📚 A-Level WJEC Philosophy: Formula & Theorem Quick Reference Handbook | A-Level WJEC 哲学：公式定理速查手册
This handbook compiles the essential logical formulas, valid inference rules, and key philosophical principles required for A-Level WJEC Philosophy. Use it as a rapid revision tool to formalise arguments, detect fallacies, and recall definitions in epistemology, ethics, and metaphysics.
本手册汇总了 A-Level WJEC 哲学所需的核心逻辑公式、有效推理规则以及关键哲学原理。将其用作快速复习工具，帮助你将论证形式化、识别谬误，并回顾认识论、伦理学和形而上学中的重要定义。
1. Propositional Logic Symbols | 命题逻辑符号
Propositional logic uses symbols to represent statements and their logical connections. Mastery of this notation is essential for formalising philosophical arguments.
命题逻辑使用符号来表示陈述及其逻辑联结。掌握这些符号是形式化哲学论证的基础。
Let P, Q, R stand for propositions. The basic symbols are: ¬ (negation/not), ∧ (conjunction/and), ∨ (disjunction/or), → (material implication/if...then), ↔ (biconditional/if and only if).
令 P, Q, R 代表命题。基本符号为：¬（否定/非）、∧（合取/且）、∨（析取/或）、→（实质蕴涵/如果...那么）、↔（双条件/当且仅当）。
Parentheses are used to avoid ambiguity. For example, P → Q ∧ R is ambiguous; write P → (Q ∧ R) or (P → Q) ∧ R to clarify scope.
用括号避免歧义。例如，P → Q ∧ R 有歧义；应写作 P → (Q ∧ R) 或 (P → Q) ∧ R 以明确作用域。
The symbol ⊢ denotes logical consequence (provability), and ⊨ denotes logical truth (entailment). Thus P, P → Q ⊢ Q means from P and P → Q, Q can be derived.
符号 ⊢ 表示逻辑后承（可证性），⊨ 表示逻辑真（蕴含）。因此 P, P → Q ⊢ Q 表示从 P 和 P → Q 可推出 Q。
| Symbol | Meaning | Example |
|---|---|---|
| ¬ | Not | ¬P |
| ∧ | And | P ∧ Q |
| ∨ | Or (inclusive) | P ∨ Q |
| → | If...then | P → Q |
| ↔ | If and only if | P ↔ Q |
| ⊢ | Provable | P ⊢ Q |
2. Logical Equivalences | 逻辑等价式
Logical equivalences allow substitution of one formula for another without changing truth values. They are the algebraic backbone of logical analysis in philosophy.
逻辑等价式允许将一个公式替换为另一个而不改变真值。它们是哲学中逻辑分析的代数基础。
Double Negation: P 𠪪P. This states that a proposition is equivalent to the negation of its negation.
双重否定律：P ≡ ¬¬P。一个命题等价于其否定的否定。
De Morgan's Laws: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q; ¬(P ∨ Q) ≡ ¬P ∧ ¬Q. They are vital for moving negations through conjunctions and disjunctions.
德摩根律：¬(P ∧ Q) ≡ ¬P ∨ ¬Q；¬(P ∨ Q) ≡ ¬P ∧ ¬Q。它们在合取与析取间移动否定词时至关重要。
Commutativity: P ∧ Q ≡ Q ∧ P; P ∨ Q ≡ Q ∨ P. Associativity: (P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R). Distributivity: P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R).
交换律：P ∧ Q ≡ Q ∧ P；P ∨ Q ≡ Q ∨ P。结合律：(P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R)。分配律：P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)。
Implication as disjunction: P → Q ≡ ¬P ∨ Q. This equivalence is often used to convert conditional statements for inspection.
蕴涵析取律：P → Q ≡ ¬P ∨ Q。这个等价式常用于将条件陈述转换以便检查。
Contrapositive: P → Q ≡ ¬Q → ¬ powerful proof strategy in philosophy, especially in arguments about necessary conditions.
逆否命题：P → Q ≡ ¬Q → ¬P。换质位法是哲学中强大的证明策略，尤其在涉及必要条件的论证中。
3. Valid Inference Rules | 有效推理规则
These are the basic building blocks of deductive argumentation. Each rule can be applied to derive conclusions from premises in a logically rigorous way.
