📚 Formula & Theorem Quick Reference Handbook for Year 10 CAIE Philosophy | Year 10 CAIE 哲学：公式定理速查手册
Philosophy may not have mathematical formulas, but it relies on rigorous structures of reasoning, definitions of key concepts, and landmark arguments that function much like theorems. This handbook distils the essential 'formulas' and 'theorems' of the CAIE Year 10 Philosophy syllabus into a clear, bilingual reference. Each entry presents a core idea, its logical structure, and its typical examination usage, helping you build a solid toolkit for explaining, analysing, and evaluating philosophical positions.
哲学虽然没有数学公式，但它的严密推理结构、核心概念定义和里程碑式的论证，就如同定理一样重要。本手册将 CAIE 十年级哲学课程中不可或缺的“公式”和“定理”浓缩为清晰的双语速查资料。每个条目都包含核心思想、逻辑结构及考试中的典型用法，助你构建一套坚固的工具包，用以解释、分析和评价各种哲学立场。
1. The Argument 'Formula': Premises → Conclusion | 论证“公式”：前提 → 结论
An argument in philosophy is a structured set of statements where one or more premises are intended to support a conclusion. The basic 'formula' is: If all premises are true, and the reasoning is valid (for deductive arguments) or strong (for inductive arguments), then the conclusion is rationally acceptable. Always identify the premises and conclusion before evaluating.
哲学中的论证是一组结构化陈述，其中一个或多个前提旨在支持一个结论。基本的“公式”是：如果所有前提为真，且推理有效（对于演绎论证）或强有力（对于归纳论证），则结论在理性上是可接受的。在评价之前，务必先识别前提和结论。
| Deductive Validity Formula | emises true → conclusion must be true (impossible for premises true and conclusion false). |
| Inductive Strength Formula | If premises true → conclusion probable. |
| Soundness = | Valid argument + all premises actually true. |
| Cogency = | Strong argument + all premises actually true. |
演绎有效性公式：如果前提为真，则结论必然为真（不可能前提真而结论假）。归纳强度公式：如果前提为真，则结论很可能为真。可靠性=有效论证+所有前提实际为真。说服力=强论证+所有前提实际为真。
2. Common Logical Fallacies: Spotting Invalid 'Formulas' | 常见逻辑谬误：识别无效的“公式”
Fallacies are flawed reasoning patterns that appear persuasive but violate the rules of good argument. Memorising their structures functions like knowing incorrect equations to avoid in mathematics. The following table summarises the must-know fallacies for Year 10.
谬误是看似有说服力但违反良好论证规则的错误推理模式。记住它们的结构就像数学中知道要避免的错误方程一样。下表总结了十年级必知的谬误。
| Ad Hominem | Attacking the person instead of the argument. Formula: "X's claim is false because X is a bad person." | 人身攻击 | 攻击人而非论点。公式：“X 的说法是假的，因为 X 是坏人。” |
| Straw Man | Misrepresenting an opponent's view to make it easier to attack. Formula: "You say A (but they said B); B is absurd; therefore A is wrong." | 稻草人 | 歪曲对方观点使其易于攻击。公式：“你说的是 A（其实他们说的是 B）；B 荒谬；所以 A 错。” |
| Begging the Question | Assuming the conclusion in the premises. Formula: "X is true because X is true." | 乞题 | 前提中已假定结论。公式：“X 为真，因为 X 为真。” |
| False Dilemma | Presenting only two options when more exist. Formula: "Either A or B; not A; therefore B" (but C, D, etc. ignored). | 假两难 | 只给出两个选项而忽略其他可能。公式：“要么 A 要么 B；非 A；故 B”（但忽略 C、D 等）。 |
3. Utilitarianism: The Greatest Happiness Calculation | 功利主义：最大幸福计算
rally right action is the one that produces the greatest balance of pleasure over pain, often summarised by the 'hedonic calculus' proposed by Jeremy Bentham. The core 'formula' affected individuals. The action with the highest net utility is the morally correct one.
功利主义是一种后果论伦理学理论，主张道德上正确的行为是能为所有受影响者产生最大快乐与痛苦之平衡的行为，常以边沁的“快乐计算法”概括。核心“公式”可表达为：正确性 = ∑ (快乐) − ∑ (痛苦)，针对所有受影响的个体。总净功利最高的行为即正确行为。
Bentham's Hedonic Calculus factors: Intensity, Duration, Certainty, Propinquity (nearness in time), Fecundity (chance of leading to more pleasure), Purity (chance of leading to pain), Extent (number of people affected).
