📚 Year 13 SQA Philosophy: Formula & Theorem Quick-Reference Handbook | SQA 13年级哲学：公式定理速查手册
Philosophy may not feature equations like physics or maths, but its core arguments and definitions can be expressed as logical structures akin to formulae. This quick-reference handbook distils the essential ‘theorems’ and argument schemas from the SQA Advanced Higher Philosophy course (Year 13) into digestible, exam-ready patterns. Each entry presents a key idea, its formal representation, and a brief explanation to aid revision.
哲学也许不像物理或数学那样有方程式，但其核心论证和定义可以表达为类似公式的逻辑结构。这本速查手册将SQA高级高等哲学课程（13年级）中的基本“定理”和论证模式提炼成易于理解、适合备考的形式。每个条目呈现一个关键思想、其形式化表述和简要说明，以辅助复习。
1. The JTB Formula: Knowledge = Justified True Belief | JTB公式：知识 = 确证的真信念
For Plato and much of the tradition, knowledge has been analysed as justified true belief. A subject S knows a proposition P if and only if P is true, S believes that P, and S is justified in believing that P. We can formalise this triad as a simple conjunction: K(S,P) ↔ True(P) ∧ Bel(S,P) ∧ Jus(S,P).
对于柏拉图和长久以来的传统，知识被分析为确证的真信念。一个认知主体S知道命题P，当且仅当P为真、S相信P，并且S相信P是有确证的。我们可以将这个三元组合形式化为一个简单的合取式：K(S,P) ↔ True(P) ∧ Bel(S,P) ∧ Jus(S,P)。
This ‘JTB formula’ serves as the baseline for epistemology. In exams, you will use it to test whether a given case counts as knowledge. Each component is necessary; missing any one defeats the claim. When writing essays, always check the truth condition first, then belief, then justification.
这个“JTB公式”是认识论的基线。在考试中，你用它来检验某个案例是否算作知识。每个要素都是必要的；缺少任何一个都会使知识主张失败。在写作论文时，务必先检查真理条件，然后信念，然后是确证。
- True(P): the world matches the proposition.
- Bel(S,P): the subject genuinely holds the proposition.
- Jus(S,P): the subject has adequate reasons or evidence.
- True(P)：世界与命题相符。
- Bel(S,P)：主体真正持有该命题。
- Jus(S,P)：主体有充足的理由或证据。
2. Gettier’s Counterexample: The JTB + X Equation | 盖梯尔反例：JTB + X 方程
Edmund Gettier demonstrated that JTB is not sufficient for knowledge by constructing cases where a justified true belief arises from luck or defective reasoning. This forces us to add a fourth condition, ‘X’, that rules out epistemic luck. The formula becomes: K(S,P) ↔ True(P) ∧ Bel(S,P) ∧ Jus(S,P) ∧ X. Candidate values for X include: no false lemmas, reliable belief-forming process, or indefeasible justification.
爱德蒙·盖梯尔通过构造一些由于运气或错误推理而产生确证真信念的案例，证明了JTB不足以构成知识。这迫使我们增加第四个条件“X”，以排除认知运气。公式变为：K(S,P) ↔ True(P) ∧ Bel(S,P) ∧ Jus(S,P) ∧ X。X的可能候选值包括：没有错误引理、可靠的信念形成过程、或不可挫败的确证。
In your revision, memorise at least one Gettier-style counterexample (e.g., Smith and Jones, the stopped clock, or the fake barn). Then show how your chosen ‘X’ blocks the counterexample. The key is to spot the epistemic gap between justification and truth.
在复习时，要记住至少一个盖梯尔式反例（例如史密斯与琼斯、停了的钟、或假谷仓）。然后展示你所选的“X”如何阻止反例。关键在于识破确证与真理之间的认知缝隙。
3. Cartesian Scepticism: The Evil Demon Theorem | 笛卡尔怀疑论：恶魔定理
Descartes’ method of doubt leads to the evil demon argument: if I cannot rule out the possibility that an all-powerful demon is deceiving me about all my sensory experiences, then my beliefs about the external world are not certain. The sceptical structure is a modus tollens: (1) If I know external-world propositions, then I can rule out the evil demon hypothesis. (2) I cannot rule it out. (3) Therefore, I do not know external-world propositions.
笛卡尔的怀疑方法导向其恶魔论证：如果我不能排除一个全能的恶魔在我所有感官经验上欺骗我的可能性，那么我关于外部世界的信念就不是确定的。其怀疑结构是一个否定后件式：(1) 如果我知道外部世界命题，那么我就能排除恶魔假设。(2) 我不能排除它。(3) 因此，我不知道外部世界命题。
Formalise this as: ◇(Demon deceives me) ∧ ¬□(I can eliminate Demon) → ¬K(I, ExternalWorld). The dream argument works similarly. The response requires either a proof of God’s non-deceiving nature (Descartes’ route) or a denial of the principle that knowledge requires elimination of all logical possibilities (fallibilism).