以下规则是演绎论证的基本构件。每条规则都可以在逻辑严格的方式下从前提推导出结论。
Ponens (MP): From P → Q and P, infer Q. It is perhaps the most widely used deductive form: 'If it rains, the ground is wet. It rains. Therefore, the ground is wet.'
肯定前件式 (MP)：从 P → Q 和 P 推出 Q。这或许是最常用的演绎形式：“如果下雨，地就湿。下雨了。所以地湿。”
Modus Tollens (MT): From P → Q and ¬Q, infer ¬P. This rule is fundamental to falsification: ' the theory is true, we observe X. We do not observe X. Therefore, the theory is not true.'
否定后件式 (MT)：从 P → Q 和 ¬Q 推出 ¬P。此规则是证伪的基础：“如果理论为真，我们会观察到 X。我们没有观察到 X。因此理论不为真。”
ical Syllogism (HS): From P → Q and Q → R, infer P → R. Disjunctive Syllogism (DS): From P ∨ Q and ¬P, infer Q.
假言三段论 (HS)：从 P → Q 和 Q → R 推出 P → R。析取三段论 (DS)：从 P ∨ Q 和 ¬P 推出 Q。
Constructive Dilemma: From (P → Q) ∧ (R → S) and P ∨ R, infer Q ∨ S. This mirrors reasoning about multiple possibilities leading to separate consequences.
构造性二难推理：从 (P → Q) ∧ (R → S) 和 P ∨ R 推出 Q ∨ S。这反映了关于多个可能性分别导致不同后果的推理。
Reductio ad Absurdum (RAA): Assume the negation of what you want to prove, derive a contradiction, and conclude the original statement. This is a cornerstone of Socratic method and ontological arguments.
归谬法 (RAA)：假设欲证命题的否定，导出一个矛盾，进而断定原命题。这是苏格拉底方法和本体论论证的基石。
4. Predicate Logic | 谓词逻辑
Predicate logic extends propositional logic by quantifying over individuals and expressing properties and relations. It is indispensable for analysing categorical syllogisms and existence claims.
谓词逻辑通过量化个体并表达属性与关系，拓展了命题逻辑。它对于分析范畴三段论和存在性断言必不可少。
The universal quantifier ∀ means 'for all'. The existential quantifier ∃ means 'there exists' or 'some'. For example, ∀x (Man(x) → Mortal(x)) says all men are mortal.
全称量词 ∀ 意为“所有”。存在量词 ∃ 意为“存在”或“有些”。例如，∀x (Man(x) → Mortal(x)) 表示所有人都会死。
The classic syllogism: ∀x (Man(x) → Mortal(x)), Man(Socrates) ⊢ Mortal(Socrates). This is an instance of Universal Instantiation (UI).
经典三段论：∀x (Man(x) → Mortal(x)), Man(Socrates) ⊢ Mortal(Socrates)。这是全称实例化 (UI) 的一个实例。
Existential Generalization (EG): From property F(a) for some individual a, infer ∃x F(x). This is crucial when an argument commits the fallacy of moving from a particular to a universal incorrectly.
存在概括 (EG)：从某个个体 a 具有性质 F(a) 推出 ∃x F(x)。当论证错误地由特称推广到全称时，这一点至关重要。
The square of opposition is formalised: A: ∀x (S(x) → P(x)), E: ∀x (S(x) → ¬P(x)), I: ∃x (S(x) ∧ P(x)), O: ∃x (S(x) ∧ ¬P(x)). A and O are contradictories, as are E and I.
对当关系可形式化为：A：∀x (S(x) → P(x))，E：∀x (S(x) → ¬P(x))，I：∃x (S(x) ∧ P(x))，O：∃x (S(x) ∧ ¬P(x))。A 与 O 是矛盾关系，E 与 I 亦然。
5. Common Fallacies | 常见谬误
Recognising fallacies is as important as knowing valid rules. Many philosophical debates pivot on detecting informal fallacies in reasoning.
识别谬误与了解有效规则同等重要。许多哲学辩论的关键在于发现推理中的非形式谬误。
, Q, therefore P. This is invalid because Q might be true for other reasons. For instance, 'If it is a dog, it has four legs. It has four legs. Therefore, it is a dog.'
