📚 Philosophy Formula & Theorem Quick Reference Manual | 哲学公式定理速查手册
This quick reference manual distils key principles, argument structures, and logical rules from the Year 13 Cambridge Philosophy syllabus into formula-like statements. Think of each entry as a ‘theorem’ that captures a foundational move in epistemology, ethics, logic, or the philosophy of mind. The aim is not to reduce philosophy to mere calculation, but to equip you with schemas for analysing arguments under timed conditions.
这本速查手册将Year 13剑桥哲学课程中的核心原理、论证结构和逻辑规则提炼为类似公式的表述。你可以把每个条目看作一条“定理”，它捕捉了知识论、伦理学、逻辑学或心灵哲学中的基础性推导。目的并非把哲学降格为单纯的计算，而是为你提供在限时条件下分析论证的清晰图式。
1. Modus Ponens | 肯定前件式
If P implies Q, and P is true, then Q must be true. Formally: (P → Q), P ∴ Q.
如果P蕴涵Q，且P为真，那么Q必然为真。形式化为：(P → Q), P ∴ Q。
This is the most straightforward valid inference rule in propositional logic. Whenever you encounter a conditional claim followed by the affirmation of its antecedent, the consequent follows deductively.
这是命题逻辑中最直接的可靠推理规则。每当你遇到一个条件断言并且其前件被肯定，后件就必然演绎地得出。
2. Modus Tollens | 否定后件式
If P implies Q, and Q is false, then P must be false. Formally: (P → Q), ¬Q ∴ ¬P.
如果P蕴涵Q，且Q为假，那么P必然为假。形式化为：(P → Q), ¬Q ∴ ¬P。
Modus tollens is essential for falsifying hypotheses. If a predicted consequence does not occur, the hypothesis that entails it is refuted. Watch out for the fallacy of denying the antecedent, which is not a valid move.
否定后件式对于否证假说至关重要。如果一个被预言的后果并未出现，那么蕴含该后果的假说即被反驳。注意不要犯否定前件的谬误，那不是一个有效的推导。
3. Hypothetical Syllogism | 假言三段论
If P implies Q, and Q implies R, then P implies R. Formally: (P → Q), (Q → R) ∴ (P → R).
如果P蕴涵Q，且Q蕴涵R，那么P蕴涵R。形式化为：(P → Q), (Q → R) ∴ (P → R)。
This rule chains conditionals together. It captures transitive reasoning and is frequently used in constructing longer deductive arguments. Ensure the middle term is identical in both premises.
此规则将条件命题串联起来。它捕捉了传递性推理，常用于构造较长的演绎论证。必须保证中项在两个前提中完全一致。
4. Disjunctive Syllogism | 选言三段论
Given a disjunction ‘P or Q’, and the negation of one disjunct, the other must be true. Formally: (P ∨ Q), ¬P ∴ Q.
给定一个选言命题“P或Q”，并否定其中一个选言支，则另一支必为真。形式化为：(P ∨ Q), ¬P ∴ Q。
This works only for inclusive ‘or’ and assumes the disjunction is exhaustive. In philosophical argument, ensure the disjuncts genuinely cover all possibilities; otherwise the inference is unsound.
这只对相容的“或”有效，并且假设选言支是穷尽的。在哲学论证中，要确保选言支真正覆盖了所有可能性；否则推理不可靠。
5. The Principle of Non-Contradiction | 不矛盾律
A proposition and its negation cannot both be true in the same sense at the same time. Formally: ¬(P ∧ ¬P).
一个命题与其否定不可能在同样意义上同时为真。形式化为：¬(P ∧ ¬P)。
This is a foundational law of classical logic. Any argument that leads to an explicit contradiction signals that at least one premise must be rejected. In reductio ad absurdum, you deliberately derive a contradiction to refute an assumption.
这是经典逻辑的基本规律。任何导出显性矛盾的论证都表明至少有一个前提必须被拒斥。在归谬法中，你有意推导出矛盾，用以驳斥某个假设。
6. Gettier Cases & JTB Definition | 盖梯尔案例与JTB知识定义
The classical analysis: S knows that P if and only if (i) P is true, (ii) S believes that P, and (iii) S is justified in believing that P. Gettier’s theorem: JTB is not sufficient for knowledge.
经典分析：S知道P，当且仅当 (i) P为真，(ii) S相信P，且 (iii) S有理由相信P。盖梯尔定理：JTB对于知识来说不是充分条件。
In a Gettier case, a belief is justified and true but depends on a false lemma or luck. For example, Smith has strong evidence that ‘the person who gets the job has ten coins in his pocket’; he gets the job and coincidentally also has ten coins, yet his belief rested on a false assumption about who would get the job. Thus, Knowledge = JTB + X, where X rules out epistemic luck.
在盖梯尔案例中，信念有理由且为真，却依赖于错误的前提或运气。例如，史密斯有强证据相信“获得职位的人口袋里有十枚硬币”；他获得了职位，且碰巧口袋里也有十枚硬币，但他的信念是基于谁会获得职位的错误假说。因此，知识 = JTB + X，其中X排除了认知运气。
7. The Categorical Imperative (Formula of Universal Law) | 定言令式（普遍法则公式）
Act only according to that maxim whereby you can at the same time will that it should become a universal law. Formally: Permissible(Act) if and only if the maxim of the action can be consistently universalized without contradiction.
只按照你同时能够意愿其成为一项普遍法则的准则去行动。形式化：一个行为是许可的，当且仅当该行为的准则能够无矛盾地被一贯地普遍化。
Contradictions can be in conception (the maxim cannot even be thought as universal law, e.g., false promising) or in the will (you cannot rationally will the universalised maxim, e.g., refusing to help others). This ‘formula’ provides a decision procedure for deontological ethics.
