📚 AS CIE Philosophy: Formula & Theorem Quick Reference Handbook | AS CIE 哲学：公式定理速查手册
Philosophy in the CIE AS syllabus may look like a free-flowing essay subject, but underpinning it are precise logical structures, argument forms, and conceptual tools that function exactly like 'formulae' in science. This quick reference handbook collects the essential propositional logic rules, valid argument forms, fallacies, and core philosophical principles you must be able to identify and apply in exam responses. Each entry is presented as a compact formula-like statement followed by its meaning and use. Keep this guide at hand when analysing exam passages, constructing your own arguments, or revising key thinkers such as Descartes and Hume. Mastering these foundations will sharpen your critical reasoning and elevate the quality of your written answers.
CIE AS 阶段的哲学看似是自由写作科目，但其底层是精确的逻辑结构、论证形式和概念工具，功能与科学中的“公式”如出一辙。这本速查手册汇集了你必须能在考试中识别和应用的基本命题逻辑规则、有效论证形式、常见谬误以及核心哲学原理。每个条目都以紧凑的公式式陈述呈现，附带含义和用法说明。无论是分析考题文本、构建自己的论证，还是复习笛卡尔和休谟等关键思想家，都可以随时翻阅这份指南。掌握这些基础将显著提升你的批判性推理能力和答题水平。
1. Modus Ponens & Modus Tollens | 肯定前件与否后件
Modus Ponens (affirming the antecedent) and Modus Tollens (denying the consequent) are the two most fundamental valid forms of deductive reasoning. Any CIE exam question that asks you to assess the logical structure of an argument will almost certainly require you to recognise these patterns.
肯定前件（Modus Ponens）和否后件（Modus Tollens）是两种最基本且有效的演绎推理形式。几乎每一道要求你评估论证逻辑结构的 CIE 考题都需要你识别这两种模式。
Modus Ponens: (P → Q), P ⊢ Q
If P implies Q, and P is true, then Q must be true. For example: If it is raining (P), the ground is wet (Q). It is raining (P). Therefore, the ground is wet (Q). This form is always truth-preserving.
如果 P 蕴含 Q，并且 P 为真，那么 Q 必然为真。例如：如果下雨（P），地面是湿的（Q）。现在下雨了（P）。因此，地面是湿的（Q）。该形式永远保存真值。
Modus Tollens: (P → Q), ¬Q ⊢ ¬P
If P implies Q, and Q is false, then P must be false. Using the same conditional: If it is raining, the ground is wet. The ground is not wet (¬Q). Therefore, it is not raining (¬P). A common exam mistake is to confuse this with the fallacy of denying the antecedent.
如果 P 蕴含 Q，并且 Q 为假，那么 P 必然为假。沿用同一个条件：如果下雨，地面是湿的。地面没有湿（¬Q）。因此，没有下雨（¬P）。考试中常见的错误是将其与否前件谬误混淆。
2. Hypothetical Syllogism & Disjunctive Syllogism | 假言三段论与选言三段论
These extended argument forms are built from the same conditional reasoning and are treated as 'theorems' of propositional logic. They frequently appear in multi-step philosophical reasoning passages.
这些扩展的论证形式建立于同样的条件句推理之上，被视为命题逻辑的“定理”。它们经常出现在多步骤的哲学推理段落中。
Hypothetical Syllogism: (P → Q), (Q → R) ⊢ (P → R)
If P implies Q, and Q implies R, then P implies R. Example: If we have free will, we are morally responsible. If we are morally responsible, punishment is justified. Therefore, if we have free will, punishment is justified. This chaining is valid.
如果 P 蕴含 Q，并且 Q 蕴含 R，那么 P 蕴含 R。例如：如果我们有自由意志，我们就有道德责任。如果我们有道德责任，惩罚就是正当的。因此，如果我们有自由意志，惩罚就是正当的。这种链接推理是有效的。
Disjunctive Syllogism: (P ∨ Q), ¬P ⊢ Q
Either P or Q. Not P. Therefore Q. This works only for an inclusive 'or' in standard logic. Be careful: if the context suggests an exclusive 'or' (one but not both), the inference remains valid in the sense that at least one disjunct holds.
