A-Level Philosophy Quick Reference: Theorems & Formulas | A-level 哲学速查手册：定理与公式

📚 A-Level Philosophy Quick Reference: Theorems & Formulas | A-level 哲学速查手册：定理与公式

In the CIE A2 Philosophy syllabus, success often depends on the ability to recognise, reconstruct and evaluate arguments with precision. This quick-reference handbook collects the central logical rules, epistemological definitions, ethical principles and metaphysical positions into a single, accessible toolkit. Each entry is presented as a memorable ‘formula’ or structured principle, paired with a concise explanation in both English and Chinese. Students can use the handbook to revise efficiently, to sharpen their analytical writing and to internalise the core moves that examiners reward.

在 CIE A2 哲学课程中，能否准确地识别、重构和评价论证往往是成功的关键。这份速查手册将核心的逻辑规则、认识论定义、伦理原则和形而上学立场汇集成一个便捷的工具箱。每一条都以容易记忆的“公式”或结构化原则呈现，并配有简洁的中英文解释。学生可以利用本手册高效复习、提高分析写作能力，并内化那些阅卷人欣赏的关键学术动作。

1. Propositional Logic: Rules of Inference | 命题逻辑：推理规则

The fundamental ‘theorems’ of philosophy are the valid forms of deductive reasoning. In propositional logic, any substitution instance of these patterns guarantees that the conclusion is true if the premises are true.

哲学最基本的“定理”便是演绎推理的有效形式。在命题逻辑中，以下模式的任何代入实例都能保证：如果所有前提为真，则结论必然为真。

Modus Ponens: P → Q, P; therefore Q. (If the premises are true, the conclusion follows necessarily.)

肯定前件式： 如果 P 则 Q，已知 P；因此 Q。（前提为真时结论必然为真。）

Modus Tollens: P → Q, ¬Q; therefore ¬P. (Denying the consequent forces denial of the antecedent.)

否定后件式： 如果 P 则 Q，非 Q；因此非 P。（否定后件必然导致否定前件。）

Hypothetical Syllogism: P → Q, Q → R; therefore P → R. (A chain of conditionals collapses into a single implication.)

假言三段论： 如果 P 则 Q，如果 Q 则 R；因此如果 P 则 R。（条件链可以压缩为一个蕴含式。）

Disjunctive Syllogism: P ∨ Q, ¬P; therefore Q. (If at least one disjunct holds and one is denied, the other must hold.)

选言三段论： P 或 Q，非 P；因此 Q。（当至少一个支命题为真而其中一个被否定时，另一个必须为真。）

Rule	Symbolic Form
Modus Ponens	P → Q, P ⊢ Q
Modus Tollens	P → Q, ¬Q ⊢ ¬P
Hypothetical Syllogism	P → Q, Q → R ⊢ P → R
Disjunctive Syllogism	P ∨ Q, ¬P ⊢ Q

These four rules are indispensable for analysing philosophical arguments, from Descartes’s cogito to critiques of the design argument.

这四条规则是分析哲学论证不可或缺的工具，从笛卡尔的“我思”到对设计论证的批判都离不开它们。

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2. Predicate Logic: Quantification Essentials | 谓词逻辑：量化要点

When arguments concern ‘all’ or ‘some’ entities, propositional logic is insufficient. Predicate logic introduces quantifiers to formalise existence and universality. Mastering these moves is critical for tackling the ontological argument and other topics.

当论证涉及“所有”或“有些”实体时，命题逻辑就不够用了。谓词逻辑引入了量词来形式化存在性和普遍性。掌握这些方法对于处理本体论论证和其他主题至关重要。

Universal Instantiation (UI): ∀x Fx ⊢ Fa (If everything has property F, then any specific individual a has F.)

全称示例化： ∀x Fx ⊢ Fa（如果一切事物都有属性 F，那么任意特定个体 a 也有 F。）

Existential Generalisation (EG): Fa ⊢ ∃x Fx (From a specific instance, we can infer that something has F.)