矛盾可以是概念上的（该准则甚至不能被思考为普遍法则，如虚假承诺），或是意愿上的（你不能理性地意愿被普遍化的准则，如拒绝帮助他人）。这一“公式”为义务论伦理学提供了一个决策程序。
8. Utilitarian Calculus | 功利计算
An action is right insofar as it tends to promote the greatest happiness of the greatest number. Formal model: Rightness(A) ∝ Σ (Pleasure) – Σ (Pain) across all affected individuals.
一个行为是正当的，当它倾向于增进最大多数人的最大幸福。形式模型：正当性(A) ∝ Σ (快乐) – Σ (痛苦)，针对所有受影响个体。
Bentham’s hedonistic calculus breaks pleasure and pain into dimensions: intensity, duration, certainty, propinquity, fecundity, purity, and extent. Contemporary consequentialism often replaces hedonistic utility with preference satisfaction, but the summative structure remains.
边沁的快乐计算将快乐与痛苦分解为强度、持续时间、确定性、远近程度、繁殖性、纯度和范围等维度。当代后果主义常以偏好满足替代快乐功利，但加总结构依然保留。
9. The Problem of Evil (Logical Form) | 恶的问题（逻辑形式）
The logical problem of evil: If an omnipotent, omnibenevolent God exists, evil would not exist. Evil does exist. Therefore, such a God does not exist. Formally: (Omnipotent ∧ Omnibenevolent) → ¬Evil; Evil; ∴ ¬(Omnipotent ∧ Omnibenevolent).
恶的逻辑问题：如果全能、全善的上帝存在，恶就不会存在。恶存在。因此，这样的上帝不存在。形式化：(全能 ∧ 全善) → ¬恶；恶；∴ ¬(全能 ∧ 全善)。
Theists typically respond either by denying that God’s omnipotence includes doing logically impossible things (so God cannot eliminate evil without removing free will) or by contesting the claim that an omnibenevolent being would necessarily eliminate all evil. The evidential problem of evil lowers the bar: the sheer quantity and distribution of evil makes God’s existence improbable.
有神论者通常回应说，上帝的全能不包括做逻辑上不可能之事（因此上帝无法在不取消自由意志的情况下消除恶），或质疑全善的存在必然消除所有恶的主张。恶的证据问题降低了门槛：恶的巨大数量和分布使得上帝的存在变得不可能。
10. Cartesian Dualism & the Interaction Problem | 笛卡尔二元论与交互问题
Substance dualism: Mind and body are distinct substances — mind is unextended thinking substance, body is extended non-thinking substance. The interaction problem: If mind and body are radically different substances, how can they causally interact?
实体二元论：心灵与身体是截然不同的实体——心灵是非广延的思维实体，身体是广延的非思维实体。交互问题：如果心灵与身体是完全不同的实体，它们如何能因果地相互作用？
Princess Elisabeth of Bohemia famously objected that contact or push-pull causation requires spatial locality, which a non-material mind lacks. Descartes’ appeal to the pineal gland merely relocates the mystery. This ‘inconceivability theorem’ challenges dualism’s explanatory power.
波希米亚的伊丽莎白公主著名地反对说，接触性或推拉式的因果作用需要空间位置，而非物质的心灵缺乏这一点。笛卡尔求助于松果腺只不过转移了谜团。这条“不可思议定理”挑战了二元论的解释力。
11. The Is-Ought Gap (Hume’s Guillotine) | 实然-应然鸿沟（休谟断头台）
No set of purely descriptive statements logically entails an evaluative or prescriptive conclusion without at least one evaluative premise. Formally: {D1, D2, …} ⊬ O, unless O is already implicit in the premises.
一组纯粹描述性的陈述，如果没有至少一个评价性前提，就不能在逻辑上蕴含一个评价性或规范性的结论。形式化为：{D1, D2, …} ⊬ O，除非O已经隐含在前提中。
Hume observed that many moral systems slide from ‘is’ to ‘ought’ without justification. For example, from ‘suffering causes pain’ we cannot directly infer ‘we ought to prevent suffering’ without a bridge principle like ‘pain is bad’. This gap is central to metaethics and prevents naturalistic fallacies.
休谟观察到，许多道德体系会在未经证成的情况下从“是”滑向“应当”。例如，从“苦难导致痛苦”不能直接推出“我们应当防止痛苦”，除非有一条“痛苦是坏的”这样的架桥原则。这一鸿沟是元伦理学的核心，可以防止自然主义谬误。
12. The Trolley Problem Decision Matrix | 电车难题决策矩阵
The trolley problem exposes the tension between consequentialist and deontological reasoning. Decision matrix for the standard bystander switch case:
电车难题暴露了后果论与义务论推理之间的张力。标准旁观者扳道岔案例的决策矩阵如下：
| Option | Utilitarian Verdict | Deontological Verdict |
| Pull lever (kill 1, save 5) | Permissible / obligatory (maximises utility) | Impermissible: uses the one as a mere means |
| Do nothing (5 die) | Impermissible (suboptimal utility) | Permissible: respecting negative duties not to harm |
This ‘theorem’ of moral inconsistency suggests that our intuitions about doing vs. allowing harm, intended vs. foreseen outcomes, and using persons as means need careful mapping. Variations like the footbridge case further refine the schemas.
这个道德不一致性的“定理”表明，我们关于“做”与“允许”伤害、意图与预见的结果、以及把人当作手段使用的直觉，需要仔细地梳理。像天桥案例这样的变体进一步精炼了这些图式。
Published by TutorHao | Philosophy Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply