📚 Year 7 CAIE Philosophy: Formula & Theorem Quick Reference Handbook | Year 7 CAIE哲学：公式定理速查手册
Philosophy may not contain mathematical formulas in the traditional sense, but it is built on powerful ‘formulae of thought’ — structured arguments, logical rules, ethical principles and famous paradoxes that act like theorems for the mind. This handbook distils the essential patterns and key ideas that every Year 7 CAIE Philosophy student should know, from Socratic questioning to the Trolley Problem, all arranged in a quick-reference format that makes revision intuitive and effective.
哲学或许没有传统意义上的数学公式，但它建立在一系列强大的“思维公式”之上——结构化的论证、逻辑规则、伦理原则和著名的悖论，如同心灵的定理。本手册浓缩了每一位 Year 7 CAIE 哲学学生应当掌握的基本模式与核心观念，从苏格拉底式追问到电车难题，全都以速查形式编排，让复习变得直观而高效。
1. What is Philosophy? | 什么是哲学？
Philosophy comes from the Greek words ‘philo’ (love) and ‘sophia’ (wisdom). It is the systematic attempt to answer fundamental questions about existence, knowledge, morality, reason and reality through critical analysis and argument rather than empirical observation alone. Philosophers do not simply collect facts; they examine the assumptions behind our beliefs, clarify concepts and construct reasoned arguments to justify conclusions. The subject is traditionally divided into branches: metaphysics (what is real?), epistemology (what can we know?), ethics (how should we live?) and logic (how should we reason?).
哲学一词来源于希腊语“philo”（爱）与“sophia”（智慧）。它是通过批判性分析与论证，而非单纯的经验观察，系统性地尝试回答关于存在、知识、道德、理性与实在的根本问题。哲学家不只收集事实；他们审视我们信念背后的假设，澄清概念，并构建有理有据的论证来证明结论。这门学科传统上分为几个分支：形而上学（何为实在？）、认识论（我们能知道什么？）、伦理学（我们应该如何生活？）和逻辑学（我们应该如何推理？）。
2. The Socratic Method | 苏格拉底方法
Socrates taught not by lecturing but by asking a series of probing questions designed to reveal contradictions in his interlocutor’s beliefs and to lead them toward deeper understanding. This technique, known as the Socratic method or elenchus, typically follows a pattern: an initial definition or claim is proposed; Socrates asks for clarification; through further questioning, an inconsistency or inadequate understanding is exposed; the claim is then revised or abandoned. The method shows that wisdom begins with recognising one’s own ignorance.
苏格拉底并非通过讲授来教学，而是提出一连串探究式的问题，旨在揭示对话者信念中的矛盾，并引导他们走向更深刻的理解。这种技术被称为苏格拉底方法或反诘法，通常遵循这样的模式：先有人提出一个定义或主张；苏格拉底要求澄清；通过进一步追问，暴露出不一致或理解不充分之处；然后原有主张被修正或放弃。该方法表明，智慧始于承认自己的无知。
3. Logic: Argument Structure | 逻辑：论证结构
An argument in philosophy consists of premises (supporting statements) and a conclusion (the statement they are meant to support). For an argument to be logically valid, the conclusion must follow necessarily from the premises; if the premises are true in a valid argument, the conclusion must be true. An argument is sound when it is both valid and its premises are actually true. The standard formulation can be written as:
哲学中的论证由前提（支持性的陈述）和结论（它们所要支持的陈述）组成。一个论证在逻辑上有效，意味着结论必然从前提中得出；在有效论证中，如果前提为真，结论必然为真。当一个论证既有效、前提又确实为真时，它就是可靠的。标准表述可以写成：
P1, P2, … Pn ⊢ C
Philosophy trains students to distinguish validity from soundness and to evaluate whether premises withstand scrutiny. A single counterexample can refute a universal premise.
哲学训练学生区分有效性与可靠性，并评估前提是否经得起推敲。一个反例就可以驳倒一个全称前提。
4. Logical Operators & Truth Tables | 逻辑运算符与真值表
Propositional logic uses symbols to represent entire statements and operators to connect them. The main operators are: conjunction ( ∧ , ‘and’), disjunction ( ∨ , ‘or’), negation ( ¬ , ‘not’), conditional ( → , ‘if… then…’), and biconditional ( ↔ , ‘if and only if’). Truth tables define the truth value of a compound statement based on the truth values of its components. For example:
命题逻辑用符号代表整个陈述，用运算符连接它们。主要的运算符有：合取（ ∧ ，‘且’）、析取（ ∨ ，‘或’）、否定（ ¬ ，‘非’）、条件（ → ，‘如果……那么……’）和双条件（ ↔ ，‘当且仅当’）。真值表根据各组成部分的真值来定义复合陈述的真值。例如：
| P | Q | P ∧ Q | P ∨ Q | P → Q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | F | T | T |
| F | F | F | F | T |
Notice that P → Q is only false when P is true and Q is false — a crucial insight for testing conditional claims. These basic tables form the backbone of rigorous philosophical analysis.
