📚 A-Level Cambridge Philosophy: Logic Cheatsheet & Quick Reference Formulas | A-Level 剑桥哲学：逻辑公式定理速查手册
Mastering logical reasoning is essential for success in the Cambridge A-Level Philosophy examination (9699). This quick reference handbook summarises the key logical connectives, rules of inference, equivalence principles, quantifier operations, and common argument forms. It is designed as a compact revision aid, enabling you to recall formal tools instantly while tackling extended essays and structured questions.
掌握逻辑推理对于在剑桥 A-Level 哲学考试 (9699) 中取得好成绩至关重要。本速查手册总结了关键的逻辑联结词、推理规则、等价原理、量词操作和常见论证形式，旨在作为紧凑的复习辅助工具，帮助您在撰写长文和结构化问题答案时即时回忆形式工具。
1. Logical Connectives & Truth Tables | 逻辑联结词与真值表
In propositional logic, simple statements are combined using connectives to form compound propositions. Understanding their truth-functional behaviour is fundamental for evaluating arguments.
在命题逻辑中，简单命题通过联结词组合成复合命题。理解它们的真值函数行为是评估论证的基础。
| Symbol | English Name | 中文名称 | Truth Rule / 真值规则 |
|---|---|---|---|
| ¬ | Negation | 否定 | Reverses truth value. ¬P is true exactly when P is false. 翻转真值。¬P 为真仅当 P 为假。 |
| ∧ | Conjunction | 合取 | True only when both conjuncts are true. 仅当两个合取支都为真时为真。 |
| ∨ | Disjunction (inclusive) | 析取（相容） | True when at least one disjunct is true. 当至少一个析取支为真时为真。 |
| → | Conditional / Implication | 条件 / 蕴涵 | False only when antecedent is true and consequent is false. P → Q is equivalent to ¬P ∨ Q. 仅当前件真而后件假时为假。P → Q 等价于 ¬P ∨ Q。 |
| ↔ | Biconditional | 双条件 | True when both sides share the same truth value. P ↔ Q is (P → Q) ∧ (Q → P). 当两边真值相同时为真。P ↔ Q 即 (P → Q) ∧ (Q → P)。 |
2. Validity & Soundness | 有效性与可靠性
An argument is valid if and only if it is impossible for the premises to be true and the conclusion false. Validity concerns the logical structure, not the actual truth of the statements. A sound argument is one that is valid and whose premises are actually true. Soundness guarantees a true conclusion.
一个论证是 有效 的，当且仅当不可能出现前提真而结论假的情况。有效性关乎逻辑结构，与命题的实际真假无关。一个 可靠 的论证是有效且所有前提皆为真的论证。可靠性确保结论为真。
When analysing philosophical texts, always distinguish between attacking validity and challenging soundness. Spotting an invalid form instantly weakens a position, while questioning the truth of a premise attacks soundness.
在分析哲学文本时，要始终区分攻击有效性和挑战可靠性。识别出一个无效的形式会立刻削弱一个立场，而质疑前提的真实性则是在攻击可靠性。
3. Basic Argument Forms | 基本论证形式
Recognising these common valid argument forms saves time and helps construct rigorous philosophical arguments. Each form is displayed with its symbolic pattern and traditional name.
识别以下常见有效论证形式可以节省时间，并有助于构建严谨的哲学论证。每种形式均给出其符号模式和传统名称。
Modus Ponens (Affirming the Antecedent): P → Q, P ⊢ Q
肯定前件式 (Modus Ponens)：P → Q，P ⊢ Q
Modus Tollens (Denying the Consequent): P → Q, ¬Q ⊢ ¬P
否定后件式 (Modus Tollens)：P → Q，¬Q ⊢ ¬P
Hypothetical Syllogism: P → Q, Q → R ⊢ P → R
假言三段论：P → Q，Q → R ⊢ P → R
Disjunctive Syllogism: P ∨ Q, ¬P ⊢ Q
选言三段论：P ∨ Q，¬P ⊢ Q
Constructive Dilemma: (P → Q) ∧ (R → S), P ∨ R ⊢ Q ∨ S
构造性二难：(P → Q) ∧ (R → S)，P ∨ R ⊢ Q ∨ S
Destructive Dilemma: (P → Q) ∧ (R → S), ¬Q ∨ ¬S ⊢ ¬P ∨ ¬R
破坏性二难：(P → Q) ∧ (R → S)，¬Q ∨ ¬S ⊢ ¬P ∨ ¬R
4. Propositional Logic Rules of Inference | 命题逻辑推理规则
Natural deduction systems employ introduction and elimination rules for each connective. These rules govern how you may introduce a connective into a proof or extract information from it.
自然演绎系统为每个联结词设置了引入和消除规则。这些规则规定了如何在证明中引入一个联结词，或从中提取信息。
∧I (Conjunction Introduction): From P and Q, infer P ∧ Q.
