📚 CAIE Year 12 Philosophy: Quick Reference Handbook of Formulae and Theorems | CAIE 12年级哲学：公式定理速查手册
This revision companion is designed to help CAIE Year 12 Philosophy students capture the logical core of major arguments, theories, and dilemmas in a compact, formula-like format. Each entry translates an influential philosophical idea into a central 'theorem' or 'formula,' enabling faster recall, sharper analysis, and clearer essay structure. The handbook covers epistemology, ethics, metaphysics, and the philosophy of religion, aligning closely with the Cambridge 9697 syllabus. Use these distillations not as a substitute for the original texts, but as a mental scaffold to organise your understanding and spark critical evaluation.
这本复习手册旨在帮助 CAIE 12 年级哲学学生以紧凑的公式化形式，掌握主要论证、理论和困境的逻辑核心。每一条目将富有影响力的哲学思想转译为中央“定理”或“公式”，以便更快速地回忆、更敏锐地分析，并构建更清晰的论文结构。本手册涵盖认识论、伦理学、形而上学和宗教哲学，与剑桥 9697 教学大纲紧密对接。请将这些提炼出的要点作为组织理解和激发批判性评价的心理脚手架，而非原典的替代品。
1. The Cogito Formula – Descartes' First Certainty | 我思公式——笛卡尔的第一确定性
Cogito, ergo sum → First Indubitable Truth
Descartes' method of radical doubt aims to raze all beliefs to the ground until he reaches something immune to even the most extravagant sceptical scenarios, including the evil demon. He discovers that while he can doubt the existence of his body, the external world, and even mathematical truths, he cannot doubt that he is doubting—and therefore thinking. From this he derives the foundational certainty: "I think, therefore I am" (Cogito, ergo sum). Every act of doubt confirms the existence of the doubter. This formula serves as the Archimedean point of his epistemology: an indubitable truth upon which a reliable system of knowledge can be rebuilt. In essay terms, the Cogito is the first premise in the rationalist project, establishing the self as a thinking substance.
笛卡尔采用彻底怀疑的方法，试图把一切信念推倒重来，直到找到某种连最夸张的怀疑情景——包括邪恶妖灵——也无法撼动的东西。他发现，尽管他可以怀疑身体的存在、外部世界的存在，甚至数学真理，但他无法怀疑自己正在怀疑，而这就意味着他正在思考。由此他得出奠基性的确定性：“我思故我在”（Cogito, ergo sum）。每一次怀疑都确证了怀疑者的存在。这一公式扮演着他认识论中阿基米德点的角色：一条不可置疑的真理，可以在此基础上重建可靠的知识体系。在论文写作中，“我思”是理性主义方案的首要前提，将自我确立为思维实体。
2. The JTB Theorem of Knowledge | 知识的三要素定理（JTB）
Knowledge = Justified True Belief (K ↔ JTB)
Rooted in Plato's Theaetetus, the standard analysis of propositional knowledge states that a subject S knows that P if and only if P is true, S believes that P, and S has adequate justification for believing that P. This tripartite definition functions like a theorem: the presence of all three conditions is taken to be necessary and sufficient for knowledge. A belief that is accidentally true or held without evidence cannot count as knowledge. For years, the JTB formula provided a neat map for structuring responses on the nature of knowledge. It also sets the stage for Gettier's famous challenge, which demonstrates that the conditions can jointly hold without yielding knowledge—suggesting the formula requires a fourth condition such as 'no false lemmas'.
源于柏拉图《泰阿泰德篇》，命题性知识的经典分析指出，主体 S 知道命题 P 当且仅当 P 为真，S 相信 P，且 S 有充分的理据支持 P。这种三元定义如同一条定理：三个条件全部满足被视为知识的充要条件。一个偶然为真或缺乏证据的信念不能算作知识。多年来，JTB 公式为构建关于知识本质的作答提供了清晰的图谱。它也为此后盖梯尔的著名挑战埋下伏笔：盖梯尔证明，即便三个条件同时满足，也未必产生知识——这意味着公式可能需要增添第四项条件，如“没有虚假前提”。
3. The Gettier Problem Contrapositive | 盖梯尔问题的反例逻辑
∃ cases where JTB ∧ ¬K → JTB is insufficient for K
Edmund Gettier's 1963 paper presents counterexamples in which a person holds a justified true belief yet intuitively lacks knowledge. A classic case: Smith has strong evidence that Jones will get the job, and he also believes (truly) that the person who will get the job has ten coins in his pocket—because he saw Jones count ten coins. Unbeknownst to Smith, he himself gets the job and also happens to have ten coins in his pocket. Smith's belief is true and justified, but it is based on a false lemma and thus does not constitute knowledge. The logical form of the Gettier problem demonstrates that the JTB conditions are jointly insufficient; the conditional 'if JTB then knowledge' is false. This forces epistemologists to supplement JTB with a robust clause that eliminates epistemic luck.
埃德蒙·盖梯尔 1963 年的论文给出了一系列反例：一个人持有确证的真信念，却在直觉上并不算拥有知识。经典情形：史密斯有充分证据相信琼斯会得到工作，并且他相信自己看见琼斯数了十个硬币，因此他相信“那个将得到工作的人口袋里有十枚硬币”。出乎意料，得到工作的是史密斯本人，而他口袋里恰好也有十枚硬币。史密斯的信念既真且得到确证，但它建立在虚假前提之上，因而不构成知识。盖梯尔问题的逻辑形式表明，JTB 条件合取并不充分；条件句“如果满足 JTB 则获得知识”为假。这迫使认识论学者为 JTB 添加一个能排除认知运气的严格条款。
4. The Hedonic Calculus – Bentham's Utility Formula | 快乐计算法——边沁的效用公式
U = f(Intensity, Duration, Certainty, Propinquity, Fecundity, Purity, Extent)
Jeremy Bentham's act utilitarianism evaluates actions by the quantity of pleasure they produce, measured by the hedonic calculus. This seven-dimensional formula weighs the value of a pleasure or pain resulting from an action according to its intensity, duration, certainty, propinquity (nearness), fecundity (likelihood of being followed by further pleasures), purity (likelihood of not being followed by pain), and extent (the number of persons affected). The principle of utility then commands us to maximise the total net pleasure across all affected individuals. Critics argue that the calculus reduces moral judgment to a crude, perhaps unworkable, arithmetic and that it fails to capture qualitative differences between pleasures—a point developed by John Stuart Mill.
边沁的行为功利主义根据行为所产生快乐的数量来评价行为，并借快乐计算法加以度量。这一七维公式从以下方面权衡某个行为所带来快乐或痛苦的值：强度、时长、确定性、远近度（距当下的远近）、丰饶度（是否可能带来更多快乐）、纯度（是否可能不伴随痛苦）以及范围（受影响的人数）。功利原则进而要求我们将所有相关个体的净快乐总量最大化。批评者指出，这一计算法将道德判断降格为粗糙的、或许无法操作的算术，也未能捕捉快乐之间的质性差异——密尔后来就这一点作了深入阐发。
5. Kant's Categorical Imperative Formulations | 康德的定言命令公式化
Formula of Universal Law: Act only on maxims that can be universalised
Formula of Humanity: Treat humanity always as an end, never merely as a means
Immanuel Kant's deontological ethics centres on the categorical imperative as the supreme principle of morality. The first formulation, the Formula of Universal Law, demands that we act only according to that maxim through which we can at the same time will that it should become a universal law. This functions as a test of consistency and rationality: if a maxim's universalisation leads to a contradiction in conception or in the will, the proposed action is morally impermissible. The second major formulation, the Formula of Humanity, commands that we treat humanity—in our own person and in others—always as an end in itself and never merely as a means. The two formulations are supposed to be equivalent expressions of the same moral law, providing analytic tools for evaluating duties such as honesty, beneficence, and the prohibition of suicide.
