📚 Philosophy Formulae & Theorems Quick Reference | 哲学公式定理速查手册
This handbook gathers the essential conceptual ‘formulae’ and argument ‘theorems’ that every AQA Year 12 Philosophy student must know. Treat these as your intellectual toolkit—each entry captures the logical structure, key definitions, and classic moves that recur in epistemology, moral philosophy, and philosophy of religion. Mastering them will turn scattered ideas into a system you can deploy with confidence in essays and exams.
本手册汇集了每一位 AQA 12 年级哲学学生必须掌握的核心概念“公式”与论证“定理”。请将它们视为你的智识工具箱——每一条目都捕捉了在认识论、道德哲学和宗教哲学中反复出现的逻辑结构、关键定义和经典论证。熟练掌握它们，能将零散的想法转化为一个体系，让你在论文和考试中充满信心地加以运用。
1. The JTB Tripartite Formula for Knowledge | 知识的JTB三元公式
Knowledge is standardly analysed as Justified True Belief (JTB). A subject S knows that proposition p if and only if: (i) p is true; (ii) S believes that p; (iii) S’s belief that p is justified. In symbolic form: Kₛp ↔ (p ∧ Bₛp ∧ Jₛp).
知识的标准分析是确证的真信念（JTB）。一个认知主体 S 知道命题 p，当且仅当：(i) p 为真；(ii) S 相信 p；(iii) S 对 p 的信念是确证的。符号形式为：Kₛp ↔ (p ∧ Bₛp ∧ Jₛp)。
This formula treats truth as objective, belief as a psychological state, and justification as having adequate evidence or reasons. The three conditions are individually necessary and jointly sufficient.
这一公式将真理视为客观的，信念视为心理状态，确证则意味着拥有充分的证据或理由。三个条件各自必要且联合充分。
2. Gettier’s Counterexample Structure | 盖梯尔反例结构
Gettier cases show that JTB is not sufficient for knowledge. The general formula: (1) S has a justified false belief q; (2) S validly deduces a true proposition p from q; (3) S’s belief that p is therefore justified and true, yet intuitively S does not know that p. The structure relies on epistemic luck where justification and truth accidentally align.
盖梯尔案例表明 JTB 并非知识的充分条件。一般结构为：(1) S 拥有一个确证的假信念 q；(2) S 从 q 有效演绎出一个真命题 p；(3) 因此 S 对 p 的信念是确证且为真的，但直觉上 S 并不知道 p。该结构依赖认知运气，确证与真理偶然对齐。
A classic instance: Smith has strong evidence that Jones owns a Ford (q). He deduces: ‘Either Jones owns a Ford or Brown is in Barcelona’ (p = q ∨ r). Unbeknownst to Smith, Jones does not own a Ford, but Brown happens to be in Barcelona. p is true and justified, yet Smith does not know it.
经典实例：史密斯有强证据表明琼斯拥有一辆福特车（q）。他演绎出：“要么琼斯拥有福特，要么布朗在巴塞罗那”（p = q ∨ r）。史密斯不知道的是，琼斯没有福特，但布朗恰好在巴塞罗那。p 为真且确证，但史密斯并不知道它。
3. Direct Realism’s Perceptual Equation | 直接实在论的知觉等式
Direct realism holds that perception gives immediate access to mind-independent physical objects. The ‘equation’ can be expressed as: Perception = Direct Acquaintance with External Object. When I see a tree, I am directly aware of the tree itself, not an intermediary.
直接实在论主张，知觉直接通达心灵独立的物理对象。其“等式”可表示为：知觉 = 与外部对象的直接亲知。当我看见一棵树时，我直接意识到树本身，而非某个中介。
This view faces the argument from illusion: if a straight stick appears bent in water, I am aware of a bent appearance, which cannot be identical to the straight stick. Thus, what I am directly aware of is not always the physical object, threatening the directness of perception.
此观点面临错觉论证的挑战：如果一根直直的棍子在水中显得弯曲，我意识到的是一个弯曲的外观，它不可能等同于笔直的棍子。因此，我直接意识到的并不总是物理对象，这威胁到知觉的直接性。
4. Indirect Realism and the Quality Distinction Formula | 间接实在论与性质区分公式
Indirect realism introduces a two-step model: Physical Object → Sense‑Data → Perceiver. The object causes non‑physical sense‑data, and we are directly aware only of these sense‑data. Locke’s formula separates qualities into primary (solidity, extension, figure, motion, number) and secondary (colour, sound, taste, etc.), where primary qualities resemble the object and secondary qualities are powers to produce ideas in us.
间接实在论引入了一个两步模型：物理对象 → 感觉材料 → 感知者。对象引起非物理的感觉材料，而我们直接意识到的只是这些感觉材料。洛克的公式将性质区分为第一性质（坚实性、广延、形状、运动、数目）和第二性质（颜色、声音、味道等），其中第一性质与对象相似，第二性质则是对象在我们心中产生观念的能力。
This leads to the veil of perception problem: if we only ever access sense‑data, how can we know that physical objects resemble our ideas? The sceptical challenge asks whether the external world beyond our perceptions can be known.
