📚 A-Level CAIE Philosophy: Formula & Theorem Quick Reference | CAIE 哲学：公式定理速查手册
This quick reference guide condenses the core logical patterns, ethical principles, argument kernels and foundational definitions from the CAIE A-Level Philosophy syllabus (9696) into formula-like entries. Treat these as ready-to-use tools for analysing arguments, constructing essays and checking coherence across epistemology, ethics, philosophy of mind and metaphysics. Each formula is presented in a standardised notation to support precise reasoning under exam conditions.
本速查手册将 CAIE A-Level 哲学大纲 (9696) 中的核心逻辑模式、伦理原则、论证内核和基础定义浓缩为公式化的条目。你可以将这些公式视为分析论证、撰写论文以及检验认识论、伦理学、心灵哲学和形而上学连贯性的现成工具。每个公式都采用标准化记法，有助于你在考试中实现精确推理。
1. Modus Ponens and Modus Tollens | 肯定前件式与否定后件式
The simplest valid argument forms underpin almost all deductive reasoning in philosophy. Modus ponens affirms the antecedent, while modus tollens denies the consequent. Both guarantee that if the premises are true, the conclusion must be true. Mastering these patterns allows you to dissect complex philosophical arguments into chains of elementary valid steps.
最简单的有效论证形式支撑着哲学中几乎所有的演绎推理。肯定前件式通过肯定前件推出后件，否定后件式则通过否定后件推出前件的否定。这两种形式都能保证：如果前提为真，结论必然为真。掌握这些模式，你就能把复杂的哲学论证拆解为一系列基本的有效步骤。
Modus Ponens: P → Q, P ⊢ Q
Modus Tollens: P → Q, ¬Q ⊢ ¬P
In CAIE examinations, you might encounter conditionals such as ‘If the mind is a distinct substance, then it is indivisible’ (Descartes). Applying modus tollens, one could reason: Mind is divisible; therefore it is not a distinct substance. Always check whether the conditional is true and whether the premise denying the consequent is justified.
在 CAIE 考试中，你可能会遇到诸如“如果心灵是一个独立的实体，那么它就是不可分的”（笛卡尔）这类条件语句。运用否定后件式可以推出：心灵是可分的；因此，心灵不是一个独立的实体。你始终需要检查条件句是否为真，以及否定后件的前提是否得到辩护。
2. Hypothetical Syllogism and Disjunctive Syllogism | 假言三段论与选言三段论
Longer deductive chains often rely on hypothetical syllogism to link conditionals, while disjunctive syllogism eliminates one option to force the other. These forms are heavily used in arguments about God, free will and the mind-body problem. Recognising them speeds up analysis and reveals hidden assumptions.
较长的演绎链条往往借助假言三段论来串联条件句，而选言三段论则通过排除一个选项来推出另一个选项。这些形式在关于上帝、自由意志和身心问题的论证中大量使用。识别它们可以加快分析速度并揭示隐藏的假设。
Hypothetical Syllogism: P → Q, Q → R ⊢ P → R
Disjunctive Syllogism: P ∨ Q, ¬P ⊢ Q
For instance, Plantinga’s free will defence often takes the shape: If God exists, there is evil only if free will brings a greater good. If free will brings a greater good, then evil is justified. Thus, if God exists, evil is justified. Disjunctive syllogism appears when eliminativism asserts: either folk psychology is true or false; folk psychology is false; therefore eliminativism.
例如，普兰丁格为自由意志的辩护常呈现出这样的结构：如果上帝存在，那么只有当自由意志带来更大的善时才会有恶。如果自由意志带来更大的善，那么恶就是正当的。因此，如果上帝存在，恶就是正当的。当取消主义主张“要么民间心理学为真，要么为假；民间心理学为假，因此取消主义成立”时，运用的正是选言三段论。
3. Inductive Strength and the Principle of the Uniformity of Nature | 归纳强度与自然齐一性原理
Inductive arguments do not aim for deductive validity but for high probability. Hume’s problem of induction highlights that all inductive inferences presuppose the principle of the uniformity of nature (PUN): the future will resemble the past. Without a rational justification for PUN, the strength of any inductive prediction is merely psychological.
归纳论证不追求演绎有效性，而是追求高概率。休谟的归纳问题指出，所有的归纳推理都以自然齐一性原理为前提：未来将与过去相似。如果没有对自然齐一性原理的理性辩护，任何归纳预测的强度都只不过是一种心理习惯。
Inductive Schema: All observed A’s have been B → The next A will probably be B (presupposes uniformity)
In CAIE philosophy, when evaluating cosmological or design arguments, you must ask whether the inference from observed regularities to an unobserved cause is inductively strong. Hume’s critique shows that such inferences rely on PUN, which itself cannot be justified without circularity. This formula reminds you to treat all induction with caution.