存在概括： Fa ⊢ ∃x Fx（从一个具体事例可以推出存在某物有 F。）

Quantifier Negation (QN): ¬∀x Fx ⇔ ∃x ¬Fx; ¬∃x Fx ⇔ ∀x ¬Fx. (Denying ‘all are F’ is equivalent to asserting ‘some are not-F’.)

量词否定规则： ¬∀x Fx ⇔ ∃x ¬Fx；¬∃x Fx ⇔ ∀x ¬Fx。（否定“所有都是 F”等价于断言“有些不是 F”。）

In Anselm’s ontological argument, the move from ‘God exists in the understanding’ to ‘God exists in reality’ critically exploits quantifier and existence-predicate logic. Cautious handling of ‘exists’ as a predicate is a typical A2 skill.

在安瑟伦的本体论论证中，从“上帝存在于理智中”到“上帝存在于现实中”的推论关键性地利用了量词和存在谓词的逻辑。谨慎处理“存在”是否可作为谓词是 A2 的典型技能。

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3. Inductive Reasoning: Probability and Strength | 归纳推理：概率与强度

Philosophy of science and everyday reasoning rely on induction, which does not guarantee truth but can confer high probability. The ‘formulas’ here measure inductive strength rather than deductive validity.

科学哲学和日常推理都依赖归纳法，它不保证结论为真，但可以赋予结论高概率。这里的“公式”衡量的是归纳强度而非演绎有效性。

Enumerative Induction: Observed: n cases of A are B; no A observed is not B. Therefore (likely) all A are B. The strength increases with n and variety of observations.

枚举归纳： 已观察到 n 个 A 是 B，且没有观察到 A 不是 B。因此（很可能）所有 A 都是 B。归纳强度随 n 的增大和观察的多样性而增强。

Bayesian Confirmation: P(H|E) = [P(E|H) × P(H)] / P(E). A hypothesis H is confirmed if the posterior probability P(H|E) exceeds the prior P(H).

贝叶斯确证： P(H|E) = [P(E|H) × P(H)] / P(E)。若后验概率 P(H|E) 大于先验概率 P(H)，则假说 H 得到确证。

Problem of Induction (Hume): The premise ‘The future will resemble the past’ cannot be justified deductively (without circularity) nor inductively (begging the question). Thus, inductive inferences rest on habit, not rational proof.

归纳问题（休谟）：“未来将类似于过去”这一前提既不能演绎地（否则循环）也不能归纳地（诉诸乞题）得到辩护。因此，归纳推理依赖于习惯而非理性证明。

These tools help evaluate arguments from design, the reliability of sense experience, and the problem of other minds.

这些工具有助于评价设计论证、感官经验的可靠性以及他心问题。

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4. Knowledge: The JTB Formula and Gettier’s Challenge | 知识：JTB 公式与盖蒂尔挑战

Since Plato, knowing that p has been standardly analysed as justified true belief (JTB). The ‘formula’ is: S knows that p iff (i) p is true, (ii) S believes that p, and (iii) S is justified in believing that p.

自柏拉图以来，“知道 p”一直被标准地分析为得到辩护的真信念（JTB）。“公式”为：S 知道 p，当且仅当 (i) p 为真，(ii) S 相信 p，且 (iii) S 有理由相信 p。

Gettier Cases: A belief can be justified and true yet still not constitute knowledge because the justification is accidentally connected to the truth. Example: Smith has strong evidence that Jones will get the job (the company president told him), and Smith also believes ‘The person who will get the job has ten coins in his pocket’ because Jones has ten coins. Unbeknownst to Smith, he himself will get the job and he himself has ten coins. Smith’s belief is true and justified, but intuitively it is not knowledge.

