chance of leading to more pleasure), purity (chance free from pain), and extent (number of people affected). The right action is that which maximises net utility across all affected beings.
边沁的行为功利主义根据所产生的快乐量与痛苦量之差来评判行为。快乐计算衡量七个因素：强度、持续时间、确定性、邻近性（远近）、丰产性（带来更多快乐的可能性）、纯度（免于痛苦的可能性）以及范围（受影响的人数）。正当行为即在所有受影响者之间最大化净功利的行为。
| Factor | Description | 因子 |
| Intensity | How strong? | 强度 |
| Duration | How long? | 持续性 |
| Certainty | How likely? | 确定性 |
| Propinquity | How soon? | 邻近性 |
| Fecundity | Leads to more pleasure? | 丰产性 |
| Purity | Free from pain? | 纯度 |
| Extent | How many affected? | 范围 |
rinciple: Right Action = Maximise Σ (Pleasure − Pain) across all sentient beings affected.
13. Kant's Categorical Imperative – Universal Law Formula | 康德的义务论——可普遍化律公式
Kant's moral philosophy rests on the categorical imperative's universalisability test: 'Act only according to that maxim whereby you can at the same time will that it should become a universal law.' A contradiction in conception (e.g. making a false promise — if universalised, the institution of promising would collapse) or a contradiction in the will (e.g. refusing to help others — one could not rationally will a world of complete indifference) reveals an action to be morally impermissible.
康德的道德哲学立足于绝对律令的可普遍化检验：“只按照你同时能够意愿它成为一项普遍法则的准则去行动。”无论是概念上的矛盾（例如作虚假承诺——若普遍化，承诺制度将不复存在）还是意志上的矛盾（例如拒绝帮助他人——一个人无法理性地意愿一个完全冷漠的世界），均可揭示某一行为在道德上不可允许。
Test: Maxime → Universal Law? If contradiction in conception OR contradiction in will → Impermissible.
14. Aquinas' Natural Law – Primary Precepts as Theorems of Practical Reason | 阿奎那自然法——实践理性的基本规条定理
Aquinas argues that practical reason apprehends self-evident natural inclinations, which constitute the primary precepts: preserve life, reproduce and educate offspring, seek God (know truth, live in society), and live in an ordered society. These are the foundational moral 'theorems' derivable from the synderesis principle: 'good is to be done and pursued, and evil is to be avoided.'
阿奎那主张，实践理性把握自明的自然倾向，这些倾向构成基本规条：保存生命、繁衍并教育后代、寻求上帝（认识真理、在社会中生活）、以及生活在有序的社会中。这些是从“良心本原”原则——“行善避恶”中推导出的根本性道德“定理”。
Rule: Good → to be done; Evil → to be avoided. Primary Precepts = derivative logical ends.
15. Pascal's Wager – The Expected Value Calculation | 帕斯卡的赌注——期望值计算
Pascal frames belief in God as a rational wager under uncertainty. If God exists and you believe, you gain infinite happiness; if God does not exist and you believe, you lose little. If you do not believe and God exists, you face infinite loss. The expected value of believing dramatically outweighs non-belief, making faith pragmatically rational. The demand is not to prove God but to prudentially commit.
帕斯卡将信仰上帝框定为不确定性下的理性赌注。若上帝存在而人相信，则获得无限幸福；若上帝不存在而人相信，损失甚微。若不信仰而上帝存在，则面临无限损失。信仰的期望值远大于不信，这使得信仰在精算上是理性的。要求并非证明上帝，而是出于明智的委身。
| God exists | God does not exist | |
| Believe | Infinite gain | Finite loss (worldly pleasures) |
| Do not believe | Infinite loss | Finite gain (worldly pleasures) |
eve) = ∞ × p + small loss; EV(disbelieve) = −∞ × p + small gain. Dominance favours belief.
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