📚 Year 13 Edexcel Philosophy: Formulae & Theorems Quick Reference | Year 13 Edexcel 哲学：公式定理速查手册
This quick-reference handbook translates central arguments, principles, and ethical theories from the Year 13 Edexcel Philosophy specification into concise logical forms and formula-style statements. Use these symbolic summaries to sharpen your analysis, structure essays, and recall the logical skeleton behind each philosophical move.
这本速查手册将 Edexcel Year 13 哲学考试中的核心论证、原则与伦理理论转化为简洁的逻辑形式与公式化表述。用这些符号性总结来锻炼你的分析力、搭建论文结构，并快速回忆每一哲学推演背后的逻辑骨架。
1. Reductio ad Absurdum and Anselm's Ontological Formula | 归谬法与安瑟伦本体论论证公式
A reductio ad absurdum shows that rejecting a claim leads to a contradiction, therefore the claim must be true. Anselm's ontological argument follows this structure by assuming the non-existence of God and deriving an impossibility.
归谬法通过指出否认一个命题会导致矛盾，从而证明该命题必为真。安瑟伦的本体论论证即遵循这一结构，先假设上帝不存在，再推出一个不可能的结果。
Assume: God (the greatest conceivable being) exists only in the mind, not in reality.
Then a greater being can be conceived — one that exists both in the mind and in reality.
But God is 'that than which nothing greater can be conceived'.
∴ Contradiction. ∴ God exists in reality.
Anselm treats existence in reality as a perfection. If the greatest conceivable being lacked real existence, we could conceive a greater one that possesses it, which violates the definition. The argument is a priori and hinges on the claim that existence is a property that can be greater or lesser.
安瑟伦将现实中的存在视为一种完满性。如果最伟大的可设想存在者缺乏现实存在，我们就能设想一个更伟大的拥有现实存在的存在者，这就违反了定义。该论证是先天的，关键在于认为存在是一种可以成就大小之别的属性。
2. Paley's Design Argument: Analogy and Inference to the Best Explanation | 佩利的设计论证：类比与最佳解释推理
William Paley's watchmaker analogy invites us to infer a designer from the complex, purpose-serving order of the world. The argument can be formulated as an inference to the best explanation, where the likelihood of order given a designer is far higher than given chance.
威廉·佩利的钟表匠类比引导我们从世界复杂而服务于目的的秩序中推出设计者。该论证可以表述为一次最佳解释推理：在有设计者的情况下出现秩序的概率远高于偶然产生秩序的概率。
Observation: The universe exhibits intricate, functional order (E).
H1 (Design): E is the product of an intelligent designer.
H2 (Chance): E is the product of blind physical processes.
P(E | H1) ≫ P(E | H2).
∴ Rational to accept H1 over H2.
Paley strengthens the analogy by appealing to the marks of design — such as means-to-ends structure — that characterise both watches and natural organisms. Critics point to disanalogies: the universe is not clearly like a single machine, and Darwin later offered a powerful H2 with natural selection.
佩利通过诉诸设计标记——例如手段目的的结构——来强化类比，这些标记既见于钟表也见于自然有机体。批评者指出不相似之处：宇宙并不明显像一台单一机器，而达尔文后来用自然选择提供了一个有力的 H2。
3. The Cosmological Argument: Principle of Sufficient Reason | 宇宙论论证：充足理由律
The cosmological argument relies on the Principle of Sufficient Reason (PSR): every contingent fact must have a full explanation. Since the universe as a whole is contingent, it too requires an explanation, which must lie in a necessary being (God).
宇宙论论证依赖于充足理由律（PSR）：每一个偶然事实都必须有充分的解释。既然宇宙整体是偶然的，它也需要一个解释，而这个解释必然存在于一个必然存在者（上帝）之中。
PSR: ∀x (Contingent(x) → ∃y (Explains(y, x))).
The universe (U) is contingent.
∴ ∃z (Necessary(z) ∧ Explains(z, U)).
Leibniz appeals to the PSR to rule out an infinite regress of contingent explainers. The entire series of contingent things, he argues, must have an ultimate reason outside itself. Contemporary versions are often inductive and do not claim absolute certainty.
莱布尼茨诉诸充足理由律以排除偶然解释项的无穷回溯。他认为，整个偶然事物的系列必须有一个外在于自身的终极理由。当代版本常是归纳性的，并不声称拥有绝对确定的结论。
4. The Logical Problem of Evil: Inconsistent Triad | 恶的逻辑问题：不一致三元组
The logical problem of evil challenges the coherence of classical theism by asserting that the propositions 'God is omnipotent', 'God is omnibenevolent', and 'Evil exists' cannot all be true simultaneously.
恶的逻辑问题挑战古典有神论的融贯性，它主张“上帝全能”、“上帝全善”和“恶存在”这三个命题无法同时为真。
Proposition triad:
1. God is omnipotent (can prevent any evil).
2. God is omnibenevolent (wants to prevent every evil).
3. Evil exists.
∴ (1 ∧ 2) → ¬3. Since 3 is true, ¬(1 ∧ 2). (At least one attribute is false.)
J. L. Mackie presents this as a formal inconsistency. The standard theistic reply is to introduce a fourth proposition — that God allows some evil for the sake of greater goods (e.g., free will) — thereby breaking the deductive contradiction and requiring the atheologian to move to an evidential form of the argument.
