📚 Philosophy Formula & Theorem Quick Reference Handbook | 哲学公式定理速查手册
This quick reference handbook distils the core arguments, logical patterns, and moral calculus devices encountered in the Year 10 CAIE Philosophy course. Each entry is presented as a 'formula' or 'theorem' to help you memorise structure, identify valid reasoning, and apply ethical decision procedures with precision.
本速查手册浓缩了 Year 10 CAIE 哲学课程中的核心论证、逻辑模板和道德运算工具。每个条目均以'公式'或'定理'形式呈现，帮助你记忆结构、识别有效推理，并精准应用伦理决策程序。
1. Modus Ponens & Modus Tollens | 肯定前件式与否定后件式
Affirming the Antecedent): If P → Q and P is true, then Q must be true. Formally: P → Q, P ⊢ Q. This is always valid; if the premises are true, the conclusion necessarily follows.
肯定前件式：如果 P → Q 且 P 为真，那么 Q 必定为真。形式：P → Q，P ⊢ Q。此形式始终有效；前提为真时，结论必然成立。
Modus Tollens (Denying the Consequent): If P → Q and Q is false (¬Q), then P must be false (¬P). Formally: P → Q, ¬Q ⊢ ¬P. It is equally valid and forms the backbone of falsification reasoning.
否定后件式：如果 P → Q 且 Q 为假 (¬Q)，那么 P 必定为假 (¬P)。形式：P → Q，¬Q ⊢ ¬P。此形式同样有效，构成证伪推理的骨架。
2. Hypothetical Syllogism & Disjunctive Syllogism | 假言三段论与选言三段论
ins conditional relationships. No truth of P, Q, or R is asserted individually; only the logical link is built.
假言三段论：由 P → Q 和 Q → R，可有效推出 P → R。它将条件关系串联起来，不单独断言 P、Q 或 R 的真值，只构建逻辑链接。
Disjunctive Syllogism: Given P ∨ Q and ¬P, we infer Q. Alternatively, given P ∨ Q and ¬Q, we infer P. This relies on the exclusive sense of 'or' in many philosophical contexts, although inclusive disjunction often needs careful qualification.
选言三段论：已知 P ∨ Q 和 ¬P，可推出 Q。或者已知 P ∨ Q 和 ¬Q，可推出 P。尽管在许多哲学语境中'或'兼容两者，但这里往往需要作排他性限定。
3. Bentham's Felicific Calculus | 边沁的幸福计算法
Jeremy Bentham proposed a quantitative method to measure pleasure and pain produced by an action. The calculus consists of seven dimensions: intensity, duration, certainty/uncertainty, propinquity (nearness), fecundity (chance of being followed by more of the same), purity (chance of not being followed by the opposite), and extent (number of people affected).
杰里米·边沁提出了一种量化方法来计算行为产生的快乐与痛苦。该计算包含七个维度：强度、持续时间、确定性/不确定性、远近（时间接近程度）、丰度（后随同类感受的概率）、纯度（不后随相反感受的概率）和范围（受影响的人数）。
xtent ...). Though not a strict mathematical equation, the goal is to maximise net pleasure over pain for the greatest number.
公式思维：总效用 = Σ (强度 × 持续时间 × 概率 × 范围 ……)。虽非严格的数学方程，但目标是为最大多数人获得最大净快乐。
4. Mill's Qualitative Utilitarianism | 穆勒的质化功利主义
J. S. Mill introduced a quality dimension: 'It is better to be a human being dissatisfied than a pig satisfied; better to be Socrates dissatisfied than a fool satisfied.' Higher pleasures (intellect, moral sentiments) carry greater weight than lower pleasures (mere bodily gratification), even if lower pleasures score higher on Bentham's dimensions.
J.S.穆勒引入了质的维度：'宁做不满足的人，不做满足的猪；宁做不满足的苏格拉底，不做满足的傻瓜。'高级快乐（理智、道德情感）比低级快乐（单纯肉体满足）权重更大，即便低级快乐在边沁维度上得分更高。
Competent judges test: those who have experienced both types of pleasure are the only reliable arbiters. The formula shifts from pure aggregation to a weighted preference system.
