📚 A-Level Edexcel Philosophy: Quick-Reference Handbook of Formulae & Theorems | A-Level Edexcel 哲学：公式定理速查手册
This handbook reframes the core arguments, conceptual analyses, and definitional structures within the Edexcel A-Level Philosophy specification as 'formulae' and 'theorems'. Just as a mathematician relies on compact symbolic expressions, students of philosophy can benefit from seeing the logical skeleton behind the Gettier problem, the Ontological Argument, the Problem of Evil, Utilitarianism, and more. Each entry offers a formalised version of a key philosophical move, followed by bilingual explanation. Use this document to revise, test your precision, and ensure you can reconstruct any argument from its essential structure.
本手册将 Edexcel A-Level 哲学大纲中的核心论证、概念分析和定义结构重塑为“公式”与“定理”。如同数学依赖简洁的符号表达式，哲学学习者可以通过看清盖梯尔问题、本体论论证、恶的问题、功利主义等背后的逻辑骨架而获益。每个条目给出一个关键哲学步骤的形式化版本，并附有中英双语解释。用这份资料来复习、检验表述的精确性，并确保你能从基本结构重构任何论证。
1. The JTB+ Formula for Knowledge | 知识的 JTB+ 公式
a no-defeater condition. The schematic representation is: K = JTB + N, where N stands for 'no undefeated defeaters' or an anti-luck condition.
知识传统上被定义为得到辩护的真信念（JTB），但反例表明该公式还需要一个无否决条件的补充。其形式化表达为：K = JTB + N，其中 N 代表“不存在未被击败的否决者”或反运气条件。
- Standard JTB: S knows that P if and only if (i) P is true, (ii) S believes that P, (iii) S is justified in believing that P. | 标准 JTB：S 知道 P，当且仅当 (i) P 为真，(ii) S 相信 P，(iii) S 相信 P 是有辩护的。
- Gettier Counterexample Structure: A justified true belief is inferred from a false but justified lemma. Formally: L is false, S justifiedly believes L, S deduces P from L, P happens to be true. Then S has JTB without knowledge. | 盖梯尔反例结构：一个得到辩护的真信念是从一个虚假但得到辩护的引理推出的。形式化：L 为假，S 有辩护地相信 L，S 从 L 推出 P，P 碰巧为真。那么 S 拥有 JTB 却无知识。
- No-False-Lemma Formula: K = JTB ∧ (the justification does not depend on any false proposition). | 无虚假引理公式：K = JTB ∧（辩护不依赖于任何假命题）。
2. The Ontological Argument Formula (Anselm) | 本体论论证公式（安瑟伦）
Anselm's argument can be compressed into a reductio ad absurdum that hinges on the definition of God as 'that than which nothing greater can be conceived'. The logical form: Assume God exists only in the understanding. That leads to a contradiction, because a being that exists in reality is greater. Therefore God must exist in reality.
安瑟伦的论证可压缩为一个归谬法，其关键在于把上帝定义为“无法设想比之更伟大之物”。逻辑形式：假设上帝只存在于理智中。由此导出矛盾，因为存在于实在中的东西更伟大。所以上帝必须存在于实在中。
- Definition: G = the being than which no greater can be conceived. | 定义：G = 无法设想比之更伟大之物。
- Premise 1: G exists in the understanding. | 前提 1：G 存在于理智中。
- Premise 2: It is greater to exist in reality than in the understanding alone. | 前提 2：存在于实在中比仅仅存在于理智中更伟大。
- Reductio tanding. Then we can conceive of G* = G plus existence in reality, which is greater than G — a contradiction. Therefore, G exists in reality. | 归谬：假设 G 仅存在于理智中。那么我们可以设想 G* = G 加上实在存在，这比 G 更伟大 —— 矛盾。所以，G 存在于实在中。
G (understanding) → Contradiction ∴ G (reality)
Thus, the formula: if the greatest conceivable being is conceivable, it must have the property of existence. | 因此，公式：如果可设想的最伟大之物是可设想的，它必然具有实存属性。
3. The Modal Ontological Argument (Plantinga) | 模态本体论论证（普兰丁格）
Plantinga's version employs possible worlds semantics. A being is maximally excellent if it has omnipotence, omniscience, and moral perfection; a being is maximally great if it has maximal excellence in every possible world. The argument runs: It is possible that a maximally great being exists. If so, a maximally great being exists in some possible world. But if a being is maximally great, it exists in every possible world. Therefore, it exists in the actual world.
