📚 Formula & Theorem Quick Reference Handbook for A-Level CCEA Philosophy | A-Level CCEA 哲学：公式定理速查手册
Philosophy at A-Level, particularly in the CCEA specification, demands not only conceptual understanding but also the ability to recognise, apply, and evaluate formal logical structures and key theoretical principles. This quick-reference handbook gathers the essential ‘formulae’ and ‘theorems’—from sentential and predicate logic notation to deontic modalities, ethical principles, and argument patterns—that every CCEA candidate should have at their fingertips. Whether you are preparing for the AS Ethics paper or the A2 Synoptic unit, this resource will help you revise the symbolic and structural foundations of the discipline.
A-Level 哲学，尤其是在 CCEA 考试局的教学大纲中，不仅要求概念理解，还要求能够识别、应用和评估形式化的逻辑结构以及核心理论原则。这本速查手册汇集了关键的“公式”与“定理”——从命题逻辑和谓词逻辑的符号、道义模态到伦理学原则和论证模式——这些都是每位 CCEA 考生应熟稔的。无论你是准备 AS 伦理学试卷，还是 A2 综合单元，这份资料都将帮助你复习哲学的符号化与结构化基础。
1. Sentential Logic Connectives & Truth Tables | 命题逻辑联结词与真值表
Sentential (or propositional) logic analyses complete statements and how they combine. Its connectives are the building blocks of symbolic reasoning. Below is a summary of the standard operators used in CCEA Philosophy.
命题逻辑分析完整陈述句及其组合方式。其联结词是符号化推理的基石。以下是 CCEA 哲学中使用的标准算符摘要。
| Symbol | Name | Truth table key |
|---|---|---|
| ¬ | Negation (not) | ¬P true iff P false |
| ∧ | Conjunction (and) | P ∧ Q true iff both true |
| ∨ | Disjunction (or) | P ∨ Q false iff both false |
| → | Conditional (if… then) | P → Q false only when P true and Q false |
| ↔ | Biconditional (iff) | P ↔ Q true when P and Q have same truth value |
The truth tables can be memorised with these fundamental rules: a conditional is only false when the antecedent is true and the consequent false; a disjunction is true unless both disjuncts are false.
真值表可通过以下基本规则记忆：条件句仅在前件真后件假时为假；析取式除非两个析取项都假，否则为真。
2. Modus Ponens, Modus Tollens & Associated Valid Forms | 肯定前件、否定后件与相关有效式
Modus ponens (MP) and modus tollens (MT) are the most fundamental valid argument forms in deductive logic. They appear frequently in CCEA examination questions requiring logical analysis.
肯定前件式（MP）和否定后件式（MT）是演绎逻辑中最基本的有效论证形式。它们在需要逻辑分析的 CCEA 试题中频繁出现。
MP: P → Q, P ⊢ Q
肯定前件式：P → Q，P，所以 Q
MT: P → Q, ¬Q ⊢ ¬P
否定后件式：P → Q，¬Q，所以 ¬P
Other useful valid forms include Hypothetical Syllogism (P → Q, Q → R ⊢ P → R), Disjunctive Syllogism (P ∨ Q, ¬P ⊢ Q), and Constructive Dilemma ((P → Q) ∧ (R → S), P ∨ R ⊢ Q ∨ S).
其他有用的有效式包括假言三段论（P → Q，Q → R ⊢ P → R）、析取三段论（P ∨ Q，¬P ⊢ Q）和构成式二难推理（(P → Q) ∧ (R → S)，P ∨ R ⊢ Q ∨ S）。
3. Predicate Logic Symbols: Quantifiers & Relations | 谓词逻辑符号：量词与关系
Predicate logic introduces internal structure to statements. The CCEA specification expects candidates to be comfortable with universal and existential quantifiers, as well as identity.