有资格的评判者检验：唯有体验过两类快乐的人才是可靠的裁决者。公式从纯粹的加总转向加权偏好体系。
5. Kant's Categorical Imperative: Formula of Universal Law | 康德的绝对命令：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.
只依据那些你可以同时愿意其成为普遍法则的准则而行动。
The test has two steps: (1) Formulate the maxim of your action as a universal law of nature. (2) Check for contradiction. A contradiction in conception (e.g., lying promise) shows a perfect duty; a contradiction in the will (e.g., never helping others) shows an imperfect duty.
检验分两步：(1) 将你的行为准则设想为一条自然普遍法则。(2) 检查矛盾。概念矛盾（如虚假承诺）揭示完全义务；意志矛盾（如从不帮助他人）揭示不完全义务。
Formula of Humanity variation: 'Act in such a way that you treat humanity, whether in your own person or in the person of another, always at the same time as an end and never simply as a means.' Both formulas are meant to be equivalent.
人性公式变体：'你要如此行动，即无论是对你自己还是对他人，总是同时把人性当作目的，而绝不仅仅当作手段。'两条公式理应是等价的。
6. Aristotle's Doctrine of the Mean | 亚里士多德的中庸之道
Virtue is a mean between two vices – one of excess and one of deficiency – relative to us, determined by reason. The formula is not arithmetic; it requires practical wisdom (phronesis). Example: Courage is the mean between Cowardice (deficiency) and Recklessness (excess).
德性是过度与不及两种恶之间的中庸，相对于我们而言，由理性决定。该公式并非算术平均，而需要实践智慧（phronesis）。例如：勇敢是怯懦（不及）与鲁莽（过度）之间的中庸。
Mean = f (situation, agent, right reason). The mean is relative to the individual and context; it is not a fixed midpoint. Other key examples: Temperance between insensibility and self-indulgence; Generosity between stinginess and wastefulness.
中庸 = f (情境、行动者、正确理性)。中庸相对于个体和情境，非固定中点。其他典型例子：节制是冷漠与放纵的中庸；慷慨是吝啬与挥霍的中庸。
7. Aquinas' Primary Precepts of Natural Law | 阿奎那自然法的基本准则
According to Thomas Aquinas, practical reason naturally apprehends five primary precepts: (1) Preservation of Life, (2) Reproduction and Education of Offspring, (3) Living in Society (Order), (4) Seeking Truth (Knowledge of God), (5) Worshipping God. These are self-evident to anyone with right reason and never change.
据托马斯·阿奎那，实践理性自然把握五条基本准则：(1) 保存生命，(2) 繁衍并教育后代，(3) 在社会中生活（秩序），(4) 追求真理（认识上帝），(5) 敬拜上帝。对有正确理性者，这些准则不证自明且永不改变。
Secondary precepts are derived from primary precepts through human reasoning and can vary in detail. The primary precepts direct us toward our natural end (telos) of flourishing (eudaimonia) in accordance with God's eternal law.
次级准则由人类理性从基本准则推演而来，细节可以变化。基本准则指引我们朝向上帝永恒法则下的自然目的（telos）——人类的繁荣（eudaimonia）。
8. The Problem of Evil: The Inconsistent Triad | 恶的问题：不一致三元组
J. L. Mackie formulated the logical problem of evil as an inconsistent triad. The three propositions cannot all be true:
J.L. Mackie 将恶的逻辑问题表述为一个不一致三元组。以下三个命题无法同时为真：
God is omnipotent (all-powerful). | 上帝全能。
God is omnibenevolent (all-good). | 上帝全善。
Evil exists. | 恶存在。
If any two are true, the third must be false. To solve the logical problem, a theist must deny or drastically qualify one of these attributes, or explain why the existence of evil is compatible with them (theodicy).