盖蒂尔案例： 一个信念可以得到辩护且为真，但仍不构成知识，因为辩护与真理之间的联系是偶然的。例如：史密斯有强有力的证据表明琼斯会得到工作（公司总裁告诉他的），并且史密斯相信“得到工作的人口袋里有十枚硬币”，因为琼斯有十枚硬币。史密斯不知道的是，他自己最终会得到工作，而且他自己口袋里也有十枚硬币。史密斯的信念为真且有辩护，但直觉上这并不是知识。

No False Lemmas Response: S knows that p iff JTB + S’s justification does not depend on any false belief. This blocks many Gettier cases but invites further counterexamples.

无假引理回应： S 知道 p，当且仅当 JTB 条件满足，且 S 的辩护不依赖于任何假信念。这堵住了许多盖蒂尔案例，但又引出了新的反例。

The JTB formula and its refinements are the starting point for any A2 epistemology essay.

JTB 公式及其修正方案是所有 A2 认识论论文的出发点。

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5. Utilitarianism: The Calculus of Happiness | 功利主义：幸福的计算

Classical act utilitarianism can be expressed through a decision formula: An action is right if and only if it maximises net utility compared to all alternative actions. Utility is typically understood as happiness or pleasure minus pain.

古典行为功利主义可以用一个决策公式来表达：一个行动是正确的，当且仅当相较于所有其他可选行动，它能使净功利最大化。功利通常被理解为幸福或快乐减去痛苦。

Bentham’s Hedonic Calculus: Consider seven dimensions: intensity, duration, certainty, propinquity, fecundity, purity, and extent. The total utility is roughly the sum over all affected individuals of (intensity × duration × certainty × …).

边沁的苦乐计算： 考虑七个维度：强度、持续时间、确定性、远近、继生性、纯粹性和范围。总功利大致是所有受影响个体的（强度 × 持续时间 × 确定性 × …）之和。

Mill’s Higher Pleasures: Quality must be weighted alongside quantity; ‘It is better to be a human being dissatisfied than a pig satisfied.’ Utilitarianism thus becomes: maximise utility, weighting higher pleasures more heavily.

密尔的高级快乐： 质量必须与数量一并考量；“做一个不满足的人胜于做一头满足的猪”。因此功利主义变为：最大化功利，且对高级快乐赋予更高的权重。

Act vs Rule Utilitarianism: Act: perform the particular action that maximises utility. Rule: follow the general rule that, if universally adopted, would maximise utility. The formulas differ in the object of evaluation.

行为功利主义与规则功利主义： 行为功利主义：采取那个能最大化功利的具体行动。规则功利主义：遵循那条如果被普遍采纳就会最大化功利的一般规则。两者的公式在评价对象上不同。

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6. Deontology: The Categorical Imperative Formula | 义务论：定言命令式

Kant’s moral philosophy offers an alternative formula for moral permissibility, independent of consequences. The core test is the Categorical Imperative, which has several formulations.

康德的道德哲学提供了一种独立于后果的道德许可性公式，其核心检验标准是定言命令式，它有几个表达形式。

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. (3) Check for contradiction in conception or in will. If contradiction arises, the action is impermissible.

普遍法则公式： “要只按照你同时能够愿意它成为一个普遍法则的那个准则去行动。”检验步骤：(1) 陈述你行动的准则。(2) 将其普遍化。(3) 检查是否产生概念上的矛盾或意愿上的矛盾。若出现矛盾，则该行动在道德上是不允许的。

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 forbids using rational agents merely as tools.

人性公式： “你要如此行动，即无论是你自身中的人性还是任何他人中的人性，都始终同时当作目的，而绝不仅仅当作手段来对待。”这禁止仅仅将理性行动者当作工具使用。

Both formulations are designed to generate perfect and imperfect duties; for example, a maxim of making a false promise cannot be universalised without contradiction, because the institution of promising would collapse.

这两种表述都旨在产生完全义务和不完全义务；例如，作出虚假承诺的准则无法被无矛盾地普遍化，因为承诺制度会因此崩溃。

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7. Mind-Body Theories: Identity and Supervenience Formulas | 心身理论：同一性与随附性公式

In the philosophy of mind, metaphysical positions can be crystallised into precise dependence or identity claims. A2 students need to manipulate these to evaluate arguments for and against physicalism.

