📚 IGCSE CAIE Philosophy: Formula and Theorem Quick Reference | IGCSE CAIE 哲学：公式定理速查手册
Philosophy may not have equations in the mathematical sense, but CAIE IGCSE Philosophy rewards students who can compress arguments, definitions and ethical principles into repeatable patterns. This quick reference collects the ‘formulae’ and ‘theorems’ most often needed in the exam, so you can move from describing ideas to applying them with precision.
哲学或许没有数学意义上的方程，但 CAIE IGCSE 哲学奖励那些能将论证、定义和伦理原则压缩为可重复模式的学生。本速查手册汇集了考试中最常用的“公式”和“定理”，帮助你从描述观点走向精确应用。
1. The Argument Formula | 论证公式
Every argument in philosophy can be reduced to a standard form: premises support a conclusion. In the exam, always write arguments as P1, P2, therefore C before evaluating them.
哲学中的每个论证都可以简化为标准形式：前提支持结论。在考试中，务必先把论证写成 P1、P2、因此 C，然后再进行评价。
P1 + P2 → C
Indicator words help you spot the parts: ‘because’, ‘since’ and ‘for’ introduce premises, while ‘therefore’, ‘thus’ and ‘so’ introduce the conclusion.
指示词有助于识别各部分：“because”、“since”和“for”引导前提，而“therefore”、“thus”和“so”引导结论。
- Example: P1: All humans are mortal. P2: Socrates is human. C: Socrates is mortal.
- 示例：P1：所有人都会死。P2：苏格拉底是人。C：苏格拉底会死。
2. Deductive Validity and Soundness | 演绎有效性与可靠性
An argument is deductively valid if the truth of the premises would guarantee the truth of the conclusion. Validity is about the structure, not the actual truth of the premises.
如果一个论证的前提为真能保证结论为真，那么该论证就是演绎有效的。有效性关乎结构，而不是前提事实上是否为真。
Valid = (Premises true ⇒ Conclusion true)
A sound argument is a valid argument with all true premises. Soundness is the gold standard: a sound argument must have a true conclusion.
可靠的论证是前提全部为真的有效论证。可靠性是黄金标准：可靠的论证其结论必然为真。
Sound = Valid + True premises
Example: P1: All birds have wings. P2: A robin is a bird. C: A robin has wings. This is valid, and if both premises are true, it is sound.
示例：P1：所有鸟都有翅膀。P2：知更鸟是鸟。C：知更鸟有翅膀。该论证有效，如果两个前提都为真，则它也是可靠的。
3. Inductive Strength and Cogency | 归纳强度与可信度
Inductive arguments do not guarantee their conclusions; they make the conclusion probable. An inductively strong argument is one where, if the premises are true, the conclusion is likely.
归纳论证不保证结论为真，而是使结论具有可能性。归纳强的论证是指：如果前提为真，结论很可能为真。
Strong = (Premises true ⇒ Conclusion probable)
A cogent argument is an inductively strong argument with all true premises. In IGCSE answers, avoid saying an inductive argument ‘proves’ its conclusion; say it ‘supports’ or ‘gives good reason for’ it.
可信的论证是前提全部为真的归纳强论证。在 IGCSE 答题中，不要说归纳论证“证明”了结论，而应说它“支持”或“为结论提供了充分理由”。
Cogent = Strong + True premises
- Example: The sun has risen every day for billions of years, therefore the sun will rise tomorrow.
- 示例：太阳每天升起已有数十亿年，因此太阳明天会升起。
4. The JTB Formula for Knowledge | 知识的 JTB 公式
The traditional tripartite definition of knowledge says that knowledge is justified true belief. All three conditions are necessary, and together they are sufficient.
传统的知识三元定义认为，知识是得到辩护的真信念。三个条件都是必要的，合起来就是充分的。
K = J + T + B
Knowledge is a belief that is true and for which the believer has justification. A lucky guess, even if true, is not knowledge because it lacks justification.
知识是一种真实且持有者有辩护的信念。即使猜对了，也不能算知识，因为缺乏辩护。
- J = Justified (has evidence or reasons)
- T = True (matches the facts)
- B = Belief (the person actually holds it)
- J = 有辩护（有证据或理由）
- T = 真（与事实相符）
- B = 信念（主体确实相信）
Gettier cases challenge JTB by showing that a person can have justified true belief but still lack knowledge due to luck. Mention this to show evaluation in essays.
