📚 A-Level CAIE Philosophy: Formula & Theorem Quick Reference | A-Level CAIE 哲学：公式定理速查手册
This quick-reference guide condenses the recurring argument structures, evaluative tests, and conceptual ‘formulae’ that CAIE Philosophy candidates need to apply under timed conditions. It is not a list of empirical laws, but a toolkit of valid patterns and key distinctions used across epistemology, ethics, philosophy of religion, and philosophy of mind.
本速查手册浓缩了 CAIE 哲学考生在限时条件下需要应用的常见论证结构、评估测试与概念“公式”。它不是经验定律清单，而是贯穿认识论、伦理学、宗教哲学与心灵哲学的有效模式与关键区分的工具箱。
1. Logical Toolkit: Validity, Soundness, Modus Ponens, Modus Tollens | 逻辑工具箱：有效性、健全性、肯定前件、否定后件
An argument is valid if and only if it is impossible for the premises to be true while the conclusion is false. Validity concerns the form of reasoning, not the actual truth of the premises.
一个论证是有效的，当且仅当不可能出现前提为真而结论为假的情况。有效性关乎推理形式，而非前提的实际真假。
A sound argument is a valid argument with all true premises. Soundness is therefore a stronger standard than validity because it requires both good form and true content.
一个健全的论证是有效且所有前提都为真的论证。因此，健全性比有效性要求更高，因为它同时要求良好的形式和真实的内容。
Valid Modus Ponens: P → Q; P; therefore Q
有效肯定前件：P → Q；P；所以 Q
Valid Modus Tollens: P → Q; ¬Q; therefore ¬P
有效否定后件：P → Q；¬Q；所以 ¬P
Two common formal fallacies must be avoided: affirming the consequent (P → Q; Q; therefore P) and denying the antecedent (P → Q; ¬P; therefore ¬Q). Both are invalid because the conclusion can be false even if the premises are true.
必须避免两种常见的形式谬误：肯定后件（P → Q；Q；所以 P）和否定前件（P → Q；¬P；所以 ¬Q）。两者都是无效的，因为即使前提为真，结论也可能为假。
2. The JTB Formula and Gettier Counterexamples | JTB 公式与盖梯尔反例
The traditional analysis of knowledge states that knowledge is justified true belief. This can be written as a three-part conjunction.
传统的知识分析认为，知识是得到辩护的真信念。这可以写成一个三部分的合取式。
Knowledge = Justified + True + Belief (JTB)
知识 = 得到辩护 + 真 + 信念（JTB）
Gettier cases show that JTB is not sufficient for knowledge. A person can have a justified true belief that is only true by luck or through a false lemma, so the belief is correct accidentally rather than because of good evidence.
盖梯尔案例表明 JTB 并不是知识的充分条件。一个人可以拥有一个得到辩护的真信念，但该信念只是碰巧为真或依赖于一个假前提，因此该信念是偶然正确，而非基于良好证据。
A common repair is the No False Lemmas condition: S knows that p only if S’s belief that p is not inferred from any false proposition. This blocks many Gettier-style cases but still faces further counterexamples.
一种常见的修补方案是“无假前提条件”：S 知道 p，仅当 S 对 p 的信念不是从任何假命题推导出来的。这能排除许多盖梯尔式案例，但仍面临更多反例。
3. Utilitarian Calculus and Hedonic Formula | 功利主义计算与快乐公式
Classical act utilitarianism says that an action is right if and only if it produces the greatest total happiness for everyone affected. This is often summarised as the principle of utility.
古典行为功利主义认为，一个行为是正确的，当且仅当它为所有受影响者带来最大总量的幸福。这通常被概括为功利原则。
Right action = action that maximises Σ utility (pleasure minus pain)
正确行为 = 使 Σ 效用（快乐减去痛苦）最大化的行为
Bentham’s hedonic calculus weighs seven factors: intensity, duration, certainty, propinquity, fecundity, purity, and extent. This makes utilitarianism a quantitative and empirical moral theory, but it is also open to objections about measuring pleasure.
边沁的快乐计算衡量七个因素：强度、持续时间、确定性、远近、继生性、纯度和范围。这使功利主义成为一种量化的经验性道德理论，但也面临快乐难以测量的反对意见。
Mill modified the theory by distinguishing higher and lower pleasures. He argued that intellectual and moral pleasures are qualitatively superior to bodily pleasures, so the formula is not merely about quantity.
密尔通过区分高级快乐和低级快乐修正了这一理论。他认为理智和道德的快乐在性质上优于肉体快乐，因此公式不仅仅关乎数量。
4. Kantian Formulae: Categorical Imperative Variants | 康德公式：定言命令的表述
Kant’s categorical imperative is a test for whether a maxim, or personal rule of action, can be willed as a universal law. The first formulation is often called the Formula of Universal Law.