康德的义务论伦理学以定言命令为最高的道德原则。第一公式，即普遍法则公式，要求我们只依照那些你同时能够愿意其成为普遍法则的准则去行动。这作为一个融贯性与合理性的测试：如果某准则的普遍化会导致构想上的矛盾或意志中的矛盾，那么所拟的行动在道德上就不被允许。第二主要公式，即人性公式，命令我们要始终将人性——无论在自己身上还是他人身上——作为目的本身来对待，而绝不单纯当作手段。这两个公式应是同一道德律的等效表达，为我们评价诸如诚实、行善和禁止自杀等义务提供了分析工具。
6. Hume's Fork – The Division of Propositions | 休谟的叉子——命题的划分
All meaningful propositions: Relations of Ideas ∨ Matters of Fact
David Hume's 'fork' declares that all objects of human reason can be divided into two exclusive and exhaustive classes: relations of ideas and matters of fact. Relations of ideas are a priori, necessary, and discoverable by pure thought—such as the truths of mathematics and logic; their denial implies a contradiction. Matters of fact are a posteriori, contingent, and known through experience; their denial is always possible without contradiction. Hume then deploys this fork as a razor: any claim that falls into neither category should be committed to the flames as mere sophistry and illusion. The fork sets the stage for his sceptical argument about induction, because causal reasoning (a matter of fact) cannot be justified by either a relation of ideas or a straightforward appeal to experience without presupposing the very principle at issue.
休谟的“叉子”宣称，人类理性的一切对象可以划分为两个互斥且全尽的类别：观念关系与事实。观念关系是先天的、必然的，仅凭纯粹思维即可发现——比如数学和逻辑的真理；否认它们会导致矛盾。事实则是后天的、偶然的，须通过经验才能知道；否认它们永远有可能而不含矛盾。休谟接着将这把叉子作为剃刀：任何不能落入这两个类别的主张都应被投入烈火，视作纯粹的诡辩和幻象。这把叉子为他关于归纳的怀疑论论证架设了舞台，因为因果推理（一项事实）既无法凭借观念关系加以证成，也无法直接诉诸经验而不预设争议中的原理本身。
7. The Problem of Induction – Hume's Sceptical Theorem | 归纳问题——休谟的怀疑定理
ithout circularity: P → Q justified by past P→Q only if we assume the uniformity of nature
Hume's problem of induction exposes the circularity at the heart of all empirical reasoning that goes beyond immediate experience. When we infer that bread will nourish us tomorrow because it has done so in the past, we rely on the principle that the future will resemble the past (the Uniformity Principle). But any attempt to prove this principle must itself be either deductive—impossible, since the denial of uniformity is conceivable without contradiction—or inductive, which would beg the question. The sceptical conclusion is that our belief in induction is not rationally justified but is instead a product of custom and habit. This 'theorem' remains one of the most pressing challenges in epistemology, provoking responses from Popper's falsificationism to Strawson's descriptive justification.
休谟的归纳问题揭示了一切超出直接经验的经验推理在核心处的循环性。当我们推断面包明天仍能滋养我们，因为过去一直如此，我们依赖的是“未来会与过去相似”这一原则（齐一性原则）。但任何试图证明该原则的做法，本身要么是演绎的——这不可能，因为对齐一性的否认是可以设想且不矛盾的——要么就是归纳的，而这恰恰犯了乞题谬误。其怀疑论结论在于：我们对归纳的信念缺乏理性证成，实则不过是习惯与习俗的产物。这条“定理”至今仍是认识论中最为紧迫的挑战之一，激起了从波普尔的证伪主义到斯特劳森的描写性证成等回应。
8. The Euthyphro Dilemma Logic | 游叙弗伦困境逻辑
(Right because God commands) ∨ (God commands because right) → Arbitrariness ∨ Independence
In Plato's Euthyphro, Socrates asks whether the holy is holy because the gods love it, or the gods love it because it is holy. Applied to divine command theory, the dilemma takes the form: Is an action right because God commands it, or does God command it because it is right? If the former, morality becomes arbitrary—God could have commanded cruelty, and that would make it right. If the latter, moral standards exist independently of God, and God's role is reduced to that of a messenger, undermining the divine command theorist's claim that God is the source of morality. This logical fork challenges meta-ethical views that ground normative properties solely in divine will, and it remains central to debates in philosophy of religion.
