📚 Year 12 OCR Philosophy: Formulae & Theorems Quick Reference Handbook | OCR 12年级哲学：公式定理速查手册
In mathematics and the sciences, a formula is a concise way to express a relationship, and a theorem is a proposition that has been proved from accepted premises. In philosophy we work with the same tools under different names: structured patterns of reasoning, argument blueprints, definitional schemas and foundational principles. This handbook collects the essential logical forms, argument templates and core philosophical principles that run through the OCR Year 12 course, presenting them as a quick-reference guide so that you can analyse arguments faster and write sharper essays.
在数学和科学中，公式是表达关系的简洁方式，定理是从公认前提出发被证明的命题。在哲学中我们使用同样的工具，只是名称不同：结构化的推理模式、论证蓝图、定义图式和基本原则。本手册汇集了贯穿 OCR 12 年级课程的基本逻辑形式、论证模板和核心哲学原则，以速查指南的形式呈现，帮助你更快地分析论证、写出更犀利的论文。
1. Valid Deductive Argument Forms | 有效演绎论证形式
A valid deductive argument is one where, if the premises are true, the conclusion must be true. Validity is about the form or structure of the argument, not about whether the premises actually are true. The following forms are the ones you will use most often when analysing and constructing philosophical arguments.
有效的演绎论证是指：如果前提为真，结论必然为真。有效性关乎论证的形式或结构，而不关乎前提实际上是否为真。以下是你分析和构建哲学论证时最常用的形式。
Modus Ponens (Affirming the Antecedent): If P then Q; P is true; therefore, Q is true. This is the workhorse of philosophical reasoning and appears in almost every essay you will write.
肯定前件式：如果 P 则 Q；P 为真；因此 Q 为真。这是哲学推理的主力形式，几乎出现在你写的每一篇论文中。
P → Q, P ⊢ Q
Modus Tollens (Denying the Consequent): If P then Q; Q is false; therefore, P is false. This form is especially useful for refuting a rival position by showing that its consequence is false.
否定后件式：如果 P 则 Q；Q 为假；因此 P 为假。这一形式在通过指出对方结论为假来反驳其立场时特别有用。
P → Q, ¬Q ⊢ ¬P
Hypothetical Syllogism (Chain Argument): If P then Q; if Q then R; therefore, if P then R. Long philosophical arguments are frequently chains of this kind, and spotting the chain helps you locate the weakest link.
假言三段论（连锁论证）：如果 P 则 Q；如果 Q 则 R；因此，如果 P 则 R。长篇哲学论证常常是这种链条，识别出链条有助于你找到最薄弱的一环。
P → Q, Q → R ⊢ P → R
Disjunctive Syllogism: Either P or Q; not P; therefore, Q. Note that this form only works when the disjunction is exclusive or at least when both options are genuinely on the table.
选言三段论：要么 P 要么 Q；非 P；因此 Q。注意这一形式只在选言为排斥性、或至少两个选项确实都被摆上台面时才成立。
P ∨ Q, ¬P ⊢ Q
Constructive Dilemma: If P then Q; if R then S; either P or R; therefore, either Q or S. This is the formal skeleton behind many 'either way, the conclusion follows' moves in ethics and philosophy of religion.
构成式二难推理：如果 P 则 Q；如果 R 则 S；要么 P 要么 R；因此，要么 Q 要么 S。许多伦理学与宗教哲学中"无论哪种情况，结论都成立"的论证，其形式骨架正是这一条。
P → Q, R → S, P ∨ R ⊢ Q ∨ S
Remember the distinction between validity and soundness: an argument is sound only if it is valid and all its premises are actually true. Soundness is what you aim for in your own reasoning; validity is what you check in someone else's.