这引出了知觉之幕的问题：如果我们永远只能通达感觉材料，我们如何得知物理对象与我们的观念相像？怀疑论挑战追问，我们能否认识知觉之外的外部世界。
5. Berkeley’s Idealist Maxim: Esse est percipi | 贝克莱唯心论格言：存在即被感知
Berkeley eliminates matter by identifying physical objects with collections of ideas. The central theorem: Esse est percipi aut percipere — to be is to be perceived or to perceive. For a sensible thing, its existence consists in being perceived; for a mind, its existence consists in perceiving.
贝克莱通过将物理对象等同于观念的集合而取消了物质。核心定理：Esse est percipi aut percipere——存在即被感知或感知。对于可感事物，其存在就在于被感知；对于心灵，其存在在于感知。
The master argument summarises this: try to conceive of an object existing unperceived. You cannot, because in the very act of conceiving you are perceiving it in your mind. Therefore, the notion of mind‑independent existence is incoherent. Consistency is maintained by God, who perceives all objects continuously.
主论证概括了这一点：试想在未被感知的情况下存在一个对象。你不能，因为在设想它的那一刻，你就在心灵中感知着它。因此，心灵独立存在的概念是不融贯的。上帝持续感知一切对象，保证了世界的连续存在。
6. Anselm’s Ontological ‘Formula’ | 安瑟伦的本体论“公式”
Anselm defines God as ‘that than which nothing greater can be conceived’ (TTWNGCBC). The ontological argument reduces to the claim that if TTWNGCBC exists only in the understanding, we can conceive of a greater being that also exists in reality — which is a contradiction. Therefore, God must exist in reality.
安瑟伦将上帝定义为“无以设想其更伟大者”（TTWNGCBC）。本体论论证可归结为：如果TTWNGCBC只存在于理解中，那么我们可以设想一个更伟大的存在者，它也存在于现实中——这便是一个矛盾。因此，上帝必然存在于现实中。
In logical form: (1) Assume God (G) exists only in the mind. (2) A being with all of G’s perfections that also exists in reality is greater than G. (3) But then G is not TTWNGCBC, contradicting the definition. (4) Hence, G exists in reality. Gaunilo’s parody uses the same form for a perfect island, challenging whether existence is a genuine predicate.
逻辑形式为：(1) 假设上帝（G）只存在于心灵中。(2) 一个拥有 G 所有完美且也存在于现实中的存在者比 G 更伟大。(3) 但这样一来 G 就不是 TTWNGCBC，与定义矛盾。(4) 因此，G 存在于现实中。高尼罗的戏仿用同样的形式为一个完美的岛屿辩护，质疑存在是不是一个真正的谓词。
7. Aquinas’ First Way: The Argument from Motion | 阿奎那的第一路：运动论证
Aquinas’ formula for cosmological argument: Everything in motion is moved by another. An infinite regress of movers is impossible. Therefore, there must exist a first unmoved mover — God. Motion here means change from potentiality to actuality.
阿奎那宇宙论论证的公式：一切运动者皆被另一者所推动。无限回溯的推动者是不可能的。因此，必须存在一个第一不动的推动者——上帝。这里的运动指的是从潜能到现实的转变。
The key principle is that something cannot be both potentially and actually the same thing in the same respect; thus, it cannot move itself. The impossibility of an infinite regress of essentially ordered movers (those that depend simultaneously on others) secures the necessity of a first mover that is pure act, without potentiality.
关键原则是，某物不可能在同一个方面同时是潜能和现实；因此，它不能推动自身。本质上序列的推动者（那些同时依赖于其他推动者的序列）不可能无限回溯，这确保了必须存在一个第一推动者，它是纯现实，没有潜能。
8. Paley’s Watchmaker Theorem (Design Argument) | 佩利的钟表匠定理（设计论证）
The teleological argument employs an analogical form: Watch : Watchmaker :: Universe : Universe‑Maker. Paley argues that a watch’s complexity and purposeful arrangement of parts imply a designer. By analogy, the vastly greater complexity and purpose observed in the natural world imply a divine designer.
目的论论证运用类比形式：钟表：钟表匠 ∷ 宇宙：宇宙创造者。佩利主张，一只钟表各部分的复杂性和目的性安排暗示了设计者。经由类比，在自然界中观察到的远为巨大的复杂性和目的性暗示着一位神圣的设计者。
Hume’s criticisms refine the debate: the analogy is weak because the universe is not clearly like a machine; even if a designer were inferred, it would not entail the God of classical theism (the ‘only so much’ principle). The argument is therefore an inductive risk, not a deductive theorem.
