📚 CIE Year 12 Philosophy Formula & Theorem Quick Reference Handbook | CIE 12年级哲学公式定理速查手册
Welcome to your go-to revision companion for CIE AS Philosophy. What mathematicians call formulas and theorems, philosophers recognise as structured arguments, essential definitions, and normative principles. This handbook reframes the core content of Year 12 into compact, memorable propositions — your intellectual toolkit for essays, structured questions, and unseen texts. Each entry pairs an English statement with its Chinese equivalent, helping you master bilingual expression and deepen your conceptual clarity. Treat these not as dry equations but as living summaries of centuries of critical thought.
欢迎使用这本专为 CIE AS 哲学设计的速查手册。数学家所说的公式与定理，在哲学家手中化为结构严密的论证、根本定义与规范性原理。本手册将 12 年级核心内容浓缩为简洁易记的命题——是你撰写论文、应对结构题与解读陌生文本的思维工具箱。每一条目均以英中配对呈现，助你掌握双语表达，深化概念理解。请将这些条目视为活的思想结晶，而非枯燥的等式。
1. The JTB Account of Knowledge | 知识的 JTB 理论
where a subject S knows a proposition p if and only if S believes p, p is true, and S is justified in believing p. This tripartite definition dominated epistemology until Gettier’s counterexamples in 1963 showed that justified true belief is not sufficient for knowledge, prompting the search for a fourth condition.
知识传统上被分析为“得到证成的真信念”。公式常写作 K = J + T + B，即主体 S 知道命题 p，当且仅当 S 相信 p，p 为真，且 S 相信 p 是有证成的。这一三元定义统治认识论，直到 1963 年盖梯尔的反例表明证成真信念并非知识的充分条件，从而引发了补充第四条件的探索。
K ≡ B(p) ∧ T(p) ∧ J(p) — needs ‘no Gettier’ clause
K ≡ B(p) ∧ T(p) ∧ J(p) —— 需附加“无盖梯尔情形”条款
2. The Tripartite Definition of Truth | 真理的三部分定义
Three major theories of truth can be expressed as rival formulas. The correspondence theory holds: a proposition p is true iff it corresponds to a fact. The coherence theory claims that p is true iff it belongs to a consistent system of beliefs. The pragmatic theory, associated with William James, asserts that p is true iff it is useful to believe it. Each theory captures a different aspect of the concept of truth and brings distinct strengths and weaknesses.
三种主要的真理理论可表示为相互竞争的公式。符合论主张：命题 p 为真当且仅当它与事实相符合。融贯论声称：p 为真当且仅当它属于一个协调的信念系统。实用论，联系威廉·詹姆士，断言：p 为真当且仅当相信它是有用的。每一种理论抓住了真理概念的不同侧向，各具优势与弱点。
Truthcorr(p) ⇔ p reflects fact; Truthcoh(p) ⇔ p ∈ coherent set; Truthprag(p) ⇔ useful(p)
符合真理(p) ⇔ p 反映事实；融贯真理(p) ⇔ p ∈ 协调集；实用真理(p) ⇔ p 有用
3. Cartesian Doubt and the Cogito | 笛卡尔式怀疑与“我思故我在”
Descartes’ method of radical doubt can be formalised as: doubt all beliefs that are not absolutely indubitable. The evil demon scenario and the dream argument leave one certainty — that I exist as a thinking thing whenever I think. This foundational truth is captured in the proposition ‘Cogito, ergo sum’, often abbreviated as the Cogito. It serves as the first principle of knowledge in Descartes’ rationalist system.
笛卡尔的彻底怀疑方法可形式化为：怀疑一切并非绝对无可置疑的信念。恶魔情景与梦境论证留下的唯一确定性是——每当我思考时，我作为思维之物而存在。这一奠基性的真理凝结在“我思故我在”的命题中，常简称为“我思”。它充当笛卡尔理性主义体系中知识的首要原理。
If I think, then I exist → (Think → Exist) is necessarily true
如果我思考，那么我存在 → (思考 → 存在) 必然为真
4. The Hypothetical and Categorical Imperatives | 假言命令与定言命令
Kant distinguishes two types of rational principles. A hypothetical imperative commands action as a means to a desired end: if you want X, you ought to do Y. Formula: O(Y) given desire(X). A categorical imperative commands an action as necessary in itself, unconditionally. Kant offers three formulations, the first being the Formula of Universal Law: act only according to that maxim through which you can at the same time will that it become a universal law.