在 CAIE 哲学中，当评价宇宙论论证或设计论证时，你必须追问：从观察到的规律推出不可观察的原因，这种归纳推理是否足够强？休谟的批判表明，这类推理依赖于自然齐一性原理，而该原理本身无法在为无法循环的情况下得到辩护。这条公式提醒你对所有归纳推理都要保持谨慎。
4. Act Utilitarianism: The Greatest Happiness Calculus | 行为功利主义：最大幸福计算
Bentham’s act utilitarianism can be expressed as a moral formula: an action is right iff it maximizes net expected utility across all affected beings. The calculus involves seven dimensions: intensity, duration, certainty, propinquity, fecundity, purity and extent. Mill later refined the theory by introducing qualitative differences in pleasures.
边沁的行为功利主义可以表达为一条道德公式：一个行为在道德上是正确的，当且仅当它为所有受影响的存在者带来了最大的净期望效用。这个计算涉及七个维度：强度、持续时间、确定性、远近度、衍生性、纯粹性和范围。密尔后来引入了快乐质的差异，从而完善了这一理论。
AU Formula: φ is right ↔ φ produces max Σ(Pleasure – Pain) for all S affected, with probability weighting
When applying this in CAIE essays, remember to consider objections: the calculation is impractical, it ignores justice and individual rights, and it can license horrendous acts if they happen to maximise utility. Rule utilitarianism attempts to overcome these by testing rules rather than acts, but the basic hedonic calculus remains the conceptual core.
在 CAIE 论文中应用这条公式时，要记得列出反对意见：计算本身不切实际，忽视了正义和个人权利，并且如果某些可怕的行为恰好能实现功利最大化，它也会允许这些行为。规则功利主义试图通过检验规则而非具体行为来克服这些困难，但基本的快乐计算仍然是其概念核心。
5. Kant’s Categorical Imperative: Formula of Universal Law | 康德的定言命令：普遍法则公式
Kant’s first formulation of the categorical imperative demands that a maxim be universalisable without contradiction. The logical structure can be captured as: act only according to that maxim whereby you can, at the same time, will that it should become a universal law. This universalisation test functions like a logical purity check on moral principles.
康德对定言命令的第一条表述要求，准则必须能经得起普遍化检验而不产生矛盾。其逻辑结构可以表述为：只依据那个你同时能够意愿它成为普遍法则的准则去行动。这种普遍化测试就像对道德原则进行一次逻辑纯度检验。
FUL: Maxim M is permissible ↔ M can be consistently willed as universal law without contradiction in conception or in will
In CAIE answers, illustrate with the false promise example: if everyone made promises they do not intend to keep, the institution of promising would collapse—a contradiction in conception. The formula helps you distinguish between perfect duties (violations involve a contradiction in conception) and imperfect duties (contradiction in will).
在 CAIE 答案中，可以用虚假承诺的例子来说明：如果每个人都做出自己无意兑现的承诺，承诺制度就会崩溃——这就是设想上的矛盾。这条公式可以帮助你区分完全义务（违反时产生设想上的矛盾）和不完全义务（产生意愿上的矛盾）。
6. Kant’s Categorical Imperative: Formula of Humanity | 康德的定言命令：人性公式
The second major Kantian test for moral worth centres on treating persons never merely as a means but always also as an end in themselves. This formula captures the intrinsic dignity of rational agents and has been hugely influential in applied ethics, from medical consent to business ethics.
康德的第二条主要道德价值检验标准围绕一个要求：永远不要只把人当作手段，而要始终同时将其视为目的本身。这一公式抓住了理性行动者的内在尊严，并在应用伦理学（从医疗知情同意到商业伦理）中产生了深远影响。
FH: Act so that you treat humanity, whether in your own person or in that of another, always as an end and never merely as a means
To deploy this formula in an exam, check whether a proposed action involves using someone’s rational agency without their full, free consent. For example, coercive contracts or manipulative advertising treat persons merely as tools for profit, violating FH. Note that we can use others as means, but not merely as means—a subtle distinction that strong essays exploit.
在考试中运用这一公式时，需要检查某个行为是否在未获得对方充分、自由同意的情况下利用了其理性能力。例如，胁迫性合同或操纵性广告将人仅仅当成获利工具，就违反了人性公式。注意，我们可以把人当作手段，但不能把人仅仅当作手段——这一细微区别是高分作文经常利用的关键。
7. Descartes’ Cogito: The Argument from Doubt | 笛卡尔的“我思”：从怀疑出发的论证
Descartes’ cogito (‘I think, therefore I am’) is often reconstructed as an immediate logical inference from the act of doubting to the existence of a thinking substance. In its most rigorous form it resists the evil demon hypothesis because even if deceived, I must exist to be deceived.
笛卡尔的“我思故我在”通常被重构为一条从怀疑活动到思维实体存在的直接逻辑推论。在其最严格的形式中，即使邪恶天才的假设也无法推翻它，因为即便我被欺骗，我也必须先存在，才能被欺骗。
Cogito Kernel: I am thinking (or doubting) → I exist (at least as a thinking thing)
Formally: ∀x (Doubt(x) → Think(x)), Think(I) → Exist(I) ∴ Exist(I)
CAIE questions on Descartes frequently ask you to analyse whether the cogito is an argument or a direct intuition. Some critics claim the inference depends on the suppressed premise ‘Whatever thinks exists’, which itself might be doubted. Treat this formula as a concise representation of the core rationalist move, but be ready to discuss the possibility that the self revealed is only a momentary thinking subject, not a persistent substance.