记住有效性与可靠性的区别：一个论证只有在既有效、其所有前提又确实为真时，才是可靠的。可靠性是你自己推理时追求的目标；有效性是你检查他人论证时检验的东西。
2. Invalid Deductive Forms & Fallacies | 无效演绎形式与谬误
Recognising invalid patterns is just as important as knowing the valid ones, because fallacies frequently appear disguised in philosophical writing, in exam stimulus material, and in your own first drafts. The following formal fallacies are the ones examiners most often test.
识别无效模式与了解有效模式同样重要，因为谬误常常伪装出现在哲学写作、考试材料和你的初稿中。以下形式谬误是考官最常考查的。
Affirming the Consequent: If P then Q; Q is true; therefore, P is true. This is invalid because Q could be true for reasons entirely independent of P. A classic example: if it has rained, the pavement is wet; the pavement is wet; therefore it has rained — but a sprinkler would do just as well.
肯定后件：如果 P 则 Q；Q 为真；因此 P 为真。这是无效的，因为 Q 可能由于与 P 完全无关的原因而为真。经典例子：如果下过雨，地面就湿；地面湿；因此下过雨——但洒水器也能造成同样的结果。
P → Q, Q ⊬ P
Denying the Antecedent: If P then Q; P is false; therefore, Q is false. Q might still be true for other reasons, so the conclusion does not follow. Together, affirming the consequent and denying the antecedent are the two mistakes you should hunt for first when evaluating a conditional argument.
否定前件：如果 P 则 Q；P 为假；因此 Q 为假。Q 仍可能因其他原因而为真，所以结论推不出来。肯定后件与否定前件是你在评价条件论证时应首先查找的两个错误。
P → Q, ¬P ⊬ ¬Q
Undistributed Middle: All A are B; all C are B; therefore, all A are C. The middle term B is never said to cover all of A or all of C, so the link collapses.
中项不周延：所有 A 都是 B；所有 C 都是 B；因此所有 A 都是 C。中项 B 从未被断定涵盖全部 A 或全部 C，所以联系崩塌。
∀x(Ax → Bx), ∀x(Cx → Bx) ⊬ ∀x(Ax → Cx)
Informal fallacies are not errors of form but errors of content, relevance or evidence. The table below lists the ones that appear most often in A Level essays.
非形式谬误不是形式上的错误，而是内容、相关性或证据上的错误。下表列出 A Level 论文中最常出现的几种。
| Fallacy | English description | 中文说明 |
| Begging the question | Assuming the conclusion among the premises. | 把结论当作前提之一来使用。 |
| False dilemma | Presenting two options when more exist. | 只给两个选项，实际上还有更多。 |
| Straw man | Refuting a weakened version of the opponent's view. | 反驳对手观点的弱化版本。 |
| Ad hominem | Attacking the person rather than the argument. | 攻击人而非攻击论证。 |
| Hasty generalisation | Drawing a broad conclusion from too little evidence. | 证据太少就下宽泛结论。 |
| Slippery slope | Claiming one step inevitably leads to an extreme outcome. | 声称一步必然导致极端后果。 |
3. Categorical Logic: Syllogisms and the Square of Opposition | 范畴逻辑：三段论与对当方阵
Categorical propositions relate two classes or categories, and they come in four standard forms. Mastering these forms lets you translate ordinary philosophical prose into a testable structure.
范畴命题关联两个类别，共有四种标准形式。掌握这些形式，你就能把日常哲学表述翻译成可检验的结构。
- A proposition (universal affirmative): All S are P. — A 命题（全称肯定）：所有 S 都是 P。
- E proposition (universal negative): No S are P. — E 命题（全称否定）：没有 S 是 P。
- I proposition (particular affirmative): Some S are P. — I 命题（特称肯定）：有些 S 是 P。
- O proposition (particular negative): Some S are not P. — O 命题（特称否定）：有些 S 不是 P。
The Square of Opposition states the logical relations between these four forms. A and O are contradictories: they cannot both be true and cannot both be false. E and I are likewise contradictories. A and E are contraries: they cannot both be true, but they can both be false. I and O are subcontraries: they cannot both be false, but they can both be true.
