📚 Cambridge Year 9 Philosophy: Quick Reference Handbook for Formulas and Theorems | 剑桥9年级哲学：公式定理速查手册
In the spirit of a science or mathematics revision guide, this handbook condenses key philosophical ideas into memorable 'formulas' and 'theorems'. Mastering these conceptual equations will help you navigate ethics, logic, metaphysics, and epistemology. Each entry offers a clear statement of the idea, followed by a plain‑English explanation and a practical example. Keep this guide close as you prepare for your Philosophy assessments at Cambridge Year 9 level.
秉承科学或数学复习指南的精神，本手册将关键的哲学思想浓缩成易记的“公式”和“定理”。掌握这些概念性方程式将帮助你应对伦理学、逻辑学、形而上学和认识论。每个条目都清楚地陈述观点，然后用通俗的语言解释并给出实例。在你准备剑桥9年级哲学评估时，请将本指南放在手边。
1. The Law of Identity (A = A) | 同一律 (A = A)
A thing is what it is. Formally, for any entity A, A is equal to A. This is the most fundamental law of thought. If a statement is true, then it is true; a cat is a cat, not a dog.
事物就是它自身。形式上，对于任何实体A，A等于A。这是最基本的思维规律。如果一个陈述为真，那么它为真；猫就是猫，而不是狗。
Example: 'A book is a book' cannot be false. Without this law, no rational argument could even begin, because terms would have no fixed meaning.
示例：“书是书”不可能为假。没有这个规律，任何理性论证都无法开始，因为词语将没有固定的含义。
Theorem 1: ∀x (x = x)
定理1：∀x (x = x)
2. The Law of Non‑Contradiction | 矛盾律
Nothing can both be and not be at the same time and in the same respect. Formally, ¬(P ∧ ¬P). You cannot truthfully say 'The door is open' and 'The door is not open' simultaneously if you are referring to exactly the same door, at the same instant, from the same perspective.
任何事物不可能在同一时间、同一方面既是又不是。形式上，¬(P ∧ ¬P)。如果你指的是完全相同的门、同一瞬间、同一角度，你就不可能真实地同时说“门开着”和“门没开着”。
Example: 'The river is frozen' and 'The river is not frozen' cannot both be true when we are talking about the same stretch of water on a particular afternoon.
示例：当我们谈论某一天下午同一段水域时，“河流结冰了”和“河流没有结冰”不可能同时为真。
3. The Law of the Excluded Middle | 排中律
Every proposition is either true or false; there is no middle ground. Formally, P ∨ ¬P. This law drives much of Western logic, but some philosophies (e.g., certain interpretations of quantum mechanics) have challenged it. In your essays, however, it works as a bedrock for clear thinking.
每个命题要么为真，要么为假，没有中间状态。形式上，P ∨ ¬P。这条定律驱动着大部分西方逻辑，但某些哲学（如量子力学的某些解释）对它提出了挑战。不过，在你的论文中，它可以作为清晰思维的基石。
Example: 'It is raining here now' must be either true or false; there is no half‑rain reality that logic can admit.
示例：“现在这里正在下雨”必须要么真要么假；逻辑上无法承认一个半真半雨的现实。
4. Descartes' Cogito Theorem | 笛卡尔的我思定理
'I think, therefore I am' (Cogito, ergo sum). This is the foundational certainty that Descartes reached after methodical doubt: even if an evil demon deceives me about everything, the fact that I am being deceived proves that I must exist as a thinking thing.
“我思故我在”。这是笛卡尔在系统怀疑之后得出的基本确定性：即使有一个邪恶的恶魔在一切事情上欺骗我，我被欺骗这一事实也证明了我作为一个思维的事物必然存在。
Formula: If D = 'I doubt', then D → (I exist).
公式：如果 D = '我怀疑'，则 D → (我存在)。
Example: You can doubt the existence of your hands, your classroom, or even the physical world, but you cannot doubt that you are doubting, and therefore you know you exist as a mind.
示例：你可以怀疑你的双手、你的教室甚至物质世界的存在，但你无法怀疑你正在怀疑，因此你知道自己作为一个心灵而存在。
5. Bentham's Hedonic Calculus | 边沁的快乐计算
Jeremy Bentham proposed a utilitarian formula for deciding the moral worth of an action: choose the act that produces the greatest happiness for the greatest number. The Hedonic Calculus scores actions by seven criteria: Intensity, Duration, Certainty, Propinquity, Fecundity, Purity, and Extent.