要么 P，要么 Q。非 P。因此 Q。这在标准逻辑中仅适用于相容性的“或”。注意：如果语境暗示不相容的“或”（二者仅一），推理在有且仅有一项成立的意义上依然有效。
These two syllogisms together with Modus Ponens and Modus Tollens form the core deductive toolkit for the AS Philosophy exam. Know them thoroughly.
这两个三段论与肯定前件、否后件共同构成了 AS 哲学考试中演绎推导的核心工具包，务必熟练掌握。
3. Truth Table Fundamentals | 真值表基础
Truth tables are the calculus of propositional logic. Mastering them allows you to test for validity, tautology, contradiction, and logical equivalence. For AS CIE, you are expected to construct and interpret truth tables for basic compound propositions.
真值表是命题逻辑的演算工具。掌握它你可以检验有效性、重言式、矛盾式以及逻辑等价。AS CIE 要求你会建构并解读基本复合命题的真值表。
The standard connectives and their truth-functional meanings are shown below. Memorise these as you would mathematical multiplication tables:
以下是标准逻辑联结词及其真值函项含义，请熟记如乘法表：
| P | Q | P ∧ Q | P ∨ Q | P → Q | ¬P |
|---|---|---|---|---|---|
| T | T | T | T | T | F |
| T | F | F | T | F | F |
| F | T | F | T | T | T |
| F | F | F | F | T | T |
A conditional P → Q is only false when P is true and Q is false. A conjunctive statement P ∧ onstrate validity by showing that whenever premises are true, the conclusion is also true.
要点：条件句 P → Q 仅在 P 真且 Q 假时为假；合取命题 P ∧ Q 仅在两者皆真时为真。用这些表格来证明有效性：展示前提都为真的每一行里，结论也为真。
4. De Morgan's Laws & Logical Equivalences | 德摩根律与逻辑等价式
De Morgan's Laws allow you to translate negations of conjunctions and disjunctions. They are essential for analysing ordinary language arguments and simplifying logical expressions. In the exam, you may be asked to rewrite statements using these equivalences.
德摩根律让你能够转化合取和析取的否定式。这对于分析日常语言论证和简化逻辑表达式至关重要。考试可能会要求你运用这些等价式改写句子。
¬(P ∧ Q) ≡ ¬P ∨ ¬Q
¬(P ∨ Q) ≡ ¬P ∧ ¬Q
Example: "It is not the case that the liar is both sincere and dishonest" becomes "Either the liar is not sincere or the liar is not dishonest." The second law transforms "It is not true that she is either a Platonist or an Aristotelian" into "She is neither a Platonist nor an Aristotelian."
例如：“说谎者并非既真诚又不诚实”可以转化为“说谎者或者不真诚，或者不（不诚实）”。第二条律则将“她不是柏拉图主义者或亚里士多德主义者，这不成立”转化为“她既不是柏拉图主义者也不是亚里士多德主义者”。
Also memorise: Double Negation (¬¬P ≡ P), Contraposition (P → Q ≡ ¬Q → ¬P), and the definition of the biconditional: P ↔ Q ≡ (P → Q) ∧ (Q → P). These are recurring 'theorems' you can cite.
务必同时记住：双重否定律（¬¬P ≡ P）、换质位法（P → Q ≡ ¬Q → ¬P）以及双条件的定义：P ↔ Q ≡ (P → Q) ∧ (Q → P)。这些都是你可以引用的常见“定理”。
5. Inductive Reasoning Standards | 归纳推理标准
Unlike deductive 'formulae', inductive reasoning is probabilistic, yet it has its own criteria of strength which act as evaluative formulae. In AS Philosophy, you will discuss the problem of induction and assess inductive arguments in science and everyday life.
与演绎“公式”不同，归纳推理是概率性的，但它有自己的强度标准，可作为评估公式。在 AS 哲学中，你将讨论归纳问题，并评估科学和日常生活中的归纳论证。
The basic structure of an inductive generalisation can be represented as:
Premise: n observed A are B.
Conclusion: All/most A are B. (where the sample is large, random, representative).
归纳概括的基本结构可以表示为：
前提：所观察到的 n 个 A 都是 B。
结论：所有/大多数 A 是 B。（其中样本应足够大、随机且具有代表性）。
Hume's problem reminds us that this inferential leap presupposes the Uniformity of Nature (UN): The future will resemble the past. But UN itself cannot be justified deductively without circularity. So inductive reasoning lacks a rational 'formula' guaranteeing truth; its strength depends on the sample's quality and the absence of counter-evidence. In your essays, always weigh these factors.