谓词逻辑引入了陈述的内部结构。CCEA 教学大纲要求考生熟练掌握全称量词、存在量词以及等同关系。
- ∀x: ‘for all x’ – universal quantifier (全称量词)
- ∃x: ‘there exists at least one x’ – existential quantifier (存在量词)
- = : identity (同一性)
For instance, ‘All humans are mortal’ becomes ∀x(Hx → Mx). ‘Some philosophers are wise’ becomes ∃x(Px ∧ Wx). Understanding the difference between → inside ∀ and ∧ inside ∃ is crucial for correct translation.
例如，“所有人都会死”符号化为 ∀x(Hx → Mx)。“有些哲学家是聪明的”符号化为 ∃x(Px ∧ Wx)。理解 ∀ 里用 → 而 ∃ 里用 ∧ 的区别对正确翻译至关重要。
4. Deontic Logic & Ethical Modalities | 道义逻辑与伦理模态词
In ethical theory, deontic operators model obligation, permission, and prohibition. Although not always formalised, CCEA ethics questions often require clarity on these concepts.
在伦理学理论中，道义算子可以塑造义务、允许和禁止等概念。尽管不一定都形式化，CCEA 伦理学试题经常要求澄清这些概念。
| Symbol | Meaning |
|---|---|
| O p | It is obligatory that p (义务 p) |
| P p | It is permitted that p (允许 p) |
| F p | It is forbidden that p (禁止 p) |
Key equivalences: O p ⇔ ¬P¬p (obligatory implies not permitted not); F p ⇔ O¬p (forbidden is obligatory not). Utilitarian and Kantian theories can be partially expressed using these modalities.
关键等价关系：O p ⇔ ¬P¬p（义务蕴含不允许非 p）；F p ⇔ O¬p（禁止即义务非 p）。功利主义和康德伦理学可部分借用这些模态词来表达。
5. The Square of Opposition & Categorical Propositions | 对当方阵与直言命题
The traditional Square of Opposition displays logical relations among the four categorical propositions: A (all S are P), E (no S are P), I (some S are P), O (some S are not P). This tool is invaluable for evaluating immediate inferences.
传统对当方阵展示了四种直言命题之间的逻辑关系：A（所有 S 是 P）、E（没有 S 是 P）、I（有些 S 是 P）、O（有些 S 不是 P）。这个工具对评价直接推论极有价值。
- A and E are contraries: cannot both be true, but can both be false.
- I and O are subcontraries: cannot both be false, but can both be true.
- A implies I, E implies O (subalternation).
- A and O are contradictories; E and I are contradictories.
- A 与 E 为上反对关系：不能同真，但可同假。
- I 与 O 为下反对关系：不能同假，但可同真。
- A 蕴涵 I，E 蕴涵 O（差等关系）。
- A 与 O 矛盾；E 与 I 矛盾。
6. Aquinas’ First and Second Way Logical Forms | 阿奎那第一、第二路的逻辑形式
Thomas Aquinas’ cosmological arguments can be reconstructed in formal terms. For CCEA, candidates should recognise the underlying logical structure of motion and causation.
托马斯·阿奎那的宇宙论论证可以用形式化语言重构。对 CCEA 考生而言，应识别运动与因果论证背后的逻辑结构。
The First Way (motion): potentiality is actualised only by what is actual; an infinite regress of movers is impossible; therefore, there must be a first unmoved mover.
第一路（运动）：潜能只能由现实激活；推动者的无限回溯不可能；因此，必有一个第一不动的推动者。
∀x (Mx → ∃y (Ay ∧ y moves x)) ∧ ¬Infinite Regress ⊢ ∃z (First Mover z ∧ ¬(∃w w moves z))
∀x (Mx → ∃y (Ay ∧ y 推动 x)) ∧ ¬无限回归 ⊢ ∃z (第一推动者 z ∧ ¬(∃w w 推动 z))
The Second Way (causation) follows a similar pattern, moving from efficient causes to a first uncaused cause.