普兰丁格的版本运用可能世界语义。一个存在者如果具有全能、全知与道德完美，则具有最大优异；一个存在者如果在每一个可能世界都具有最大优异，则具有最大伟大。论证如下：可能存在一个具有最大伟大的存在者。如果是这样，那么它在某个可能世界存在。但如果一个存在者是最大伟大的，它就在每一个可能世界存在。因此，它在现实世界存在。
- Formula: ◇∃x Mx → □∃ maximally great being exists, then it is necessary that it exists, given S5 modal logic). | 公式：◇∃x Mx → □∃x Mx（若一个最大伟大的存在者可能存在，则它必然存在，基于 S5 模态逻辑）。
- Support for possibility premise: No logical inconsistency has been shown in the concept of maximal greatness. | 对可能性前提的支持：最大伟大的概念没有显示出逻辑不一致。
- Critique: One can equally assert ◇¬∃x Mx (it is possible that no maximally great being exists), leading to the opposite necessity. | 批评：人们同样可以断定 ◇¬∃x Mx（可能不存在最大伟大的存在者），从而推出相反的必然性。
4. The Teleological Argument as Likelihood Ratio | 目的论论证的似然比公式
Design arguments, especially those using the fine-tuning of the universe, can be formalised using Bayes' theorem. The evidential force is captured by the ratio: P(E|Design) / P(E|Chance). If this ratio is much greater than 1, the evidence E strongly favours design over chance.
设计论证，尤其是那些利用宇宙精细调谐的论证，可借助贝叶斯定理形式化。其证据力由比值 P(E|Design) / P(E|Chance) 捕捉。如果该比值远大于 1，则证据 E 强烈支持设计假说而非偶然假说。
- Bayesian Formula: P(H|E) = [P(E|H) × P(H)] / P(E). | 贝叶斯公式：P(H|E) = [P(E|H) × P(H)] / P(E)。
- Comparative version: P(Design|E) / P(Chance|E) = [P(E|Design) / P(E|Chance)] × [P(Design) / P(Chance)]. | 比较版本：P(Design|E) / P(Chance|E) = [P(E|Design) / P(E|Chance)] × [P(Design) / P(Chance)]。
- Key move: The proponent argues that P(E|Chance) is vanishingly small, while P(E|Design) is not similarly low, making the likelihood ratio enormous. | 关键步骤：支持者主张 P(E|Chance) 极小，而 P(E|Design) 没有这么低，使得似然比极大。
- Hume's objection formalised: The design inference is a weak analogy; we lack multiple universes to establish a baseline probability, so the likelihoods are inscrutable. | 休谟反对意见的形式化：设计推论是一个薄弱的类比；我们缺乏多个宇宙来建立基准概率，因此似然值不可知。
5. The Logical Problem of Evil – Inconsistency Formula | 恶的逻辑问题 – 不一致性公式
God is wholly good, Evil exists} is logically inconsistent. The formula can be stated as: ☐(Omnipotent & Wholly good → ¬Evil) ∧ ☐(Evil exists) → ¬nt & Wholly good). The atheist argues that the theist must give up at least one attribute.