边沁的快乐计算因子：强度、持续时间、确定性、邻近性（时间远近）、丰产性（带来更多快乐的机会）、纯度（带来痛苦的机会）、范围（受影响的人数）。
Act vs Rule Utilitarianism: Act utilitarianism applies the formula directly to each individual action; rule utilitarianism applies it to general rules (follow the rule that, if generally adopted, maximises utility).
行为功利主义 vs 规则功利主义：行为功利主义将公式直接用于每个具体行为；规则功利主义则用于一般规则（遵循若被普遍采纳会最大化功利的规则）。
4. Kantian Deontology: The Categorical Imperative Formulation | 康德道义论：定言令式的公式化
Immanuel Kant's ethics centres on duty and the categorical imperative, which functions as a moral 'formula' to test whether an action is permissible. Unlike utilitarianism, the rightness of an action does not depend on consequences but on whether the maxim (personal principle) could become a universal law.
康德的伦理学以义务和定言令式为核心，后者就像一条检验行为是否被允许的道德“公式”。与功利主义不同，行为的正确性不取决于后果，而取决于行为的准则（个人原则）能否成为普遍法则。
First Formulation (Formula of Universal Law): "Act only according to that maxim whereby you can at the same time will that it should become a universal law." Test: 1) Formulate the maxim of your action. 2) Universalise it – imagine everyone doing it. 3) Check for contradiction: (a) contradiction in conception (the action would be impossible in a world where everyone did it) or (b) contradiction in will (you cannot rationally will such a world). If contradiction, action is impermissible.
第一公式（普遍法则公式）：“只按照你同时能够愿意它成为一条普遍法则的那个准则去行动。”检验步骤：1) 表述你行为的准则。2) 将其普遍化——想象每个人都这样做。3) 检查矛盾：(a) 构想中的矛盾（在每个人都这样做的世界里该行为不可能存在）或 (b) 意愿中的矛盾（你无法理性地意愿这样的世界）。如果有矛盾，该行为就是不被允许的。
Second Formulation (Formula of Humanity): "Act in such a way that you treat humanity, whether in your own person or in the person of another, always at the same time as an end, never merely as a means." This forbids using people merely as tools for your own goals.
第二公式（人性公式）：“你要这样行动，永远不把你自身或他人人格中的人性仅仅当作手段，而要同时当作目的。”它禁止仅仅将人用作达到自己目的的工具。
5. Virtue Ethics: The Golden Mean Theorem | 美德伦理学：中道定理
Aristotle's virtue ethics does not provide a rule-based formula but a character-centred 'theorem' ition to choose the intermediate between excess and deficiency relative to the situation. The 'formula' For example, courage is the mean between cowardice (deficiency) and recklessness (excess).
亚里士多德的美德伦理学不提供基于规则的公式，而是以品格为中心的“中道定理”。美德是一种在具体情况中选取过度与不及之间的居间者的气质。其“公式”为：美德 = 两个极端之恶的中庸。例如，勇敢是怯懦（不足）和鲁莽（过度）的中道。
| Sphere of Action | Deficiency (Vice) | Virtue (Mean) | Excess (Vice) |
| Fear and Confidence | Cowardice | Courage | Recklessness |
| Pleasure and Pain | Insensibility | Temperance | Licentiousness |
| Giving/Spending Money | Stinginess | Liberality | Prodigality |
The virtuous person does not calculatedly apply the mean but perceives what is appropriate through practical wisdom (phronesis). Moral development comes from habituation – repeatedly performing virtuous acts until they become second nature.
有美德之人并非计算中道，而是通过实践智慧（phronesis）感知恰当的行为。道德的养成来自习惯化——反复做出有德的行为，直至成为第二天性。
6. The Problem of Evil: Logical Disproof ‘Formula’ | 恶的问题：逻辑反驳“公式”
The logical problem of evil is one of the most powerful arguments in philosophy of religion, functioning as a formal disproof of the existence of a tri-omni God. The 'formula' can be set out as a deductive argument:
恶的逻辑问题是宗教哲学中最有力的论证之一，它作为一个形式化的反证，旨在推翻全能、全善、全知之三全上帝的存在。其“公式”可表述为一个演绎论证：
1) If God exists, then God is omnipotent, omniscient, and omnibenevolent.
2) An omnipotent being can prevent all evil.
3) An omniscient being knows when evil occurs.
4) An omnibenevolent being would want to prevent all evil.
5) Evil exists.
∴ God (as defined) does not exist.
