📚 Year 13 CCEA Philosophy: Formula & Theorem Quick Reference Handbook | CCEA Year 13 哲学公式定理速查手册
Philosophy at Year 13 under CCEA requires rigorous logical analysis. Just like mathematics, many philosophical arguments can be expressed in formal structures resembling formulas or theorems. This handbook distills key arguments from Philosophy of Religion and Ethics into concise logical forms, providing a quick revision tool for A2 students. Each section treats a core argument or principle as a theorem, with premises and conclusions laid out symbolically and in plain English.
CCEA 的 13 年级哲学课程要求严谨的逻辑分析。如同数学一样，许多哲学论证可以用类似公式或定理的形式化结构表达。本手册将宗教哲学与伦理学中的关键论点提炼为简洁的逻辑形式，为 A2 学生提供快速复习工具。每一节将一个核心论证或原理视为一条定理，用符号和清晰的语言列出前提与结论。
1. Anselm’s Ontological Argument | 安瑟伦的本体论论证
Anselm defines God as ‘that than which nothing greater can be conceived’. His argument can be rendered as a deductive proof where the very concept of God entails existence. The formal structure shows that if we understand the definition, and accept that existence in reality is greater than existence in the understanding alone, then God must exist in reality.
安瑟伦将上帝定义为’无法设想有比之更伟大的存在者’。他的论证可以呈现为一个演绎证明，即上帝的概念本身就蕴含着存在。其形式结构表明，如果我们理解这一定义，并接受存在于现实中比仅存在于理智中更伟大，那么上帝必然存在于现实中。
- P1: God is the being than which no greater can be conceived (Definition).
- P2: God exists in the understanding (the Fool understands the phrase).
- P3: Existing in reality is greater than existing only in the understanding.
- P4: If God existed only in the understanding, we could conceive of a greater being (one that also exists in reality).
∴ God exists in reality.
Form: (P1 ∧ P2 ∧ P3) → C
Gaunilo famously objected by parodying the argument with a ‘perfect island’, claiming the same logic would force its existence. Anselm replied that the argument applies only to a necessary being, of which God alone qualifies.
高尼洛以’完美岛屿’的戏仿提出了著名反驳，声称同样的逻辑会迫使其存在。安瑟伦回应说，该论证只适用于必然存在者，而只有上帝符合这一资格。
2. Aquinas’ Cosmological Argument (The First Two Ways) | 阿奎那的宇宙论论证（前两路）
Aquinas’ Five Ways begin with empirical observations and argue to a First Cause or Prime Mover. The first two ways can be expressed as causal chains that cannot regress infinitely because an infinite regress would yield no starting point, and without a first member the series would not exist.
阿奎那的五路论证始于经验观察，并推论至第一因或第一推动者。前两路可以表示为因果链，它不能无限回溯，因为无限回溯将没有起点，而没有第一个成员，整个系列就不会存在。
Way 1: Motion/Change
Way 1: Motion/Change:
- P1: Some things in the world are in motion (change).
- P2: Whatever is moved is moved by another.
- P3: This mover cannot go on to infinity, because then there would be no first mover, and thus no subsequent motion.
∴ There must be a First Mover, unmoved by any other: God.
Way 2: Efficient Causality
Way 2: Efficient Causality:
- P1: We perceive an order of efficient causes.
- P2: Nothing can be the efficient cause of itself (it would have to precede itself).
- P3: If the chain of efficient causes went on infinitely, there would be no first cause, and thus no subsequent causes.
∴ There must be a First Efficient Cause, uncaused by any other: God.
The logical form relies on the impossibility of an actual infinite series in hierarchically ordered causes. Hume and Russell later challenged the very premise that ‘every event has a cause’ applies to the universe as a whole.
其逻辑形式依赖于在层级有序的原因中，实际无限序列的不可能性。后来休谟与罗素挑战了’每个事件都有一个原因’这一前提能否适用于整个宇宙。
3. William Paley’s Teleological Design Argument | 威廉·帕雷的目的论设计论证
Paley uses an analogical inference: a watch implies a watchmaker because of its complexity and purpose. Similarly, the world exhibits complexity and purpose, therefore it implies a divine Designer. The argument can be structured with an analogy principle and an inference to the best explanation.