休谟的批评深化了辩论：类比是薄弱的，因为宇宙并不明显地像一架机器；即便推出设计者，也不蕴含古典有神论的上帝（“只推及此”原则）。因此，该论证是一种归纳冒险，而非演绎定理。
9. The Logical Problem of Evil: The Inconsistent Triad | 恶的逻辑问题：不相容三元组
The logical problem of evil presents an inconsistent set of propositions: (1) God is omnipotent; (2) God is perfectly good; (3) Evil exists. If God is willing and able to prevent evil, evil would not exist. Yet it does, so at least one of these propositions must be false. Theism seems committed to all three, generating a contradiction.
恶的逻辑问题呈现了一个不相容的命题组：(1) 上帝全能；(2) 上帝全善；(3) 恶存在。如果上帝愿意并且能够阻止恶，恶便不会存在。然而恶确实存在，因此这三个命题中至少有一个为假。有神论似乎承诺这三个命题，从而产生矛盾。
The free will defence counters by adding a fourth proposition: a world with morally significant free will and no evil is logically impossible for God to create. Evil is thus the necessary price of a greater good. This blocks the logical contradiction by showing the triad is not strictly inconsistent.
自由意志辩护通过添加第四命题来对抗：一个拥有道德的显著自由却无恶的世界，上帝在逻辑上不可能创造。因此恶是更大善的必要代价。这通过表明三元组并非严格不相容，而化解了逻辑矛盾。
10. Utilitarianism’s Greatest Happiness Formula | 功利主义的最大幸福公式
Classical act utilitarianism runs on the Greatest Happiness Principle: an action is right insofar as it tends to maximise overall happiness, defined as pleasure minus pain. For any set of available actions {A₁, A₂, …}, the morally obligatory action is that which yields max Σ (pleasure − pain) across all sentient beings affected.
古典行为功利主义运行于最大幸福原则之上：一个行动在倾向于最大化总体幸福的程度上才是正确的，幸福被定义为快乐减去痛苦。对于任何一组可用行动 {A₁, A₂, …}，道德上义务的行动就是给所有受影响的众生带来最大 Σ (快乐 − 痛苦) 的那个。
Bentham’s hedonic calculus measures pleasure by seven factors: intensity, duration, certainty, propinquity, fecundity, purity, and extent. Mill later refined the theory by distinguishing higher and lower pleasures, insisting that ‘it is better to be a human being dissatisfied than a pig satisfied’.
边沁的快乐计算法通过七个因子衡量快乐：强度、持续时间、确定性、邻近性、丰富性、纯度和范围。穆勒后来通过区分高级快乐和低级快乐完善了理论，坚称“宁做不满的人，不做满足的猪”。
11. Kant’s Categorical Imperative: Universal Law Formula | 康德的绝对命令：普遍法则公式
The first formulation states: Act only according to that maxim whereby you can at the same time will that it should become a universal law. In decision‑checking form: (1) State your maxim. (2) Universalise it as a law of nature. (3) Test whether the maxim can be conceived without contradiction and can be willed without contradiction.
第一公式陈述为：只按照你同时能够意愿其成为一条普遍法则的那个准则去行动。在决策检验形式中：(1) 陈述你的准则。(2) 将其普遍化为一条自然法则。(3) 检验该准则是否能无矛盾地被设想，且能无矛盾地被意愿。
A perfect duty arises if the maxim cannot even be conceived as a universal law (e.g., false promising leads to the contradiction that the institution of promising would collapse). An imperfect duty arises if, although conceivable, it cannot be willed without contradicting the rational will (e.g., neglecting talents).
如果准则甚至无法被设想为一条普遍法则，则产生一个完全义务（例如，虚假承诺会导致承诺制度崩溃的矛盾）。如果虽可设想，但无法在理性意愿中无矛盾地被意愿，则产生一个不完全义务（例如，荒废才能）。
12. Aristotle’s Virtue Ethics: The Doctrine of the Mean | 亚里士多德美德伦理学：中庸之道
Virtuous character is achieved by hitting the mean between excess and deficiency relative to us, determined by practical reason (phronesis). The formula is not arithmetic but contextual: for any action‑emotion, there is a vice of excess and a vice of deficiency, with virtue lying at the appropriate intermediate.
具备美德的人格是通过达到相对于我们而言的、过度与不及之间的中道而实现的，由实践智慧（phronesis）决定。公式并非算术的，而是语境性的：对于任何行动或情感，都存在一个过度的恶和一个不及的恶，美德则处于恰当的中间状态。
Examples: courage is the mean between rashness and cowardice; generosity between prodigality and meanness. The mean is not a fixed mathematical midpoint but a flexible, situation‑sensitive target that the phronimos (practically wise person) discerns.
例子：勇敢是鲁莽与怯懦的中道；慷慨是挥霍与吝啬的中道。中道并非一个固定的数学中点，而是一个灵活、对情境敏感的目标，由拥有实践智慧的人（phronimos）辨识。
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