1) 如果上帝存在，则上帝是全能的、全知的、全善的。
2) 全能者能阻止一切恶。
3) 全知者知晓一切恶的发生。
4) 全善者愿阻止一切恶。
5) 恶存在。
∴ 上帝（如此定义）不存在。
The evidential problem of evil is an inductive variant: the sheer amount and distribution of gratuitous evil makes God's existence highly improbable.
恶的证据问题是一个归纳变体：大量的、无端的恶使得上帝的存在极度不可能。
7. The Design Argument: Analogy and Inference to the Best Explanation | 设计论证：类比与最佳解释推理
The design argument (teleological argument) can be thought of as an inductive formula: If an object exhibits complex, purposeful arrangement of parts (like a watch), we infer an intelligent designer. Natural objects (e.g., the eye) exhibit even greater complexity and purpose; therefore, the universe probably has a designer (God).
设计论证（目的论论证）可看作一个归纳公式：如果一个物体表现出复杂且各部分有目的的组织（如手表），我们就推断有一个智能设计者。自然物体（如眼睛）展现出甚至更大的复杂性和目的性；因此，宇宙很可能有一个设计者（上帝）。
Paley's Watchmaker Argument structure: Watch : watchmaker :: Universe : Universe-Maker. The analogy supports the claim that as the watch's complexity implies a watchmaker, the universe's complexity implies a divine designer. Hume's criticisms: weak analogy (the universe is not sufficiently like a watch to warrant the inference); alternative explanations (chance, necessity, evolution).
佩利的手表匠论证结构：手表 : 手表匠 :: 宇宙 : 宇宙制造者。类比支持如下主张：正如手表的复杂性意味着有手表匠，宇宙的复杂性意味着有神圣设计者。休谟的批评：类比薄弱（宇宙与手表并不足够相似以担保推断）；替代解释（偶然、必然性、进化）。
8. Mind-Body Problem: Substance Dualism vs Materialism 'Equations' | 心身问题：实体二元论与唯物主义“方程”
Philosophy of mind revolves around the relationship between mental states and physical states. Substance dualism (Descartes) can be expressed as: Mind ≠ Body; they are two fundamentally different substances – thinking substance (res cogitans) and extended substance (res extensa). The mind is private, non-spatial, and indivisible; the body is public, spatial, and divisible.
心灵哲学围绕着心理状态与物理状态之间的关系。实体二元论（笛卡尔）可表述为：心 ≠ 身；它们是两种根本不同的实体——思维实体（res cogitans）和广延实体（res extensa）。心灵是私密的、非空间的和不可分割的；身体是公共的、空间的和可分割的。
Interaction Problem: If mind and body are wholly different substances, how do they causally interact? This is the main objection to dualism.
交互问题：如果心灵和身体是完全不同的实体，它们如何因果交互？这是对二元论的主要反对。
Materialist theories offer identity 'formulas': Mind = Brain (type identity theory – each type of mental state is identical to a type of brain state). Or Mind = Function (functionalism – mental states are defined by their causal roles, not by their physical makeup; the mind is software running on the brain's hardware).
唯物主义理论提供同一性“公式”：心灵 = 大脑（类型同一论——每种类型的心理状态等同于一种类型的大脑状态）。或心灵 = 功能（功能主义——心理状态由其因果角色定义，不由其物理构成定义；心灵是运行在大脑硬件上的软件）。
9. Free Will and Determinism: Consequence Argument 'Formula' | 自由意志与决定论：后果论证“公式”
The debate over free will often uses the consequence argument to show that if determinism is true, no one has free will. The 'theorem':
1) No one has a choice about the past or the laws of nature.
2) If determinism is true, the present state of the universe is a necessary consequence of the past + laws.
3) No one has a choice about the necessary consequences of things they have no choice about.
∴ No one has a choice about any present action (no free will).
关于自由意志的辩论常用后果论证来表明，如果决定论为真，则无人拥有自由意志。其“定理”为：
1) 没有人对过去或自然法则有选择。
2) 如果决定论为真，宇宙的当下状态是过去+法则的必然结果。
3) 没有人对自己无法选择之物的必然结果有选择。
∴ 没有人对当前任何行为有选择（无自由意志）。
Compatibilism challenges premise 3: free will does not require the ability to do otherwise in a deterministic sense, only that the action flows from one's own desires without external coercion. Libertarianism (in the free will sense) denies determinism, asserting that agents have a special causal power.
