📚 IGCSE Cambridge Philosophy: Formula & Theorem Quick Reference Guide | IGCSE 剑桥哲学：公式定理速查手册
This guide condenses the key logical forms, epistemic principles, ethical tests and philosophical arguments encountered in IGCSE Cambridge Philosophy into a fast revision manual. It is not a substitute for full essays, but a checklist of formulae you must be able to recognise, explain and evaluate.
本手册将 IGCSE 剑桥哲学中常考的逻辑形式、认识论原则、伦理学检验标准和哲学论证浓缩为快速复习手册。它不是完整论文的替代品，而是你必须能够识别、解释和评价的公式清单。
1. Proposition, Premise and Conclusion | 命题、前提与结论
In philosophy, an argument is not a quarrel; it is a structured set of propositions. A proposition is a statement that can be true or false. The premises are the supporting propositions, and the conclusion is the proposition they are intended to establish.
在哲学中，论证不是争吵，而是一组结构化的命题。命题是可以为真或为假的陈述。前提是支持性命题，结论是它们旨在确立的命题。
Standard form: P₁, P₂, … ∴ C
The symbol ∴ means “therefore”. For example: P₁: All humans are mortal. P₂: Socrates is human. ∴ Socrates is mortal.
符号 ∴ 表示 “因此”。例如：P₁：所有人类都会死。P₂：苏格拉底是人类。∴ 苏格拉底会死。
Any essay in philosophy begins by identifying these elements. If a passage has no conclusion, it is not an argument; it is merely a collection of statements.
任何哲学论文都从识别这些要素开始。如果一段话没有结论，它就不是论证，而只是一组陈述。
2. Modus Ponens and Modus Tollens | 肯定前件与否定后件
Two valid argument forms appear constantly in IGCSE questions. Modus ponens affirms the antecedent; modus tollens denies the consequent.
两个有效的论证形式在 IGCSE 题目中经常出现。肯定前件式肯定前件；否定后件式否定后件。
Modus Ponens: P → Q; P; ∴ Q
Modus Tollens: P → Q; ¬Q; ∴ ¬P
Example of modus ponens: If it rains, the ground is wet. It rains. Therefore the ground is wet. Example of modus tollens: If it rains, the ground is wet. The ground is not wet. Therefore it does not rain.
肯定前件式示例：如果下雨，地面就湿。下雨了。因此地面湿。否定后件式示例：如果下雨，地面就湿。地面不湿。因此没有下雨。
These forms are valid because the truth of the premises guarantees the truth of the conclusion. Every counterexample attempt will fail.
这些形式是有效的，因为前提的真实性保证了结论的真实性。任何试图提出反例的尝试都会失败。
3. Common Formal Fallacies | 常见形式谬误
Two invalid forms are often mistaken for the valid ones. Affirming the consequent and denying the antecedent produce conclusions that do not follow with certainty.
两种无效形式经常被误认为有效形式。肯定后件和否定前件得出的结论并不具有必然性。
| Name | Form | Status |
|---|---|---|
| Affirming the consequent | P → Q; Q; ∴ P | Invalid |
| Denying the antecedent | P → Q; ¬P; ∴ ¬Q | Invalid |
Example: If it rains, the ground is wet. The ground is wet. Therefore it rains. This is invalid because a sprinkler could have wet the ground. Similarly, if it rains, the ground is wet; it does not rain; therefore the ground is not wet is invalid because the ground might be wet for another reason.
示例：如果下雨，地面就湿。地面湿。因此下雨了。这是无效的，因为洒水器也可能弄湿地面的。同样，如果下雨，地面就湿；没有下雨；因此地面不湿，这也是无效的，因为地面可能因其他原因而湿。
In exam answers, identify the form first, then explain why the conclusion does not necessarily follow.
在考试答案中，先识别形式，然后解释为什么结论并不必然推出。
4. Necessary and Sufficient Conditions | 必要与充分条件
Conditional statements express necessary and sufficient relationships. Understanding these relationships is essential for analysing definitions and causal claims.
条件句表达了必要和充分关系。理解这些关系对于分析定义和因果主张至关重要。
If P → Q, then P is sufficient for Q; Q is necessary for P.
For example: Being a dog is sufficient for being a mammal. Being a mammal is necessary for being a dog. You cannot be a dog without being a mammal, but being a mammal does not guarantee being a dog.
例如：是狗是成为哺乳动物的充分条件。是哺乳动物是成为狗的必要条件。不可能是狗却不是哺乳动物，但成为哺乳动物并不能保证是狗。
A common exam task is to state whether a condition is necessary, sufficient, both, or neither. Check the direction of the arrow and test counterexamples.
常见的考试任务是说明某个条件是必要的、充分的、两者兼有，还是两者都不是。检查箭头的方向并用反例加以检验。
5. The JTB Formula and Gettier Problem | JTB 公式与葛梯尔问题
The traditional analysis of knowledge is called justified true belief, often abbreviated as JTB. According to this formula, a person knows a proposition if and only if three conditions are met.