在心灵哲学中，各种形而上学立场可以凝练成精确的依赖或同一性主张。A2 学生需要运用这些命题来评价支持和反对物理主义的论证。

Type Identity Theory: For every type of mental state M, there is a type of physical state P such that M = P. Formula: ∀M ∃P (M = P). Pain just is C-fibre firing, for example.

类型同一论： 对于每一种心理状态类型 M，都存在一种物理状态类型 P，使得 M = P。公式：∀M ∃P (M = P)。例如，疼痛就是 C-纤维激活。

Token Identity: Every token mental event is identical with some token physical event, but types need not match. Formula: ∀m ∃p (m = p), where m ranges over particular mental events and p over particular physical events.

个例同一论： 每一个个别心理事件都与某个个别物理事件相同一，但类型无需匹配。公式：∀m ∃p (m = p)，其中 m 代表个别心理事件，p 代表个别物理事件。

Supervenience: A-properties supervene on B-properties iff there can be no difference in A without a difference in B. Mental properties supervene on physical properties: if two beings are physically indistinguishable, they cannot differ mentally. This is weaker than identity but still a physicalist claim.

随附性： A 属性随附于 B 属性，当且仅当 A 属性若无 B 属性上的差异便不可能有所不同。心理属性随附于物理属性：如果两个存在者物理上不可区分，那么它们在心理上也不可能不同。这比同一性主张弱，但仍是一种物理主义主张。

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8. Free Will: Consequence Argument and Compatibilism Formulas | 自由意志：后果论证与相容论公式

The free will debate often turns on formal patterns that connect determinism to moral responsibility. The most famous ‘theorem’ is the Consequence Argument, which attempts to show that determinism is incompatible with free will.

自由意志的争论常常围绕那些连接决定论与道德责任的形式化模式展开。最著名的“定理”是后果论证，它试图表明决定论与自由意志不可相容。

Consequence Argument (van Inwagen): If determinism is true, then every fact about our actions is a consequence of the remote past and the laws of nature. But nobody has a choice about the remote past or the laws. Therefore, nobody has a choice about any of their actions. Symbolically: □((P₀ ∧ L) → A), N(P₀), N(L) ⊢ N(A), where N means ‘nobody has a choice about’.

后果论证（范·因瓦根）： 如果决定论为真，那么我们行动的每一个事实都是遥远过去和自然法则的后果。但没有人能够选择遥远过去或自然法则。因此，没有人能够选择自己的任何行动。符号化：□((P₀ ∧ L) → A), N(P₀), N(L) ⊢ N(A)，其中 N 表示“没有人能够选择……”。

Classical Compatibilism: An action is free if it is caused by the agent’s own desires and beliefs without external coercion. Formula: Free action = action caused by internal psychological states (even if those states are themselves determined). Free will is thus compatible with determinism.

古典相容论： 一个行动是自由的，如果它由行动者自身的欲望和信念引起，且没有受到外部强制。公式：自由行动 = 由内在心理状态引起的行动（即使这些心理状态本身也是被决定的）。因此，自由意志与决定论是相容的。

Hierarchical Compatibilism (Frankfurt): A person acts freely when they act on a desire they want to have (a second-order volition). The formula distinguishes mere desire-satisfaction from authentic, endorsed motivation.

层级相容论（法兰克福）： 当一个人按照他想要拥有的欲望（即二阶意愿）行动时，他就是自由地行动。该公式区分了单纯的欲望满足与被认同的真实动机。

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9. Arguments for God’s Existence: Ontological Formula | 上帝存在的论证：本体论公式

The ontological argument remains a staple of A2 philosophy of religion. Its ‘formula’ is an a priori deduction from the concept of God to God’s necessary existence. Anselm’s original version can be formalised for analysis.