康德的定言命令是检验一个准则（个人行动规则）能否被意愿为普遍法则的标准。第一种表述通常被称为“普遍法则公式”。
Act only on that maxim which you can at the same time will that it should become a universal law
只按照你同时能够愿意它成为一条普遍法则的准则去行动
The Formula of Humanity states that we must never treat humanity, whether in our own person or in the person of any other, merely as a means but always also as an end.
人性公式指出，我们绝不能把人——无论是自己还是他人——仅仅当作手段，而必须始终同时当作目的。
A maxim fails the universal law test if it generates a contradiction in conception, such as breaking a promise when convenient, or a contradiction in willing, such as refusing to help others in distress.
如果一条准则产生构想上的矛盾（例如方便时就违背诺言）或意愿上的矛盾（例如拒绝帮助困境中的他人），它就无法通过普遍法则检验。
5. Virtue Ethics: Function and the Mean | 美德伦理学：功能与中道
Aristotle argues that the good for human beings is eudaimonia, often translated as flourishing or happiness, which is achieved by performing the human function well in accordance with reason.
亚里士多德认为，人类的善是 eudaimonia，通常翻译为繁荣或幸福，它通过按照理性出色地履行人类功能来实现。
Eudaimonia = activity of the soul in accordance with virtue over a complete life
Eudaimonia = 灵魂在整个一生中合乎美德的实现活动
Virtue is a settled disposition lying in a mean relative to us between two vices: one of excess and one of deficiency. Courage, for example, is the mean between cowardice and recklessness.
美德是一种稳定的品质，处于相对于我们而言的两种恶之间的中道：过度与不及。例如，勇敢是怯懦与鲁莽之间的中道。
Unlike act-based theories, virtue ethics focuses on character and practical wisdom, or phronesis, rather than on calculating consequences or following universal rules.
与基于行为的理论不同，美德伦理学关注品格和实践智慧（phronesis），而不是计算后果或遵循普遍规则。
6. Ontological Argument Logical Structure | 本体论论证的逻辑结构
Anselm’s ontological argument defines God as that than which nothing greater can be conceived. It then compares existence in the understanding with existence in reality.
安瑟伦的本体论论证将上帝定义为“无法设想比之更伟大的存在”。然后它将仅存在于理解中的存在与存在于现实中的存在进行比较。
God = the greatest conceivable being; existence in reality is greater than existence in the understanding alone; therefore God exists in reality
上帝 = 可设想的最伟大存在；存在于现实比仅存在于理解中更伟大；因此上帝存在于现实中
Gaunilo’s island objection uses the same structure to show that the argument would prove the existence of a perfect island, which is absurd. This suggests that the argument’s form can be misapplied.
高尼罗的岛屿反驳使用同样的结构来表明该论证会证明一个完美岛屿的存在，而这是荒谬的。这表明该论证的形式可能被误用。
Kant’s objection is that existence is not a real predicate. Adding existence to the concept of God does not enlarge the concept, so the argument fails to establish a being in reality.
康德的反对意见是：存在不是真实的谓词。给上帝的概念加上存在并不会扩大该概念，因此该论证无法在现实中确立一个存在者。
7. Cosmological Argument and Contingency Formula | 宇宙论论证与偶然性公式
The cosmological argument begins from the existence of the universe or contingent things and argues to a necessary first cause or sufficient reason. Aquinas’s versions rely on motion, causation, and contingency.
宇宙论论证从宇宙或偶然事物的存在出发，论证一个必然的第一因或充足理由。阿奎那的版本依赖于运动、因果关系和偶然性。
Every contingent thing needs a cause; the universe is contingent; therefore the universe needs a necessary cause
每个偶然事物都需要原因；宇宙是偶然的；因此宇宙需要一个必然原因
The Kalam version states: whatever begins to exist has a cause; the universe began to exist; therefore the universe has a cause. This cause is then identified with God.
卡拉姆版本指出：凡开始存在的事物都有原因；宇宙开始存在；因此宇宙有原因。这个原因随后被认定为上帝。
Hume and Russell object that we cannot apply the principle of causation beyond our experience, and even if the universe has a first cause, this does not show it is the God of classical theism.
休谟和罗素反对说，我们不能将因果原则应用于经验之外；即使宇宙有第一因，也不能表明它就是古典有神论的上帝。
8. Teleological Argument and Design Inference | 目的论论证与设计推理
Paley’s watch analogy argues that the complex and purposeful arrangement of parts in the natural world indicates a designer, just as a watch implies a watchmaker.
佩利的钟表类比论证说，自然界中复杂且有目的的部分排列表明有一位设计者，就像钟表意味着钟表匠一样。
Complex order + purpose → intelligent design → designer (God)
复杂秩序 + 目的 → 智能设计 → 设计者（上帝）
Hume’s criticisms include the weakness of analogy, the possibility of multiple designers, imperfect design in nature, and the problem of evil. These make the design inference less secure.
休谟的批评包括类比过于薄弱、可能存在多个设计者、自然界中的不完美设计以及恶的问题。这些都使设计推理不够可靠。
Darwinian evolution offers an alternative explanation: natural selection can produce the appearance of design without a designer. This is a powerful modern objection to the teleological argument.
