📚 A-Level CIE Philosophy: Quick Reference Handbook for Key Formulas and Theorems | A-Level CIE 哲学：公式定理速查手册
This handbook is not about mathematical formulas, but about the logical structures, argument patterns, and evaluative criteria that function as the ‘theorems’ of philosophical analysis. Designed specifically for CIE A-Level Philosophy candidates, it distils the essential conceptual machinery you must command in the examination hall. Each entry is a compact, precision-tooled instrument for deconstructing stimulus passages and constructing high-band essays.
本手册并非关于数学公式，而是关于逻辑结构、论证模式与评估标准——这些正是哲学分析中的“定理”。本书专为 CIE A-Level 哲学考生设计，提炼了你在考场上必须掌握的核心概念工具。每一个条目都是一件精密打磨的工具，用于解构刺激材料段落并构建高分论文。
1. The Tripartite Definition of Knowledge (JTB) | 知识的三元定义（JTB）
Knowledge is classically analysed as Justified True Belief. A subject S knows that proposition P if and only if: (i) P is true; (ii) S believes that P; (iii) S is justified in believing that P. This structure serves as the baseline model against which all epistemological theories are measured. In essays, always articulate it explicitly before introducing Gettier-style objections, for it frames the entire debate.
知识经典地被分析为“合理的真信念”。主体 S 知道命题 P，当且仅当：(i) P 为真；(ii) S 相信 P；(iii) S 有理由相信 P。这一结构是衡量所有认识论理论的基准模型。在论文中，务必在引入盖蒂尔式反驳前明确阐述它，因为它框定了整个辩论。
K = J + T + B (where K is defeasible upon Gettier cases)
K = J + T + B （在盖蒂尔案例下可废止）
2. Gettier Cases and the Defeasibility Condition | 盖蒂尔案例与可废止性条件
Gettier cases demonstrate that JTB is not sufficient for knowledge by presenting scenarios where a belief is justified and true, yet only accidentally so. The standard formula for a Gettier-style counterexample is: justified false belief + deductive inference + accidental truth = justified true belief without knowledge. Whenever you analyse a definition of knowledge, apply this pattern to test its resilience. The ‘No False Lemmas’ response adds a fourth condition: S’s belief must not be inferred from any false proposition.
盖蒂尔案例表明，JTB 对于知识而言并非充分条件，它通过呈现这样的情景——某个信念既合理又为真，但仅仅是出于偶然——来达成这一目的。盖蒂尔式反例的标准公式为：合理的虚假信念 + 演绎推理 + 偶然为真 = 缺乏知识的合理真信念。每当你分析一个知识定义时，就用这一模式来检验其稳健性。“无错误引理”回应添加了第四个条件：S 的信念不得从任何虚假命题推出。
- Pattern: Justification → False Lemma → Deduction → Accidental Truth → No Knowledge
- 模式：合理性 → 错误引理 → 演绎 → 偶然为真 → 非知识
3. Reliabilism: The Causal-Production Formula | 可靠主义：因果生成公式
Reliabilism replaces justification with reliable cognitive processes. Goldman’s early causal theory states: S knows that P if and only if the fact that P is causally connected in an appropriate way with S’s believing that P. The core theorem is that belief-forming mechanisms must have a high truth-ratio in the actual world and in relevant counterfactual situations. In your examination responses, diagram the causal chain from fact to belief, then scrutinise it for deviant causal pathways that produce true belief without knowledge (e.g., the fake barn case).
可靠主义用可靠的认知过程取代了合理性。戈德曼早期的因果理论声称：S 知道 P，当且仅当 P 这一事实以恰当的方式因果地关联着 S 对 P 的相信。其核心定理是，信念形成机制在现实世界以及相关反事实情景中，都必须具有高真值比。在考试作答时，图示从事实到信念的因果链条，然后仔细审查是否存在产生真信念而非知识的异常因果路径（例如假谷仓案例）。
4. Virtue Epistemology: The Credit Principle | 德性知识论：功劳原则
Virtue epistemology formulates knowledge as true belief arising from intellectual virtue. The governing principle is that S knows that P if and only if S’s cognitive success (true belief) is creditable to S’s cognitive agency—S believes truly because S exercised intellectual virtue. The key evaluative formula: Success from Ability = Knowledge. When success is due to luck rather than ability, knowledge is absent. Apply this in essays by asking whether the truth of the belief is best explained by the agent’s competence or by situational luck.