杰里米·边沁提出了一个功利主义的公式来决定行为的道德价值：选择能够为最大多数人带来最大幸福的行为。快乐计算通过七个标准给行为打分：强度、持续时间、确定性、临近性、生产性、纯度和范围。
Utility ≈ Σ (Pleasure units × Number of people affected)
效用 ≈ Σ (快乐单位 × 受影响的人数)
Example: If you have to choose between buying a single expensive chocolate bar for yourself (high intensity, short duration) and buying ten smaller sweets to share with friends (lower individual pleasure, but wider extent), the calculus might favour sharing.
示例：如果你必须在给自己买一块昂贵巧克力（高强度、短持续时间）和买十块小糖果与朋友分享（个人快乐较低，但范围更大）之间做出选择，计算可能会支持分享。
Remember: For Bentham, all pleasures are equal in quality; pushpin is as good as poetry. This is one of the most debated 'formulas' in ethics.
记住：对边沁来说，所有快乐在质量上都是平等的；图钉游戏和诗一样好。这是伦理学中最具争议的“公式”之一。
6. Kant's Categorical Imperative Formula | 康德的绝对命令公式
Immanuel Kant offered a moral formula that does not depend on consequences. The Categorical Imperative in its universal‑law formulation says: 'Act only according to that maxim whereby you can at the same time will that it should become a universal law.' In other words, before you act, ask: 'What if everyone did this?'
伊曼努尔·康德提供了一个不依赖后果的道德公式。绝对命令的普遍法则表述是：“只按照你同时能够希望它成为一条普遍法则的准则去行动。”换句话说，在行动之前，问一问：“如果每个人都这么做会怎样？”
CI Test: Maxim M → Can M be universalized without contradiction? If NO, M is impermissible.
绝对命令测试：准则 M → M 能否毫无矛盾地普遍化？如果不能，M 就是不被允许的。
Example: Lying. If lying were a universal law, nobody would trust anyone's words, so the very practice of lying would become self‑defeating. Lying therefore fails the test and is morally wrong, regardless of how much good it might produce in a single case.
示例：说谎。如果说谎是一条普遍法则，就没有人会相信任何人的话，于是说谎行为本身就会自毁。因此说谎通不过测试，在道德上是错误的，不论它在个别情况下能带来多少好处。
7. Aristotle's Virtue as the Mean Formula | 亚里士多德的中庸之道公式
For Aristotle, virtue is a settled disposition to choose the mean between two extremes (excess and deficiency). This 'mean' is relative to the individual and the situation. The formula is not a simple arithmetic average but a balance that practical wisdom (phronesis) uncovers.
对亚里士多德来说，美德是一种选择两极（过度与不足）之间中庸的稳定品质。这个“中庸”是相对于个人和情境的。此公式并非简单的算术平均值，而是一种由实践智慧（phronesis）发现的平衡。
Virtue = (Excess + Deficiency)/2 (adjusted by phronesis)
美德 = (过度 + 不足)/2 （由实践智慧调整）
Example: Courage is the mean between cowardice (deficiency) and rashness (excess). In battle, exactly where the mean lies depends on factors like your skill and the risk; a trained soldier can risk more without being rash.
示例：勇气是懦弱（不足）与鲁莽（过度）之间的中庸。在战斗中，中庸的具体位置取决于你的技能和风险等因素；训练有素的士兵可以承受更大风险而不至于鲁莽。
8. Hume's Fork: The Analytic‑Synthetic Divide | 休谟的叉子：分析命题与综合命题的区分
David Hume declared that all meaningful statements fall into two classes: relations of ideas (analytic truths, known by pure logic or definition) and matters of fact (synthetic truths, known through experience). Anything else is 'sophistry and illusion'. This fork is a key tool for testing philosophical claims.
大卫·休谟宣称，所有有意义的陈述分为两类：观念的关系（分析真值，通过纯粹逻辑或定义可知）和事实问题（综合真值，通过经验可知）。其他一切皆是“诡辩和幻想”。这把叉子是检验哲学主张的关键工具。
Statement Type: Analytic (true by definition) ∨ Synthetic (true by observation) → Meaningful; else → Meaningless
陈述类型：分析命题（依定义为真） ∨ 综合命题（依观察为真） → 有意义；否则 → 无意义
Example: 'All bachelors are unmarried' is analytic; 'The cat is on the mat' is synthetic. A claim like 'God is love' would need to be examined: is it a definition or a report of empirical fact? If neither, Hume would consign it to the flames.
示例：“所有单身汉都未婚”是分析命题；“猫在垫子上”是综合命题。像“上帝是爱”这样的主张就需要审视：它是一个定义，还是一个经验事实的报告？如果都不是，休谟就会将其付之一炬。
9. Ockham's Razor: The Principle of Parsimony | 奥卡姆剃刀：简约原理
'Entities should not be multiplied beyond necessity.' This methodological rule, attributed to William of Ockham, urges us to prefer simpler theories when all else is equal. In philosophy essays, it helps you avoid unnecessary complexity and remain focused on the most straightforward explanation.