麦基将此表述为一个形式上的不一致。神学家通常的回应是引入第四个命题——上帝为了更大的善（例如自由意志）而允许某些恶存在——从而打破演绎矛盾，迫使无神论者转而使用恶的证据论证形式。
5. Kant's Categorical Imperative: Universal Law and Humanity Formulae | 康德的绝对命令：普遍法则公式与人性公式
Kant offers several formulations of the Categorical Imperative, the supreme principle of morality. The first formula tests whether a maxim can be willed as a universal law without contradiction; the second demands that we always treat humanity as an end, never merely as a means.
康德给出了绝对命令的几种公式化表述，这是道德的最高原则。第一条公式检验一个准则是否能无矛盾地被意愿为普遍法则；第二条则要求我们始终将人性作为目的，而绝不仅仅当作手段。
Formula of Universal Law:
Act only on that maxim M such that you could consistently will M as a universal law.
Permissible action ⇔ ◇ Universalised(M) without contradiction in conception or volition.
Formula of Humanity:
Act so that you use humanity, whether in your own person or in the person of another, always at the same time as an end, never merely as a means.
For Kant, the capacity for rational agency gives humans an unconditional inner worth or dignity. Lying, for instance, fails the universal law test because it would undermine the very practice of promising, and it treats the listener solely as a tool for the liar's purpose.
对康德而言，理性行动者的能力赋予了人类无条件的内心价值或尊严。例如，说谎不能通过普遍法则的检验，因为说谎会破坏承诺这一实践本身，并且它将听者仅仅作为说谎者达成目的的工具。
6. Utilitarian Calculus: Greatest Happiness Principle | 功利主义计算式：最大幸福原则
Classical Utilitarianism holds that the rightness of an action is determined by its tendency to promote happiness. Bentham's felicific calculus quantifies pleasure and pain along several dimensions, while Mill refines the quality of pleasures.
古典功利主义主张，一个行为的正确性取决于它增进幸福的倾向。边沁的快乐计算式从多个维度量化快乐与痛苦，而密尔则对快乐的品质进行了精细区分。
Principle of Utility (Bentham):
Rightness of action A ∝ ∑ (pleasure − pain) for all sentient beings affected.
Calculus variables: Intensity, Duration, Certainty, Propinquity, Fecundity, Purity, Extent.
Mill's formula shifts the emphasis to higher and lower pleasures: 'It is better to be a human being dissatisfied than a pig satisfied.' The overarching imperative remains: choose the action that maximises the net balance of happiness over unhappiness for the greatest number.
密尔的公式将重心转向高等和低等快乐：“做一个不满足的人胜于做一只满足的猪。”普遍的律令仍然是：选择能为最大多数人带来最大净幸福余额的行为。
7. Aristotle's Function Argument and Doctrine of the Mean | 亚里士多德的功能论证与中道学说
Aristotle identifies the human good with eudaimonia (flourishing). His function argument claims that the good of a human lies in performing the characteristic human function — rational activity — excellently, i.e., in accordance with virtue.
亚里士多德将人的善等同于 eudaimonia（繁荣/幸福）。他的功能论证主张，人的善在于出色地——也就是合乎德性地——实现人所特有的功能，即理性活动。
Function Argument:
(rational activity of soul), then the human good = F performed in accordance with virtue.
Eudaimonia = activity of soul in accordance with the best and most complete virtue.
Virtue, for Aristotle, is a settled disposition lying in a mean relative to us, determined by practical wisdom. The 'mean' is not an arithmetic midpoint but the appropriate response between excess and deficiency: courage is the mean between rashness and cowardice.
对亚里士多德而言，德性是一种习惯性的品质，处于相对于我们而言的中道，由实践智慧决定。“中道”不是算术中点，而是过度与不及之间的恰当回应：勇敢是鲁莽与怯懦之间的中道。
8. Moore's Open Question Argument and the Naturalistic Fallacy | 摩尔的开放问题论证与自然主义谬误
G. E. Moore argues that 'good' is a simple, non-natural property that cannot be identified with any natural property. Any attempt to define 'good' in natural terms commits the naturalistic fallacy.
摩尔主张，“善”是一种简单的、非自然的属性，不能等同于任何自然属性。任何试图用自然术语定义“善”的做法都犯了自然主义谬误。
Open Question Formula:
For any natural or metaphysical property N, the question 'Is what is N good?' remains open.
Good ≠ N (for any N).
The naturalistic fallacy: If 'good' is defined by N, then 'Is N good?' would be a closed, tautologous question. It is not.
Moore's argument challenges ethical naturalism (e.g., Mill's equation of good with pleasure) and maintains that 'good' is sui generis. Critics question whether the open question test must always be decisive and whether non-natural properties are metaphysically mysterious.
摩尔论证挑战了伦理自然主义（例如密尔将善等于快乐），并坚持“善”自成一类。批评者质疑开放问题论证是否总具有决定性，以及非自然属性是否在形而上学上显得神秘。
9. The Verification Principle: Ayer's Criteria of Meaning | 证实原则：艾耶尔的意义标准
Logical positivism, as developed by A. J. Ayer, holds that a statement is factually meaningful only if it is either analytic (true by definition) or empirically verifiable. Metaphysical and religious statements that fail this test are held to be literally meaningless.
逻辑实证主义，由艾耶尔发展，主张一个陈述在事实上是有意义的，当且仅当它要么是分析命题（按定义即真），要么是经验上可证实的。不能通过这一检验的形而上学和宗教陈述
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