休谟的问题提醒我们，这一推理跳跃预设了“自然的齐一性”（UN）：未来将与过去相似。但齐一性本身无法在不陷入循环的情况下得到演绎辩护。因此归纳推理缺乏保证真理的理性“公式”；其强度取决于样本质量和反例的缺失。在写作论文时，务必权衡这些因素。
6. Common Fallacies: The Invalid Formulae | 常见谬误：无效的“公式”
Fallacies are persuasive but logically incorrect patterns. Recognising them is as vital as knowing the valid forms. They often hide in AS exam source material alongside a request to 'evaluate the reasoning'.
谬误是具有说服力但在逻辑上错误的模式。识别它们与了解有效形式同样重要。它们经常潜伏在 AS 考题的阅读材料中，并伴有“评价推理”的要求。
Affirming the Consequent: (P → Q), Q ⊢ P (INVALID)
If the burglar entered through the window, there would be footprints. There are footprints. Therefore, the burglar entered through the window. The conclusion does not follow because footprints could have another cause.
如果盗贼通过窗户进入，就会留下脚印。现场有脚印。因此盗贼是通过窗户进入的。该结论不成立，因为脚印可能有其他原因。
Denying the Antecedent: (P → Q), ¬P ⊢ ¬Q (INVALID)
If it is a cat, it is a mammal. It is not a cat. Therefore, it is not a mammal. Obviously false; other mammals exist.
如果它是猫，它就是哺乳动物。它不是猫。因此它不是哺乳动物。显然错误，因为其他哺乳动物也存在。
Other exam favourites include: circular reasoning (petitio principii), ad hominem, false dilemma, equivocation, and the appeal to nature. When you spot a fallacy, explicitly state which logical 'formula' it violates—this impresses examiners.
考试中常见的其他谬误包括：循环论证、人身攻击、虚假二难、偷换概念和诉诸自然。当你发现一个谬误时，明确说出它违反了哪一条逻辑“公式”，这会给阅卷人留下深刻印象。
7. Theories of Truth as Conceptual Formulae | 真理理论作为概念公式
Three classical theories of truth function like meta-formulae that tell us how to 'assign' truth to statements. You must compare them in the epistemology section of the syllabus.
三种经典的真理理论就像元公式，告诉我们如何给陈述“赋予”真值。你需要在认识论部分对它们进行比较。
- Correspondence Theory: A proposition P is true iff P corresponds to a fact in the world. Formulaic test: P is true ↔ there exists a fact F such that P matches F.
- 符合论：命题 P 为真，当且仅当 P 与世界中的某个事实相符合。公式化检验：P 为真 ↔ 存在事实 F，使 P 与 F 相符。
- Coherence Theory: P is true iff P coheres with a system of mutually supportive beliefs. Formula: P is true ↔ consistency with the total belief set.
- 融贯论：P 为真，当且仅当 P 与一组相互支持的信念系统融贯。公式：P 为真 ↔ 与整体信念集合一致。
- Pragmatic Theory: P is true iff belief in P proves useful and works in practice. Formula: P is true ↔ holding P leads to successful predictions or satisfaction.
- 实用论：P 为真，当且仅当相信 P 在实践中被证明有用并有效。公式：P 为真 ↔ 持有 P 能带来成功的预测或满足。
Use these succinct formulations when explaining why a certain epistemological claim is considered true or false by a particular philosopher. They are your reference theorems for the exam.
解释某位哲学家为何认为某个认识论断定为真或假时，使用这些简明的表述。它们就是你考试中的参考定理。
8. Descartes' Methodological Doubt & Cogito | 笛卡尔的方法论怀疑与“我思”
Descartes' argument in the Meditations can be distilled into a logical sequence resembling a proof. Understanding this chain is non-negotiable for AS CIE Philosophy candidates.
笛卡尔在《第一哲学沉思集》中的论证可以蒸馏为一个类似证明的逻辑序列。对于 AS CIE 哲学考生而言，理解这个链条是没有商量余地的。
Doubt Formula: If a belief is susceptible to any possible doubt, treat it as false.