恶的逻辑问题声称集合 {上帝全能，上帝全善，恶存在} 在逻辑上不一致。该公式可表述为：☐(全能 ∧ 全善 → ¬恶) ∧ ☐(恶存在) → ¬(全能 ∧ 全善)。无神论者主张有神论者至少得放弃其中一个属性。
- Mackie's formulation: If God is omnipotent, he can prevent all evil; if wholly good, he would want to prevent all evil; but evil exists. Therefore, such a God does not exist. | 麦基的表述：如果上帝全能，他能防止一切恶；如果全善，他会愿意防止一切恶；但恶存在。因此，这样的上帝不存在。
- Plantinga's free will defence formalised: It is logically possible that a world containing creatures with morally significant free will, and the evil they cause, is better than any world without such freedom. The inconsistency is thus not necessary, because a wholly good God might permit evil for the sake of a greater good that cannot be achieved without the possibility of evil. | 普兰丁格自由意志辩护的形式化：存在这样一个可能世界，其中包含拥有道德意义上重大自由的受造物及其所造成的恶，它比任何缺乏这种自由的世界更好。因此这种不一致并非必然，因为全善的上帝可能为了一个更大的善而容许恶，这个善若无恶的可能性便无法达成。
- Formula for defence: (Omnipotent & Wholly good) is compatible with Evil if there is a morally sufficient reason for permitting evil. | 辩护公式：（全能 ∧ 全善）与恶是相容的，如果存在一个道德上充分的理由允许恶。
6. The Evidential Problem of Evil – Probability Formula | 恶的证据问题 – 概率公式
The evidential problem does not claim logical inconsistency but holds that the existence of gratuitous evil makes God's existence highly improbable. Using a probabilistic formula: P(OmniGod | Evil) << 1, given that P(Evil | OmniGod) is very low, and P(Evil | ¬OmniGod) is much higher. Rowe's argument focuses on cases of intense suffering that seem pointless.
证据问题不声称逻辑不一致，而是主张无端恶的存在使得上帝存在的概率极低。使用概率公式：鉴于 P(Evil | OmniGod) 非常低，而 P(Evil | ¬OmniGod) 高得多，有 P(OmniGod | Evil) << 1。罗的论证聚焦于那些看似毫无意义的剧烈痛苦。
- Rowe's fawn: E = a fawn suffers horribly in a forest fire with no one to witness it. P(E| rs extremely low, because an omnibenevolent being would not allow pointless suffering. P(E|Atheism) is not similarly low. | 罗的幼鹿案例：E = 一头幼鹿在森林大火中遭受可怕的痛苦，无人目睹。P(E|OmniGod) 显得极低，因为一个全善的存在不会容许无意义的苦难。P(E|Atheism) 没有这么低。
- Sceptical theist response: We are not in a position to judge P(E|OmniGod), because we cannot know whether there is a greater good justifying E. Thus the low probability premise is unwarranted. | 怀疑论有神论回应：我们无法判断 P(E|OmniGod)，因为我们不可能知道是否存在一个更大的善为 E 提供理由。因此低概率前提无根据。
7. The Hedonic Calculus – Utilitarian Decision Formula | 快乐计算 – 功利主义决策公式
Bentham's act utilitarianism can be expressed as a summation formula over all sentient beings affected, with variables for intensity, duration, certainty, propinquity, fecundity, and purity. The right action maximises net pleasure.
边沁的行为功利主义可表达为一个对所有受影响的感知存在者的总和公式，其变量包括强度、持续时间、确定性、远近度、丰产性和纯度。正确的行为使净快乐最大化。
- Basic formula: Utility(Action) = Σ (Pleasureᵢ – Painᵢ) for all affected individuals i. | 基本公式：效用(行动) = Σ (快乐ᵢ – 痛苦ᵢ)，针对所有受影响的个体 i。
- Hedonic Calculus parameters e I = intensity, D = duration, C = certainty, P = propinquity, F = fecundity, Pu = purity. | 快乐计算参数：总快乐 = f(I, D, C, P, F, Pu)，其中 I = 强度，D = 持续时间，C = 确定性，P = 远近度，F = 丰产性，Pu = 纯度。
- Mill's qualitative revision formula ality × Quantity) of pleasures. Higher pleasures (intellectual) are weighted more heavily. | 密尔的质性修正公式：效用 = Σ (品质 × 数量) 的快乐。高级快乐（智识的）被赋予更大权重。
8. The Categorical Imperative – Universalisation Test Formula | 定言命令 – 普遍化检验公式
Kant's first formulation of the Categorical Imperative can be stated as: Act only on that maxim whereby you can at the same time will that it should become a universal law. The formula involves a contradiction test: if universalising a maxim leads to a contradiction in conception or in the will, the act is forbidden.
