📚 Philosophy Formula & Theorem Quick Reference | 哲学公式定理速查手册
This quick reference handbook collects the most essential argument forms, ethical principles, and philosophical maxims that appear in the Cambridge IGCSE Philosophy syllabus. Each entry is presented as a formula or theorem to help you recall its structure quickly during revision. Mastering these logical patterns and conceptual tools will strengthen your ability to analyse arguments, evaluate moral theories, and construct well‑reasoned essays.
本速查手册汇集了剑桥 IGCSE 哲学课程中最关键的论证形式、伦理原则和哲学格言。每个条目都以公式或定理的方式呈现，帮助你在复习时快速回想其结构。掌握这些逻辑模式与概念工具，将提升你分析论证、评估道德理论以及构建严谨论文的能力。
1. Modus Ponens (Affirming the Antecedent) | 肯定前件式
Modus ponens is a valid deductive argument form: if a conditional statement is true and its antecedent is true, then the consequent must be true. It is one of the most commonly used inference patterns in philosophy.
肯定前件式是一种有效的演绎论证形式：如果条件陈述为真，并且其前件为真，那么后件必然为真。这是哲学中最常用的推理模式之一。
P → Q, P ⊢ Q
The symbol ‘→’ means ‘if … then …’, and ‘⊢’ indicates logical entailment. For example: If it is raining (P), the ground is wet (Q). It is raining (P). Therefore, the ground is wet (Q).
符号‘→’表示‘如果……那么……’，‘⊢’表示逻辑推导。例如：如果下雨(P)，地面就湿(Q)。现在下雨(P)。所以地面湿(Q)。
2. Modus Tollens (Denying the Consequent) | 否定后件式
Modus tollens is another fundamental valid form. If a conditional is true and its consequent is false, the antecedent must be false. This form is often used to challenge claims by showing that their consequences do not hold.
否定后件式是另一种基本有效形式。如果条件为真且其后件为假，则前件必为假。这个形式常被用来通过表明后果不成立来质疑某个主张。
P → Q, ¬Q ⊢ ¬P
The symbol ‘¬’ denotes negation. For instance: If the theory is correct (P), the observation should occur (Q). The observation does not occur (¬Q). Therefore, the theory is not correct (¬P).
符号‘¬’表示否定。例如：如果理论正确(P)，观测就应该出现(Q)。观测没有出现(¬Q)。因此理论不正确(¬P)。
3. Hypothetical Syllogism | 假言三段论
Hypothetical syllogism allows us to chain conditionals together. If a first condition implies a second, and the second implies a third, then the first implies the third. This pattern creates a logical chain of consequences.
假言三段论允许我们把条件陈述链接起来。如果第一个条件蕴含第二个，第二个蕴含第三个，那么第一个就蕴含第三个。这个模式建立起一条逻辑的推理链。
P → Q, Q → R ⊢ P → R
Example: If we have free will (P), then we are morally responsible (Q). If we are morally responsible (Q), then punishment is justified (R). Therefore, if we have free will (P), punishment is justified (R).
例子：如果我们有自由意志(P)，那么我们就有道德责任(Q)。如果我们有道德责任(Q)，那么惩罚就是正当的(R)。因此，如果我们有自由意志(P)，惩罚就是正当的(R)。
4. Disjunctive Syllogism | 选言三段论
A disjunctive syllogism works with an ‘either … or …’ statement. If at least one of two options is true, and one option is known to be false, the remaining option must be true. It is a simple but powerful elimination rule.
选言三段论处理‘要么……要么……’的陈述。如果两个选项中至少有一个为真，并且已知其中一个为假，那么剩下的那个必定为真。这是一条简单却有力的排除规则。
P ∨ Q, ¬P ⊢ Q
‘∨’ reads as ‘or’. In a philosophical debate: Either the mind is identical to the brain (P) or the mind is non‑physical (Q). The mind is not identical to the brain (¬P). Therefore, the mind is non‑physical (Q).
‘∨’读作‘或’。在哲学辩论中：要么心灵等同于大脑(P)，要么心灵是非物质的(Q)。心灵不等同于大脑(¬P)。因此心灵是非物质的(Q)。
5. Reductio ad Absurdum (Proof by Contradiction) | 归谬法
Reductio ad absurdum disproves a claim by showing that it leads to a contradiction. We assume the claim for the sake of argument, demonstrate that a contradiction follows, and then conclude that the original claim is false. The contradiction typically takes the form Q ∧ ¬Q.
归谬法通过表明某个主张会导致矛盾来反驳它。为了论证，我们暂且假定该主张成立，证明会推导出一个矛盾，然后得出原主张为假的结论。矛盾通常表现为 Q ∧ ¬Q 的形式。
If P → (Q ∧ ¬Q), then ⊢ ¬P
In practice: Suppose the statement ‘all truths are relative’ (P) is true. If so, the statement ‘there are no absolute truths’ would be an absolute truth, which generates a contradiction. Hence, ¬P.
实际应用中：假定‘所有真理都是相对的’(P)为真。那么，‘不存在绝对真理’这一陈述本身就成了一个绝对真理，这就产生了矛盾。因此，¬P。
6. Principle of Utility (Hedonic Calculus) | 功利原则（快乐计算）
Jeremy Bentham’s principle of utility states that the morally right action is the one that produces the greatest happiness for the greatest number. The hedonic calculus allows us to weigh pleasures and pains according to several factors.
杰里米·边沁的功利原则主张，道德上正确的行为就是能为最大多数人带来最大幸福的行为。快乐计算让我们可以根据几个因素来权衡快乐与痛苦。
Net Utility = ∑ (Pleasure − Pain) over all affected individuals
Bentham identified seven dimensions to measure any pleasure or pain:
边沁确定了衡量任何快乐或痛苦的七个维度：
Intensity – how strong the feeling is.