怀疑公式：如果一个信念可受任何可能的怀疑，就暂时视其为假。
From this, he deploys the dream argument and the evil demon to undermine all sensory and even mathematical beliefs. The turning point is the cogito: 'I think, therefore I am' (Cogito ergo sum). Its logical core: the very act of doubting one's own existence proves that there must be a thinking subject doing the doubting. Structure:
由此出发，他运用梦境论证和恶魔论证来破坏所有感官甚至数学信念。转折点是“我思故我在”（Cogito ergo sum）。其逻辑内核：对自身存在进行怀疑这一行为本身，就证明必然有一个正在怀疑的思维主体存在。结构如下：
thinking. → If I am thinking, I exist. → I am doubting (undeniable). → Therefore, I exist.
如果我在怀疑，那么我在思考。→ 如果我在思考，我即存在。→ 我在怀疑（不可否认）。→ 因此，我存在。
This deductive kernel is presented as an indubitable truth that survives even the evil demon challenge. A common exam question asks you to evaluate whether the cogito is a syllogism or an immediate intuition. Either way, treat this as a foundational 'theorem' of rationalist epistemology.
这一演绎内核作为不可怀疑的真理而呈现，即便邪恶恶魔的挑战也无法撼动它。常见的考题要求你评价“我思”究竟是一个三段论还是一种直接的直观。无论哪种，都请将其视为理性主义认识论的基础“定理”。
9. Hume's Fork: The Division of Knowledge | 休谟之叉：知识的划分
David Hume's fork is a conceptual axiom that slices all meaningful propositions into two types. It functions as a formula for determining what counts as genuine knowledge in empiricist philosophy.
大卫·休谟的“叉子”是一个概念公理，将所有有意义的命题切分为两类。它作为一条公式，决定了在经验主义哲学中什么可以算作真正的知识。
Hume's Fork: All objects of human reason are either Relations of Ideas or Matters of Fact.
休谟之叉：人类理性的一切对象要么是观念的关系，要么是事实。
- Relations of Ideas: Demonstrable by thought alone, denial leads to contradiction. (e.g., 'All triangles have three angles'; 'Bachelors are unmarried'.) These are a priori and analytic. Formula: Truth is known by logic and definition.
- 观念的关系：仅凭思考即可证明，否定会陷入矛盾。（如“所有三角形都有三个角”；“单身汉是未婚的”。）这些是先天的和分析的。公式：真理由逻辑和定义得知。
- Matters of Fact: Known only through experience, the contrary is logically possible. (e.g., 'The sun will rise tomorrow'.) These are a posteriori and synthetic. Formula: Truth is established by empirical evidence and causality.
- 事实：只能通过经验得知，其反面是逻辑上可能的。（如“太阳明天会升起”。）这些是后天的和综合的。公式：真理由经验证据和因果关系确立。
Hume then applies this fork critically: if a book contains neither abstract reasoning about quantity/number nor experimental reasoning about fact, then commit it to the flames. In the exam, use this fork to test whether a claim—say, about the existence of God or substance—has cognitive content. It is the ultimate 'filtering theorem'.
休谟接着运用这把叉子进行批判：如果一本书既不包含关于量或数的抽象推理，也不包含关于事实的实验推理，那就将它付之一炬吧。在考试中，请用这把叉子检验例如关于上帝或实体存在的声称是否具有认知内容。它是终极的“过滤定理”。
10. Ethical Theories: Decision Procedures as Formulae | 伦理学理论：作为公式的决策程序
Normative ethical theories can be expressed as decision-making formulae. In AS CIE you compare Utilitarianism, Deontology, and Virtue Ethics. Treating them as 'formulae' helps you make precise comparisons.
规范伦理学理论可以表达为决策公式。在 AS CIE 中你需要比较功利主义、义务论和德性伦理学。将它们当作“公式”来对待，有助于进行精确的比较。
Act Utilitarianism Formula: An action is right iff it maximises expected aggregate happiness/pleasure over pain.
行为功利主义公式：一个行为是正确的，当且仅当它在总体上最大化预期的幸福/快乐超过痛苦。
Kantian Categorical Imperative (Formula of Universal Law): Act only on that maxim whereby you can at the same time will that it should become a universal law without contradiction.
康德定言命令（普遍法则公式）：只依据那个你同时能够意愿它成为一条普遍法则的准则而行动，且不产生矛盾。
Aristotelian Virtue Formula: An action is right iff it is what a virtuous agent, acting in character, would do in the circumstances (the Doctrine of the Mean).