本体论论证一直是 A2 宗教哲学的核心内容。它的“公式”是从上帝的概念到上帝的必然存在的一种先天演绎。安瑟伦的原始版本可以被形式化以便分析。

Anselm’s Argument (Proslogion):
(1) God is by definition ‘that than which nothing greater can be conceived’ (TTWNGCBC).
(2) TTWNGCBC exists at least in the understanding.
(3) If TTWNGCBC exists only in the understanding and not in reality, then a greater being could be conceived (one that exists in reality).
(4) But that is impossible, because nothing greater than TTWNGCBC can be conceived.
(5) Therefore, TTWNGCBC must exist in reality as well as in the understanding.

安瑟伦的论证（《宣讲》）：
(1) 根据定义，上帝是“不能设想有比之更伟大的存在者”（TTWNGCBC）。
(2) TTWNGCBC 至少在理解中存在。
(3) 如果 TTWNGCBC 仅在理解中存在而在现实中不存在，那么就可以设想一个更伟大的存在者（一个在现实中也存在的）。
(4) 但这是不可能的，因为不能设想有比 TTWNGCBC 更伟大的存在者。
(5) 因此，TTWNGCBC 必定在现实中和理解中都存在。

Modal Version (Plantinga):
(1) A being has maximal excellence if it is omnipotent, omniscient, and wholly good in every possible world.
(2) A being has maximal greatness if it has maximal excellence in every possible world.
(3) It is possible that a maximally great being exists. (◇∃x Mx)
(4) If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
(5) But if a maximally great being exists in some possible world, then by the definition of maximal greatness, it exists in every possible world.
(6) Therefore, a maximally great being exists in the actual world.

模态版本（普兰丁格）：
(1) 如果一个存在者在所有可能世界中都是全能、全知且全善的，那么它具有最大卓越性。
(2) 如果一个存在者在所有可能世界中都具有最大卓越性，那么它具有最大伟大性。
(3) 可能存在一个具有最大伟大性的存在者。(◇∃x Mx)
(4) 如果可能存在一个具有最大伟大性的存在者，那么在某个可能世界中存在这样一个存在者。
(5) 但根据最大伟大性的定义，如果这样一个存在者在某个可能世界中存在，那么他在所有可能世界中都存在。
(6) 因此，一个具有最大伟大性的存在者在现实世界中存在。

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10. Political Philosophy: Principles of Distributive Justice | 政治哲学：分配正义原则

In political philosophy, rival theories of justice can be expressed as distributive formulas. These abstract principles guide the evaluation of social institutions.

在政治哲学中，不同的正义理论可以表述为分配公式。这些抽象原则指导着对社会制度的评价。

Rawls’s Difference Principle: Social and economic inequalities are to be arranged so that they are both (a) to the greatest benefit of the least advantaged, and (b) attached to offices and positions open to all under conditions of fair equality of opportunity. The ‘formula’ priorities the position of the worst-off group: an inequality is just only if it improves the absolute position of the least advantaged compared to any feasible alternative arrangement.

罗尔斯的差别原则： 社会和经济的不平等应这样安排，使它们 (a) 最有利于最不利者，并且 (b) 在公平的机会平等的条件下，职务和地位向所有人开放。该“公式”将最不利群体的地位置于优先：一种不平等是正义的，仅当相较于任何可行的替代安排，它都改善了最不利者的绝对地位。

Nozick’s Entitlement Theory: A distribution is just if it arises through just acquisition (initial acquisition of unowned things) or just transfer (voluntary exchange). No pattern or end-state formula is required; justice is purely historical. Formula: D₁ is just if all holdings in D₁ were acquired justly; otherwise, rectification is needed.

诺齐克的资格理论： 一种分配是正义的，如果它产生于正义的获取（无主物的最初获取）或正义的转让（自愿交换）。不需要任何模式化或目的状态公式；正义纯粹是历史的。公式：分配 D₁ 是正义的，当且仅当 D₁ 中的所有持有都是正义地获取的；否则就需要矫正。

Utilitarian Justice: The just distribution is the one that maximises total or average utility across society. This can conflict strongly with individual rights, which is a key objection. Formula: choose distribution D that maximises ΣU(i).

