📚 Year 13 WJEC Philosophy: Quick Reference Formulas & Theorems | WJEC 13年级哲学：公式定理速查手册
This handbook gathers the essential formal tools and conceptual schemas you will encounter in the Year 13 WJEC Philosophy course. From truth‑table algorithms to the quantifier rules of predicate logic, from argument patterns to ethical decision procedures, every formula is presented side‑by‑side in English and Chinese to strengthen your exam revision.
本手册汇集了你在WJEC 13年级哲学课程中会遇到的核心形式工具与概念图式。从真值表算法到谓词逻辑的量词规则，从论证模式到伦理决策流程，每条公式都以英中对照呈现，帮助你在备考中迅速查阅、加深理解。
1. Propositional Logic: Basic Connectives | 命题逻辑：基本联结词
Propositional logic analyses the truth‑functional structure of statements using five standard connectives. Each connective is defined by its characteristic truth table. Memorising their symbolic forms and English translations is essential for translating arguments into logical notation.
命题逻辑使用五种标准联结词分析陈述的真值函项结构。每个联结词由其真值表定义。熟记它们的符号和英文对应表达，是将论证转化为逻辑符号的关键。
- Negation: ¬p — 'not p', 'it is not the case that p'. 否定：¬p — “并非p”“p不成立”。
- Conjunction: p ∧ q — 'p and q', 'p but q'. 合取：p ∧ q — “p并且q”“p但是q”。
- Disjunction: p ∨ q — 'p or q' (inclusive). 析取：p ∨ q — “p或者q”（相容或）。
- Conditional: p → q — 'if p then q', 'p implies q', 'p only if q'. 条件：p → q — “如果p那么q”“p蕴涵q”“p仅当q”。
- Biconditional: p ↔ q — 'p if and only if q', 'p is necessary and sufficient for q'. 双条件：p ↔ q — “p当且仅当q”“p是q的充要条件”。
2. Truth Tables for Compound Statements | 复合命题真值表
A truth table exhaustively displays the truth value of a compound formula under every possible assignment of truth values to its atomic components. Use 2ⁿ rows for n distinct atomic propositions. The main column determines whether the formula is a tautology, contradiction, or contingency.
真值表穷尽地显示一个复合公式在其原子命题的每种赋值下的真值。若有n个不同原子命题，则需要2ⁿ行。主列决定该公式是重言式、矛盾式还是偶然式。
| p | q | p → q | ¬p ∨ q |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |
p → q is logically equivalent to ¬p ∨ q
p → q 逻辑等价于 ¬p ∨ q
3. Logical Equivalences | 逻辑等价式
Logical equivalences are pairs of formulas that always share the same truth value. They are indispensable for simplifying arguments or proving theorems within a natural deduction system.
逻辑等价式是指真值始终相同的成对公式。在自然演绎系统中，它们对于化简论证或证明定理不可或缺。
- Double Negation: ¬¬p ⇔ p. 双重否定：¬¬p ⇔ p。
- De Morgan's Laws: ¬(p ∧ q) ⇔ (¬p ∨ ¬q) ; ¬(p ∨ q) ⇔ (¬p ∧ ¬q). 德摩根律：¬(p ∧ q) ⇔ (¬p ∨ ¬q) ; ¬(p ∨ q) ⇔ (¬p ∧ ¬q)。
- Commutation: (p ∧ q) ⇔ (q ∧ p); (p ∨ q) ⇔ (q ∨ p). 交换律：(p ∧ q) ⇔ (q ∧ p); (p ∨ q) ⇔ (q ∨ p)。
- Association: (p ∧ (q ∧ r)) ⇔ ((p ∧ q) ∧ r); same for ∨. 结合律：(p ∧ (q ∧ r)) ⇔ ((p ∧ q) ∧ r); ∨ 同理。
- Distribution: p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r); p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r). 分配律：p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r); p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r)。
- Contraposition: (p → q) ⇔ (¬q → ¬p). 换质换位：(p → q) ⇔ (¬q → ¬p)。
- Exportation: ((p ∧ q) → r) ⇔ (p → (q → r)). 输出律：((p ∧ q) → r) ⇔ (p → (q → r))。
4. Important Logical Theorems & Implications | 重要逻辑定理与蕴涵式
Beyond equivalences, certain strict implications capture the fundamental moves in deductive reasoning. These theorems are either axioms or derivable truths in classical propositional logic.