第二路（因果）遵循相似的模式，从动力因推进到第一非被动的因。
7. Hedonic Calculus & Utilitarian Formula | 快乐计算与功利主义公式
Jeremy Bentham’s hedonic calculus provides a quantitative decision procedure for utilitarian ethics. While not a mathematical formula in the modern sense, its seven criteria act as a guideline for measuring pleasure and pain.
杰里米·边沁的快乐计算为功利主义伦理学提供了一套量化决策程序。虽然并非现代意义上的数学公式，但其七项标准起到了衡量快乐与痛苦的指导作用。
The seven criteria: intensity, duration, certainty, propinquity, fecundity, purity, and extent. The total utility of an action could be expressed as the sum of pleasure units minus pain units, weighted by these factors.
七项标准：强度、持续时间、确定性、邻近性、丰产性、纯度和范围。一个行动的总功利可表示为快乐单位减去痛苦单位之和，并以上述因素加权。
Net Utility = Σ (Pleasure × Intensity × Duration × Certainty × …) – Σ (Pain × Intensity × …)
净功利 = Σ (快乐 × 强度 × 持续时间 × 确定性 × …) – Σ (痛苦 × 强度 × …)
Candidates should be able to explain how this ‘formula’ is used to evaluate moral actions, and its criticisms from Mill and others.
考生应能解释这个“公式”如何用于评估道德行为，以及密尔等人对它的批评。
8. Kant’s Categorical Imperative Formulations | 康德的定言命令公式
Immanuel Kant provides several formulations of the Categorical Imperative, which function as ethical ‘laws’. The CCEA specification focuses on the Formula of Universal Law and the Formula of Humanity.
伊曼努尔·康德给出了定言命令的几种表述，它们起到道德“法则”的作用。CCEA 教学大纲重点关注普遍法则公式和人性公式。
The 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.’ Logically, this requires that the maxim be conceivable without contradiction.
普遍法则公式：“只按照你同时能够意愿它成为一项普遍法则的准则去行动。”从逻辑上，这要求该准则在无矛盾的情况下是可设想的。
The 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 and never simply as a means.’ This enshrines respect for rational agency.
人性公式：“你的行动，要把你自己人身中的人性，和其他人身中的人性，在任何时候都同样看作是目的，永远不能只看作是手段。”这体现了对理性主体的尊重。
9. Virtue Ethics: The Doctrine of the Mean | 德性伦理学：中庸之道
Aristotle’s doctrine of the mean is often presented as a formula for moral virtue: virtue is a mean between two extremes of excess and deficiency, relative to us, determined by reason.
亚里士多德的中庸之道常被呈现为一条道德德性的公式：德性就是两个极端（过度与不及）之间的中道，相对于我们而言，由理性决定。
For any given sphere of action or feeling, there is a corresponding virtue that lies between two vices. The table below illustrates some examples required by CCEA.
对于任何给定的行动或情感领域，都存在一个对应的德性，它处于两种恶习之间。下表展示了 CCEA 要求的几个例子。
| Sphere | Deficiency | Virtue (Mean) | Excess |
|---|---|---|---|
| Fear & confidence | Cowardice | Courage | Rashness |
| Pleasure & pain | Insensibility | Temperance | Self-indulgence |
| Giving & taking money | Meanness | Liberality | Prodigality |
Note that not every action or passion admits of a mean; some are intrinsically base (e.g., adultery, murder).
注意并非所有行动或情感都有中道；有些本身就是鄙恶的（如通奸、谋杀）。
10. The Free Will Defence Logical Schemata | 自由意志辩护的逻辑框架
Alvin Plantinga’s free will defence against the logical problem of evil can be formalised. It aims to show that the existence of God and the existence of evil are not logically incompatible, given the possibility of free will.