传统的知识分析被称为被证成的真信念，通常缩写为 JTB。根据这个公式，一个人知道一个命题当且仅当满足三个条件。
S knows that P iff: (i) P is true; (ii) S believes that P; (iii) S is justified in believing P.
Gettier cases show that JTB is not sufficient. For example, Smith has strong evidence that Jones owns a Ford, so Smith justifiably believes the true proposition “Someone in the office owns a Ford”. However, Jones does not own a Ford; Smith’s evidence is misleading, but the proposition is true because Smith himself owns a Ford. Smith has JTB, yet we do not say he knows.
葛梯尔案例表明 JTB 并不充分。例如，史密斯有充分证据证明琼斯拥有一辆福特车，因此史密斯有理由相信真命题 “办公室里有人拥有福特车”。然而，琼斯并没有福特车；史密斯的证据具有误导性，但这个命题为真，因为史密斯自己拥有一辆福特车。史密斯拥有 JTB，但我们并不说他拥有知识。
Many exam answers require a definition of JTB plus one Gettier-style counterexample to show why philosophers add a fourth condition, such as “no false lemmas”.
许多考试答案要求给出 JTB 的定义，并加上一个葛梯尔式反例，以说明哲学家为什么会增加第四个条件，比如 “没有错误引理”。
6. Utilitarian Calculus | 功利主义计算
Utilitarianism is a consequentialist ethical theory. The right action is the one that produces the greatest overall happiness or utility. Bentham proposed a quantitative measure of pleasure and pain.
功利主义是一种后果主义伦理学理论。正确的行为是产生最大总体幸福或效用的行为。边沁提出了对快乐和痛苦的量化衡量。
U(act) = Σ pleasure − Σ pain
The formula means: calculate the total pleasure produced by an action, subtract the total pain, and choose the action with the highest positive result. Mill later distinguished higher and lower pleasures, arguing that intellectual pleasures count for more than bodily ones.
这个公式的意思是：计算一个行为产生的总快乐，减去总痛苦，然后选择正数值最高的行为。密尔后来区分了高级快乐和低级快乐，认为理智的快乐比肉体的快乐更重要。
In exam answers, apply the calculus to a moral dilemma, then evaluate it by considering problems such as measuring utility, predicting consequences, and ignoring individual rights.
在考试答案中，将这种计算应用于道德困境，然后通过考虑衡量效用、预测后果和忽视个人权利等问题来评价它。
7. Kant’s Categorical Imperative Tests | 康德的绝对命令检验
Kant’s deontological ethics rejects consequences as the basis of morality. Instead, the moral worth of an action depends on whether its maxim can pass the categorical imperative tests.
康德的义务论伦理学拒绝将后果作为道德的基础。相反，一个行为的道德价值取决于其准则能否通过绝对命令的检验。
Formula 1 (Universal Law): Act only on that maxim through which you can at the same time will that it should become a universal law.
Formula 2 (Humanity): Act so that you treat humanity, whether in your own person or in that of another, always as an end and never merely as a means.
For example, making a false promise to repay a loan fails the first test: if everyone made false promises, the institution of promising would collapse. Using a friend only to gain money fails the second test because the friend is treated merely as a means.
例如，虚假承诺还款无法通过第一个检验：如果每个人都做出虚假承诺，承诺制度就会崩溃。仅仅为了得到钱而利用朋友无法通过第二个检验，因为朋友被仅仅当作手段。
When answering Kant questions, always state the test, apply it to the maxim, and explain why the maxim passes or fails.
在回答康德问题时，始终要陈述检验标准，将其应用于准则，并解释该准则通过或失败的原因。
8. Aristotelian Virtue and the Golden Mean | 亚里士多德美德与中道
Aristotle’s virtue ethics focuses on character rather than rules or consequences. A virtue is a stable disposition to act well, and it lies in a mean between two extremes of excess and deficiency.
亚里士多德的美德伦理学关注品格，而不是规则或后果。美德是一种稳定地做出良好行为的品质，它处于过度和不足两个极端之间的中道。
Virtue = the mean between excess and deficiency, relative to the person and situation.
For example, courage is the mean between cowardice (deficiency) and rashness (excess). Generosity is the mean between stinginess and wastefulness. The mean is not a mathematical average; it requires practical wisdom, phronesis, to find the right response.
例如，勇敢是怯懦（不足）和鲁莽（过度）之间的中道。慷慨是吝啬和挥霍之间的中道。中道不是数学上的平均值；它需要实践智慧，即 phronesis，来找到正确的回应。
In evaluation questions, contrast virtue ethics with rule-based theories: it provides no clear formula for action, but it explains moral motivation and development.