康德区分两类理性原则。假言命令要求行为作为达至所欲目的之手段：若要 X，则应当做 Y。公式：在欲望(X)下，应当(Y)。定言命令无条件地将行为关连为自身必然的。康德提供三条表述，第一项为普遍法则公式：只依据那条你同时能意愿它成为普遍法则的格准而行动。
Categorical ImperativeFUL: Act only on maxims that can be universalised without contradiction
定言命令普遍法则形式：只依凭可无矛盾地普遍化的格准行动
5. Bentham’s Hedonic Calculus | 边沁的苦乐计算
Bentham quantifies happiness through the hedonic calculus, a decision-making algorithm designed to measure the pleasure and pain produced by an action. Seven criteria are considered: intensity, duration, certainty, propinquity, fecundity, purity, and extent. The total net pleasure is computed for each individual affected, and the act with the greatest overall surplus is deemed right. This calculus underpins the principle of utility.
边沁通过苦乐计算将幸福量化，这是一种旨在衡量行为所产生快乐与痛苦的决策算法。需考量七项标准：强度、持续时间、确定性、邻近性、丰产性、纯度与范围。针对每位受影响者计算净快乐总量，总盈余最大的行为即被判定为正当。这一计算支撑着功利原则。
Net Utility = Σ (Pleasure × criteria) − Σ (Pain × criteria)
净功利 = Σ (快乐 × 各项标准) − Σ (痛苦 × 各项标准)
6. The Principle of Utility | 功利原则
Mill refines Bentham’s principle into a deceptively simple theorem: actions are right in proportion as they tend to promote happiness, wrong as they tend to produce the reverse of happiness. Happiness is intended pleasure and the absence of pain. Mill introduces qualitative distinctions, arguing that higher pleasures (intellectual and moral) must be given greater weight because competent judges prefer them, even when accompanied by less quantity.
密尔将边沁的原理提炼为一个看似简单的定理：行为正当与否与其促进幸福的倾向成比例，反之则错误。幸福即预期的快乐与痛苦之缺失。密尔引入质的区分，主张高级快乐（智识与道德的）需赋予更大权重，因为合格的裁判官宁愿追求这些快乐，即使数量较少。
Rightness ∝ (∑ higher pleasures × quality weight + ∑ lower pleasures)
正当性 ∝ (∑ 高级快乐 × 质权重 + ∑ 低级快乐)
7. The Is-Ought Problem | 实然-应然问题
Hume famously notes that moral philosophers often slide from propositions about what is the case to propositions about what ought to be the case without explanation. This logical gap, known as the is-ought problem, can be expressed as the inference from descriptive premises to a normative conclusion being invalid without a bridging premise that itself contains an ought. It challenges naturalistic ethical theories that try to derive morality from facts.
休谟著名地指出，道德哲学家常悄然从“是”的陈述滑向“应当”的陈述而未加说明。这一逻辑鸿沟，即实然-应然问题，可表述为：从描述性前提推出规范性结论的推理无效，除非存在一条本身包含“应当”的桥接前提。它挑战了试图从事实推出道德的自然主义伦理理论。
¬(Descriptive set ⊢ Normative) without a hidden ‘ought’ premise
无隐含“应当”前提则 ¬(描述集 ⊢ 规范)
8. Aristotle’s Doctrine of the Mean | 亚里士多德的中庸之道
Aristotle’s virtue ethics presents virtue as a mean relative to us, located between two extremes of excess and deficiency. For any given sphere of feeling or action, there is an intermediate state that constitutes virtue. Fear and confidence: courage is the mean between cowardice (excess of fear) and rashness (deficiency of fear). The formula is not arithmetic but relative, determined by practical reason (phronesis).