在柏拉图《游叙弗伦篇》中，苏格拉底问道：虔敬是因为被众神所爱而虔敬，还是被众神所爱因为虔敬？将此应用到神命论上，困境的形式如下：一个行为是因为上帝命令它才正当，还是上帝因为该行为本身正当才命令它？若是前者，道德便成为任意——上帝完全可以命令残忍之事，那它就成了正义。若是后者，道德标准便独立于上帝而存在，上帝的功用降格为信使，这就削弱了神命论者关于上帝是道德本源的主张。这一逻辑分叉对将规范性属性完全奠基于神意之上的后设伦理学观点构成挑战，并在宗教哲学论辩中始终居于核心。
9. Free Will and Determinism – The Consequence Argument | 自由意志与决定论——后果论证
(Determinism ∧ No choice about past ∧ No choice about laws) → No choice about actions
Peter van Inwagen's Consequence Argument is a classic formulation of the incompatibilist position that free will and determinism cannot coexist. If determinism holds, then every fact about an agent's present actions is a necessary consequence of the remote past and the laws of nature. The agent has no choice about the remote past, nor about the laws of nature. By a valid transfer principle—roughly, if we have no choice about P, and P entails Q, then we have no choice about Q—it follows that the agent has no choice about her present actions. The argument turns the abstract thesis of determinism into a direct threat to moral responsibility. Compatibilists, such as David Hume and more recent Frankfurt-style theorists, must then show that the sort of free will worth wanting does not require the ability to do otherwise in the sense demanded by the consequence argument.
范·因瓦根的后果论证是“自由意志与决定论不相容”这一立场的经典表述。如果决定论为真，那么对于行动者当下的行为，一切事实都是遥远过去和自然律的必然后果。行动者无法对遥远过去作出选择，也无法对自然律作出选择。根据一条有效的传递原则——大致而言，如果我们对 P 没有选择，且 P 必然蕴含 Q，那么我们对 Q 也没有选择——便可推出，行动者对自己的现时行为没有选择。这一论证将抽象的决定论命题转变成了对道德责任的直接威胁。兼容论者，如休谟和晚近的法兰克福式理论家，则需要证明，那种值得欲求的自由意志并不要求后果论证所预设的那种“另样可能”的能力。
10. The Mind-Body Problem – Leibniz's Law and the Identity Thesis | 身心问题——莱布尼茨律与同一性论题
(A = B) → ∀P (PA ↔ PB); mental states have qualia, physical states do not → Mental ≠ Physical
Leibniz's Law states that if two things are identical, then any property of one is a property of the other. The mind-type identity theory claims that mental states are strictly identical to brain states. However, critics invoke Leibniz's Law to argue for dualism: mental states have subjective, qualitative properties—often called qualia—that seem irreducible to physical descriptions. For instance, the sensation of seeing red has a 'what-it-is-like' character that is not captured by any purely physical account of neural firings. Therefore, if the mental has a property that the physical lacks, they cannot be identical. This argument, sometimes called the knowledge argument or the explanatory gap, remains a cornerstone of property dualism and panpsychism, pressuring physicalists to provide reductive or eliminative accounts.
莱布尼茨律规定，如果两个事物是同一的，那么凡可归属于其中一个的任何属性，也必然归属于另一个。心理-大脑同一论主张，心理状态与大脑状态严格同一。然而，批评者援引莱布尼茨律来支持二元论：心理状态具有主观的、质的属性——常被称为感质（qualia）——而这些属性似乎无法还原为物理描述。例如，看见红色的感觉具有一种“作为什么样子”的特性，这是任何针对神经放电的纯粹物理说明所无法捕捉的。因此，如果心理拥有物理所缺乏的属性，它们就不可能同一。这一论证有时被称为知识论证或解释鸿沟，至今仍是属性二元论和泛心论的基石，也迫使物理主义者提供还原或消除性的说明。
Published by TutorHao | Philosophy Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图