若任意两条为真，则第三条必为假。要解决逻辑问题，有神论者必须否定或大幅限定其中某一属性，或解释为何恶的存在与它们相容（神义论）。
9. 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' (TTWNGCBC). The argument has a reductio ad absurdum structure:
安瑟伦将上帝定义为'无法设想有比之更伟大者'。论证采用归谬法结构：
Assume God exists only in the understanding but not in reality. | 假定上帝仅存在于理智中而不在现实中。
We can conceive of a being exactly like God but also existing in reality. | 我们可以设想一个与上帝完全相似、但也在现实中存在的存在者。
Existing in reality is greater than existing in the understanding alone. | 存在于现实中比仅存在于理智中更伟大。
Thus we can conceive of a being greater than God – contradiction. | 这样我们就设想了比上帝更伟大的存在者——矛盾。
∴ God must exist in reality.
∴ 上帝必定存在于现实中。
Gaunilo's reply used the perfect island parody: the logic would equally 'prove' a perfect island exists, suggesting a flaw.
高尼罗用完美之岛进行仿拟回应：该逻辑同样可以'证明'完美之岛存在，揭示了缺陷。
10. Pascal's Wager Decision Matrix | 帕斯卡的赌注决策矩阵
Pascal frames belief in God as a rational bet under uncertainty. The payoff matrix can be summarised:
帕斯卡将信仰上帝框定为不确定性下的理性赌注。收益矩阵可总结为：
| Decision | 决策 | God exists | 上帝存在 | God does not exist | 上帝不存在 |
|---|---|---|
| Believe | 信仰 | Infinite gain (Heaven) | 无限收益（天堂） | Finite loss (earthly pleasures) | 有限损失（世俗快乐） |
| Not believe | 不信仰 | Infinite loss (Hell) | 无限损失（地狱） | Finite gain (earthly pleasures) | 有限收益（世俗快乐） |
Expected value reasoning: the expected utility of believing dominates because infinity multiplied by any finite probability is still infinity. Critics challenge the many-gods objection and the assumption that one can simply decide to believe.
期望值推理：信仰的期望效用占优，因为无穷大乘以任何有限概率仍为无穷大。批评者提出多神反驳，并质疑人能简单决定相信这一假设。
11. Euthyphro Dilemma | 游叙弗伦两难
Plato's dialogue presents a dilemma: 'Is the pious loved by the gods because it is pious, or is it pious because it is loved by the gods?' In monotheistic terms: Is something good because God commands it, or does God command it because it is good?
柏拉图对话中的两难：'虔诚是因为被众神所爱而虔诚，还是因为虔诚所以被众神所爱？'在一神论语境中：某事物是善的因为上帝命令它，还是上帝命令它因为它是善的？
If good because God commands it, then morality becomes arbitrary – God could command cruelty and it would be good. If God commands something because it is already good, then goodness exists independently of God, challenging divine sovereignty.
如果因为上帝命令才善，那么道德变得任意——上帝若命令残忍，残忍也成了善。如果上帝因某事物已为善而命令它，那么善独立于上帝而存在，挑战了上帝的主权。
12. The Is–Ought Gap (Hume's Law) | 实然—应然鸿沟（休谟法则）
David Hume observed that many moral arguments illicitly move from descriptive statements (is) to prescriptive conclusions (ought) without justification. No set of purely factual premises entails a moral conclusion.
大卫·休谟观察到许多道德论证未经辩护便从描述性陈述（实然）滑向规范性结论（应然）。任何一组纯粹事实性前提都不能逻辑地蕴含道德结论。
Formally: from premises containing no 'ought', no 'ought'-conclusion can be validly deduced. This challenges natural law theories that attempt to derive morality from human nature, unless a bridging premise is supplied.
形式上：从不含'应然'的前提中，无法有效演绎出'应然'结论。这对试图从人性推导道德的自然法理论构成了挑战，除非补充一座桥梁前提。
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