📚 Year 13 AQA Philosophy: Formula & Theorem Quick Reference Handbook | AQA哲学公式定理速查手册
Philosophy is often seen as a discipline of arguments rather than calculations. However, many of the most powerful philosophical moves can be crystallised into formulaic structures, logical schemas, and theorem-like principles. This handbook compiles the essential ‘formulas’ and ‘theorems’ that every Year 13 AQA Philosophy student should have at their fingertips. From the hedonic calculus to the structure of the Gettier problem, these compact formulations allow you to dissect complex texts, construct precise exam answers, and recognise patterns across epistemology, moral philosophy, metaphysics of mind, and philosophy of religion.
哲学常被视为一门侧重论证而非计算的学科。然而，许多最具力量的哲学推进都可以凝结为公式化的结构、逻辑图式和类定理的原理。本手册汇集了每位Year 13 AQA哲学学生都应烂熟于心的核心“公式”与“定理”。从快乐计算法到盖梯尔问题的结构，这些精练的表述能帮助你剖析复杂文本，构建严谨的考试答案，并在认识论、道德哲学、心灵形而上学和宗教哲学中识别出贯穿的模式。
1. Logical Inference Rules | 逻辑推理规则
Valid deductive arguments are the engine of philosophical reasoning. The following inference rules function as the logical ‘formulas’ that guarantee the truth of the conclusion if the premises are true. Modus ponens (affirming the antecedent) states: If P then Q, P is true, therefore Q must be true. Its form is P → Q, P ⊢ Q. Modus tollens (denying the consequent) is equally vital: If P then Q, not-Q, therefore not-P (P → Q, ¬Q ⊢ ¬P). These patterns underpin countless a priori arguments in the AQA syllabus, from Descartes’ trademark argument to versions of the cosmological argument.
有效的演绎论证是哲学推理的引擎。下列推理规则发挥着逻辑“公式”的作用，只要前提为真，就能保证结论为真。肯定前件式（modus ponens）陈述为：如果 P 则 Q，P 为真，因此 Q 必然为真。其形式为 P → Q，P ⊢ Q。否定后件式（modus tollens）同样关键：如果 P 则 Q，非 Q，因此非 P（P → Q，¬Q ⊢ ¬P）。这些模式支撑着AQA大纲中无数的先验论证，从笛卡尔的商标论证到宇宙论论证的某些版本。
Hypothetical syllogism chains conditionals: P → Q, Q → R ⊢ P → R. Disjunctive syllogism works with ‘or’: P ∨ Q, ¬P ⊢ Q. A reductio ad absurdum assumes the negation of what you want to prove, derives a contradiction, and thereby establishes the original claim. Recognising these forms allows you to test validity quickly.
假言三段论链接条件句：P → Q，Q → R ⊢ P → R。析取三段论处理“或”：P ∨ Q，¬P ⊢ Q。归谬法假设所要证明之命题的否定，推导出矛盾，从而确立原命题。识别这些形式能让你快速检验有效性。
2. The Felicific Calculus (Hedonic Calculus) | 快乐计算（幸福微积分）
Bentham’s act utilitarianism proposes a quasi-mathematical procedure to determine the moral worth of an action. The felicific calculus considers seven dimensions of pleasure and pain produced by an act: intensity, duration, certainty, propinquity (nearness), fecundity (chance of being followed by similar sensations), purity (chance of not being followed by opposite sensations), and extent (the number of people affected). The net utility can be expressed as the sum over all affected individuals of (intensity × duration × certainty factor …) adjusted by positive and negative valences.
边沁的行为功利主义提出了一种准数学程序，用以衡量一个行为的道德价值。快乐计算法考量行为所产生的快乐与痛苦的七个维度：强度、持续时间、确定性、远近（迫近性）、丰产性（被同类感觉跟随的机会）、纯度（不被相反感觉跟随的机会）和广度（受影响的人数）。净效用可以表达为所有受影响个体的（强度 × 持续时间 × 确定性因子……）之总和，并根据正负价值进行调整。
Though Bentham never provided a single numerical formula, the essential imperative is: the right action is the one that maximises the total net pleasure over pain. Critics challenge the commensurability of different pleasures and the practical impossibility of performing such a calculation before every act. Mill later refined the theory by distinguishing higher and lower pleasures, adding a qualitative weight: ‘higher pleasure’ > ‘lower pleasure’.