相容论挑战前提3：自由意志不要求有在决定论意义上做其他事的能力，只要求行为源于自己的欲望且无外在强制。自由意志论（在自由意志意义上）否认决定论，断言行动者具有特殊的因果力量。
10. Knowledge: The JTB 'Formula' and the Gettier Problem | 知识：JTB“公式”与盖梯尔问题
The traditional analysis of knowledge offers a simple 'formula': Knowledge = Justified True Belief (JTB). S knows that P if and only if: (i) P is true, (ii) S believes that P, and (iii) S is justified in believing that P. This was widely accepted until Gettier cases showed the formula is insufficient.
传统的知识分析提供了一个简单的“公式”：知识 = 得到辩护的真信念 (JTB)。S 知道 P 当且仅当：(i) P 为真，(ii) S 相信 P，且 (iii) S 相信 P 是有理由的。这被广泛接受，直到盖梯尔案例表明该公式不足。
A Gettier case is one where someone has a justified true belief but the belief is true by luck, so intuitively it is not knowledge. Example: Smith has strong evidence that Jones owns a Ford (he always has); thus he believes "Someone in the office owns a Ford." Unbeknownst to Smith, Jones sold his Ford, but by coincidence, another colleague, Brown, owns a Ford. Smith's belief is true and justified, but not knowledge.
盖梯尔案例是指某人拥有一个得到辩护的真信念，但该信念因为碰巧为真，因此直觉上不是知识。例如：史密斯有有力证据表明琼斯拥有一辆福特车（琼斯一直如此）；于是他相信“办公室里有人拥有一辆福特车”。史密斯不知道的是，琼斯已卖掉福特车，但巧合的是另一同事布朗刚好有一辆福特车。史密斯的信念是真且得到辩护的，但不是知识。
Responses include adding a 'no false lemmas' condition (the justification must not rest on any false belief) or a causal connection requirement.
回应包括增加“无假前提”条件（辩护不得依赖于任何错误信念）或因果联系要求。
11. The Social Contract: Sovereignty Formula | 社会契约：主权公式
Political philosophy often uses the idea of a social contract, conceived by thinkers like Thomas Hobbes and John Locke, to justify political authority. Hobbes' 'formula' for escaping the state of nature is: All individuals agree to "I authorise and give up my right of governing myself to this man, or to this assembly of men, on this condition, that thou give up thy right to him, and authorise all his actions in like manner." This creates the Leviathan – an absolute sovereign.
政治哲学常用社会契约的理念来证成政治权威，由霍布斯和洛克等思想家构思。霍布斯的逃离自然状态的“公式”是：所有个体同意“我授权并放弃自我统治的权利给这个人或这个集体，条件是你们也同样放弃权利给他，并授权他所有行为。”这创造出了“利维坦”——绝对主权者。
Locke's formula is more limited: people contract to form a government that protects their natural rights (life, liberty, property). If the government violates these rights, the contract is broken and citizens may revolt.
洛克的公式则更有限：人们缔约成立政府以保护自然权利（生命、自由、财产）。如果政府侵犯这些权利，契约被破坏，公民有权革命。
Hobbes: Without Leviathan → war of all against all; With absolute sovereign → security.
.Locke: Without government → inconvenience (but not war); With limited government → protection of rights.
霍布斯：无利维坦 → 一切人反对一切人的战争；有绝对主权者 → 安全。
洛克：无政府 → 不便（但非战争）；有限政府 → 权利受保护。
12. The Is-Ought Gap: Hume's Law | 实然-应然鸿沟：休谟法则
David Hume famously noted that many moral arguments suddenly shift from factual 'is' statements to normative 'ought' statements without justification. This has been formulated as a logical principle: One cannot derive an 'ought' from an 'is' alone. Any moral conclusion that claims to follow solely from descriptive facts commits the naturalistic fallacy unless a normative premise is supplied.
大卫·休谟著名地指出，许多道德论证会突然毫无理由地从事实性“是”陈述跳转到规范性“应当”陈述。这被总结为一条逻辑原则：不能仅从“是”推导出“应当”。任何声称仅从描述性事实得出的道德结论，除非提供了规范性前提，都犯了自然主义谬误。
Example of violation: "Nature gives humans certain desires; therefore humans ought to fulfil those desires." This is invalid unless we add a normative premise such as "We ought to follow nature."
违反的例子：“自然赋予了人类某些欲望；因此人类应当满足这些欲望。”除非加上诸如“我们应当遵循自然”这样的规范性前提，否则无效。
This 'theorem' is crucial for evaluating ethical theories. Utilitarianism cannot deduce 'we ought to maximise pleasure' from the mere fact that humans seek pleasure without a bridging normative principle.
这条“定理”对评价伦理理论至关重要。功利主义不能仅从人类追求快乐的事实推导出“我们应当最大化快乐”，除非搭一个规范性的桥接原则。
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