德性知识论将知识表述为源自理智德性的真信念。指导原则是：S 知道 P，当且仅当 S 的认知成功（真信念）可归功于 S 的认知能力——S 因运用了理智德性而真确地相信。关键评估公式：源于能力的成功 = 知识。当成功源于运气而非能力时，知识便缺席。在论文中运用此点，要追问信念之真，最好由主体的能力来解释，还是由情境运气来解释。
5. The Argument from Illusion and Sense-Data | 幻觉论证与感觉材料
The Argument from Illusion is structured as a deductive proof for sense-data. Premise 1: In illusions and hallucinations, we are directly aware of something. Premise 2: That something cannot be a physical object (since it lacks the properties the object actually has). Conclusion: We are directly aware of sense-data, not mind-independent objects. The critical theorem holds that perceptual experience essentially involves a mental intermediary. When evaluating, attack the ‘Common Kind Assumption’—the claim that veridical and non-veridical experiences share the same fundamental nature.
幻觉论证被构建为一个支持感觉材料的演绎证明。前提 1：在幻觉与幻象中，我们直接觉察到某物。前提 2：该物不可能是物理对象（因为它缺少该对象实际具有的属性）。结论：我们直接觉察到的是感觉材料，而非心灵独立的对象。批判性定理认为，知觉经验本质上包含一个心理中介。进行评估时，要攻击“共同种类假设”——即声称真实经验与非真实经验享有相同的基本性质。
6. Direct Realism and the Relational Formula | 直接实在论与关系公式
Direct realism asserts that perception is a direct relation to mind-independent objects and their properties. The formula is: Perceptual experience = direct acquaintance with physical objects. Disjunctivism, a sophisticated variant, redefines the logical structure: veridical perception and hallucination are fundamentally different kinds of mental states, even when subjectively indistinguishable. The evaluation hinges on whether disjunctivism can account for the phenomenological similarity without collapsing into the Common Kind Assumption. In essays, test direct realism against the causal argument and the time-lag argument.
直接实在论断言，知觉是与心灵独立对象及其属性的直接关系。公式为：知觉经验 = 与物理对象的直接亲知。析取论作为一种复杂的变体，重新定义了逻辑结构：真实知觉与幻觉是根本不同的心理状态种类，即便在主观上不可区分。评估关键在于，析取论能否解释现象上的相似性，而不陷入共同种类假设。在论文中，要用因果论证和时间滞后论证来检验直接实在论。
7. Utilitarianism: The Hedonic Calculus | 功利主义：快乐计算法
Bentham’s act utilitarianism proposes a quantitative metric for moral action: the Hedonic Calculus. While not a numeric formula in the strict sense, it mandates evaluating actions along seven dimensions: Intensity, Duration, Certainty, Propinquity, Fecundity, Purity, and Extent. The decision procedure is to sum the pleasure and pain produced for all affected by the action, and the right action maximises net utility. Mill’s qualitative reformulation distinguishes higher and lower pleasures—a theorem often expressed as: ‘It is better to be a human being dissatisfied than a pig satisfied.’
边沁的行为功利主义提出了一种量化道德行动的度量：快乐计算法。虽然在严格意义上并非一个数值公式，但它要求沿七个维度评估行动：强度、持续时间、确定性、邻近性、丰富性、纯度与范围。决策程序是，对所有受影响者所产生的快乐与痛苦进行加总，正当的行动就是最大化净效用的行动。密尔的质化重述区分了高级快乐与低级快乐——这条定理常被表述为：“宁做不满的人，不做满足的猪。”
Right action = MAX (∑ Pleasure – ∑ Pain) across all affected beings
正当行动 = MAX (∑ 快乐 – ∑ 痛苦) 覆盖所有受影响的存在者
8. Kantian Deontology: The Categorical Imperative Formulae | 康德式义务论：定言命令公式
Kant provides not one but three major formulations of the Categorical Imperative, each functioning as a decisive test for moral permissibility. Formula of Universal Law: ‘Act only on that maxim through which you can at the same time will that it become a universal law.’ The contradiction test has two branches: contradiction in conception (a perfect duty is violated) and contradiction in the will (an imperfect duty is violated). Formula of Humanity: ‘Act so that you treat humanity, whether in your own person or in that of another, always as an end, never merely as a means.’ These formulae are the core evaluative toolkit for any Kantian examination question.
康德提供了并非一个而是三个定言命令的主要表述，每个表述都可作为道德可允许性的决定性检验。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。”矛盾检验有两个分支：构想中的矛盾（违反完全义务）和意愿中的矛盾（违反不完全义务）。人性公式：“你要这样行动，永远都把你自身中的人性以及他人身中的人性，同时用作目的，而绝不仅仅用作手段。”这些公式是任何康德式考题的核心评估工具箱。
9. Aristotle’s Function Argument and the Doctrine of the Mean | 亚里士多德的功能论证与中道学说
Aristotle’s ergon argument deduces the human good by identifying the characteristic function (ergon) of human beings. The theorem proceeds: (1) The human function is rational activity of the soul; (2) A thing’s good resides in performing its function excellently; (3) Therefore, the human good is rational activity performed in accordance with virtue (arête). The Doctrine of the Mean then specifies that moral virtue is a disposition to choose the intermediate relative to us, between excess and deficiency. The formula is: Virtue = the mean determined by practical wisdom (phronesis). Never present the mean as a mechanical midpoint; it is an Aristotelian ‘mean relative to us’.