对当方阵说明了这四种形式之间的逻辑关系。A 与 O 是矛盾关系：不能同真，也不能同假。E 与 I 同样是矛盾关系。A 与 E 是反对关系：不能同真，但可以同假。I 与 O 是下反对关系：不能同假，但可以同真。
A ⊣ O, E ⊣ I (contradictories); A ⊥ E (contraries); I ⊥ O (subcontraries)
A categorical syllogism contains exactly three terms — major, minor and middle — and exactly three propositions. A syllogism is valid only if the middle term is distributed in at least one premise. Distribution rules are the quickest way to test validity without drawing diagrams.
范畴三段论恰好包含三个词项——大项、小项和中项——以及三个命题。只有当中项至少在一个前提中周延，三段论才有效。周延规则是不画图就能检验有效性的最快方法。
- Universal propositions (A, E) distribute their subject term. — 全称命题（A、E）周延其主项。
- Negative propositions (E, O) distribute their predicate term. — 否定命题（E、O）周延其谓项。
- No conclusion may contain a term that was undistributed in the premises. — 结论中不得出现前提里未周延的词项。
In essays, categorical analysis is most useful when a philosopher makes a sweeping claim, such as 'all knowledge derives from experience'. Translating that into A-form exposes exactly what would have to be true for the claim to hold, and what a single counterexample would do to it.
在论文中，当哲学家作出宽泛断言（如"一切知识都来自经验"）时，范畴分析最为有用。把它翻译成 A 命题形式，就能暴露出该主张成立所需的条件，以及一个反例会造成什么后果。
4. Propositional Logic: Connectives and Truth Tables | 命题逻辑：联结词与真值表
Propositional logic treats whole statements as units and builds complex statements out of five connectives. Knowing the exact truth conditions of each connective is the foundation of all formal evaluation in the course.
命题逻辑把整个陈述当作单位，用五个联结词构建复合陈述。掌握每个联结词的真值条件，是本课程全部形式化评价的基础。
- Negation ¬P: true exactly when P is false. — 否定 ¬P：当且仅当 P 为假时为真。
- Conjunction P ∧ Q: true only when both conjuncts are true. — 合取 P ∧ Q：仅当两个合取支都为真时为真。
- Disjunction P ∨ Q: false only when both disjuncts are false. — 析取 P ∨ Q：仅当两个析取支都为假时为假。
- Conditional P → Q: false only when P is true and Q is false. — 条件 P → Q：仅当 P 真而 Q 假时为假。
- Biconditional P ↔ Q: true when both sides share the same truth value. — 双条件 P ↔ Q：两侧真值相同时为真。
The conditional is the connective students most often misread. A conditional says nothing about the truth of its antecedent; it only rules out one combination, namely true antecedent with false consequent. This is why 'if P then Q' is true whenever P is false, a result known as vacuous truth.
条件句是学生最容易误读的联结词。条件句并不断言前件的真假，它只排除一种组合：前件真而后件假。这就是为什么当 P 为假时"如果 P 则 Q"为真，这一结果被称为空真。
| P | Q | P → Q | P ↔ Q |
| T | T | T | T |
| T | F | F | F |
| F | T | T | F |
| F | F | T | T |
A tautology is a formula true under every assignment of truth values; a contradiction is false under every assignment; a contingency is true under some and false under others. De Morgan's laws let you move negation through brackets, and they are the standard tool for showing that two formulations are logically equivalent.
重言式是在所有真值赋值下都为真的公式；矛盾式是在所有赋值下都为假的公式；偶然式在某些赋值下真、在另一些赋值下假。德摩根定律让你把否定移入括号内，是证明两种表述逻辑等值的标准工具。
¬(P ∧ Q) ≡ ¬P ∨ ¬Q ¬(P ∨ Q) ≡ ¬P ∧ ¬Q
Finally, distinguish the turnstile from the double turnstile: ⊢ means 'is provable from', while ⊨ means 'is a logical consequence of'. For the propositional systems you meet at this level the two coincide, but the concepts are not the same.