除了等价式外，某些严格的蕴涵关系刻画了演绎推理中的基本步骤。这些定理在经典命题逻辑中要么是公理，要么是可推出的真命题。
Law of Non‑Contradiction: ¬(p ∧ ¬p)
不矛盾律：¬(p ∧ ¬p)
Law of Excluded Middle: p ∨ ¬p
排中律：p ∨ ¬p
Modus Ponens Theorem: (p ∧ (p → q)) → q
肯定前件定理：(p ∧ (p → q)) → q
Modus Tollens Theorem: (¬q ∧ (p → q)) → ¬p
否定后件定理：(¬q ∧ (p → q)) → ¬p
Hypothetical Syllogism: ((p → q) ∧ (q → r)) → (p → r)
假言三段论：((p → q) ∧ (q → r)) → (p → r)
5. Rules of Inference | 推理规则
Rules of inference justify the step‑by‑step derivation of conclusions from premises. In a natural deduction proof, each line must be a premise, an assumption, or the result of applying one of these rules to earlier lines.
推理规则为从前提逐步推出结论提供依据。在自然演绎证明中，每一行必须是前提、假设，或是将下列规则之一应用于先前行所得的结果。
- Modus Ponens (MP): From p → q and p, infer q. 肯定前件：由 p → q 和 p 推出 q。
- Modus Tollens (MT): From p → q and ¬q, infer ¬p. 否定后件：由 p → q 和 ¬q 推出 ¬p。
- Hypothetical Syllogism (HS): From p → q and q → r, infer p → r. 假言三段论：由 p → q 和 q → r 推出 p → r。
- Disjunctive Syllogism (DS): From p ∨ q and ¬p, infer q. 选言三段论：由 p ∨ q 和 ¬p 推出 q。
- Simplification (Simp): From p ∧ q, infer p (or q). 简化律：由 p ∧ q 推出 p（或 q）。
- Conjunction (Conj): From p and q, infer p ∧ q. 合取引入：由 p 和 q 推出 p ∧ q。
- Addition (Add): From p, infer p ∨ q. 附加律：由 p 推出 p ∨ q。
- Constructive Dilemma (CD): From (p → q) ∧ (r → s) and p ∨ r, infer q ∨ s. 构造性二难：由 (p → q) ∧ (r → s) 和 p ∨ r 推出 q ∨ s。
6. Rules of Replacement | 替换规则
Replacement rules allow substituting a logically equivalent expression for any part of a compound formula. Unlike inference rules, they can be applied to sub‑formulas and work in both directions.
替换规则允许将逻辑等价的表达式替换复合公式中的任一部分。与推理规则不同，它们可作用于子公式并且可双向使用。
- Double Negation (DN): p ⇔ ¬¬p. 双重否定：p ⇔ ¬¬p。
- De Morgan's Laws (DeM): ¬(p ∧ q) ⇔ ¬p ∨ ¬q. 德摩根律：¬(p ∧ q) ⇔ ¬p ∨ ¬q。
- Commutation (Comm): p ∨ q ⇔ q ∨ p. 交换律：p ∨ q ⇔ q ∨ p。
- Association (Assoc): p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r. 结合律：p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r。
- Distribution (Dist): p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r). 分配律：p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r)。
- Contraposition (Contra): (p → q) ⇔ (¬q → ¬p). 换质换位：(p → q) ⇔ (¬q → ¬p)。
- Implication (Impl): (p → q) ⇔ (¬p ∨ q). 蕴涵律：(p → q) ⇔ (¬p ∨ q)。
- Equivalence (Equiv): (p ↔ q) ⇔ ((p → q) ∧ (q → p)). 等价律：(p ↔ q) ⇔ ((p → q) ∧ (q → p))。
- Exportation (Exp): ((p ∧ q) → r) ⇔ (p → (q → r)). 输出律：((p ∧ q) → r) ⇔ (p → (q → r))。
- Tautology (Taut): p ∨ p ⇔ p; p ∧ p ⇔ p. 重言律：p ∨ p ⇔ p; p ∧ p ⇔ p。
7. Predicate Logic: Quantifiers & Rules | 谓词逻辑：量词与规则
When statements involve 'all' or 'some', predicate logic extends propositional logic with quantifiers. The universal quantifier ∀xPx reads 'for every x, Px'. The existential quantifier ∃xPx reads 'there exists an x such that Px'.
