Year 11 CCEA Philosophy: Formula & Theorem Quick Reference Handbook | CCEA 11年级哲学：公式定理速查手册

📚 Year 11 CCEA Philosophy: Formula & Theorem Quick Reference Handbook | CCEA 11年级哲学：公式定理速查手册

Philosophy often feels abstract, but beneath every great argument lies a hidden structure — a kind of ‘formula’ or ‘theorem’ that shapes reasoning. For CCEA Year 11 Philosophy, mastering these logical skeletons is key to analysing arguments about God, ethics, evil, and human freedom. This handbook presents each core philosophical argument as a clear, step‑by‑step formula, complete with definitions, test conditions, and counter‑arguments. Use it as a rapid revision tool to lock in the logic before your exam.

哲学常常显得抽象，但在每一个伟大论证的背后，都隐藏着一种结构——一种塑造推理的“公式”或“定理”。对于 CCEA 11 年级哲学课程，掌握这些逻辑骨架是分析关于上帝、伦理、恶与人类自由等论证的关键。本手册将每一个核心哲学论证呈现为清晰、逐步推导的公式，并配有定义、检验条件和反驳论点。把它当作快速复习工具，在考试前锁定逻辑链条。

1. The Cosmological Argument Formula | 宇宙论证公式

The Cosmological Argument claims that the existence of the universe demands a first cause. Aquinas presented this as a chain of dependent causes. The formula captures the movement from contingent existence to a necessary being.

宇宙论证声称宇宙的存在需要一个第一因。阿奎那将其呈现为一条依赖因果链。该公式捕捉了从偶然存在到必然存在的推导过程。

Premise 1: Everything that exists has a cause or reason for its existence (Principle of Sufficient Reason).
前提1： 每一存在的事物都有其存在的原因或理由（充足理由律）。
Premise 2: The universe exists.
前提2： 宇宙存在。
Premise 3: An infinite regress of causes is impossible (it would lead to no ultimate explanation).
前提3： 无限回溯的因果链是不可能的（它将导致没有终极解释）。
Conclusion: Therefore, there must be a first, uncaused cause, which is God.
结论： 因此，必然存在一个第一的、无因的起因，这就是上帝。

∀x ∃y (C(y, x)) ∧ ¬∞Regress → ∃!G

任何事物都有其原因 ∧ 不可能无限回溯 → 存在唯一的第一因上帝

Objection: Hume argues we cannot apply the principle of cause from within the universe to the universe as a whole. Russell treats the universe as a brute fact. Yet the formula remains the standard map of the argument for revision.

反驳：休谟认为我们不能把宇宙内部的因果原则应用于宇宙整体；罗素把宇宙视为纯粹事实。然而，该公式仍然是复习时论证的标准路线图。

2. The Teleological Argument Formula | 设计论证公式

William Paley’s watchmaker analogy drives the Teleological Argument: complexity with purpose implies a designer. The formula isolates the analogical reasoning and the inference to the best explanation.

威廉・佩利的钟表匠类比推动着设计论证：带有目的的复杂性暗示着一位设计者。该公式分离出类比推理和最佳解释推理。

P1: The universe exhibits complexity, order, and purpose (e.g., eye, laws of physics).
P1： 宇宙展现出复杂性、秩序与目的（例如眼睛、物理定律）。
P2: Complexity with purpose in human artefacts (watch) is always the product of intelligent design.
P2： 人工制品（钟表）中带有目的的复杂性总是智慧设计的产物。
P3: Similar effects suggest similar causes (analogical principle).
P3： 相似的结果暗示相似的原因（类比原则）。
C: Therefore, the universe is probably the product of an intelligent designer (God).
C：因此，宇宙很可能是一位智慧设计者（上帝）的产物。

Complexity_order × Purpose → Design_inferred

秩序化的复杂性 × 目的 → 推断的设计

Hume’s objections: an imperfect world suggests an imperfect designer; the universe could be the result of chance over infinite time. Darwin’s evolution by natural selection offers an alternative non‑intelligent explanation.

休谟的反驳：不完美的世界暗示着一位不完美的设计者；宇宙可能是无限时间中偶然的结果。达尔文的自然选择进化论提供了另一种非智慧的解释。

3. The Moral Argument Formula | 道德论证公式

Kant argued that the existence of objective moral law points to a moral lawgiver. The formula traces the move from universal duty to a postulate of God.