盖梯尔案例挑战了 JTB 理论，表明一个人可以有得到辩护的真信念，但仍因运气而缺乏知识。在论文中提及这一点可展示评价能力。
5. Utilitarian Calculus | 功利主义计算
Utilitarianism is a consequentialist theory: the right action is the one that produces the greatest happiness for the greatest number. The basic formula weighs pleasure against pain.
功利主义是一种后果论理论：正确的行为就是为最大多数人带来最大幸福的行为。其基本公式是权衡快乐与痛苦。
H = Σ(pleasure) − Σ(pain)
Act utilitarianism applies the calculation to each individual action. Rule utilitarianism applies it to general rules, asking which rules tend to maximise happiness if followed.
行为功利主义将计算应用于每个具体行为。规则功利主义则将计算应用于普遍规则，追问哪些规则在被遵守时倾向于使幸福最大化。
| Act Utilitarianism | Judges each act directly by H = ΣP − ΣPain. |
| Rule Utilitarianism | Judges rules by their tendency to maximise H. |
| 行为功利主义 | 直接用 H = Σ快乐 − Σ痛苦 评价每个行为。 |
| 规则功利主义 | 根据规则是否倾向于使 H 最大化来评价规则。 |
In essays, remember that Bentham’s version treats all pleasures as equal, while Mill distinguishes higher and lower pleasures.
在论文中要记住，边沁的版本认为所有快乐都平等，而密尔区分了高级快乐和低级快乐。
6. Kantian Categorical Imperative Formulae | 康德定言命令公式
Kant’s ethics is deontological: the rightness of an action depends on the motive and the principle behind it, not on consequences. The categorical imperative has two main formulations.
康德的伦理学是义务论的：行为的对错取决于动机和背后的原则，而不是后果。定言命令有两种主要表述。
Universal Law: Act only on that maxim which you can will to become a universal law.
The first formula tests whether the rule behind your action could be consistently applied by everyone. If it leads to a contradiction, the action is wrong.
第一公式检验你行为背后的准则是否能被所有人一致地应用。如果导致矛盾，该行为就是错误的。
Humanity Formula: Treat humanity, whether in yourself or others, always as an end and never merely as a means.
The second formula forbids using people only as tools. It grounds the idea of human dignity and respect.
第二公式禁止仅仅把人当作工具。它奠定了人的尊严和尊重观念。
- Example: Lying fails the universal law test because if everyone lied, trust and language would collapse.
- 示例：说谎不能通过普遍法则检验，因为如果人人说谎，信任和语言就会崩溃。
7. Free Will and Determinism | 自由意志与决定论
Determinism claims that every event, including human actions, has a prior cause. If determinism is true, it is hard to see how we could have acted otherwise.
决定论声称每个事件，包括人类行为，都有先前的原因。如果决定论为真，我们很难看出自己本可以不那样做。
Determinism: Every event E has a cause C.
Free will is usually defined as the ability to do otherwise. Incompatibilists say free will and determinism cannot both be true; compatibilists say freedom means acting according to one’s own desires without external coercion.
自由意志通常被定义为有能力做出不同选择。不相容论者认为自由意志和决定论不能同时为真；相容论者认为自由意味着按照自己的欲望行动而不受外部强制。
Free will = Agent could have done otherwise
- Hard determinism: determinism is true, so free will is an illusion.
- Libertarianism: free will exists, so determinism is false.
- Compatibilism: free will and determinism are compatible.
- 强决定论：决定论为真，因此自由意志是幻觉。
- 意志自由论：自由意志存在，因此决定论为假。
- 相容论：自由意志与决定论相容。
8. Cosmological Argument Structure | 宇宙论论证结构
The cosmological argument reasons from the existence of the universe to the existence of God as its first cause or necessary ground. It relies on the principle that nothing comes from nothing.
宇宙论论证从宇宙的存在推理出上帝作为第一因或必然根据的存在。它依赖的原则是：无中不能生有。
P1: Everything contingent has a cause.
P2: The universe is contingent.
C: The universe has a cause (God).
A contingent being is one that could have failed to exist. The argument says an infinite chain of contingent causes is impossible, so there must be a necessary being.
偶然存在物是指那些本可以不存在的存在物。该论证认为偶然原因的无限链条是不可能的，因此必定有一个必然存在者。
Evaluation: the argument may commit the fallacy of composition, because even if every part of the universe has a cause, the universe as a whole may not need one.