∧I (合取引入)：由 P 和 Q 推出 P ∧ Q。
∧E (Conjunction Elimination): From P ∧ Q, infer P (or Q).
∧E (合取消去)：由 P ∧ Q 推出 P（或 Q）。
∨I (Disjunction Introduction): From P, infer P ∨ Q (for any Q).
∨I (析取引入)：由 P 可推出 P ∨ Q（无论 Q 是什么）。
∨E (Disjunction Elimination): If you have P ∨ Q, and both P ⊢ R and Q ⊢ R, then infer R.
∨E (析取消去)：若已有 P ∨ Q，且可分别从 P 推出 R 和从 Q 推出 R，则可推出 R。
→E (Conditional Elimination, Modus Ponens): From P → Q and P, infer Q.
→E (蕴涵消除，即肯定前件)：由 P → Q 和 P 推出 Q。
→I (Conditional Introduction): If assuming P allows you to derive Q, then infer P → Q, discharging the assumption.
→I (蕴涵引入)：若假设 P 能推导出 Q，则可推出 P → Q，同时撤销假设。
↔I (Biconditional Introduction): From P → Q and Q → P, infer P ↔ Q.
↔I (双条件引入)：由 P → Q 和 Q → P 推出 P ↔ Q。
↔E (Biconditional Elimination): From P ↔ Q, infer P → Q (or Q → P).
↔E (双条件消除)：由 P ↔ Q 推出 P → Q（或 Q → P）。
¬I (Negation Introduction, Reductio ad Absurdum): If assuming P leads to a contradiction (Q ∧ ¬Q), infer ¬P, discharging P.
¬I (否定引入，归谬法)：若假设 P 导致矛盾 (Q ∧ ¬Q)，则可推出 ¬P，同时撤销假设。
¬E (Negation Elimination): From a contradiction, infer any proposition. Ex falso quodlibet.
¬E (否定消除)：由矛盾可推出任何命题。从假命题可得任意结论。
5. Propositional Logic Equivalence Rules | 命题逻辑等价规则
These rules allow you to replace logically equivalent expressions within a proof. Use them to transform sentences into more convenient forms.
这些规则允许在证明中将逻辑等价的表达式互相替换。利用它们可将语句转化为更方便的形式。
Double Negation (DN): ¬¬P ≡ P
双重否定律：¬¬P 等价于 P
De Morgan's Laws (DeM): ¬(P ∧ Q) ≡ ¬P ∨ ¬Q; ¬(P ∨ Q) ≡ ¬P ∧ ¬Q
德摩根律：¬(P ∧ Q) 等价于 ¬P ∨ ¬Q；¬(P ∨ Q) 等价于 ¬P ∧ ¬Q
Commutation (Com): P ∧ Q ≡ Q ∧ P; P ∨ Q ≡ Q ∨ P
交换律：P ∧ Q 等价于 Q ∧ P；P ∨ Q 等价于 Q ∨ P
Association (Assoc): (P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R); same for ∨
结合律：(P ∧ Q) ∧ R 等价于 P ∧ (Q ∧ R)；∨ 同理
Distribution (Dist): P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R); P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)
分配律：P ∧ (Q ∨ R) 等价于 (P ∧ Q) ∨ (P ∧ R)；P ∨ (Q ∧ R) 等价于 (P ∨ Q) ∧ (P ∨ R)
Implication (Imp): P → Q ≡ ¬P ∨ Q
蕴涵等价：P → Q 等价于 ¬P ∨ Q
Contraposition (Contra): P → Q ≡ ¬Q → ¬P
假言易位：P → Q 等价于 ¬Q → ¬P
Exportation (Exp): (P ∧ Q) → R ≡ P → (Q → R)
输出律：(P ∧ Q) → R 等价于 P → (Q → R)
Tautology (Taut): P ∧ P ≡ P; P ∨ P ≡ P
重言律：P ∧ P 等价于 P；P ∨ P 等价于 P
6. Predicate Logic: Quantifier Rules | 谓词逻辑：量词规则
Predicate logic extends propositional logic with quantifiers that range over domains of objects. When applying these rules, you must observe careful restrictions on variable names to avoid fallacies.
谓词逻辑通过量词在对象域上进行扩展，运用这些规则时必须严格遵守变量名称的限制，以避免谬误。
∀E (Universal Elimination): From ∀x P(x), infer P(a) for any constant a in the domain.
∀E (全称消去)：由 ∀x P(x) 可推出 P(a)，其中 a 为域内任意常元。
∀I (Universal Introduction): If you can prove P(a) for an arbitrary constant a that does not appear in any undischarged assumption, then infer ∀x P(x).