康德的定言命令第一表述可陈述为：只按照你同时愿意它成为一条普遍法则的准则去行动。该公式包含一个矛盾检验：如果普遍化一条准则导致概念上的矛盾或意志的矛盾，该行为就是被禁止的。
- CI1 formalised: Action A is permissible if and only if the maxim M can be consistently universalised without contradiction. | CI1 形式化：行动 A 是许可的，当且仅当准则 M 能无矛盾地被一贯地普遍化。
- Contradiction in conception: The universalised maxim undermines the very action it describes. Example: universalising 'make a false promise when in need' would destroy the institution of promising, making false promising impossible. | 概念矛盾：普遍化后的准则破坏了它自身所描述的行动。例子：普遍化“当需要时做出虚假承诺”会毁掉承诺制度，使得虚假承诺不可能。
- Contradiction in the will: The universalised maxim goes against something a rational agent necessarily wills. Example: a maxim of not helping others in distress cannot be willed universally because any rational agent wills the assistance of others when in need. | 意志矛盾：普遍化后的准则违背了理性行动者必然意愿的东西。例子：不在他人危难时提供帮助的准则不能被意愿为普遍法则，因为任何理性行动者在自己需要时都会意愿他人的帮助。
9. The Mind-Body Supervenience Formula | 心-身随附性公式
In Philosophy of Mind, supervenience is a key formal concept. Mental properties supervene on physical properties if and only if there cannot be a mental difference without a physical difference. This can be expressed as: For any two individuals, if they are physical duplicates, they must be mental duplicates.
在心灵哲学中，随附性是一个关键的形式概念。心理属性随附于物理属性，当且仅当不可能有心理差异而无物理差异。这可以表达为：对于任意两个个体，如果它们是物理上的复制品，它们必然在心理上也是复制品。
- Strong supervenience formula: ☐∀x∀M (Mx → ∃P (Px ∧ ☐∀ , there is a physical property P such that x has P, and necessarily anything with P has M. | 强随附性公式：☐∀x∀M (Mx → ∃P (Px ∧ ☐∀y (Py → My)))。即，必然地，对于任何心理属性 M，如果 x 具有 M，则存在一个物理属性 P，使得 x 具有 P，且必然地任何具有 P 的东西都具有 M。
- Zombie argument against physicalism: If physicalism is true, then zombies (physical duplicates without consciousness) are impossible. But zombies are conceivable. Using the conceivability-to-possibility principle, zombies are possible, so physicalism is false. | 反对物理主义的僵尸论证：如果物理主义为真，那么僵尸（物理复制品但没有意识）就是不可能的。但僵尸是可设想的。运用可设想性到可能性的原则，僵尸是可能的，因此物理主义为假。
- Physicalist formula = Functional states, reducible to physical realisers via identity theory or functionalism. | 物理主义公式：心理状态 = 功能状态，可通过同一论或功能主义还原为物理实现者。
10. The Tripartite Theory of Personal Identity – Psychological Continuity Formula | 人格同一性三分理论 – 心理连续性公式
Locke's memory criterion and its modern descendants can be expressed as a chain of overlapping psychological connections. A person X at time t1 is identical to a person Y at time t2 if and only if there is sufficient psychological continuity between them, typically realised by direct memory connections, intentions, and character traits.