帕雷使用类比推理：手表因其复杂性和目的性而暗示手表匠的存在。同理，世界展现出复杂性与目的性，因此暗示一位神圣的设计师。该论证可以用类比原则和最佳解释推论来构建结构。
- P1: A watch shows marks of design (complex parts arranged for a purpose).
- P2: Such marks compel us to infer an intelligent watchmaker.
- P3: The natural world displays similar marks of design—e.g., the eye, the planetary system—but on a far grander scale.
- P4: Similar effects imply similar causes.
∴ The natural world implies an intelligent Designer, God.
Paley emphasises that even if we had never seen a watch being made, the inference would still hold—illustrating that the argument does not depend on experiencing the Designer directly. Hume’s criticisms via Philo target the weakness of the analogy and the problem of evil, arguing the world resembles a vegetable more than a machine.
帕雷强调，即使我们从未见过手表被制造，这一推论也依然成立——说明该论证并不依赖于直接经验到设计师。休谟通过斐洛之口批评了类比的脆弱性和恶的问题，认为世界更像是植物而非机器。
4. Kant’s Moral Argument for God’s Existence | 康德的道德论证
Kant did not propose a direct argument from morality to God in the same way as others, but his postulate of practical reason serves as a moral argument. It can be formalised as: the moral law requires the summum bonum (highest good) to be attainable, which in turn requires the existence of God and immortality. Simplified as a syllogism:
康德并没有像其他人那样提出直接的从道德到上帝的论证，但他实践理性的公设形成了一种道德论证。它可以形式化为：道德律要求至善可实现，这反过来又要求上帝与灵魂不朽。简化为三段论：
- P1: Moral duty demands that the highest good (perfect virtue and happiness combined) is possible.
- P2: Human reason alone cannot guarantee this harmony; only a supreme moral intelligence (God) can coordinate virtue and happiness.
- P3: ‘Ought implies can’—what we are morally obligated to pursue must be achievable.
∴ God exists as the necessary condition for the summum bonum.
Some contemporary moral arguments focus on objective moral values: If objective moral duties exist, then a transcendent moral lawgiver must exist. The CCEA specification often includes the form: P1: Objective moral laws exist. P2: Objective moral laws require a moral lawgiver. C: Therefore, God exists.
当代某些道德论证聚焦于客观道德价值：如果客观道德义务存在，则必须存在一位超越的道德立法者。CCEA 大纲常包含此形式：P1：客观道德律存在。P2：客观道德律要求一位道德立法者。C：因此，上帝存在。
5. Mackie’s Logical Problem of Evil | 麦基的恶的逻辑问题
J.L. Mackie formulated the problem of evil as an inconsistent triad: God is omnipotent, God is wholly good, and evil exists. The three cannot all be true. The argument is logically decisive if one grants the implicit assumptions.
J.L. 麦基将恶的问题构建为不一致三角：上帝全能、上帝全善、恶存在。三者不能同时为真。如果接受隐含的假设，该论证在逻辑上是决定性的。
- P1: God is omnipotent (can do anything logically possible).
- P2: God is wholly good (always eliminates evil as far as it can).
- P3: Evil exists.
- P4: A good being always eliminates evil as far as it can.
- P5: There are no limits to what an omnipotent being can do (apart from logic).
From P1-P5: Contradiction → ∴ An omnipotent, wholly good God does not exist.
Symbolic: (Omni ∧ Good ∧ Evil) ⊢ ⊥
The free will defence attempts to break the inconsistency by adding the premise that genuine free will is a great good that requires the possibility of evil. Plantinga argues that a world containing free creatures who sometimes go wrong may be a greater good than a world of automata, thus God’s goodness is compatible with evil.
自由意志辩护试图通过添加一个前提打破不一致性：真正的自由意志是极大的善，它要求恶的可能性。普兰丁格论证说，包含有时会犯错的自由受造物的世界，可能比自动机器的世界更善，因此上帝的善与恶是相容的。
6. Hume’s Critique of Miracles | 休谟对奇迹的批判
Hume defines a miracle as a violation of a law of nature by a particular volition of the Deity. His argument against belief in miracles relies on a principle of evidential probability: testimony for a miracle can only be credible if its falsehood would be more miraculous than the event itself.