CAIE 中关于笛卡尔的问题经常要求你分析“我思”究竟是一个论证还是一种直接的直观。有些批评者认为，这个推论依赖于一个被隐含的前提——“凡思考者必存在”，而这个前提本身也可以被怀疑。把这条公式当作对核心理性主义步骤的简洁表达，同时也要准备好讨论被揭示的自我仅仅是一个瞬间的思维主体，而非一个持久的实体。
8. Ontological Argument: The Logical Kernel of Anselm and Descartes | 本体论论证：安瑟尔谟与笛卡尔的逻辑内核
Ontological arguments try to derive God’s existence from the very concept of God. Anselm’s version compares existing in the understanding alone with existing in reality, while Descartes argues that existence is a perfection contained in the idea of a supremely perfect being. Both can be formalised to test their logical validity.
本体论论证试图从上帝的概念本身推导出上帝的存在。安瑟尔谟的版本比较了“只存在于理解中”和“也存在于现实中”，而笛卡尔则主张存在是一种包含在最完满存在者观念中的完满性。两者都可以被形式化，以便检验其逻辑有效性。
Anselmian Schema: God =df that than which nothing greater can be conceived; existing in reality is greater than existing merely in the understanding; therefore God must exist in reality.
Descartes’ Perfection Formula: The idea of SUPREMELY PERFECT BEING includes all perfections; existence is a perfection; ∴ the supremely perfect being necessarily exists.
In CAIE essays, use the formula to pinpoint where objections bite. Kant famously objects that existence is not a real predicate that adds to the concept of a thing. Gaunilo’s parody island challenges the assumption that a concept can generate existence. Always link the formula back to the symmetric critique: just as the perfect island cannot be brought into existence by its definition, neither can God.
在 CAIE 论文中，用这些公式来精准定位反驳的攻击点。康德著名地反对说，存在不是一个可以增加事物概念的真实谓词。高尼罗的完美岛戏仿则挑战了“概念可以产生存在”这一假设。始终要把公式联系到对称性批判上：正如完美岛屿不能因其定义就真实存在，上帝也不能。
9. Tripartite Definition of Knowledge and the Gettier Problem | 知识的三元定义与盖梯尔问题
The traditional, pre-Gettier analysis of knowledge defines knowledge as justified true belief. This definition functions as a formula against which counterexamples are tested. Gettier cases show that justified true belief is not sufficient for knowledge, leading to various responses such as the no-false-lemma condition or reliabilism.
在盖梯尔之前，传统的知识分析将知识定义为有辩护的真信念。这个定义就像是用来检验反例的公式。盖梯尔案例表明，有辩护的真信念并不足以构成知识，这引发了各种回应，比如没有虚假预设的条件，或是可靠主义。
Traditional K=JTB: S knows that p iff (i) p is true, (ii) S believes that p, (iii) S is justified in believing that p.
In the CAIE syllabus, the Gettier problem appears both in epistemology papers and as part of broader questions on philosophical method. The formula reminds you to check for epistemic luck: if the chain of justification passes through a false premise, or if true belief is arrived at accidentally, knowledge may be absent despite satisfying JTB. This is a powerful diagnostic tool for any knowledge claim.
在 CAIE 大纲中，盖梯尔问题不仅出现在认识论部分，也作为更广泛的哲学方法问题的一部分。这条公式提醒你要检查认知运气：如果辩护链条中经过了一个假的前提，或者真信念是偶然获得的，那么即使满足了 JTB，也可能不构成知识。这是剖析任何知识主张的有力诊断工具。
10. Type Identity Theory: Mental States as Brain States | 类型同一论：心理状态即大脑状态
The mind-brain identity theory holds that mental states are numerically identical to physical states of the central nervous system. It is presented as a contingent identity, analogous to scientific discoveries like ‘water is H₂O’. This formulation acts as a bridge law that connects mental vocabulary with neuroscientific vocabulary.
心脑同一论主张，心理状态与中枢神经系统的物理状态在数量上是同一的。这种同一被呈现为一种后天偶然的同一性，类似于“水是 H₂O”的科学发现。这个表述充当了连接心理词汇和神经科学词汇的桥梁定律。
Type Identity: For any type of mental state M, there is a type of neural state N such that M = N (contingent identity).
When evaluating identity theory in CAIE, use this formula to clarify the difference between token and type identity, and to consider objections: multiple realizability (the same mental type can be realised by different neural types across species), and the explanatory gap (knowing all the physical facts does not seem to convey what it is like to have the experience). The formula highlights both the theory’s parsimony and its vulnerability.
在 CAIE 中评价同一论时，用这个公式来厘清类型同一与殊型同一的区别，并由此思考反对意见：多重可实现性（同一个心理类型可以在不同物种中由不同的神经类型实现）和解释的鸿沟（知道了所有物理事实似乎仍不能传达拥有该体验是什么样子）。这条公式既展示了理论的简洁性，也揭示了其脆弱之处。
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