当陈述涉及“所有”或“有些”时，谓词逻辑通过量词扩展命题逻辑。全称量词 ∀xPx 读作“对所有 x，Px”。存在量词 ∃xPx 读作“存在 x 使得 Px”。
- Universal Instantiation (UI): From ∀xPx, infer Pa (where a is any constant). 全称例示：由 ∀xPx 推出 Pa（其中 a 是任意常项）。
- Existential Generalisation (EG): From Pa, infer ∃xPx. 存在概括：由 Pa 推出 ∃xPx。
- Universal Generalisation (UG): From an arbitrary instance Pa, infer ∀xPx, provided a does not occur in any assumption. 全称概括：从任意实例 Pa 推出 ∀xPx，前提是 a 不在任何假设中出现。
- Existential Instantiation (EI): From ∃xPx, infer Pa for a new constant, not used earlier. 存在例示：由 ∃xPx 推出 Pa，其中 a 是一个之前未使用过的新常项。
- Quantifier Negation (QN): ¬∀xPx ⇔ ∃x¬Px ; ¬∃xPx ⇔ ∀x¬Px. 量词否定：¬∀xPx ⇔ ∃x¬Px ; ¬∃xPx ⇔ ∀x¬Px。
8. Common Valid Argument Forms | 常见有效论证形式
Beyond the basic inference rules, standard argument forms appear repeatedly in philosophical writing. Recognising them helps you formalise an opponent's reasoning and assess its validity.
除基本推理规则外，标准论证形式在哲学写作中反复出现。识别这些形式有助于你将对手的推理形式化并评估其有效性。
- Modus Ponens: If P then Q. P. Therefore Q. 肯定前件：如果 P 则 Q。P。所以 Q。
- Modus Tollens: If P then Q. Not Q. Therefore not P. 否定后件：如果 P 则 Q。非 Q。所以非 P。
- Hypothetical Syllogism: If P then Q. If Q then R. Therefore if P then R. 假言三段论：如果 P 则 Q。如果 Q 则 R。所以如果 P 则 R。
- Disjunctive Syllogism: P or Q. Not P. Therefore Q. 选言三段论：P 或 Q。非 P。所以 Q。
- Constructive Dilemma: If P then Q, and if R then S. P or R. Therefore Q or S. 构造性二难：若 P 则 Q，且若 R 则 S。P 或 R。所以 Q 或 S。
- Reductio ad Absurdum: Assume ¬P, derive a contradiction, therefore P. 归谬法：假设 ¬P，导出矛盾，所以 P。
9. Common Fallacies (Formal & Informal) | 常见谬误（形式与非形式）
A fallacy is a persuasive but invalid pattern of reasoning. Formal fallacies violate logical structure; informal fallacies rely on content or context. Being able to name and diagnose a fallacy is a core skill in WJEC Philosophy.
谬误是一种有说服力但无效的推理模式。形式谬误违反逻辑结构；非形式谬误依赖于内容或语境。能命名并诊断谬误是WJEC哲学的核心技能。
- Affirming the Consequent: If P then Q. Q. Therefore P. 肯定后件：如果 P 则 Q。Q。所以 P。（无效）
- Denying the Antecedent: If P then Q. Not P. Therefore not Q. 否定前件：如果 P 则 Q。非 P。所以非 Q。（无效）
- Begging the Question: Circular reasoning where the conclusion is assumed in the premises. 乞题：循环推理，结论已被预设在前提出中。
- False Dilemma: Presenting only two options when more exist. 假两难：仅提供两个选项，但实际上存在更多可能。
- Ad Hominem: Attacking the person instead of the argument. 人身攻击：攻击个人而非论证本身。
- Straw Man: Misrepresenting an opponent's position to make it easier to refute. 稻草人：歪曲对方立场，以便更容易反驳。
10. Philosophy of Religion: Key Arguments in Syllogistic Form | 宗教哲学：关键论证的三段论形式
Many classical arguments for the existence of God can be reconstructed as deductive syllogisms. Formalising them reveals hidden premises and logical gaps.