亚里士多德德性公式：一个行为是正确的，当且仅当它是一个有德之人在相同情境中出于品格而会做出的行为（中庸之道）。
In application questions on euthanasia or lying, deploy the appropriate formula: calculate the balance of pleasure for Utilitarianism, test the maxim's universalisability for Kant, or ask what the courageous/honest person would do for Virtue Ethics. This approach demonstrates analytical rigour.
在处理有关安乐死或说谎的应用题时，请运用相应的公式：用功利主义计算快乐与痛苦的平衡，用康德检验准则的可普遍化，或者用德性伦理学追问勇敢/诚实的人会怎么做。这一方法展示了分析的严谨性。
11. The Argument from Illusion & Sense-Data | 幻觉论证与感觉材料
In the philosophy of perception, the argument from illusion functions as a 'proof' that we cannot directly perceive physical objects. It is a recurrent theme in CIE AS epistemology exam questions.
在感知哲学中，幻觉论证充当了一条“证明”，表明我们无法直接感知物理对象。这是 CIE AS 认识论考题中反复出现的主题。
The Schema: (1) In illusion or hallucination, I am conscious of something. (2) That something is not a physical object. (3) The subjective experience is qualitatively indistinguishable from veridical perception. (4) Therefore, in all perception I am directly aware only of sense-data, not mind-independent objects.
论证图式：(1) 在幻觉中我意识到某物。(2) 该物不是物理对象。(3) 主观体验在质上与真实知觉无从区分。(4) 因此，在所有知觉中我直接意识到的仅仅是感觉材料，而非心灵独立的对象。
The logical flow resembles a substitution: if the internal mental state is identical in both delusive and veridical cases, then the common object of awareness must be internal sense-data. Critiques of this argument often target premise (3): does indistinguishable phenomenology really entail the same object? Think of this as a test of your ability to dissect deductive arguments in the exam.
其逻辑流程类似替代：如果内在的心理状态在错觉和真实案例中完全一致，那么所有觉知的共同对象必定是内在的感觉材料。对此论证的批评常针对前提(3)：不可区分的现象特征真的能推出对象相同吗？把它当作检验你在考试中解剖演绎论证能力的试金石。
12. The Tripartite Definition of Knowledge & Gettier Cases | 知识的三元定义与葛梯尔案例
The classic 'formula' ). AS candidates must state this precisely and explain how Gettier-style counterexamples challenge its sufficiency.
葛梯尔问题之前，经典的知识“公式”是：知识 = 已证成的真信念（JTB）。AS 考生必须精确陈述这一定义，并解释葛梯尔式的反例如何挑战其充分性。
JTB Formula: S knows that P iff (i) P is true; (ii) S believes that P; (iii) S is justified in believing that P.
JTB 公式：S 知道 P，当且仅当 (i) P 为真；(ii) S 相信 P；(iii) S 的相信是有证成的。
A Gettier case follows this structure: a justified belief that happens to be true through luck or coincidental connection. Example: Smith has strong evidence that 'Jones owns a Ford' (because he sees Jones driving a Ford daily). From this, Smith justifiably deduces 'Either Jones owns a Ford or Brown is in Barcelona', even though Smith has no idea where Brown is. It turns out Jones does not own a Ford—it was a rental—but coincidentally Brown is indeed in Barcelona. Smith's belief is justified, true, but intuitively not knowledge because the truth is not appropriately connected to the justification. The 'formula' thus needs a fourth condition to block epistemic luck. Being able to write this out formally shows mastery of epistemological rigour.
一个葛梯尔案例遵循以下结构：一个已证成的信念碰巧因运气或巧合而为真。例如：史密斯有强证据相信“琼斯有一辆福特车”（因为他天天看到琼斯开福特）。史密斯由此证成地推导出“要么琼斯有一辆福特，要么布朗在巴塞罗那”，尽管他对布朗行踪一无所知。结果琼斯并没有福特车——那是租来的——但巧合的是布朗的确在巴塞罗那。史密斯的信念有证成且为真，但直觉上不是知识，因为真值与证成之间缺乏恰当联系。因此该“公式”需要一个第四条件来排除认知运气。能够正式写出这一分析，展现了你对认识论严谨性的掌握。
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