达尔文进化论提供了另一种解释：自然选择可以在没有设计者的情况下产生看似设计的外观。这是对目的论论证的一个强有力的现代反驳。
9. Problem of Evil: Logical and Evidential Forms | 恶的问题：逻辑与证据形式
The logical problem of evil claims that the existence of an omnipotent, omnibenevolent God is logically incompatible with the existence of evil. The Inconsistent Triad shows this tension clearly.
恶的逻辑问题声称，全能、全善的上帝的存在与恶的存在在逻辑上不相容。不一致三元组清楚地显示了这种张力。
God is omnipotent; God is omnibenevolent; evil exists → logical contradiction?
上帝全能；上帝全善；恶存在 → 逻辑矛盾？
Plantinga’s free will defence argues that it is possible that God cannot create a world containing moral good without also allowing moral evil, because free creatures may choose wrongly. This blocks the logical contradiction.
普兰丁格的自由意志辩护论证说，上帝可能无法创造一个既包含道德善又不允许道德恶的世界，因为自由的受造物可能选择作恶。这消除了逻辑矛盾。
The evidential problem of evil is weaker but more challenging: the sheer quantity and distribution of suffering makes God’s existence improbable, even if not logically impossible.
恶的证据问题较弱但更具挑战性：苦难的巨大数量和分布使上帝的存在变得不太可能，即使在逻辑上并非不可能。
10. Mind-Body Positions and Identity Formula | 身心立场与同一性公式
Substance dualism holds that the mind and body are distinct substances: mental properties are non-physical and cannot be reduced to physical states. Descartes’s conceivability argument supports this view.
实体二元论认为心灵和身体是两种不同的实体：心理属性是非物理的，不能被还原为物理状态。笛卡尔的可设想性论证支持这一观点。
Physicalism, by contrast, says that everything is physical. The identity theory claims that mental states are identical with brain states, so pain just is C-fibre firing.
相比之下，物理主义认为一切都是物理的。同一论声称心理状态与大脑状态同一，所以疼痛就是 C-纤维的激活。
Type identity: mental state M = brain state B
类型同一：心理状态 M = 大脑状态 B
Functionalism defines mental states by their causal roles rather than their physical make-up. This allows multiple realizability: the same mental state can be realised by different physical systems.
功能主义根据因果角色而非物理构成来定义心理状态。这允许多重可实现性：同一种心理状态可以由不同的物理系统实现。
Leibniz’s law states that if X is identical to Y, then X and Y share all properties. Critics use mental-physical property differences to argue against strict identity.
莱布尼茨定律指出，如果 X 与 Y 同一，那么 X 和 Y 共享所有属性。批评者使用心理与物理属性之间的差异来反驳严格同一性。
11. Compatibilism and Free Will Formula | 相容论与自由意志公式
Hard determinism argues that if determinism is true, then no one has free will, because every action is the inevitable result of prior causes. Libertarianism rejects determinism and affirms free will.
硬决定论认为，如果决定论为真，那么没有人拥有自由意志，因为每个行为都是先前原因的必然结果。自由意志论则拒绝决定论并肯定自由意志。
Compatibilism, held by Hobbes and Hume, redefines free will so that it is compatible with determinism. A free action is one caused by the agent’s own beliefs and desires without external constraint.
霍布斯和休谟所持的相容论重新定义自由意志，使其与决定论相容。一个自由行为是由行动者自身的信念和欲望引起、且没有外部强制约束的行为。
Free action = caused by one’s own desires + absence of external compulsion
自由行为 = 由自己的欲望引起 + 不存在外部强制
The key dispute is whether free will requires alternative possibilities or merely voluntary action. Compatibilists accept the latter, while incompatibilists demand the former.
关键争论在于自由意志是否要求替代可能性，还是仅仅要求自愿行为。相容论者接受后者，而不相容论者要求前者。
12. Political Philosophy: Social Contract and Principles of Justice | 政治哲学：社会契约与正义原则
Social contract theory explains political obligation as arising from a hypothetical or actual agreement among individuals to form a state. Hobbes, Locke, and Rousseau give different versions of this contract.
社会契约论将政治义务解释为来自个人之间组成国家的假设或实际协议。霍布斯、洛克和卢梭给出了不同版本的契约。
Rawls’s original position is a thought experiment in which rational agents choose principles of justice behind a veil of ignorance, not knowing their own social position or natural talents.
罗尔斯的原初状态是一个思想实验：理性行动者在无知之幕背后选择正义原则，而不知道自己的社会地位或自然天赋。
Justice = equal basic liberties + fair equality of opportunity + difference principle
正义 = 平等的基本自由 + 公平的机会平等 + 差别原则
The difference principle permits social and economic inequalities only if they benefit the least advantaged members of society. This distinguishes Rawls’s theory from strict egalitarianism and libertarianism.
差别原则允许社会和经济不平等，但前提是它们必须使社会中最不利的成员受益。这使罗尔斯的理论区别于严格的平等主义和自由至上主义。
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