注意 P → Q 只有在 P 真而 Q 假时才为假——这是检验条件性主张的关键洞见。这些基本真值表构成了严谨哲学分析的主干。
5. Modus Ponens & Modus Tollens | 肯定前件与否定后件
Two fundamental valid argument forms appear again and again in philosophical reasoning. Modus ponens (affirming the antecedent) has the structure: If P then Q; P; therefore, Q. Modus tollens (denying the consequent) is: If P then Q; not Q; therefore, not P. Expressed symbolically:
两种基本的有效论证形式在哲学推理中反复出现。肯定前件（modus ponens）的结构为：如果 P 则 Q；P 成立；故 Q 成立。否定后件（modus tollens）则为：如果 P 则 Q；非 Q；故非 P。用符号表达：
Modus Ponens: P → Q, P ⊢ Q
Modus Tollens: P → Q, ¬Q ⊢ ¬P
Avoid the fallacies of affirming the consequent (P → Q, Q, therefore P) and denying the antecedent (P → Q, ¬P, therefore ¬Q). Both are invalid because there could be alternative reasons for Q to be true or for P to be false.
避免肯定后件谬误（P → Q，Q，故 P）和否定前件谬误（P → Q，¬P，故 ¬Q）。这两者都是无效的，因为可能存在其他使 Q 为真或使 P 为假的原因。
6. Fallacies: Formal and Informal | 谬误：形式与非形式
A fallacy is a defect in reasoning that undermines an argument’s logical strength. Formal fallacies, like those mentioned above, arise from errors in logical structure. Informal fallacies occur because of problems with content, language or relevance. Common examples include: ad hominem (attacking the person instead of the argument), straw man (misrepresenting an opponent’s view to make it easier to attack), begging the question (assuming what you are trying to prove), and false dilemma (presenting only two options when more exist). Spotting fallacies is a core critical thinking skill.
谬误是推理中的缺陷，会削弱论证的逻辑力量。形式谬误，如上所述，源于逻辑结构的错误。非形式谬误则因内容、语言或相关性问题而产生。常见的例子包括：人身攻击（攻击人而非论证）、稻草人谬误（歪曲对方观点以便更容易攻击）、乞题（假设你试图证明的结论）和假两难（只呈现两个选项，而实际上存在更多）。识别谬误是批判性思维的核心技能。
7. Plato’s Theory of Forms | 柏拉图的理型论
Plato proposed that beyond the physical world we perceive with our senses lies a realm of perfect, eternal and unchanging ‘Forms’ or ‘Ideas’. For any property we notice in the world — beauty, justice, largeness — there is a corresponding Form that is the pure essence of that property. Particular objects in the sensory world ‘participate’ in these Forms but are imperfect copies. The Allegory of the Cave illustrates this: prisoners mistaking shadows for reality are like us relying on sensory experience alone, while the philosopher who escapes the cave glimpses the Form of the Good and true understanding.
柏拉图提出，在我们感官所感知的物理世界之外，存在一个由完美、永恒且不变的“理型”或“理念”构成的领域。对于我们在世界中注意到的任何属性——美、正义、大——都有一个相应的理型，它是该属性的纯粹本质。感官世界中的具体事物“分有”这些理型，但只是不完美的摹本。洞穴比喻说明了这一点：将影子误以为现实的囚徒就像仅依赖感官经验的我们，而逃出洞穴的哲学家则瞥见了善的理型与真实的理解。
8. Aristotle’s Virtue Ethics | 亚里士多德的美德伦理学
Aristotle shifted the ethical question from ‘What should I do?’ to ‘What kind of person should I be?’ His answer centres on eudaimonia (often translated as flourishing or happiness) and the cultivation of virtues. A virtue is a character trait that lies at the mean between two extremes (vices): courage is the mean between cowardice and recklessness; generosity between stinginess and wastefulness. The virtuous person acts for the right reasons, at the right time, and feels pleasure in acting well. This practical wisdom (phronesis) is developed through habit and experience, not just by learning rules.