“没必要不增加实体。”这条方法论规则出自奥卡姆的威廉，它敦促我们在其他条件相同时选择更简单的理论。在哲学论文中，它帮助你避免不必要的复杂性，并专注于最直接的解释。
If T1 explains the same data as T2 but posits fewer objects/properties, choose T1.
如果 T1 解释了与 T2 相同的数据但假设了更少的对象/属性，则选择 T1。
Example: When explaining why your pen disappeared, the hypothesis 'It fell behind the desk' is simpler than 'An invisible elf stole it and then returned it to a parallel dimension.' Ockham's razor favours the fallen‑pen theory.
示例：在解释你的笔为什么不见了时，“它掉在桌子后面了”的假说比“一个隐形的精灵偷走了它并把它送到了平行维度”更简单。奥卡姆剃刀支持掉笔理论。
10. Plato's Theory of Forms: The Reality Equation | 柏拉图的形式理论：现实方程
For Plato, what we see around us are imperfect copies of eternal, perfect Forms (or Ideas). The physical cat is a shadow of the Form of Cat. True knowledge is knowledge of the Forms, not of the shifting material world. This can be expressed as a relationship between appearance and reality.
对柏拉图来说，我们周围看到的是永恒、完美的形式（或理念）的不完美复制品。实体的猫是猫的形式的影子。真正的知识是关于形式的知识，而不是关于变化不定的物质世界的知识。这可以表达为现象与实在之间的关系。
Realness of X ∝ (Participation in the Form of X)
X 的实在性 ∝ (对 X 的形式的分布程度)
Example: A beautifully drawn circle on the whiteboard is very round, but it is still a less real circle than the Form of Circularity, which is perfectly round. The more a thing partakes of the Form, the more real it is in Plato's scheme.
示例：白板上画得漂亮的圆已经很圆了，但它仍然不如圆的形式那么真实，后者是完美的圆。在柏拉图的体系中，一个事物分有形式越多，它就越真实。
11. Berkeley's Immaterialism: Esse Est Percipi | 贝克莱的非物质论：存在即被感知
Bishop George Berkeley argued that material substance does not exist; the only things that exist are minds and their ideas. His famous formula is 'Esse est percipi' – to be is to be perceived. An object is simply the bundle of its sensible qualities, and God ensures continuity when no human perceives it.
乔治·贝克莱主教认为物质实体并不存在；唯一存在的是心灵及其观念。他著名的公式是“Esse est percipi”——存在即被感知。物体只不过是其可感性质的集合，上帝确保当无人感知时它仍然存在。
Existence of tree = Perception of tree (by finite mind or by God)
树的存在 = 对树的感知（由有限心灵或上帝感知）
Example: If you leave the room, the table ceases to exist as a material thing and remains only as an idea in the mind of God. Your daily experience does not change, but Berkeley's formula radically rewrites what you take to be reality.
示例：如果你离开房间，桌子作为物质的东西便不复存在，它只作为上帝心中的观念继续存在。你的日常经验并未改变，但贝克莱的公式彻底重写了你所以为的现实。
12. Russell's Paradox and the Limits of Sets | 罗素悖论与集合的界限
Bertrand Russell discovered a logical puzzle that challenged set theory: consider the set of all sets that are not members of themselves. If it is a member of itself, then by definition it must not be; and if it is not, then it must be. This paradox shows that not every well‑defined condition defines a set, which is a crucial caution for philosophical analysis.
伯特兰·罗素发现了一个挑战集合论的逻辑难题：考虑所有不包含自身的集合的集合。如果它是自身的成员，那么根据定义它就不能是；如果它不是，那么它又必须是。这个悖论表明，并非每个明确的条件都定义一个集合，这对哲学分析是一个重要的警示。
Let R = {x | x ∉ x}. Does R ∈ R? If yes → contradiction; if no → contradiction.
设 R = {x | x ∉ x}。R ∈ R 吗？若成立 → 矛盾；若不成立 → 矛盾。
Example: The 'barber who shaves all those who do not shave themselves' is the popular version. Does the barber shave himself? If he does, he shouldn’t; if he doesn’t, he should. In your philosophy work, this formula reminds you to check for self‑reference when defining concepts.
示例：“只给那些不自己刮脸的人刮脸的理发师”是通俗版本。理发师给自己刮脸吗？如果他刮，他就不该刮；如果他不刮，他就该刮。在你的哲学工作中，这个公式提醒你在定义概念时要检查自我指涉。
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