康德论证，客观道德律的存在指向一位道德立法者。该公式追踪从普遍义务到上帝公设的推导过程。

P1: If objective moral duties exist, they require a supreme moral authority.
P1： 如果客观道德义务存在，它们就要求一个最高的道德权威。
P2: Objective moral duties do exist (e.g., we feel absolute obligation to tell the truth).
P2： 客观道德义务确实存在（例如，我们感到说实话的绝对义务）。
P3: The source of this authority cannot be imperfect humans.
P3： 这一权威的来源不能是不完美的人类。
C: Therefore, God exists as the foundation of moral law.
C：因此，上帝作为道德律的基础而存在。

Duty_objective → Law_universal → Lawgiver_divine

客观义务 → 普遍法则 → 神圣立法者

The Euthyphro dilemma challenges this: is something good because God commands it, or does God command it because it is good? This creates a logical tension within the formula.

尤息弗罗悖论对此提出挑战：某事是善的是因为上帝命令它，还是上帝命令它因为它是善的？这在公式内部制造了逻辑张力。

4. The Problem of Evil Logical Structure | 恶的问题逻辑结构

The logical problem of evil argues that the existence of evil is incompatible with an all‑powerful, all‑good God. The formula sets out the inconsistent triad.

恶的逻辑问题论证，恶的存在与全能全善的上帝是不相容的。该公式列出了不相容的三元组。

G1: God is omnipotent (able to prevent all evil).
命题1： 上帝是全能的（能够阻止所有恶）。
G2: God is omnibenevolent (would want to prevent all evil).
命题2： 上帝是全善的（会想要阻止所有恶）。
G3: Evil exists.
命题3： 恶存在。
Inconsistency: G1, G2, and G3 cannot all be true. If G1 and G2 hold, evil must not exist. But evil does exist, so either God is not powerful, not good, or does not exist.
不相容性： G1、G2 和 G3 无法同时成立。如果 G1 与 G2 成立，恶必不存在。但恶确实存在，因此要么上帝不强大，要么不善，要么不存在。

(Omni_potent ∧ Omni_good) → ¬Evil, yet Evil → ¬(Omni_potent ∧ Omni_good)

(全能 ∧ 全善) → 无恶，然而恶存在 → 并非(全能 ∧ 全善)

The Free Will Defence, evil as privation, and the soul‑making theodicy are attempts to break the inconsistency by adding premises (e.g., moral freedom justifies some evil).

自由意志辩护、恶是善的缺失以及灵魂塑造神正论，都是通过添加前提（例如道德自由可以证成某些恶）来尝试打破不相容性。

5. Utilitarianism: The Greatest Happiness Principle | 功利主义：最大幸福原则

Jeremy Bentham’s act utilitarianism treats the moral quality of an action as a function of its consequences. The formula is a hedonic calculus.

边沁的行为功利主义将行为的道德品质视为其后果的函数。该公式是一个快乐计算法。

Principle: The right action is the one that maximises pleasure and minimises pain for the greatest number.
原则： 正确的行为是能为最大多数人带来最大快乐、最小痛苦的行为。
Hedonic Calculus variables: Intensity, Duration, Certainty, Propinquity, Fecundity, Purity, Extent.
快乐计算变量： 强度、持续时间、确定性、远近性、继生性、纯度、范围。

Moral Value = Σ (Pleasure_x − Pain_x) for all affected x

道德价值 = Σ (快乐ₓ − 痛苦ₓ) 对每个受影响者 x

Mill’s rule utilitarianism adjusts the formula by distinguishing higher and lower pleasures, moving the calculus away from pure quantity to quality. Common exam scenarios test this formula against promises, justice, and individual rights.

穆勒的规则功利主义通过区分高级快乐和低级快乐调整了公式，将计算从纯粹的数量转向质量。常见考题会测试这一公式在承诺、正义和个体权利情境中的应用。

6. Situation Ethics: Agape Calculus | 情境伦理：圣爱算法

Joseph Fletcher’s situation ethics posits that the only absolute rule is agape (self‑giving love). The formula operates as a middle way between legalism and antinomianism.

