📚 Year 12 Cambridge Philosophy: Formula & Theorem Quick Reference | 剑桥 AS 哲学：公式定理速查手册
Studying Cambridge AS Philosophy (9489) involves mastering logical structures, ethical calculations, and conceptual clarifications that often behave like 'formulas' or 'theorems'. This quick-reference handbook distils the core arguments and principles from ethics, epistemology, philosophy of religion, and philosophy of mind into concise symbolic forms and memorable rules. Use it to sharpen analysis and build chains of reasoning under timed conditions.
学习剑桥 AS 哲学(9489) 需要掌握逻辑结构、伦理计算和概念澄清——这些经常表现为 “公式” 或 “定理”。这本速查手册将从伦理学、认识论、宗教哲学和心灵哲学中提炼核心论证与原则，化为简洁的符号形式和易于记忆的规则。用它来强化分析、在限时考试中构建推理链。
1. Logical Inference Rules | 逻辑推理规则
Philosophical arguments are built on deductive inference rules. The most common valid forms you will encounter are Modus Ponens, Modus Tollens, Hypothetical Syllogism and Disjunctive Syllogism. Any argument that instantiates these forms is valid.
哲学论证建立在演绎推理规则之上。最常见的有效形式是肯定前件、否定后件、假言三段论和选言三段论。任何实例化这些形式的论证都是有效的。
Modus Ponens: P → Q, P ⊢ Q
Modus Tollens: P → Q, ¬Q ⊢ ¬P
Hypothetical Syllogism: P → Q, Q → R ⊢ P → R
Disjunctive Syllogism: P ∨ Q, ¬P ⊢ Q
An argument is deductively valid if and only if it is impossible for the premises to be true and the conclusion false. Validity concerns the structure, not the actual truth of the premises.
一个论证是演绎有效的，当且仅当前提真而结论假是不可能的。有效性关乎结构，而非前提的实际真值。
| Theorem of Validity: Valid(Arg) ↔ ¬◇(Premises true ∧ Conclusion false) |
This theorem helps you evaluate the formal strength of philosophical arguments across the syllabus.
这一定理可用于评估大纲中所有哲学论证的形式力度。
2. Descartes' Cogito Theorem | 笛卡尔“我思”定理
In the Meditations, Descartes seeks an indubitable foundation for knowledge. Through methodic doubt, he arrives at the certainty of his own existence as a thinking thing. The Cogito is the first principle of his rationalist system.
在《沉思集》中，笛卡尔寻求不可怀疑的知识基础。通过普遍怀疑，他达到作为思维者的自我存在的确定性。“我思”是其理性主义体系的“第一原理”。
Dubito → Cogito → Sum
(I doubt → I think → I am)
The Cogito cannot be doubted even by an evil demon, because the very act of doubting confirms one's existence as a thinking substance. This yields a criterion for truth: 'whatever I perceive very clearly and distinctly is true', though this later requires God's veracity.
即使有恶魔欺骗，“我思”也无法怀疑，因为怀疑这一行动本身就确证了我作为思维实体的存在。这进而给出真理标准：“凡是我极清晰、极分明地知觉到的都是真的”，尽管这个标准最终还需要上帝的诚实来担保。
3. Hume's Fork and the Copy Principle | 休谟的叉子与复制原则
David Hume's empiricism rests on two powerful tools. 'Hume's Fork' divides all objects of human reason into Relations of Ideas (analytic, a priori) and Matters of Fact (synthetic, a posteriori). Any claim that fits neither category should be 'committed to the flames'.
休谟的经验论建立在两个有力工具之上。“休谟之叉”把人类理性的对象分为观念关系（分析、先天）和事实（综合、后天）。任何不属于这两类的断言都应“付之一炬”。
Knowledge = Relations of Ideas ∪ Matters of Fact
The Copy Principle states that all simple ideas are derived from corresponding simple impressions. Where there is no impression, terms are meaningless.
复制原则断言所有简单观念都来源于对应的简单印象。没有印象，对应的词语就没有意义。
∀ simple idea ξ, ∃ impression ε such that ξ = copy(ε)
These principles underpin Hume's critiques of causation, the self, and the argument from design.
这些原则支撑了休谟对因果性、自我和设计论证的批判。
4. The Problem of Induction Theorem | 归纳问题定理
Hume reveals that inductive reasoning cannot be rationally justified without circularity. We assume the future will resemble the past (the Principle of the Uniformity of Nature, UN), but any attempt to prove UN uses induction itself.
休谟揭示出归纳推理无法被非循环地辩护。我们预设未来将类似于过去（自然齐一性原则），但任何证明这一原则的尝试本身都在使用归纳。
Induction justified ⇔ UN justified
UN justified ⇒ Induction already assumed (circular)
Thus, there is no non-circular rational warrant for inductive inferences. This 'scandal of philosophy' challenges all empirical science and everyday prediction.
因此，归纳推理没有非循环的理性保证。这个“哲学的丑闻”挑战了一切经验科学和日常预测。
5. Kant's Categorical Imperative Formulas | 康德定言命令公式
Kant's ethical system is built around the Categorical Imperative, which commands unconditionally. He offers three primary formulations; the two directly examined at AS are the Formula of Universal Law and the Formula of Humanity.
康德的伦理体系围绕定言命令建立，它无条件地颁布命令。他给出了三个主要公式；AS 考试直接涉及的是普遍法则公式和人性公式。
Universal Law: Act only according to that maxim whereby you can at the same time will that it should become a universal law.