功利主义正义观： 正义的分配就是那个能使社会总体或平均功利最大化的分配。这可能会与个体权利产生严重冲突，这是一个关键的反对意见。公式：选择能使 ΣU(i) 最大化的分配 D。

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11. Meta-Ethics: The Open Question and Naturalistic Fallacy | 元伦理学：开放问题与自然主义谬误

Moore’s meta-ethical argument is a logical ‘formula’ designed to show that ‘good’ cannot be defined in naturalistic terms. It is essential for understanding the is-ought gap.

摩尔的元伦理学论证是一个逻辑“公式”，旨在表明“善”不能用自然主义的术语来定义。这对理解“是-应”鸿沟至关重要。

Open Question Argument: For any proposed naturalistic definition of ‘good’ (e.g., good = pleasure), it remains an open, intelligible question whether that natural property really is good. If the definition were correct, the question would be closed (like ‘Is a bachelor unmarried?’). Because it is open, the definition fails. Moore concludes that goodness is a simple, non-natural property.

开放问题论证： 对于任何关于“善”的自然主义定义（例如，善 = 快乐），问“这个自然属性真的是善的吗？”仍然是一个开放的、可理解的问题。如果定义是正确的，这个问题就应该是闭合的（就像“单身汉是未婚的吗？”一样）。因为这个问题是开放的，所以该定义是失败的。摩尔得出结论：善是一种简单的、非自然的属性。

Naturalistic Fallacy: Any attempt to deduce an ‘ought’ from an ‘is’ without the aid of an evaluative premise commits the naturalistic fallacy. The ‘formula’ is a bridging principle: you cannot validly infer a normative conclusion from purely descriptive premises alone. Hume’s Law: no set of factual statements entails a value statement.

自然主义谬误： 任何试图在没有评价性前提辅助的情况下从“是”推出“应”的做法都犯了自然主义谬误。这里的“公式”是一个衔接原则：你不能仅从纯粹描述性的前提中有效地推出规范性的结论。休谟法则：任何事实陈述的集合都不能蕴含价值陈述。

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12. Aesthetic Value: The Formula of Disinterestedness | 审美价值：无利害性公式

Kant’s aesthetics provides a formula for pure judgements of taste, which are central to A2 discussions of aesthetic objectivity and subjectivity.

康德的美学提供了一种纯粹鉴赏判断的公式，这在 A2 关于审美客观性和主观性的讨论中处于中心地位。

Kant’s Formula of Taste: A pure judgement of beauty is (1) disinterested (not concerned with the object’s real existence), (2) universal (we claim that everyone ought to agree), (3) purposive without purpose (the object appears designed but we ascribe no determinate purpose to it), and (4) necessary (based on a common sense that grounds the necessity of agreement).

康德的鉴赏公式： 一个纯粹的美的判断是 (1) 无利害的（不关心对象的实际存在），(2) 普遍的（我们声称每个人都应该同意），(3) 无目的的合目的性（对象看起来像是被设计的，但我们不把任何特定目的赋予它），(4) 必然的（基于一种奠定同意之必然性的共通感）。

Antinomy of Taste: Thesis: ‘Judgements of taste are not based on concepts, for otherwise they would be open to dispute (decision by proofs).’ Antithesis: ‘Judgements of taste are based on concepts, for otherwise we could not even contend about them (lay claim to necessary agreement).’ The resolution is that aesthetic judgements are based on an indeterminate concept (the supersensible substrate of humanity).

鉴赏的二律背反： 正题：“鉴赏判断不是基于概念的，否则就可以允许争论（通过证明来裁决）了。”反题：“鉴赏判断是基于概念的，否则我们甚至不能对它们产生争执（要求必然的同意）。”解决方法是：审美判断基于一个不确定的概念（人类的超感官基底）。

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