弗莱彻的情境伦理假定唯一的绝对规则是圣爱（舍己的爱）。该公式作为律法主义与反律法主义之间的中道运作。

Core axiom: Agape love is the ultimate criterion; rules are illuminators, not absolute.
核心公理： 圣爱是终极标准；规则是启示者而非绝对。
Decision procedure: (i) Identify the problem; (ii) Apply the four working principles (pragmatism, relativism, positivism, personalism); (iii) Calculate the most loving outcome for those involved.
决策程序： (i) 识别问题；(ii) 应用四项工作原则（实用主义、相对主义、实证主义、人本主义）；(iii) 为相关方计算出最具有爱心的结果。

Action_right ← argmax_A Agape(A)

正确行为 ← 使圣爱最大化的行动 A

Critics warn that the formula is too subjective: what one person deems most loving, another might not. The lack of a measurable unit for love makes the ‘calculation’ problematic.

批评者警告该公式过于主观：一个人认为最有爱心的事，另一个人可能不这么看。爱缺乏可衡量的单位，使得这种“计算”充满问题。

7. Kantian Ethics: The Categorical Imperative Test | 康德伦理学：绝对命令检验

Kant’s ethics is deontological: the moral worth of an action lies in its maxim, not its consequences. The formula centres on the universalisability test.

康德的伦理学是义务论的：一个行为的道德价值在于其准则而非后果。该公式以可普遍化检验为核心。

Formulation 1 (Formula of Universal Law): Act only according to that maxim whereby you can at the same time will that it should become a universal law.
公式一（普遍法则公式）： 只依据那些你同时愿意其成为普遍法则的准则去行动。
Test: (i) Formulate your maxim; (ii) Universalise it; (iii) Check for contradiction in conception or in will. If contradiction, the act is impermissible.
检验： (i) 形成你的准则；(ii) 将其普遍化；(iii) 检查是否存在观念上的矛盾或意志上的矛盾。如有矛盾，该行为不被允许。

Maxim_permissible ⟷ Univeralisable_{no contradiction}

可允许的准则 ↔ 可普遍化且无矛盾

The second formulation (treat humanity always as an end) adds a dignity calculator: never use a person merely as a means. This pair of formulas is at the heart of many CCEA moral dilemmas.

第二公式（始终把人当作目的）增加了一个尊严计算器：绝不仅仅把人当作手段。这一对公式是许多 CCEA 道德困境的核心。

8. Religious Experience as Evidence: Inductive Inference | 宗教体验作为证据：归纳推理

Religious experiences are often used as evidence for God through an inductive argument. The formula treats them as perceptions of a divine reality, analogous to sense perceptions.

宗教体验常被用作通过归纳论证证明上帝的证据。该公式将其视为对神圣实在的感知，类似于感官知觉。

Principle of Credulity (Swinburne): If a person seems to perceive something, it is reasonable to believe it exists unless there are defeaters.
轻信原则（斯温伯恩）： 如果一个人似乎感知到某事物，那么相信其存在是合理的，除非有击败理由。
Formula: Experiences_{numinous/mystical} + absence of overriding defeaters → Justified belief in God.
公式： 神圣/神秘体验 + 无压倒性击败理由 → 对上帝的合理信念。

E_x(perception_divine) ∧ ¬D → ∴ P(God)_increased

个体x感知神圣 ∧ 无击败理由 → ∴ 上帝存在的概率增加

Challenges: experiences are diverse and contradictory; psychological or neurological explanations can defeat the inference. The formula shines in the cumulative case but struggles with verification.

挑战：体验多样且相互矛盾；心理或神经学的解释可以击败这一推理。该公式在累积论证中表现良好，但在验证性上存在困难。

9. The Nature of God: Omnipotence Paradox Formula | 上帝的本质：全能悖论公式

The concept of omnipotence generates logical puzzles. Can God create a stone so heavy that He cannot lift it? The formula maps the paradoxical reasoning to test definitions of omnipotence.

全能的概念会产生逻辑谜题。上帝能否创造一块重到他举不起来的石头？该公式描绘了悖论推理，用以检验全能的定义。

Paradox structure: If God can create such a stone → there is a stone He cannot lift → He is not omnipotent. If God cannot create it → there is something He cannot do → He is not omnipotent.
悖论结构： 如果上帝能创造这样一块石头 → 存在他举不起来的石头 → 他非全能。如果上帝不能创造 → 存在他做不到的事 → 他非全能。
Resolution attempt: Omnipotence = ability to do all logically possible things. Creating a stone an omnipotent being cannot lift is logically impossible, just as creating a square circle.
解决方案尝试： 全能 = 能够做一切逻辑上可能的事。创造一块全能者举不起来的石头就像画一个方的圆，是逻辑上不可能的。

Omnipotence_coherent ≡ ∀p (Possible_logical(p) → CanDo_God(p))

一贯的全能 ≡ 对所有 p：若 p 逻辑可能 → 上帝能做 p

This refined formula preserves God’s power while respecting logical boundaries, a key nuance for essay answers on divine attributes.