尽管边沁从未给出单一的数字公式，但其核心律令是：正确的行为就是那个能最大化净快乐总量而非痛苦总量的行为。批评者质疑不同快乐之间的可通约性，以及在每个行为前进行此种计算的实际不可能性。密尔后来通过区分高级快乐和低级快乐来完善该理论，增加了质的权重：“高级快乐” > “低级快乐”。
3. Kant’s Categorical Imperative Formulae | 康德的绝对命令诸公式
Kant’s supreme principle of morality can be formulated in several equivalent ways, functioning as ethical ‘theorems’ that test maxims. The Formula of Universal Law: ‘Act only according to that maxim whereby you can at the same time will that it should become a universal law.’ This generates a contradiction-in-conception or contradiction-in-will test. The Formula of Humanity: ‘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.’ A permissible maxim must pass both universalisability and respect-for-persons criteria.
康德的道德最高原则可以用几种等价的方式加以表述，它们发挥着检验准则的伦理“定理”功能。普遍法则公式：“只按照你同时能够愿意它成为一条普遍法则的那个准则去行动。”这会产生概念矛盾或意愿矛盾的检验。人性公式：“你要如此行动，即无论是你人格中的人性，还是任何他人人格中的人性，你始终都将其当作目的，而绝不仅仅当作手段。”一条被允许的准则必须同时通过可普遍化和尊重人格的检验。
The Formula of Autonomy and the Kingdom of Ends further require that we act as legislating members of a universal realm of rational beings. In exam terms, the ‘formula’ for applying Kant: (1) State the maxim, (2) universalise it, (3) look for contradictions in conception or will, (4) check whether any rational being is used merely as a means. The Trolley Problem illustrates a clash: pushing the fat man uses him as a mere means (impermissible), while pulling a lever arguably does not.
自律公式与目的王国公式进一步要求我们作为普遍理性存在者王国的立法成员而行动。在考试中，应用康德的“公式”为：（1）陈述准则，（2）将其普遍化，（3）寻找概念或意愿中的矛盾，（4）检查是否有理性存在者被仅仅当作手段。电车难题展示了一种冲突：推下胖子是利用他作为单纯手段（不被允许），而拉动操纵杆则可能并非如此。
4. The JTB Definition of Knowledge | 知识的JTB定义
For decades, the tripartite definition served as the standard ‘theorem’ of knowledge: 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. Symbolically, Knowledge = Justified True Belief. Each condition is taken to be individually necessary and jointly sufficient. The truth condition rules out false beliefs; the belief condition requires the subject to actually hold the proposition; the justification condition demands that the belief is not a lucky guess.
数十年来，三元定义一直作为知识的标准“定理”：S 知道 p，当且仅当（i）p 为真，（ii）S 相信 p，并且（iii）S 有理由地相信 p。符号化表示为：知识 = 被确证的真信念。每个条件被认为单独必要且联合充分。真之条件排除了虚假信念；信念条件要求主体实际持有该命题；确证条件要求该信念并非幸运的猜测。
This formula is crucial for understanding the classic response to scepticism as well as the stage-setting for Gettier. In AQA epistemology, you must be able to state this structure precisely and apply it to examples such as ‘the stopped clock’ or ‘sheep in the field’ to test whether each condition is satisfied.
这一公式对于理解对怀疑论的经典回应以及盖梯尔问题的铺垫至关重要。在AQA认识论中，你必须能够精确地陈述这一结构，并将其应用于如“停走的钟”或“田野里的绵羊”等例子，以检验每个条件是否得到满足。
5. The Gettier Problem Structure | 盖梯尔问题结构
Gettier cases demonstrate that justified true belief is not sufficient for knowledge. The general ‘theorem’ for constructing a Gettier case: Step 1: Take a justified but false belief (inferential or perceptual). Step 2: Form a new true belief by deducing a disjunction or by relying on a background fact that coincidentally makes the conclusion true. Step 3: The belief is now true and justified, yet it is clearly not knowledge because the truth arises from luck, not from the justification. The standard form: S has strong evidence for a false proposition f; S infers a true proposition t from f (e.g., ‘Either Jones owns a Ford or Brown is in Barcelona’); t is true, but S’s justification rests on the false f.
盖梯尔案例表明，有确证的真信念并不足以构成知识。构建盖梯尔案例的一般“定理”：第一步：获取一个得到了确证但却为假的信念（无论是推理性的还是知觉性的）。第二步：通过推论出一个析取命题，或凭借一个碰巧使结论为真的背景事实，形成一个为真的新信念。第三步：该信念现在既真实又得到了确证，但它显然不是知识，因为其真实性源于运气，而非源于确证。标准形式为：S 拥有支持假命题 f 的有力证据；S 从 f 推论出一个真命题 t（例如，“要么琼斯有一辆福特车，要么布朗在巴塞罗那”）；t 为真，但 S 的确证却建立在虚假的 f 之上。
The ‘formula’ for solving the Gettier problem has generated numerous responses: adding a ‘no false lemmas’ condition, requiring causal connection, or shifting to reliabilism. In your AQA essays, being able to outline the exact inferential structure of a Gettier case shows analytical maturity.