亚里士多德的德性伦理将德性呈现为相对于我们的中道，处于过度与不及两个极端之间。对任何感受或行动的领域而言，存在一种构成德性的中间状态。恐惧与信心：勇敢是懦弱（恐惧过度）与鲁莽（恐惧不及）之间的中道。此公式非算术的，而是相对的，由实践智慧（phronesis）决定。
Virtue = Meanrelative to agent (Excess, Deficiency) defined by phronesis
德性 = 相对于行动者的中道 (过度，不及)，由实践智慧界定
9. The Euthyphro Dilemma | 游叙弗伦困境
In Plato’s dialogue, Socrates poses a question that can be formulated as a logical fork: is the pious loved by the gods because it is pious, or is it pious because it is loved by the gods? Applied to morality, the dilemma asks: does God command what is good because it is intrinsically good, or is something good simply because God commands it? The first horn seems to make goodness independent of God; the second makes morality arbitrary.
在柏拉图对话中，苏格拉底抛出一个可化约为逻辑岔路的问题：虔诚是因为诸神喜爱所以虔诚，还是因为它是虔诚的所以诸神喜爱？应用于道德，困境问道：上帝命令善是因为善本身是好的，还是某事物之所以是善的仅仅因为上帝命令它？第一岔路似乎让善独立于上帝；第二岔路使道德变成任意的。
Dilemma: Good → (independent of God’s will) or Good = (merely God’s will)
困境：善 → (独立于神意) 或 善 = (仅是神意)
10. Descartes’ Trademark Argument | 笛卡尔的商标论证
The trademark argument posits a causal principle like a theorem: the idea of an infinite, perfect God within a finite imperfect being must have been caused by something possessing at least as much formal reality. A finite mind cannot conjure the idea of infinity by itself; thus, the idea is a ‘trademark’ left by God on his creation. The argument offers an a posteriori proof of God’s existence rooted in the nature of ideas.
商标论证提出一项因果原理如同定理：有限不完善存在者心中的无限完善之上帝观念，必然由一个至少拥有同等形式实在的原因所刻印。有限心灵无法凭空创造无限观念；因此，这观念是上帝刻在其受造物上的“商标”。这一论证基于观念性质，从后天路径证明上帝存在。
Cause ≥ Effect (formal reality): Idea of ∞ → Cause must be ∞
原因 ≥ 结果（形式实在）：∞ 的观念 → 原因必须是 ∞
11. Paley’s Design Argument as a Theorem | 佩利的设计论证作为定理
Paley’s analogical design argument can be structured as: if an object exhibits complexity, purpose, and adaptation of parts to an end, it is most reasonably believed to be the product of an intelligent designer. The watch implies a watchmaker; by analogy, the universe, with its order and intricate design, implies a Divine Designer. Critics shatter the analogy by highlighting the disanalogies between artifacts and natural organisms.
佩利的类比设计论证可结构化为：若一个物体展现复杂性、目的性与部件之适配，则最合理地相信其为有智能设计者所造。手表暗含制表匠；类比而言，宇宙及其秩序和精妙设计暗含一位神圣设计者。批评者通过强调人造物与自然有机体之间的不类比之处，击碎了这一类比。
Complexity + Teleology + Adaptation ⇒ Designer (analogical inference)
复杂性 + 目的性 + 适配 ⇒ 设计者（类比推理）
12. The Problem of Evil as a Logical Formula | 恶的问题作为逻辑公式
The logical problem of evil sets up an inconsistent triad. If God is omnipotent, omniscient, and omnibenevolent, then evil should not exist. Yet evil exists. The theist must show there is no contradiction, typically by augmenting the definition of God’s attributes or denying the reality of evil. The formula reveals a classic a priori challenge to theistic belief.
恶的逻辑问题构造了一个不一致的三元组。如果上帝全能、全知且全善，那么恶不应存在。然而恶确实存在。有神论者必须证明这里并无矛盾，通常通过补充上帝属性的定义或否认恶的实在性。这一公式揭示了有神论信仰面临的一个经典的先天挑战。
(Omnipotent ∧ Omniscient ∧ Omnibenevolent) → ¬Evil [But Evil exists]
(全能 ∧ 全知 ∧ 全善) → ¬恶 [然而恶存在]
Published by TutorHao | Philosophy Revision Series | aleveler.com
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