肯定后件：P → Q，Q，因此 P。这是无效的，因为 Q 可能出于其他原因而为真。例如，“如果是狗，那么有四条腿。它有四条腿。所以它是狗。”
Denying the Antecedent: P → Q, ¬P, therefore ¬Q. Equally invalid: 'If it rains, the match is cancelled. It didn't rain. Therefore, the match wasn't cancelled.' (Perhaps it was cancelled for a different reason.)
否定前件：P → Q，¬P，因此 ¬Q。同样无效：“如果下雨，比赛就取消。没下雨。所以比赛没取消。”（也许因为其他原因取消。）
usion is assumed in one of the premises. Formal structure: P → P, or circular definitions.
丐题（循环论证）：结论已隐含在某前提中。形式结构：P → P，或循环定义。
False Dilemma: Forcing a choice between two alternatives when more exist: P ∨ Q, but actually R is possible. This is often seen in overly simplistic ethical debates.
假二分：在有更多选项时强制二择一：P ∨ Q，但实际上 R 可能存在。这在过于简单化的伦理辩论中常见。
6. Epistemology: Definitions & Principles | 认识论：定义与原则
Epistemology investigates the nature, sources, and limits of knowledge. The tripartite definition and Gettier cases can be formalised symbolically to highlight logical structure.
认识论探究知识的本质、来源与限度。三要素定义与盖梯尔案例可用符号形式化，以凸显其逻辑结构。
Classical tripartite definition: S knows that P iff (i) P is true, (ii) S believes that P, (iii) S is justified in believing that P. As a formula: K(S, P) ↔ True(P) ∧ Bel(S, P) ∧ Jus(S, P).
经典的三要素定义：S 知道 P 当且仅当 (i) P 为真，(ii) S 相信 P，(iii) S 有理由相信 P。公式表达为：K(S, P) ↔ True(P) ∧ Bel(S, P) ∧ Jus(S, P)。
is justified in believing P (a false lemma); Q is true but accidental. Formally, Jus(S, P) ∧ (P → Q) ∧ ¬P ∧ True(Q) does not yield knowledge.
盖梯尔问题：满足三条件但似乎缺乏知识的情形。结构：P 蕴涵 Q (P → Q)；S 有理由相信 P（虚假引理）；Q 为真但出于偶然。形式上，Jus(S, P) ∧ (P → Q) ∧ ¬P ∧ True(Q) 并不产生知识。
Reliabilism: S knows P iff S' ognitive process. Notation: Rel(Process) → K(S, P) if the process outputs P and P is true.
可靠主义：S 知道 P 当且仅当 S 的真信念由一个可靠的认知过程产生。符号化：若过程输出 P 且 P 为真，则 Rel(Process) → K(S, P)。
cal argument: If the evil demon scenario is possible, then S cannot know empirical truths. ◇(Evil demon deceives) ∧ (¬□(¬demons)) → ¬K(S, external world). The modal operator □ stands for necessity, ◇ for possibility.
怀疑论论证：如果恶魔境况是可能的，那么 S 不能知道经验真理。◇(恶魔欺骗) ∧ (¬□(¬恶魔)) → ¬K(S, 外部世界)。模态算子 □ 代表必然，◇ 代表可能。
7. Moral Philosophy: Formal Principles | 道德哲学：形式化原则
Normative theories can be expressed as logical schemas. Utilitarianism and Kantian ethics offer paradigms that can be captured in semi-formal notation.
规范伦理理论可表达为逻辑模式。功利主义与康德伦理学提供了可用半形式化符号表示的理论范式。
es utility. U(A) > U(B) for all alternatives B implies Ought(A). More formally, Right(A) ↔ ∀B (U(A) ≥ U(B)).
行为功利主义：一个行动是正确的，当它最大化功利。对于所有可选行动 B，U(A) > U(B) 蕴含应当做 A。更形式化地：Right(A) ↔ ∀B (U(A) ≥ U(B))。
Kant' w. Maxims can be tested for contradiction in conception or will. Contradiction → Impermissible.