阿尔文·普兰丁格针对恶的逻辑问题的自由意志辩护可以被形式化。其目的在于表明，鉴于自由意志的可能性，上帝存在与恶的存在在逻辑上并非不相容。
The core claim: a world containing creatures who are significantly free (and freely perform more good than evil) is more valuable than a world containing no free creatures at all. An omnipotent God cannot causally determine a free being to do only good without violating that freedom.
核心主张：一个包含具有显著自由的受造物（且自由地行善多于作恶）的世界，比一个完全没有自由受造物的世界更有价值。全能上帝不可能在因果上决定一个自由存在者只行善而不侵犯其自由。
Possibly (God exists & Evil exists) → ¬Necessarily (God → ¬Evil)
可能存在（上帝存在且恶存在）→ 并非必然（如果上帝存在则没有恶）
Plantinga’s argument uses the transworld depravity thesis to show that it is possible that every creaturely essence suffers from transworld depravity, making it infeasible for God to actualise a sinless, free world.
普兰丁格的论证运用跨世界堕落论题，表明可能每一个受造本质都患有跨世界堕落，使得上帝无法现实化一个无罪且自由的世界。
11. Hume’s Fork & Verification Principle | 休谟的叉子与证实原则
Hume’s distinction between relations of ideas and matters of fact is a foundational ‘formula’ for empiricist epistemology. The later Logical Positivists transformed this into the verification principle, which CCEA candidates must critique.
休谟对观念的关系与事实的区分是经验主义认识论的一项基础“公式”。后来的逻辑实证主义者将其转化为证实原则，CCEA 考生必须对此进行批判。
Hume’s Fork: all meaningful propositions are either analytic (true by definition, e.g., ‘All bachelors are unmarried’) or synthetic and empirically verifiable. Claims not fitting these categories are ‘sophistry and illusion’.
休谟的叉子：所有有意义的命题要么是分析的（根据定义为真，如“所有单身汉都未婚”），要么是综合的且可通过经验证实。不符合这些类别的论断即是“诡辩和幻想”。
The verification principle: a statement is cognitively meaningful only if it is either analytic or empirically verifiable. This self-referentially undercuts itself, as the principle itself is neither analytic nor verifiable.
证实原则：一个陈述在认知上有意义，当且仅当它是分析的或经验上可证实的。这自我指涉地削弱了自身，因为该原则本身既非分析，也非可证实。
12. Gettier Cases & JTB Formula | 盖梯尔案例与 JTB 公式
The traditional analysis of knowledge as justified true belief (JTB) can be expressed as a biconditional: S knows that p iff (i) p is true, (ii) S believes that p, (iii) S is justified in believing that p. Edmund Gettier’s counterexamples show this formula is not sufficient.
将知识分析为得到辩护的真信念的传统（JTB）可以表达为一个双条件句：S 知道 p，当且仅当 (i) p 为真，(ii) S 相信 p，(iii) S 有理由相信 p。埃德蒙·盖梯尔的反例表明该公式并不充分。
Gettier’s famous case: Smith has strong evidence for ‘Jones owns a Ford’ and deduces ‘Jones owns a Ford or Brown is in Barcelona’. Unbeknownst to Smith, Jones’s Ford was just sold, yet coincidentally Brown is indeed in Barcelona. Smith has a justified true belief but not knowledge.
盖梯尔的著名案例：史密斯有强证据相信“琼斯拥有一辆福特”，并由此推出“琼斯拥有一辆福特或者布朗在巴塞罗那”。史密斯不知道，琼斯的福特刚刚被卖掉，然而碰巧布朗确实在巴塞罗那。史密斯拥有得到辩护的真信念，却不是知识。
JTB: Kp ↔ (p ∧ Bp ∧ Jp) — but Gettier: ∃ cases where (p ∧ Bp ∧ Jp) ∧ ¬Kp
JTB：Kp ↔ (p ∧ Bp ∧ Jp) ——但盖梯尔：存在某些情况使得 (p ∧ Bp ∧ Jp) ∧ ¬Kp
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