∀I (全称引入)：若能证明对于任意专名 a 有 P(a)，且 a 不出现在任何未撤销的假设中，则可推出 ∀x P(x)。
∃I (Existential Introduction): From P(a), infer ∃x P(x).
∃I (存在引入)：由 P(a) 推出 ∃x P(x)。
∃E (Existential Elimination): If you have ∃x P(x) and, from P(a) for a fresh constant a that does not appear elsewhere, you can derive Q, then infer Q, discharging the assumption P(a).
∃E (存在消去)：若已有 ∃x P(x)，并且在一个不含其他假设的新常元 a 下从 P(a) 能推出 Q，则可推出 Q 并撤销假设 P(a)。
7. Common Fallacies to Avoid | 常见谬误
Invalid arguments often resemble valid forms superficially. Being able to name and diagnose these fallacies is vital for evaluative essays.
无效论证常常表面上类似有效形式。能够指认和诊断这些谬误对评估性论文至关重要。
Affirming the Consequent: P → Q, Q ⊢ P (invalid). Do not confuse with Modus Ponens.
肯定后件：P → Q，Q ⊢ P（无效）。切勿与肯定前件相混淆。
Denying the Antecedent: P → Q, ¬P ⊢ ¬Q (invalid).
否定前件：P → Q，¬P ⊢ ¬Q（无效）。
Begging the Question (Petitio Principii): Assuming the truth of the conclusion in one's premises.
循环论证（窃题）：在前提中已假定结论为真。
False Dilemma: Presenting two options as exhaustive when they are not.
假二难推理：将并非穷尽的两种选择当作唯一可能性。
Equivocation: Using a term in different senses within the same argument.
偷换概念：在同一论证中对一个词项使用不同含义。
8. Key Theorems & Laws | 重要定理与定律
These logical truths are ubiquitous in philosophical reasoning and can be cited in proofs or essays as established tautologies.
以下逻辑真理在哲学推理中无处不在，可作为已确证的重言式在证明或文章中引用。
Law of Excluded Middle (LEM): P ∨ ¬P is always true. No third truth-value is allowed.
排中律：P ∨ ¬P 恒真。不存在第三真值。
Law of Non-Contradiction (LNC): ¬(P ∧ ¬P) is a logical truth.
矛盾律：¬(P ∧ ¬P) 是逻辑真理。
Double Negation Theorem: ¬¬P ↔ P.
双重否定定理：¬¬P ↔ P。
Peirce's Law: ((P → Q) → P) → P. A classic tautology not provable in intuitionistic logic.
皮尔士定律：((P → Q) → P) → P。直觉主义逻辑中无法证明的经典重言式。
Principle of Explosion (Ex Contradictione Quodlibet): (P ∧ ¬P) → Q. From a contradiction anything follows.
爆炸原理（由矛盾可得任意命题）：(P ∧ ¬P) → Q。从矛盾可以推出任何结论。
9. Proof Strategies | 证明策略
Constructing formal proofs requires strategic thinking. A handful of heuristics can guide you when searching for a derivation.
构建形式证明需要策略性思考。若干启发式方法可以指导你寻找推导过程。
P and try to derive Q for →I. If you are stuck with a disjunction in the premises, attempt ∨ dependently. When you suspect that no direct proof is possible, use reductio ad absurdum (¬ nective of the goal because it often determines which introduction rule you will use last.
当目标是蕴涵 P → Q 时，假设 P 并尝试推出 Q 以使用 →I。若前提中有析取而感到卡住，可尝试使用 ∨E，分别从每个析取支证明目标。对于双条件，需分别证明两个方向。当怀疑不存在直接证明时，可使用归谬法 (¬I)：假设目标的否定并推出矛盾。时刻检视目标的主要联结词，因为它往往决定最终将使用哪种引入规则。
In predicate logic proofs, always introduce new constants when eliminating existential quantifiers, and when introducing a universal quantifier, make sure the constant is completely arbitrary. Avoid using the same constant in conflicting roles.
在谓词逻辑证明中，消去存在量词时总要引入新常元，而在引入全称量词时需确保该常元完全任意。避免让同一常元扮演冲突角色。
10. Glossary of Symbols | 符号词汇表
A quick reference to the logical symbols used throughout this handbook and in exam papers.
本手册及试卷中使用的逻辑符号速查表。
| Symbol | Meaning | 中文 |
|---|---|---|
| ¬ | Negation (not) | 否定 |
| ∧ | Conjunction (and) | 合取 |
| ∨ | Disjunction (or) | 析取 |
| → | Material implication (if...then) | 实质蕴涵 |
| ↔ | Material biconditional (if and only if) | 实质双条件 |
| ∀ | Universal quantifier (for all) | 全称量词 |
| ∃ | Existential quantifier (there exists) | 存在量词 |
| ⊢ | Syntactic entailment / derivability | 语法后承 / 可推导性 |
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