康德绝对命令（普遍法则公式）：只依据你同时愿意它成为一项普遍法则的准则而行动。准则可通过概念上或意愿上的矛盾来检验。矛盾 → 不允许。
The contradiction in conception for false promising: Maxim: 'When in need, make a false promise to obtain money.' m cannot be universalised without contradiction: the universalised maxim implies the negation of its own efficacy. Formally, Universalise(M) → ¬Practicable(Promising), yet M depends on promising. Hence, ¬Permissible(M).
虚假承诺在概念上的矛盾：准则：“当需要时，做一个假承诺以获取金钱。” 普遍化后：人人都做假承诺。承诺制度将崩溃，因为承诺不再被相信。因此准则无法无矛盾地普遍化：普遍化的准则蕴含着承诺制度功效的否定，而该准则又依赖于承诺制度。因此，不允许(M)。
the bad effect is not intended as an end or means. Symbolically: Permissible(A) ↔ (Good(A) ∧ Bad(A)) ∧ ¬Intended_as_End_or_Means(Bad(A)).
双重效应原则：若一种行为兼具好与坏的效应，且坏效应并非作为目的或手段而被意图，则该行为是允许的。符号化：Permissible(A) ↔ (Good(A) ∧ Bad(A)) ∧ ¬Intended_as_End_or_Means(Bad(A))。
8. Philosophy of Religion: Ontological Argument | 宗教哲学：本体论证明
Anselm's ontological argument is a prime example of a priori reasoning that can be expressed in modal logic. Its logical core continues to be debated.
安瑟伦的本体论证明是先验推理的典型例子，可以用模态逻辑表达。其逻辑核心至今备受争议。
Anselm' be conceived. Let G = God. Define: G = df the greatest conceivable being. In modal terms: ◇(G exists) ∧ □(G has all perfections).
安瑟伦的定义：上帝是那无法设想有比之更伟大者的存在者。令 G = 上帝。定义：G = 可设想的最伟大者。用模态术语表示：◇(G 存在) ∧ □(G 具有所有完善性)。
The logical move: If G exists only in the understanding, something greater that also exists in reality can be conceived. So if G is that than which nothing greater can be conceived, G must exist in reality. Formally: ¬∃x (x > G). If G exists only in mind, then ∃y (y > G), contradiction. Hence, G exists in reality.
逻辑步骤：如果 G 只存在于理智中，那么仍可设想一在现实中亦存在的更伟大者。因此，如果 G 是无法设想有比之更伟大者，则 G 必定在现实中存在。形式化：¬∃x (x > G)。若 G 仅存在于思想中，则 ∃y (y > G)，矛盾。因此 G 在现实中存在。
Gaunilo's parody: the greatest conceivable island. The parodist argues that the same form would prove the existence of a perfect island, absurdly. The reply: 'island' is not a kind that admits of intrinsic maximum, whereas 'being' does. The logical form can be generalised, challenging whether the predicate 'is a greatest conceivable F' is coherent for every F.
高尼罗的仿效：最完美的岛屿。仿效者认为同一形式也能证明完美岛屿的存在，这是荒谬的。回应：“岛屿”这类事物不容许内在的最大值，而“存在”则不然。该逻辑形式可被普遍化，进而质疑谓词“是可设想的最伟大之 F”是否对每一个 F 都融贯。
Plantinga' ible world (since maximal greatness entails necessary existence). Therefore, it exists in the actual world. Formal: ◇□G → □G (axiom S5). From ◇□G, deduce □G, hence G.
普兰丁格的模态版本：可能存在一位最大化伟大的存在者。若一最大化伟大者存在于某一可能世界，则它存在于所有可能世界（因为最大化伟大包含必然存在）。因此，它存在于现实世界。形式化：◇□G → □G（S5公理）。由◇□G推出□G，进而得到G。
9. Philosophy of Mind: Identity Theory & Functionalism | 心灵哲学：同一论与功能主义
The mind-body problem generates positions whose logical structure can be clarified by quantifiers and modal operators. Keep these formal distinctions in mind.
心身问题产生各种立场，其逻辑结构可通过量词与模态算子加以澄清。请记住这些形式上的区分。
Identity Theory: Every mental state type is identical to some physical state type. Formal: ∀M ∃ ive claim that permits the a posteriori identification of mental events with brain events.