在评价性问题中，将美德伦理学与基于规则的理论进行对比：它没有提供明确的行为公式，但它解释了道德动机和发展。
9. Cosmological and Teleological Arguments | 宇宙论与目的论论证
Arguments for the existence of God often appear in IGCSE questions. The cosmological argument reasons from the existence or beginning of the universe to a first cause. The teleological argument reasons from apparent design to a designer.
关于上帝存在的论证经常出现在 IGCSE 问题中。宇宙论论证从宇宙的存在或开端推出第一因。目的论论证从明显的设计推出设计者。
Cosmological: Everything that begins to exist has a cause; the universe began to exist; ∴ the universe has a cause.
Teleological: The universe shows order, purpose, and complexity; order and purpose are best explained by a designer; ∴ the universe has a designer.
Hume’s criticisms include: we have no experience of universe-making, the analogy between the universe and human artefacts is weak, and the argument does not establish the God of classical theism. The cosmological argument also faces the question “Who caused God?”.
休谟的批评包括：我们没有创造宇宙的经验，宇宙与人工制品之间的类比很弱，而且该论证不能确立古典有神论的上帝。宇宙论论证还面临 “谁导致了上帝” 的问题。
For exam success, state the premises clearly, then evaluate whether the conclusion follows and whether the God identified matches the desired divine attributes.
为了在考试中取得好成绩，要清晰地陈述前提，然后评价结论是否推出，以及所确定的上帝是否符合所期望的神圣属性。
10. Problem of Evil Logical Formula | 恶的问题逻辑公式
The logical problem of evil argues that the existence of an all-powerful, all-knowing, and all-good God is incompatible with the existence of evil. The argument can be expressed as a valid deductive form.
恶的逻辑问题认为，一个全能、全知、全善的上帝的存在与恶的存在是不相容的。该论证可以表达为有效的演绎形式。
If God (omnipotent, omniscient, omnibenevolent) exists, then evil would not exist.
Evil exists. ∴ God (with those properties) does not exist.
Responses include the free will defence: God allows evil because free will is a greater good, and it is logically impossible to create free beings who always choose good. Another response is the soul-making theodicy: suffering develops moral virtues such as courage and compassion.
回应包括自由意志辩护：上帝允许恶，因为自由意志是更大的善，而创造总是选择善的自由存在者在逻辑上是不可能的。另一种回应是塑造灵魂的神义论：苦难培养了勇气和同情等道德美德。
In essays, distinguish between moral evil (caused by humans) and natural evil (caused by nature), and evaluate whether any theodicy can fully explain both.
在论文中，要区分道德恶（由人类造成）和自然恶（由自然造成），并评价是否有任何神义论能充分解释两者。
11. Free Will and Determinism | 自由意志与决定论
The free will debate examines whether human actions are causally determined and whether moral responsibility requires free will. Hard determinism accepts the incompatibility and denies free will.
自由意志之争考察人类行为是否被因果决定，以及道德责任是否需要自由意志。硬决定论接受不相容性并否认自由意志。
Hard determinism: If determinism is true, then no action is free. Determinism is true. ∴ No action is free.
Compatibilism, by contrast, defines freedom as acting according to one’s own desires without external coercion, even if those desires are themselves determined. Libertarianism rejects determinism and claims that agents have genuine indeterministic choice.
相反，相容论将自由定义为按照自己的欲望行动而不受外部强制，即使这些欲望本身是被决定的。自由意志论拒绝决定论，并声称行动者拥有真正非决定性的选择。
Exam questions often ask whether moral responsibility is possible if determinism is true. Use the formula to explain the incompatibilist position, then contrast with compatibilist counterarguments.
考试问题经常问，如果决定论为真，道德责任是否可能。使用该公式解释不相容论立场，然后与相容论的反驳进行对比。
12. Falsification and Scientific Method | 证伪与科学方法
In the philosophy of science, Popper argued that the mark of a scientific theory is falsifiability, not verifiability. A theory is scientific if there is a possible observation that would show it to be false.
在科学哲学中，波普尔认为科学理论的标志是可证伪性，而不是可证实性。如果存在一种可能的观察能够证明一个理论为假，那么该理论就是科学的。
Scientific theory ↔ Falsifiable: there exists a possible observation O such that if O occurs, the theory is false.
For example, Einstein’s prediction that light bends near the sun was falsifiable because a different observation would have refuted it. Freud’s psychoanalysis, Popper claimed, could explain any behaviour and was therefore unfalsifiable and unscientific.
例如，爱因斯坦关于光线在太阳附近弯曲的预测是可证伪的，因为不同的观察结果会反驳它。波普尔声称，弗洛伊德的精神分析可以解释任何行为，因此不可证伪，也不科学。
Candidates should be able to explain why falsifiability matters, and also to evaluate criticisms: some sciences use induction, and some unfalsifiable theories may still be meaningful.
考生应能解释为什么可证伪性很重要，并能评价批评意见：有些科学使用归纳法，有些不可证伪的理论可能仍然有意义。
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