休谟将奇迹定义为由于神明的特定意志而对自然法则的违反。他反对相信奇迹的论证依赖于一条证据概率原则：只有当证词的虚假比所证事件本身更神奇时，对奇迹的证词才可信。
- P1: A miracle is a violation of the laws of nature.
- P2: The laws of nature are established by firm and unalterable experience.
- P3: The evidence in favour of any miracle testimony is the testimony itself.
- P4: Wise persons proportion their belief to the evidence.
- P5: It will always be more probable that the testimony is false (through error or deceit) than that a law of nature has been violated.
∴ No testimony is sufficient to establish a miracle unless its falsehood would be an even greater miracle.
Hume also adds practical arguments: no miracle has been attested by a sufficient number of educated, sensible witnesses; human nature enjoys the sensational; and miracle stories flourish in ignorant and barbarous nations. The CCEA exam expects students to weigh these criticisms against the cumulative case for particular miracles.
休谟还补充了实践论证：没有任何奇迹得到足够数量受过教育且有理智的证人证实；人性喜欢耸人听闻；奇迹故事在无知野蛮的民族中盛行。CCEA 考试期望学生能权衡这些批评与支持特定奇迹的累积论证。
7. Religious Language: Verification and Falsification | 宗教语言：验证与虚假原则
The logical positivists’ verification principle states that a statement is meaningful only if it is either analytically true or empirically verifiable. The falsification principle, associated with Antony Flew, asserts that a statement is meaningful only if it is falsifiable—i.e., there is some evidence that would count against it.
逻辑实证主义者的验证原则指出，一个陈述只有当它要么是分析真，要么在经验上可验证时才有意义。由安东尼·弗卢提出的虚假原则断言，一个陈述只有当它可被证伪时才有意义——即存在某些可能否定它的证据。
Verification: Meaningful(S) ↔ (Analytic(S) ∨ Empirically verifiable(S))
Falsification: Meaningful(S) ↔ Falsifiable(S) [some possible observation O would show S is false]
Religious claims such as ‘God loves us’ seem unfalsifiable; believers allow nothing to count against them, leading Flew to argue they ‘die the death of a thousand qualifications.’ Hare responded with the concept of ‘bliks’—unfalsifiable but meaningful frameworks that shape one’s life. Mitchell argued that religious claims are genuinely assertions that can be challenged by evidence but are held by faith, akin to trusting a partisan in ambiguous circumstances.
像’上帝爱我们’这样的宗教宣称似乎不可证伪；信徒不允许任何证据来否定它们，弗卢因此认为它们经历了’千般限定的死亡’。黑尔以’blik’（生命定向的基本态度）概念回应——它们虽不可证伪但富有意义，塑造着一个人的生活。米切尔则论证说，宗教宣称是真正的断言，可受证据挑战，但凭信心持守，犹如在模糊情境中信任一个抵抗者。
8. Aquinas’ Natural Law Ethics | 阿奎那的自然法伦理
Natural Law, as developed by Aquinas, is based on the idea that God created the world with an inherent order, accessible by reason. The first precept is self-evident and functions as the foundational formula: ‘Good is to be done and pursued, and evil is to be avoided.’ From this, secondary precepts are derived by reflecting on human nature.
阿奎那发展的自然法基于这样的观念：上帝创造的世界具有内蕴的秩序，理性可以把握。第一条原则是不证自明的，并作为基础公式发挥作用：’善应行而行，恶应避而避。’由此，通过反思人性推导出次要训诫。
First Principle: Bonum faciendum et prosequendum, et malum vitandum.
Formula: Primary Precepts → Human Nature → Right Action
The primary precepts reflect the natural inclinations: preserve life, reproduce and educate offspring, live in society, seek truth, and worship God. Any action that fulfils these precepts is good; an action is evil if it contravenes them. The doctrine of double effect is applied when an action has two effects, one good and one bad.
主要训诫反映自然倾向：保存生命、生育与教育后代、过社会生活、探求真理、崇拜上帝。任何实现这些训诫的行动就是善的；如果违背它们，行动就是恶的。双重效果学说适用于一个行动同时具有好坏两种效果的情况。
| Primary Precept | Natural Inclination |
| Preservation of life | Self-preservation |
| Reproduction and educationPublished by TutorHao | Year 13 哲学 Revision Series | aleveler.com
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