Humanity: Act so that you treat humanity, whether in your own person or in the person of another, always as an end, never merely as a means.
The Universal Law formula generates a contradiction test: if universalising a maxim leads to a contradiction in conception or in will, the action is forbidden. The Humanity formula forbids using rational beings as mere tools.
普遍法则公式产生矛盾检测：如果普遍化一个准则会导致“概念中的矛盾”或“意志中的矛盾”，则行为被禁止。人性公式禁止把理性存在者仅仅当作工具使用。
6. The Utilitarian Calculus | 功利主义计算
Classical utilitarianism, developed by Jeremy Bentham and John Stuart Mill, holds that the morally right action is the one that produces the greatest overall net utility – pleasure minus pain.
由边沁和密尔发展的古典功利主义认为，道德上正确的行为就是能产生最大净功利——快乐减去痛苦——的行为。
Net Utility = Σ Pleasure – Σ Pain
Bentham proposed a 'felicific calculus' with seven dimensions: intensity, duration, certainty, propinquity, fecundity, purity and extent. The formula, applied to each action, yields a quantitative comparison.
边沁提出“快乐计算”的七个维度：强度、持续时间、确定性、远近、继生性、纯净性和范围。将这个计算公式应用于每个行为，可以得出定量比较。
U(A) = (Iₚ×Dₚ×Cₚ×Pᵣ×Fₑ×Pᵤ×Eₚ) – (Iₙ×Dₙ×Cₙ×Pᵣ×Fₑ×Pᵤ×Eₙ)
Mill later introduced qualitative distinctions: 'It is better to be a human being dissatisfied than a pig satisfied.' Higher pleasures (intellectual, aesthetic) carry more weight.
密尔后来引入质的区分：“做一个不满足的人比做一头满足的猪更好”。高级快乐（理智的、审美的）拥有更大的分量。
7. Aristotle's Doctrine of the Mean | 亚里士多德中道公式
For Aristotle, moral virtue is a state of character concerned with choice, lying in a mean relative to us, determined by reason. Virtue is the midpoint between two corresponding vices: excess and deficiency.
对亚里士多德来说，道德德性是一种与选择相关的品质，处于相对于我们而言的中道，由理性决定。德性是其对应的两个恶——过度与不及——的中点。
Virtue = Mean(Deficiency, Excess)
| Vice of Deficiency (不及) | Virtue (胜性) | Vice of Excess (过度) |
|---|---|---|
| Cowardice | Courage | Rashness |
| Stinginess | Generosity | Wastefulness |
| Self-deprecation | Truthfulness | Boastfulness |
The mean is not arithmetic but relative to the individual's circumstances and capacities, and is identified through practical wisdom (phronesis).
中道不是算术平均数，而是相对于个人的处境和能力，并借由实践智慧（明智）来识别。
8. The JTB + Gettier Theorem | JTB 与盖梯尔问题定理
The traditional tripartite analysis defines knowledge as justified true belief (JTB). However, Edmund Gettier's 1963 paper presented cases where someone has a justified true belief that intuitively is not knowledge, shaking this long-held view.
传统的三元分析把知识定义为被证成的真信念（JTB）。然而，盖梯尔在1963年的论文中提出了一些反例，在其中某人拥有被证成的真信念，但直觉上并不是知识，动摇了这一长期观点。
Traditional: K = JTB ∧ (B is true)
Gettier Theorem: JTB ≠ K
Gettier cases typically involve false lemmas or luck. Responses include adding a 'no false lemmas' condition, a causal connection, or a reliability condition. The post-Gettier formula becomes:
盖梯尔案例通常涉及虚假引理或运气。回应方式包括添加“无虚假引理”条件、因果联系或可靠性条件。经过盖梯尔修正后的公式变为：
K = JTB + X (where X = no false lemmas ∨ reliable process ∨ safety condition)
This theorem is essential for epistemic analysis in the syllabus.
这个定理对于大纲中的认识论分析至关重要。
9. The Design Argument Theorem | 目的论论证定理
The teleological argument (argument from design) infers the existence of an intelligent creator from the apparent purpose and complexity of the natural world. Paley's watchmaker analogy is its classic presentation.
目的论论证（设计论证）从自然界明显的秩序与复杂性推论出一位智慧创造者的存在。佩利的“制表匠类比”是其经典表述。
Complexity ∧ Purpose → Designer
Watch : Watchmaker :: Universe : Universe-maker
Hume's critique (through Philo) attacks the analogy: the universe is not sufficiently similar to a watch; the argument cannot establish a single, all-powerful, or benevolent designer. Darwinian evolution offers a non-intentional explanation for biological complexity.
休谟的批判（借斐洛之口）攻击了这一类比：宇宙与手表并不足够相似；该论证无法确立一位单一的、全能的或仁慈的设计者。达尔文进化论为生物复杂性提供了一种非意图性的解释。
10. The Problem of Evil Logical Formula | 恶的问题的逻辑公式
The logical problem of evil asserts an inconsistency in holding that an omnipotent, omnibenevolent, and omniscient God exists alongside the existence of evil. Either God cannot prevent evil, or does not know about it, or does not want to prevent it – then He is not the God of classical theism.
逻辑版的恶的问题断言：相信一位全能、全善、全知的上帝存在，同时又承认恶的存在，这两者是不一致的。上帝要么不能阻止恶，要么不知有恶，要么不愿阻止恶——那这就不是古典有神论中的上帝。
If (Omnipotent ∧ Omnibenevolent ∧ Omniscient) then ¬∃ evil.