类型同一论：每一种心理状态类型都等同于某种物理状态类型。形式化：∀M ∃P (M = P)。这是一种还原论主张，容许后天地将心理事件等同于大脑事件。
here is some physical event p such that m = p, without requiring type correlation. Formal: ∀m (Mental(m) → ∃p (Physical(p) ∧ m = p)).
个例同一论：对每一个具体的心理事件 m，存在某个物理事件 p 使得 m = p，但不要求类型上的相关。形式化：∀m (Mental(m) → ∃p (Physical(p) ∧ m = p))。
Functionalism: Mental states are defined by their causal roles, i.e., by their relations to sensory inputs, behavioural outputs, and other mental states. A mental state M can be multiply realised: ∃P1, P2,... such that M is realised by P1 in one organism and P2 in another. This blocks reduction to a single physical type.
功能主义：心理状态由其因果角色定义，即由它们与感觉输入、行为输出以及其他心理状态的关系来定义。一个心理状态 M 可被多重实现：存在 P₁, P₂,... 使得 M 在某一有机体由 P₁ 实现，在另一有机体由 P₂ 实现。这阻碍了还原为单一物理类型。
cal facts about colour, but when she sees red for the first time, she learns something new—what it is like to see red. Let P = all physical facts, Q = what-it's-like facts. The argument's structure: (□(∀f ∈ P) Known(Mary, f)) ∧ ¬Known(Mary, Q) → Q ∉ P. So physicalism is false.
反物理主义的知识论证：玛丽知道关于颜色的一切物理事实，但当她在现实世界中第一次看到红色时，她学到了新东西——看见红色的感受质。令 P = 所有物理事实，Q = 感受质事实。论证结构：(□(∀f ∈ P) 玛丽知道 f) ∧ ¬玛丽知道 Q → Q 不属于 P。因此物理主义为假。
10. Scientific Reasoning: Hypothetico-Deductive Model | 科学推理：假说演绎模型
The hypothetico-deductive (H-D) method and Bayesian confirmation are formalised approaches to testing hypotheses in philosophy of science.
假说演绎方法 (H-D) 与贝叶斯确证是科学哲学中检验假设的形式化进路。
ce an observational prediction O. If O is observed, H is confirmed (though not proved). If ¬O is observed, H is falsified. Formally: (H ∧ C) → O; ¬O ∴ ¬H ∨ ¬C. This showcases Popper's emphasis on falsifiability.
H-D 模式：由假设 H 与初始条件 C 演绎出观察预测 O。若观察到 O，则 H 被确证（但未被证明）。若观察到 ¬O，则 H 被证伪。形式化：(H ∧ C) → O；¬O ∴ ¬H ∨ ¬C。这体现了波普尔对可证伪性的强调。
Bayesian confirmation: The degree of belief in H given evidence E is updated using Bayes' theorem: P(H|E) = [P(E| H)] / P(E). Philosophy students should note that prior probabilities are often subjective.
贝叶斯确证：根据证据 E 更新对假设 H 的信念度，使用贝叶斯定理：P(H|E) = [P(E|H) × P(H)] / P(E)。哲学专业学生应注意，先验概率往往是主观的。
P(E) = P(E|H)P(H) + P(E|¬H)P(¬ experience arguments: the probability of the existence of a religious experience given God's existence vs God's non-existence affects the posterior probability.
总概率定律：P(E) = P(E|H)P(H) + P(E|¬H)P(¬H)。这一公式用于宗教经验论证：给定上帝存在与不存在时宗教体验出现的概率，会影响后验概率。
Problem of induction (Hume): Can we justify that the future will resemble the past? The principle of uniformity of nature (PUN) seems to be presupposed in every inductive inference. Represented: Inductive inference from n observed instances to all instances is valid only if PUN holds. But PUN cannot be proved without circularity.
归纳问题（休谟）：我们能否证明未来与过去相似？自然的齐一性原则 (PUN) 似乎被每一个归纳推理所预设。可表示为：从 n 个已观察的实例到所有实例的归纳推理，只在 PUN 成立时有效。但 PUN 本身无法被非循环地证明。
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