最后要区分单转向符与双转向符：⊢ 意为"可从中证明"，而 ⊨ 意为"是其逻辑后承"。在你这个阶段接触的命题系统中两者重合，但概念并不相同。
5. Necessary and Sufficient Conditions | 充分必要条件
Almost every definitional debate in philosophy turns on necessary and sufficient conditions. Learning to state them precisely is worth more marks than almost any other single skill in the course.
哲学中几乎每一次定义之争都围绕充分必要条件展开。学会精确表述它们，比课程中几乎任何其他单项技能都更能得分。
P is sufficient for Q when the presence of P guarantees Q. In symbols, P → Q. P is necessary for Q when Q cannot occur without P, so Q → P, or equivalently ¬P → ¬Q.
P 是 Q 的充分条件：P 出现即保证 Q。符号表示为 P → Q。P 是 Q 的必要条件：没有 P 就不可能有 Q，即 Q → P，等价于 ¬P → ¬Q。
P sufficient for Q ⇔ P → Q P necessary for Q ⇔ Q → P ⇔ ¬P → ¬Q
Being a bachelor is sufficient for being unmarried, but it is not necessary, since a person can be unmarried without being a bachelor. The classic test for a proposed necessary condition is to find a case where the condition fails but the target still occurs; the classic test for a proposed sufficient condition is to find a case where the condition holds but the target does not.
"单身汉"是"未婚"的充分条件，但不是必要条件，因为一个人可以未婚而不是单身汉。检验某个必要条件是否成立，经典方法是找一个条件不满足但目标仍出现的情况；检验某个充分条件是否成立，则找一个条件满足但目标未出现的情况。
This apparatus explains why definitions fail. A definition of knowledge as justified true belief claims that truth, belief and justification are jointly sufficient and individually necessary. Gettier's counterexamples attack the sufficiency half of that claim, while sceptical scenarios attack the necessity of justification.
这套工具解释了定义为何会失败。把知识定义为"被证成的真信念"，等于声称真、信念与证成三者合起来构成充分条件、且各自是必要条件。盖蒂尔的反例攻击的是充分性那一半，而怀疑论场景攻击的是证成的必要性。
6. Defining Knowledge: The Tripartite Analysis and Gettier | 知识的定义：三元分析与盖蒂尔
The traditional analysis of propositional knowledge comes from Plato's Theaetetus and is usually written as a three-part formula. S knows that P if and only if three conditions are jointly satisfied.
关于命题知识的传统分析源自柏拉图的《泰阿泰德篇》，通常写成一个三部分公式。S 知道 P，当且仅当以下三个条件同时满足。
K(S, P) ⇔ T(P) ∧ B(S, P) ∧ J(S, P)
- Truth condition: P is true. You cannot know what is false. — 真之条件：P 为真。你无法知道虚假的东西。
- Belief condition: S believes that P. Knowledge implies belief. — 信念条件：S 相信 P。知识蕴含信念。
- Justification condition: S is justified in believing that P. — 证成条件：S 有理由相信 P。
Gettier's 1963 paper presents cases in which all three conditions hold but intuition says the subject does not know. In the standard example, Smith has strong evidence that Jones will get the job and that Jones has ten coins in his pocket; Smith therefore believes 'the man who will get the job has ten coins in his pocket'. It turns out that Smith himself gets the job and, unknown to him, also has ten coins in his pocket. The belief is true, believed and justified, yet it seems to be true only by luck.
盖蒂尔 1963 年的论文给出了三个条件都满足、但直觉认为主体并不知情的案例。在标准例子中，史密斯有强证据认为琼斯会得到这份工作、且琼斯口袋里有十枚硬币；
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