可将其形式化为：◇(恶魔欺骗我) ∧ ¬□(我能排除恶魔) → ¬K(我, 外部世界)。梦论证与之类似。回应要么需要证明上帝的非欺骗本性（笛卡尔的路线），要么否定知识要求排除所有逻辑可能性的原则（可谬论）。
4. Rationalism vs Empiricism: The Origin Formula | 理性主义与经验主义：起源公式
The dispute between rationalists and empiricists concerns the foundation of human knowledge. Rationalism asserts the existence of innate ideas and a priori knowledge: ∃K such that K is obtained independently of sense experience. Empiricism insists that all knowledge stems from experience: ∀K, K originates in sense perception and reflection upon it.
理性主义与经验主义之间的争论关涉人类知识的根基。理性主义断言存在着天赋观念和先天知识：∃K 使得 K 是独立于感官经验而获得的。经验主义则坚持所有知识源于经验：∀K，K 起源于感官知觉及其反思。
You can represent the two positions as competing theses: Rationalism: K = f(Innate Ideas + Reason) ≠ f(Senses). Empiricism: K = f(Sense Data + Reflection). Key arguments include Plato’s Meno, Leibniz’s veins of marble, Hume’s copy principle, and Locke’s attack on innate ideas. When revising, align each thinker with the formula that best captures their position.
你可以将这两种立场表示为相互竞争的论旨：理性主义：K = f(天赋观念 + 理性) ≠ f(感官)。经验主义：K = f(感觉材料 + 反思)。核心论证包括柏拉图的《美诺篇》、莱布尼茨的大理石纹路、休谟的摹写原则和洛克对天赋观念的攻击。复习时将每一位思想家与最切合其立场的公式对应起来。
5. Utilitarian Calculus: The Greatest Happiness Principle | 功利主义计算：最大幸福原则
Bentham’s utilitarianism provides a decision procedure: the morally right act is the one that produces the greatest happiness for the greatest number. Happiness is measured on a single scale of pleasure minus pain. The utility of an action U(A) = Σ(Pleasureᵢ – Painᵢ) over all affected individuals i. Right action = argmax U(A).
边沁的功利主义提供了一个决策程序：道德上正确的行动就是为最大多数人带来最大幸福的行动。幸福在单一的快乐减去痛苦尺度上加以度量。一个行动的功利值 U(A) = Σ(快乐ᵢ – 痛苦ᵢ)，对所有受影响的个体i求和。正确的行动 = argmax U(A)。
Mill refined the calculus by introducing qualitative dimensions of pleasure, distinguishing higher intellectual pleasures from lower bodily ones. Thus the formula gains a weighting factor: U = Σ(w × Quality-adjusted Pleasure – Pain). Critics point to the ‘tyranny of the majority’ and the demandingness of impartial calculation.
密尔通过引入快乐的质性维度改进了功利计算，区分了高级的理智快乐与低级的身体快乐。于是公式获得了一个权重因子：U = Σ(w × 质性调整后的快乐 – 痛苦)。批评者则指出“多数人的暴政”以及不偏不倚计算的苛求性。
6. Kant’s Categorical Imperative: The Universal Law Formulation | 康德的定言命令：普遍法则公式
Kant’s first formulation of the categorical imperative commands: ‘Act only according to that maxim whereby you can at the same time will that it should become a universal law.’ This yields a test for right action. For any proposed maxim M, ask: (1) Can M be conceived as a universal law without contradiction? (contradiction in conception) (2) Can M be willed as a universal law without contradiction in the will? If either fails, acting on M is forbidden.
康德的第一条定言命令公式要求：“只按照你同时能够意愿它成为一条普遍法则的准则去行动。”这提供了一个检验正确行动的方法。对任何拟议的准则M，问：(1) 能否在构设上无矛盾地将M设想为一条普遍法则？（构想矛盾）(2) 能否在意愿上无矛盾地将M意愿为一条普遍法则？（意愿矛盾）若有一者失败，则依M行动是被禁止的。
We can formalise the test: Permissible(M) ↔ Consistent(Universalised M) ∧ Willable(Universalised M). Classic examples: a lying promise fails the conception test; not developing one’s talents fails the willing test. In essays, apply this formula to moral dilemmas and contrast it with consequentialism.