评价：该论证可能犯了合成谬误，因为即使宇宙的每个部分都有原因，宇宙整体也未必需要一个原因。
9. Teleological Argument Formula | 目的论论证公式
The teleological argument, or design argument, moves from observed order and purpose in the world to the existence of an intelligent designer. Paley’s watch analogy is the classic illustration.
目的论论证（设计论证）从世界中观察到的秩序和目的推出智能设计者的存在。佩利的钟表类比是经典说明。
P1: The universe shows order, purpose and complexity.
P2: Order and purpose usually come from a designer.
C: The universe probably has a designer (God).
The argument is inductive: it claims the designer hypothesis is the best explanation for features like the human eye or the laws of physics.
该论证是归纳性的：它声称设计者假说是对人类眼睛或物理定律等特征的最佳解释。
Hume’s criticisms: the analogy between the universe and a machine is weak, and even if a designer exists, the argument does not prove the God of classical theism.
休谟的批评：宇宙与机器之间的类比是薄弱的，而且即使设计者存在，该论证也不能证明古典有神论的上帝。
10. The Logical Problem of Evil Formula | 恶的逻辑问题公式
The logical problem of evil argues that the existence of evil is logically incompatible with an omnipotent, omnibenevolent God. This is a key challenge to theism.
恶的逻辑问题认为，恶的存在与全能、全善的上帝在逻辑上不相容。这是对有神论的关键挑战。
P1: If God is omnipotent, He can prevent evil.
P2: If God is omnibenevolent, He wants to prevent evil.
P3: Evil exists.
C: Therefore, an omnipotent and omnibenevolent God does not exist.
If God has both the power and the will to stop evil, evil should not exist. Since evil does exist, at least one of God’s attributes must be false.
如果上帝既有能力又有意愿阻止恶，恶就不应该存在。既然恶确实存在，上帝的至少一个属性必定为假。
The free will defence replies that God allows evil because free will is a greater good, and moral evil results from human misuse of freedom.
自由意志辩护回应说，上帝允许恶存在是因为自由意志是更大的善，而道德恶来自人类对自由的误用。
11. Ethical Relativism and Absolutism | 伦理相对主义与绝对主义
Ethical relativism claims that moral truths are relative to culture or individual preference. What is right in one society may be wrong in another, and there is no universal moral standard.
伦理相对主义声称道德真理相对于文化或个人偏好。在一个社会中对的事在另一个社会中可能是错的，不存在普遍的道德标准。
Relativism: Moral truth = Dependent on culture or person
Ethical absolutism holds that at least some moral principles are universally true, regardless of culture or opinion. Absolutists often point to principles like ‘do not torture the innocent’ as non-negotiable.
伦理绝对主义认为至少有些道德原则是普遍为真的，无论文化或意见如何。绝对主义者常以“不要折磨无辜者”等原则为例，认为它们不可妥协。
Absolutism: Moral truth = Universal and objective
- Relativism strength: explains cultural diversity.
- Relativism weakness: cannot criticise harmful practices such as slavery.
- Absolutism strength: provides a basis for moral criticism and human rights.
- Absolutism weakness: may ignore cultural context.
- 相对主义优点：解释了文化多样性。
- 相对主义缺点：无法批评奴隶制等有害习俗。
- 绝对主义优点：为道德批评和人权提供基础。
- 绝对主义缺点：可能忽视文化背景。
12. Mind-Body Identity and Dualism | 心身同一论与二元论
In philosophy of mind, dualism claims the mind and body are two distinct substances. Descartes argued that the mind is non-physical and can exist without the body.
在心灵哲学中，二元论声称心灵和身体是两种不同的实体。笛卡尔认为心灵是非物质的，可以脱离身体存在。
Dualism: Mind ≠ Body (two substances)
Identity theory claims that mental states are identical to brain states. On this view, pain is nothing over and above a certain pattern of neural activity.
同一论声称心理状态等同于大脑状态。根据这种观点，疼痛不过是某种神经活动模式，别无其他。
Identity theory: Mind = Brain
A common exam question asks you to evaluate dualism: its strength is that it fits our intuition that consciousness is private and non-physical, but its weakness is the interaction problem — how can a non-physical mind cause physical movements?
常见的考题要求评价二元论：其优点是符合我们关于意识是私密且非物理的直觉，但其缺点是交互问题——非物理的心灵如何引起物理运动？
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