这一精炼公式在尊重逻辑界限的同时保留了上帝的大能，是探讨神圣属性论文题的关键细微之处。

10. Miracles: Hume’s Defeater Formula | 奇迹：休谟的击败公式

David Hume argued that we should never rationally believe a miracle report. His formula weighs the evidence for natural law against the evidence for testimony.

大卫・休谟论证，我们绝不应理性地相信奇迹报告。他的公式权衡自然律的证据与证词的证据。

Definition of miracle: A violation of a law of nature by a divine agent.
奇迹定义： 由神圣行事者对自然律的违反。
Hume’s maxim: No testimony is sufficient to establish a miracle unless the testimony is of such a kind that its falsehood would be more miraculous than the fact it tries to establish.
休谟准则： 任何证词都不足以确立一个奇迹，除非该证词的虚假比它试图确立的事实本身更奇迹。

P(Miracle|Testimony) < P(Natural law) → Reject claim

在给定证词下奇迹的概率 < 自然律的概率 → 拒绝声称

The formula practically demands that we always reject miracles because the probability of a law violation is overwhelmingly low. Critics like Swinburne adjust the formula by including prior probability of God’s existence.

该公式在实践中要求我们总是拒绝奇迹，因为违反自然律的概率低得压倒一切。像斯温伯恩这样的批评者通过纳入上帝存在的先验概率来调整该公式。

11. Free Will Defence Against Evil | 自由意志辩护反驳恶

Alvin Plantinga’s free will defence argues that it is logically possible that God could not create a world with free moral agents who always choose the good. The formula uses modal logic to show the logical problem of evil fails.

阿尔文・普兰丁格的自由意志辩护论证，在逻辑上，上帝不可能创造一个其中自由道德主体总是选择善的世界。该公式使用模态逻辑来表明逻辑恶问题是失败的。

Key premise: Free will is a great good that requires the genuine possibility of evil.
关键前提： 自由意志是一种巨大的善，它要求具有真实的恶的可能性。
Transworld depravity: Possibly, every feasible free creature would sin in some world. Thus, God cannot actualise a perfect free world if He creates free beings.
跨世界堕落： 可能每一个可行的自由受造物都会在某些世界中犯罪。因此，如果上帝创造自由存在者，他就不能实现一个完美的自由世界。

◊(Free Will ∧ ¬Moral Evil) is false → (Omni_God ∧ Evil) is logically consistent

“自由意志且无道德恶”在逻辑上不可能 → (全能的上帝与恶存在) 在逻辑上相容

The formula shifts the debate from the logical problem (inconsistency) to the evidential problem (amount of evil), which remains a tougher challenge.

该公式将辩论从逻辑问题（不相容）转移到证据问题（恶的数量），后者仍是一个更棘手的挑战。

12. The Logic of Determinism and Moral Responsibility | 决定论与道德责任的逻辑

If all events are caused by prior states, free will seems an illusion. This formula examines the compatibility between causal determinism and the moral responsibility required for ethics.

如果所有事件都由先前的状态所决定，自由意志似乎是一种幻觉。该公式检验因果决定论与伦理学所需的道德责任之间的兼容性。

Hard determinism: Determinism true → No free will → No moral responsibility.
硬决定论： 决定论为真 → 无自由意志 → 无道德责任。
Libertarianism: Determinism false → Agents have free will → Moral responsibility intact.
自由意志论（非决定论）： 决定论为假 → 行动者拥有自由意志 → 道德责任完好。
Compatibilism: Determinism may be true, but free will = acting according to one’s desires without external coercion → Moral responsibility still applies.
相容论： 决定论可能为真，但自由意志 = 按照自己的欲望行动而不受外部强迫 → 道德责任仍然适用。

Responsibility ≡ (Action_voluntary) ∧ (Could Have Done Otherwise)_conditional

责任 ≡ (自愿行动) ∧ (有条件的可替代选择)

In CCEA philosophy exams, this formula helps structure answers on whether sin, punishment, and reward can be justified if God foreknows all actions.

在 CCEA 哲学考试中，这一公式有助于构建关于如果上帝预知一切行为，罪、惩罚和奖赏能否被证成的答案。

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Year 11 CCEA Philosophy: Formula & Theorem Quick Reference Handbook | CCEA 11年级哲学：公式定理速查手册