我们可以将检验形式化为：Permissible(M) ↔ Consistent(普遍化的M) ∧ Willable(普遍化的M)。经典例子：假承诺通不过构想检验；不发展自身才能通不过意愿检验。在论文中，要将这个公式应用到道德困境，并与后果论进行对比。
7. Virtue Ethics: The Golden Mean Theorem | 美德伦理学：中庸定理
Aristotle defines moral virtue as a state of character concerned with choice, lying in a mean relative to us, determined by reason. Virtue is the intermediate between excess and deficiency in emotions and actions. A quick schema: for any sphere of action, let the continuum be [Deficiency, Mean, Excess]. Virtue = Mean.
亚里士多德将道德德性定义为一种与选择相关的品质状态，存在于相对于我们而言的中道里，并由理性来规定。德性就是情感与行动上过度与不及之间的适中。一个简易图式：对任何行动领域，设定连续体 [不及，中道，过度]。德性 = 中道。
For example, courage is the mean between cowardice (deficiency) and rashness (excess); generosity is the mean between stinginess and wastefulness. The mean is not an arithmetic average but a dynamic target determined by practical wisdom (phronesis). When evaluating an action, ask: does it hit the mean for that situation?
例如，勇敢是怯懦（不及）与鲁莽（过度）之间的中道；慷慨是吝啬与挥霍之间的中道。中道并非算术平均，而是由实践智慧（phronesis）决定的动态目标。在评价一个行动时，要问：它是否击中了该情境的中道？
8. Mind-Body Dualism: The Divisibility Argument | 心身二元论：可分性论证
Descartes argues for the real distinction between mind and body using the divisibility (or ‘indivisibility’) argument. Premise 1: The body is by nature divisible, for any part of a body can be separated. Premise 2: The mind is indivisible; I cannot distinguish parts within myself as a thinking thing. Premise 3: If two things have different properties, they cannot be identical (Leibniz’s Law). Conclusion: Therefore, mind ≠ body.
笛卡尔用可分性（或“不可分性”）论证来主张心灵与身体的真实区分。前提1：身体在本性上是可分的，因为身体的任一部分都可以被分离出去。前提2：心灵是不可分的；作为思维之物，我无法在自身内区分出部分。前提3：如果两个事物具有不同的性质，它们就不能是同一的（莱布尼茨律）。结论：因此，心灵 ≠ 身体。
Formalise: Property(Body, divisible) ∧ Property(Mind, indivisible) → Body ≠ Mind. The strength lies in the clear and distinct perception of these essences. Responses include the ‘mind-as-software’ objection and modern neuroscience’s challenge to indivisibility (split-brain cases). For top marks, discuss the Cartesian circle and the interaction problem.
形式化：Property(身体, 可分) ∧ Property(心灵, 不可分) → 身体 ≠ 心灵。其力量在于对这些本质的清楚明晰的感知。回应包括“心灵作为软件”的反驳，以及现代神经科学对不可分性的挑战（裂脑人案例）。要拿到高分，要讨论笛卡尔循环和交互难题。
9. Physicalism: Mind-Brain Identity Theory | 物理主义：心脑同一论
Mind-brain identity theory asserts that mental states just are brain states: for any type of mental state M, there is a type of physical state B such that M = B. This is type-type identity. For instance, pain is identical to C-fibre firing. The formula: ∀M ∃B (M ≡ B).
心脑同一论断言，心灵状态就是大脑状态：对任何类型的心灵状态M，都存在一种类型的物理状态B，使得M = B。这就是类型-类型同一。例如，疼痛等同于C-纤维放电。公式：∀M ∃B (M ≡ B)。
This view elegantly dissolves the interaction problem by eliminating a separate mental substance. However, Hilary Putnam’s multiple realizability argument challenges the formula: the same mental state (e.g., pain) can be realized by different physical states across species (humans, octopuses, Martians). Thus M cannot be strictly identical to one type of B. The identity theorist may retreat to token-token identity: Mtoken = Btoken, but without type reduction.
这一观点通过取消独立的心理实体而优雅地消解了交互难题。然而，希拉里·普特南的多重可实现性论证挑战了这个公式：同一个心灵状态（如疼痛）可以在不同物种（人类、章鱼、火星人）中由不同的物理状态实现。因此M不能严格同一于单一类型的B。同一论者可以退回到殊型-殊型同一：M殊型 = B殊型，但不作类型还原。
10. Free Will & Determinism: The Consequence Argument | 自由意志与决定论：后果论证
Peter van Inwagen’s Consequence Argument crystallises the incompatibilist intuition that determinism rules out free will. Let P be any true proposition about human action, L be the laws of nature, and P0 be the state of the universe in the remote past. Determinism entails: □((P0 ∧ L) → P). Since no one has (or ever had) a choice about P0 or L, and no one can change the consequence of what they have no choice about, no one has a choice about P. Hence, no free will.