许多关于上帝存在的经典论证可被重构为演绎三段论。形式化这些论证能揭示隐含前提与逻辑缺口。
Ontological Argument (Anselm): (1) God is that than which nothing greater can be conceived. (2) That which exists in reality is greater than that which exists only in the mind. (3) Therefore God must exist in reality. 本体论论证（安瑟尔谟）：(1) 上帝是不可设想比之更伟大的存在。(2) 实际存在比仅存在于心中更伟大。(3) 所以上帝必须实际存在。
Cosmological Argument (Aquinas, Second Way): (1) Everything that exists has a cause. (2) An infinite regress of causes is impossible. (3) Therefore there must be a first uncaused cause (God). 宇宙论论证（阿奎那，第二路）：(1) 一切存在物皆有因。(2) 原因的无限回溯是不可能的。(3) 所以必有一个第一无因之因（上帝）。
11. Ethical Theories: Decision Procedures | 伦理理论：决策流程
Ethical theories often prescribe a decision procedure — a step‑by‑step method for determining the moral status of an action. Formulating them algorithmically clarifies their internal logic and exposes testable consequences.
伦理理论通常规定一个决策程序——一种判定行为道德地位的逐步方法。用算法语言表述它们，可以澄清其内在逻辑并揭示可检验的推论。
- Act Utilitarianism: (1) List available actions. (2) For each action, sum the net pleasure/pain for all affected. (3) Choose the action that maximises net pleasure. 行为功利主义：(1) 列出可能的行为。(2) 对每个行为，计算所有受影响者的净快乐/痛苦总和。(3) 选择净快乐最大化之行为。
- Kant's Categorical Imperative (Formula of Universal Law): (1) Formulate the maxim of your action. (2) Universalise it: can you will it as a universal law without contradiction? (3) If yes, the action is permissible; if no, it is forbidden. 康德定言命令（普遍法则公式）：(1) 形成你行为的准则。(2) 将其普遍化：你能否不自相矛盾地意愿它成为一条普遍法则？(3) 若能，该行为是可允许的；若不能，则是被禁止的。
- Virtue Ethics: (1) Identify the relevant virtue(s) for the situation. (2) Determine the mean between excess and deficiency. (3) Act as the virtuous agent would act. 德性伦理学：(1) 确定与该情境相关的德性。(2) 确定过度与不足之间的中道。(3) 像有德者那样行动。
12. Key Philosophical Principles | 关键哲学原则
Some principles function as starting points or regulative ideals in philosophical argument. Though not 'formulas' in a mathematical sense, they have crisp definitions worth memorising.
有些原则在哲学论证中充当起点或规范理想。虽然并非数学意义上的“公式”，但它们精炼的定义值得熟记。
- Ockham's Razor: Entities should not be multiplied beyond necessity. — Do not posit more entities than are needed to explain the phenomena. 奥卡姆剃刀：如无必要，勿增实体。——不要设立比解释现象所需更多的实体。
- Hume's Law (Is–Ought Gap): No set of descriptive premises can validly yield a normative conclusion. 休谟法则（实然–应然鸿沟）：任何一组描述性前提都无法有效地得出规范性结论。
- Principle of Charity: Interpret an opponent's argument in its strongest, most rational form before evaluating it. 宽容原则：在评价对手论证前，先以其最强、最合理的形式来解读它。
- Principle of Sufficient Reason: For every fact or truth, there must be a sufficient reason why it is so and not otherwise. 充足理由律：对任何事实或真理，必有一个充足理由说明它为何如此而非别样。
- Falsifiability (Popper): A scientific theory must make predictions that could in principle be contradicted by observation. 可证伪性（波普尔）：一个科学理论必须作出原则上可被观察所反驳的预测。
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