强度——感受有多强。
Duration – how long it lasts.
持续时间——它持续多久。
Certainty – how likely it is to occur.
确定性——它发生的可能性多大。
Propinquity – how soon it will occur.
临近性——它多久后会到来。
Fecundity – the chance it will be followed by further pleasures.
丰度——它引发更多快乐的机会。
Purity – the chance it will not be followed by pain.
纯度——它不引发痛苦的机会。
Extent – the number of people affected.
范围——受影响的人数。
7. The Categorical Imperative (Formula of Universal Law) | 绝对命令（普遍法则公式）
Immanuel Kant’s Categorical Imperative provides a test for moral duties. The Formula of Universal Law requires us to act only on maxims that we could consistently will to become universal laws for all rational beings.
康德的绝对命令为道德义务提供了一种检验。普遍法则公式要求我们只按照那些我们能够一贯地愿意其成为所有理性存在者的普遍法则的准则去行动。
Act only according to that maxim whereby you can at the same time will that it should become a universal law.
For a maxim to pass the test, it must be logically conceivable as a universal law and not lead to a contradiction in will. Lying, for example, fails because if everyone lied, the practice of truth‑telling would collapse, making lying itself impossible.
一个准则要能通过检验，必须能被逻辑地设想为一条普遍法则，并且不会导致意愿上的矛盾。例如，说谎通不过检验，因为如果人人都说谎，诚实的实践就会崩溃，说谎本身也就变得不可能。
8. Descartes’ Cogito (‘I think, therefore I am’) | 笛卡尔我思故我在
René Descartes’ Cogito is the foundational certainty that emerges from his method of radical doubt. Even if he is deceived by an evil demon, the very fact that he is thinking proves his existence as a thinking thing.
笛卡尔的‘我思故我在’是从他彻底怀疑的方法中得出的基础确定性。即使他被邪恶的精灵欺骗，他正在思考这一事实本身就证明了他作为一个思考之物的存在。
I doubt → I think → I exist (Dubito, ergo cogito, ergo sum)
The logical structure can be reconstructed: If I am doubting, then I am thinking. If I am thinking, then I must exist as a thinking substance. The Cogito is immediate and self‑verifying; it cannot be doubted while being thought.
逻辑结构可以重构为：如果我在怀疑，那么我就在思考。如果我在思考，那么我必然作为思考的实体而存在。我思故我在是直接且自明的；在思考它的时候，它不可能被怀疑。
9. Occam’s Razor (The Principle of Parsimony) | 奥卡姆剃刀（简约原则）
Occam’s Razor is a methodological principle that advises against multiplying entities beyond necessity. When confronted with competing explanations, we should prefer the one that makes the fewest assumptions, all other things being equal.
奥卡姆剃刀是一项方法论原则，告诫我们如无必要，勿增实体。当面对相互竞争的解释时，在其他条件相同的情况下，我们应当选择那个作出最少假设的解释。
Entia non sunt multiplicanda praeter necessitatem.
In philosophical debate, the principle helps to eliminate redundant concepts. For instance, if natural processes can explain the origin of the universe, adding a supernatural creator involves an extra assumption that may be shaved away by the razor.
在哲学辩论中，这项原则有助于剔除冗余概念。例如，如果自然过程能够解释宇宙的起源，那么加上一个超自然的造物主就多了一个额外假设，可能会被剃刀剃掉。
10. The Golden Mean (Aristotle’s Doctrine of the Mean) | 中庸之道（亚里士多德中道学说）
Aristotle’s ethical theory holds that moral virtue is a mean between two vices: one of excess and one of deficiency. The mean is relative to the individual and the situation, determined by practical wisdom (phronesis).
亚里士多德的伦理学认为，道德德性是两种恶习之间的中道：一种是过度，另一种是不及。中道是相对于个体和情境而言的，由实践智慧(phronesis)决定。
Virtue = the intermediate state between excess and deficiency, guided by right reason
Courage is the mean between cowardice (deficiency) and recklessness (excess). Generosity lies between stinginess and wastefulness. The doctrine is not a mechanical formula but a flexible guideline for cultivating character.
勇敢是怯懦（不及）与鲁莽（过度）之间的中道。慷慨位于吝啬与挥霍之间。该学说并非机械公式，而是培养品格的灵活指导。
11. Hume’s Is‑Ought Gap (Hume’s Law) | 休谟的‘是‑应该’鸿沟（休谟法则）
David Hume observed that moral philosophers often move imperceptibly from descriptive ‘is’ statements to prescriptive ‘ought’ statements without justification. The is‑ought gap asserts that normative conclusions cannot be logically derived from purely factual premises.
大卫·休谟注意到，道德哲学家常常不知不觉地从描述性的‘是’陈述跳到规定性的‘应该’陈述，而不提供理由。‘是‑应该’鸿沟断言，规范性的结论不能从纯粹的事实性前提中逻辑地推导出来。
From factual premises alone, no evaluative conclusion follows. ¬(Descriptive premises ⊢ Normative conclusion)
For example, from the fact that people tend to seek pleasure, one cannot logically conclude that people ought to seek pleasure without adding an evaluative premise such as ‘pleasure is good’. Hume’s insight grounds the distinction between facts and values.
例如，从‘人们倾向于追求快乐’这一事实，不能逻辑地推出‘人们应当追求快乐’，除非加上‘快乐是好的’这样的评价性前提。休谟的洞见奠定了事实与价值区分的根基。
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