洛克的记忆标准及其现代后继者可以表达为一串重叠的心理联系链。在时间 t1 的人 X 与时间 t2 的人 Y 是同一的，当且仅当他们之间有足够的心理连续性，通常由直接的记忆联系、意图和性格特征来实现。
- Psychological continuity formula: X(t1) = Y(t2) iff ∃ a chain of psychological states P₁, P₂, … ch that each link is strongly connected to the next, and there is no branching competitor. | 心理连续性公式：X(t1) = Y(t2) 当且仅当存在一个心理状态链 P₁, P₂, …, Pₙ，使得每一环强连接到下一环，且没有分支竞争者。
- Reid's brave officer objection formalised: If identity is transitive, and memory links the general to the officer, and the officer to the boy, but not the general to the boy, then Locke's theory violates transitivity. The formula must be revised to overlapping chains. | 里德的勇敢军官反对意见形式化：如果同一性是传递的，且记忆将将军连接到军官，军官连接到男孩，但将军不直接记得男孩，那么洛克理论违背了传递性。公式必须修正为重叠链条。
- Non-branching clause: Psychological continuity must not take a 'branching' form, ensuring that at most one future candidate counts as the same person. | 无分支条款：心理连续性不得呈现“分支”形式，确保最多只有一个未来候选者算作同一个人。
11. The Euthyphro Dilemma – Value Ontology Formula | 尤西弗罗困境 – 价值本体论公式
The dilemma is a disjunctive formula that challenges divine command theory: Is an action morally right because God commands it, or does God command it because it is morally right? The first horn makes morality arbitrary; the second makes morality independent of God.
该困境是一个选言公式，挑战神圣命令理论：一个行动是道德上对的因为上帝命令它，还是上帝命令它因为它是道德上对的？第一角使道德成为任意的；第二角使道德独立于上帝。
- Formal dilemma: (Right = Commanded) ∨ ommanded = Right). If Right = Commanded, then any command, even to torture, would be right — ive result. If Commanded = Right, then there is a standard of rightness independent of God's will, which limits God's sovereignty. | 形式困境：（对 = 被命令）∨（被命令 = 对）。如果对 = 被命令，那么任何命令，哪怕是去折磨，都会是对的 —— 一个反直觉的结果。如果被命令 = 对，那么就存在一个独立于上帝意志的对错标准，这限制了上帝的主权。
- Modified divine command theory formula: Right = what a loving God commands. This incorporates God's essential goodness to avoid arbitrariness. | 修正的神圣命令理论公式：对 = 一位慈爱的上帝所命令的。这融合了上帝的本质良善以避免任意性。
12. The Trolley Problem – Double Effect Formula | 电车难题 – 双重效果公式
The Doctrine of Double Effect (DDE) provides a formula to distinguish permissible from impermissible actions when harm is foreseen: An action that causes serious harm is permissible if the harm is merely a foreseen but not intended side-effect, and there is a proportionately grave reason.
双重效果原则 (DDE) 提供了一个公式，用以区分在预见伤害时何为允许、何为不允许：一个造成严重伤害的行为若是允许的，当且仅当该伤害仅仅是可预见但非意图的副作用，并且存在相称重大的理由。
- DDE formula: Act A is permissible if (i) A itself is not intrinsically wrong, (ii) the good effect is intended, (iii) the bad effect is not intended as a means, (iv) the good effect outweighs the bad effect. | DDE 公式：行动 A 是允许的，如果 (i) A 本身并非内在地错误，(ii) 好的效果是被意图的，(iii) 坏的效果不是被意图作为手段，(iv) 好的效果超过坏的效果。
- Trolley case application: In 'switch' scenario, turning the trolley to kill one as a foreseen side-effect of saving five satisfies DDE; in 'bridge' scenario, pushing a fat man uses him as a means, violating condition (iii). | 电车案例应用：在“扳道岔”情境中，扳动电车杀死一人作为拯救五人的可预见副作用，满足 DDE；在“桥”情境中，推下胖子是把他当作手段，违反了条件 (iii)。
Permissible: Means ≠ Harm; Impermissible: Means = Harm
This formal contrast encapsulates the core of the double effect reasoning and its application to ethical dilemmas. | 这组形式化对比概括了双重效果推理的核心及其在伦理困境中的应用。
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