解决盖梯尔问题的“公式”催生了众多回应：增添“无虚假引理”条件、要求因果联结，或转向可靠主义。在你的AQA论文中，能够勾勒出盖梯尔案例的确切推理结构，将展示出分析上的成熟度。
6. Hume’s Problem of Induction | 休谟的归纳问题
Hume articulated the problem that our inductive inferences lack rational justification – a sceptical ‘theorem’ about the limits of reason. Inductive reasoning moves from observed instances to unobserved instances, relying on the assumption that nature is uniform (the Principle of Uniformity of Nature, or PUN). The ‘vicious circularity formula’: we cannot justify PUN (or any inductive conclusion) by demonstration (it is not a necessary truth) nor by probable reasoning (any such reasoning already presupposes that the future will resemble the past). Any attempt to use induction to justify induction begs the question.
休谟清晰地表述了这样一个问题：我们的归纳推理缺乏理性的辩护——一个关于理性界限的怀疑论“定理”。归纳推理从已观察到的实例推到未观察的实例，依赖于自然具有齐一性这一假设（自然齐一性原则，PUN）。“恶性循环公式”为：我们无法通过演绎证明（因为PUN不是必然真理），也无法通过概然推理来辩护PUN（或任何归纳结论），因为任何此类推理都已经预设了未来与过去相似。任何试图用归纳来为归纳辩护的做法都属于乞题谬误。
This poses a foundational challenge for scientific knowledge and for the theist’s design argument, which depends on inductive generalisation from order to a designer. The structure: (1) All inductive arguments depend on PUN. (2) PUN cannot be rationally justified. (3) Therefore, no inductive belief is rationally justified. This formula recurs throughout the AQA epistemology and philosophy of religion units.
这对科学知识以及依赖从秩序到设计者的归纳概括的设计论证构成了根本挑战。其结构为：（1）所有归纳论证都依赖于PUN。（2）PUN无法得到理性辩护。（3）因此，没有任何归纳信念是理性上有辩护的。这一公式在AQA认识论和宗教哲学单元中反复出现。
7. Descartes’ Method of Doubt & the Cogito | 笛卡尔的怀疑方法与“我思”
Descartes employed a systematic method to find an indubitable foundation for knowledge. His method resembles a logical algorithm: (1) Withhold assent from any proposition that can be doubted; (2) Deploy three waves of doubt – sensory illusion, dreaming, and the evil demon; (3) Search for a proposition that survives even the most radical sceptical scenario. The Cogito emerges as the first certainty: ‘I think, therefore I am’ (Cogito, ergo sum). Even if a demon deceives me, I must exist to be deceived. This can be expressed as the necessary truth: If I am thinking, then I exist.
笛卡尔采用了一套系统方法来为知识寻找不可动摇的基础。他的方法类似于一个逻辑算法：（1）对任何可怀疑的命题暂停判断；（2）发动三波怀疑——感官幻觉、梦境和邪恶魔鬼；（3）寻找一个即使在最激进的怀疑情景下也屹立不倒的命题。我思作为第一个确定性浮现出来：“我思，故我在”（Cogito, ergo sum）。即便有恶魔在欺骗我，为了被欺骗，我也必须存在。这可以表达为必然真理：如果我正在思考，那么我存在。
The Cogito is not an inference from thinking to existence, but a direct intuition of one’s own existence as a thinking thing. The logical form: For any property F, if I am doubting/thinking, I exist. This provides a model of foundationalist justification. The ‘clear and distinct perception’ rule then becomes the criterion for further knowledge: whatever I perceive very clearly and distinctly is true.
我思并非从思考到存在的推论，而是对自己作为一个思考者的存在的直接直观。其逻辑形式为：对于任何属性 F，若我正在怀疑/思考，则我存在。这提供了一种基础主义辩护的模型。随后，“清晰明确感知”的规则便成为进一步知识的标准：凡是我十分清楚、分明地感知到的东西就是真的。
8. Descartes’ Conceivability Argument for Substance Dualism | 笛卡尔实体二元论的可设想性论证
Descartes’ modal argument for the real distinction between mind and body is a powerful ‘theorem’ in metaphysics of mind. The argument utilises Leibniz’s Law (the indiscernibility of identicals): if a = b, then any property of a is a property of b. Its contrapositive is the discernibility of distincts: if there is a property that a has and b lacks, then a ≠ b. Descartes claims: (1) I can clearly and distinctly conceive of myself as a thinking, non-extended thing existing without my body. (2) Whatever I can clearly and distinctly conceive is possible. (3) Therefore, it is possible that I (mind) exist without my body. (4) If it is possible that mind exists without body, then mind has the property ‘can exist without body’ while body lacks that property (it cannot exist without itself). (5) Hence, by Leibniz’s Law, mind ≠ body (substance dualism).