亚里士多德将伦理问题从“我应该做什么？”转向“我应该成为什么样的人？”他的答案围绕eudaimonia（通常译为繁荣或幸福）和美德的培养展开。美德是一种性格特质，处于两个极端（恶）之间的中道：勇敢是怯懦与鲁莽之间的中道；慷慨是吝啬与挥霍之间的中道。有德之人出于正确的理由，在恰当的时机行动，并以良好行为为乐。这种实践智慧（phronesis）通过习惯和经验培养，而不仅仅是学习规则。
9. Utilitarianism: The Greatest Happiness Principle | 功利主义：最大幸福原则
Jeremy Bentham and John Stuart Mill developed utilitarianism, a consequentialist theory stating that the right action is the one that produces the greatest happiness for the greatest number. This ‘Greatest Happiness Principle’ can be thought of as a moral calculus: an action’s moral worth is measured by its utility — the net balance of pleasure over pain. Bentham proposed a hedonic calculus considering intensity, duration, certainty and extent of pleasures. Mill later distinguished higher (intellectual) pleasures from lower (bodily) ones, arguing that ‘it is better to be a human being dissatisfied than a pig satisfied.’
杰里米·边沁和约翰·斯图尔特·密尔发展了功利主义，这是一种后果论理论，主张正确的行为是能为最大多数人带来最大幸福的行为。这一“最大幸福原则”可以被视为一种道德演算：一个行为的道德价值由其效用——即快乐减去痛苦后的净余额——来衡量。边沁提出了苦乐计算法，考量快乐的强度、持续时间、确定性和范围。密尔后来区分了高级（理智的）快乐和低级（肉体的）快乐，主张“宁做不满足的人，不做满足的猪。”
10. Kant’s Categorical Imperative | 康德的定言命令
Immanuel Kant rejected consequences as the basis of morality and instead rooted ethics in reason and duty. His core principle, the Categorical Imperative, comes in several formulations. The first formulation, the Universal Law Formula, states: “Act only according to that maxim whereby you can at the same time will that it should become a universal law.” This demands consistency: if the rule you act on cannot rationally be willed for everyone to follow, the action is impermissible. A second formulation, the Humanity Formula, insists we must always treat humanity, whether in ourselves or others, as an end in itself and never merely as a means.
伊曼努尔·康德拒绝将后果作为道德的基础，而将伦理学植根于理性与义务。他的核心原则——定言命令——有多种表述形式。第一表述，即普遍法则公式，指出：“只依据那些你同时能够意愿其成为普遍法则的准则去行动。”这要求一致性：如果你据以行动的规则不能被理性地意愿为每个人都遵守，则该行为是不被允许的。第二表述，即人性公式，强调我们必须始终将人性——无论是我们自己身上的还是他人身上的——作为目的本身，而绝不仅仅当作手段。
11. Knowledge and Justified True Belief | 知识与被证成的真信念
A classic definition in epistemology, often traced back to Plato, holds that knowledge is justified true belief. According to this tripartite account, a person S knows a proposition P if and only if: (i) P is true; (ii) S believes that P; and (iii) S is justified in believing that P. For example, you know that London is the capital of the United Kingdom because it is true, you believe it, and you have reliable evidence (maps, official sources). Edmund Gettier later challenged this definition with cases where justified true belief seems not to count as knowledge, prompting the search for additional conditions.
认识论中的一个经典定义，常追溯至柏拉图，认为知识是被证成的真信念。根据这一三要素说明，一个人S知道命题P，当且仅当：（i）P为真；（ii）S相信P；且（iii）S有正当理由相信P。例如，你知道伦敦是英国的首都，因为这是真的，你相信它，并且你有可靠的证据（地图、官方资料）。埃德蒙·盖梯尔后来用一些例子挑战了这一定义，在这些例子中，被证成的真信念似乎不算作知识，这促使人们寻找额外的条件。
12. The Trolley Problem | 电车难题
Philippa Foot introduced the Trolley Problem to explore the moral difference between doing and allowing harm. In the standard scenario, a runaway trolley is heading toward five workers who will be killed if it continues. You can pull a lever to divert the trolley onto a side track where one worker stands. Most people say it is permissible to pull the lever, sacrificing one to save five. In another version (the Footbridge case), you must push a large person off a bridge to stop the trolley — and here, most judge the push to be wrong, even though the numbers are the same. The contrast reveals that intentions, directness of harm and using someone ‘merely as a means’ matter greatly in our moral thinking, linking directly to Kant’s ethics and utilitarian calculations.
菲利帕·福特提出电车难题，旨在探讨作为伤害与放任伤害之间的道德区别。在标准场景中，一辆失控的电车冲向五名工人，如果继续前进他们将被撞死。你可以扳动道岔，将电车引向侧轨，而侧轨上有一名工人。多数人认为扳动道岔、牺牲一人救五人是被允许的。在另一个版本（天桥案例）中，你必须将一个大个子推下天桥来阻止电车——而在这里，多数人判断推人是错误的，尽管人数相同。这一反差揭示出意图、伤害的直接性以及将某人“仅仅当作手段”在我们的道德思维中极其重要，这与康德的伦理学和功利主义计算直接相关。
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