亚里士多德的功能（ergon）论证通过确定人类的特有功能，推导出属人之善。该定理的展开：(1) 人的功能是灵魂合乎理性的活动；(2) 一物的善在于出色地行使其功能；(3) 因此，属人之善是合乎德性（arête）的理性活动。中道学说进而明确，道德德性是一种选择相对于我们而言的居间者——在过度与不及之间——的倾向。公式为：德性 = 由实践智慧（phronesis）所确定的中道。绝不要把中道描述为机械的中点；它是亚里士多德式的“相对于我们的中道”。
10. The Mind-Body Problem: Substance Dualism’s Interaction Formula | 心身问题：实体二元论的交互公式
Cartesian substance dualism posits two radically distinct substances: res cogitans (thinking substance, mind) and res extensa (extended substance, body). The defining thesis is that mind and body are distinct and can exist independently, grounded in the argument from clear and distinct perception. The conceptual formula is: I can clearly and distinctly conceive of X existing without Y, therefore X and Y are distinct. The Princess Elisabeth objection—the interaction problem—challenges how an unextended, non-spatial substance can causally interact with extended matter. This objection functions as a permanent evaluative lever in all Cartesian examination questions.
笛卡尔式的实体二元论设定了两种截然不同的实体：思执（res cogitans，思维实体，心灵）与广执（res extensa，广延实体，身体）。其定义性论点是，心灵与身体是分离的，且能够独立存在，这基于清楚分明的知觉论证。概念公式为：我能清楚分明地设想 X 在没有 Y 的情况下存在，因此 X 与 Y 是分离的。伊丽莎白公主的异议——交互作用问题——挑战了这样一个难题：一个无广延、非空间性的实体，如何能够与有广延的物质因果地交互作用。在所有笛卡尔式考题中，这一异议都作为一个恒久不变的评估杠杆。
11. The Problem of Evil: The Logical and Evidential Theorems | 恶的问题：逻辑定理与证据定理
The logical problem of evil asserts an inconsistency within classical theism: God is omnipotent, wholly good, and omniscient; yet evil exists. If God knows of evil, can prevent it, and wants to prevent it, evil cannot exist—yet it does. The free will defence offers the formula: A world containing free creatures who sometimes do evil is more valuable than a world with no free creatures. The evidential problem applies a probabilistic structure: Given the sheer quantity and distribution of apparently pointless suffering (gratuitous evil), the existence of a theistic God is highly improbable. In essays, the evaluative move must address whether we are in a position to assert that any particular evil is truly gratuitous.
恶的逻辑问题断言，古典有神论内部存在不一致：上帝是全能的、全善的、全知的；然而恶却存在。如果上帝知道恶，能够阻止恶，并且想要阻止恶，恶就不可能存在——然而它确实存在。自由意志辩护提供了如下公式：一个包含有时作恶的自由受造物的世界，比一个没有自由受造物的世界更有价值。恶的证据问题应用了一种概率结构：鉴于表面上无意义的苦难（无端之恶）的巨量与分布，一个有神论上帝的存在是高度不可几的。在论文中，评估的关键步骤必须处理这样一个问题：我们是否有资格断言，任何特定的恶都是真正无端的。
12. The Ontological Argument: Anselm’s Logical Structure | 本体论论证：安瑟尔谟的逻辑结构
Anselm’s ontological argument proceeds as a reductio ad absurdum. Premise: God is ‘that than which nothing greater can be conceived’. Anselm argues that if such a being existed only in the understanding and not in reality, we could conceive of a greater being—one that exists in reality as well. Since this contradicts the premise (nothing greater can be conceived), God must exist in reality. Gaunilo’s ‘perfect island’ parody challenges the inference from conceivability to actual existence. Kant’s objection is definitive: existence is not a predicate or a perfection; it is the positing of a subject with all its predicates. The formula: ‘Being is obviously not a real predicate.’
安瑟尔谟的本体论论证以一个归谬法展开。前提：上帝是“那无法设想有比之更大者的存在者”。安瑟尔谟论证道，如果这样一个存在者仅仅存在于理智中而不存在于现实中，我们就能设想一个更伟大的存在者——一个也存在于现实中的。由于这矛盾于前提（无法设想有比之更大者），上帝必然存在于现实中。高尼罗的“完美岛屿”戏仿，挑战了从可设想性到现实存在的推论。康德的反对是决定性的：存在不是一个谓词或一种完善性；它是对一个主词连同其全部谓词的设定。公式：“是（存在）显然不是一个真正的谓词。”
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