笛卡尔关于心灵与身体之间实在区分的模态论证是心灵形而上学中一个有力的“定理”。该论证运用了莱布尼茨律（不可分辨的同一性）：若 a = b，则 a 的任何属性也是 b 的属性。其逆否命题是异者的可分辨性：若存在某个属性为 a 所有而 b 所无，则 a ≠ b。笛卡尔主张：（1）我能清楚明确地设想自己是一个思考的、非广延的东西，在没有我身体的情况下存在。（2）凡我能清楚明确设想的，就是可能的。（3）因此，我（心灵）在没有身体的情况下存在是可能的。（4）若心灵无需身体而存在是可能的，那么心灵具有“可无身体而存在”的属性，而身体缺乏该属性（身体不能无自身而存在）。（5）因此，根据莱布尼茨律，心灵 ≠ 身体（实体二元论）。
The critical response often focuses on the move from conceivability to possibility: what we can conceive may be limited by our ignorance of essential natures. The masked man fallacy illustrates that epistemic possibility does not entail metaphysical possibility. This formulaic structure is the backbone of many essays on dualism.
批判性回应往往聚焦于从可设想性到可能性的跨越：我们所能设想的可能受限于我们对本质属性的无知。蒙面人谬误表明，认知的可能性并不蕴含形而上学的可能性。这一公式化结构是许多关于二元论论文的脊梁。
9. Pascal’s Wager Expected Value | 帕斯卡赌注的期望值
Pascal’s Wager frames belief in God as a rational decision-theoretic gamble. The core ‘formula’ employs expected value: EV(believe) = (Probability of God × Infinite reward) + (Probability of no God × Finite loss). Since the potential reward of theistic belief is an infinite happiness (eternal salvation), even a tiny probability makes the expected value infinite: EV = ∞ × p + (−c) × (1−p) = ∞. The expected value of not believing is merely finite, making belief the rational choice according to decision theory. This is often presented in a decision matrix: if God exists, believer gets infinite gain, non-believer infinite loss; if God does not exist, both have finite outcomes. The dominance or expected value approach counsels belief.
帕斯卡的赌注将对上帝的信仰框定为一种理性的决策-理论赌局。其核心“公式”运用期望值：EV（信仰）=（上帝存在的概率 × 无限奖赏）+（上帝不存在的概率 × 有限损失）。由于有神论信仰的潜在回报是无限的幸福（永恒救赎），哪怕概率极小也会使期望值变成无穷大：EV = ∞ × p + (−c) × (1−p) = ∞。不信的期望值只是有限的，因此根据决策理论，信仰是理性的选择。这常以决策矩阵的形式呈现：若上帝存在，信徒获得无限收益，非信徒无限损失；若上帝不存在，双方结果皆为有限。基于优势策略或期望值的方法均建议选择信仰。
Objectors deploy the ‘many gods’ challenge (which god?), the impossibility of genuine belief by fiat, and the idea that a just God might not reward pragmatic betting. Nevertheless, the wager’s algebraic structure remains a memorable ‘formula’ for the rationality of religious commitment.
反对者提出“多神”挑战（该信仰哪一位神？）、通过命令产生真诚信仰的不可能性，以及公正的上帝可能不会奖赏实用主义赌注的观点。尽管如此，赌注的代数结构仍然是关于宗教委身之合理性的一个令人难忘的“公式”。
10. Bayesian Confirmation Theory | 贝叶斯确认理论
Bayesian epistemology formalises how evidence supports hypotheses. The theorem: P(H|E) = [P(E|H) × P(H)] / P(E), where P(H) is the prior probability of the hypothesis, P(E|H) the likelihood of the evidence given the hypothesis, and P(E) the total probability of the evidence. In philosophy of religion, Bayes’ theorem is used to assess arguments like the design argument: does the evidence of order and fine-tuning raise the probability of the God hypothesis? The likelihood ratio P(E|H1)/P(E|H2) measures how strongly evidence favours one hypothesis over another.
贝叶斯认识论将证据对假设的支持程度形式化。定理表达为：P(H|E) = [P(E|H) × P(H)] /
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