彼得·范·因瓦根的后果论证凝练了这样一种不相容论的直觉：决定论排除了自由意志。令P为关于人的行动的任一真命题，L为自然律，P0为遥远过去的宇宙状态。决定论蕴涵：□((P0 ∧ L) → P)。既然没有任何人对P0或L拥有（或曾拥有）选择，且没有人能够改变他们对其没有选择之事的后果，因此没有人对P拥有选择。所以，不存在自由意志。
The formula is a transfer-of-powerlessness principle: if □(X → Y) and we lack power over X, we lack power over Y. Incompatibilists wield this to show that determinism and moral responsibility conflict. Compatibilists attack either the □(X → Y) interpretation or the notion of ‘having a choice about’ the past and laws.
公式是一个无力传递原则：如果□(X → Y)且我们对X无力，则我们对Y也无力。不相容论者运用这一原则来表明决定论与道德责任是冲突的。相容论者则或攻击□(X → Y)的释义，或攻击对过去和自然律“拥有选择”的概念。
11. Compatibilism: Conditional Analysis Formula | 相容论：条件分析公式
Compatibilists such as David Hume and A.J. Ayer propose that an action is free not in the sense of being uncaused, but in the sense that if the agent had chosen otherwise, she could have done otherwise. The formula: S does A freely iff (i) A was caused by S’s own desires and beliefs, and (ii) if S had chosen to do B, S would have done B. This reduces freedom to a counterfactual conditional: Free(A) ↔ (Chosen(B) □→ Done(B)) ∧ Internal Causation.
大卫·休谟和A.J.艾耶尔等相容论者提出，一个行动是自由的，并不是在于它没有原因，而是在于：如果行动者当初选择了别的做法，她就能做出别的行动。公式：S自由地做A，当且仅当 (i) A是由S自身的欲望和信念所引起的，且 (ii) 如果S选择了做B，S就会做B。这将自由还原为一个反事实条件句：Free(A) ↔ (Chosen(B) □→ Done(B)) ∧ 内在原因。
This analysis aims to preserve moral responsibility within a deterministic world. Harry Frankfurt’s higher-order desire model (wanting one’s will) and the ‘freedom of action’ vs ‘freedom of will’ distinction refine the formula. Critics, like Peter van Inwagen, argue that conditional analysis fails because the ability to do otherwise is not the same as ‘would have done otherwise if chosen’, since the agent’s choice itself may be determined.
这一分析意在决定论世界中保留道德责任。哈里·法兰克福的高阶欲望模型（希望自己的意志成为意志）以及“行动自由”与“意志自由”的区分，进一步精致化了这一公式。批评者如范·因瓦根主张，条件分析失败了，因为能够做出别的行动并不等于“若选择别的就会做出别的”，因为行动者自身的选择可能也是被决定的。
12. Ontological Argument: Anselm’s Proof Structure | 本体论论证：安瑟尔谟的证明结构
Anselm defines God as ‘that than which nothing greater can be conceived’. The argument proceeds: (1) God exists in the understanding (even the Fool agrees). (2) If God exists only in the understanding, we can conceive of a greater being — one that exists in reality as well. (3) But this contradicts God’s definition, for nothing greater can be conceived. (4) Therefore, God must exist in reality.
安瑟尔谟将上帝定义为“可设想的无与伦比的伟大存在者”。论证展开为：(1) 上帝存在于理智之中（连愚顽人也认可）。(2) 如果上帝仅存在于理智中，我们就能设想一个更伟大的存在者——一个既在理智中也在现实中存在的存在者。(3) 但这与上帝的定义矛盾，因为不可能设想出更伟大者。(4) 因此，上帝必然存在于现实中。
Formalise: God = ιx ¬∃y (y > x). Suppose God exists only in intellect. Then ∃z (z > God) — namely, God-existing-in-intellect-and-reality. This leads to a contradiction. Hence, by reductio, God exists in intellect and reality. Gaunilo’s ‘perfect island’ parody tests the argument’s form, and Kant’s objection that existence is not a predicate challenges the premise that ‘existing in reality’ adds greatness.
形式化：上帝 = ιx ¬∃y (y > x)。假设上帝只存在于理智中。那么∃z (z > 上帝)——也即，存在于理智且存在于现实中的上帝。这导致矛盾。因此，通过归谬法，上帝既存在于理智中也存在于现实中。高尼罗的“完美岛屿”仿拟检验了论证的形式，而康德“存在不是谓词”的反对则挑战了“存在于现实中会增加伟大性”的前提。
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