📚 Year 12 OCR Philosophy Quick Reference: Formulas & Theorems | Year 12 OCR 哲学：公式定理速查手册
Philosophy may seem like a discipline of extended prose, yet at its core lie precise logical structures, definitions, and inferential patterns that function like mathematical formulae. This quick reference handbook distills key arguments from the OCR Year 12 specification — covering philosophy of religion, ethics, and philosophy of mind — into compact theorems, logical forms, and principle statements. Each entry provides a schematic formulation followed by a concise explanation, designed to help you memorise and deploy these foundational ideas with clarity in essays and examinations. Treat these pages as your ready reckoner for the conceptual machinery that drives philosophical analysis.
哲学看似一门以长篇散文为主的学科，但其核心却有着精确的逻辑结构、定义和推理模式，它们的作用类似于数学公式。本速查手册从 OCR Year 12 考纲中提炼了宗教哲学、伦理学和心灵哲学的关键论证，将其浓缩为紧凑的定理、逻辑形式和原则陈述。每个条目都提供了模式化的表述并附有简明解释，旨在帮助您记忆并在论文和考试中清晰运用这些基础观念。请将本手册视为您开展哲学分析的概念工具箱。
1. Anselm's Ontological Argument Theorem | 安瑟伦的本体论论证定理
Anselm defines God as 'that-than-which-nothing-greater-can-be-conceived' and argues that existence in the understanding and in reality is greater than existence in the understanding alone. Therefore, if God existed only in the mind, we could conceive a greater being — one that exists in reality. By reductio ad absurdum, the non-existence of God is impossible.
安瑟伦将上帝定义为“那无法设想有更伟大者”，并论证存在于理解中且存在于现实中要比仅存在于理解中更伟大。因此，如果上帝只存在于心灵中，我们就能设想一个更伟大的存在——一个现实存在的存在。通过归谬法，上帝不存在是不可能的。
God =ₑₓ that-than-which-nothing-greater-can-be-conceived
(Existence in re ∧ in intellectu) > (Existence in intellectu only) → God must exist in re.
Key move: Existence is treated as a predicate that adds greatness. Gaunilo's 'Lost Island' parody challenges this move by showing that the same reasoning could 'prove' the existence of any perfect island, suggesting the argument illicitly leaps from concept to reality.
关键步骤：存在被当作一个增添伟大性的谓词。高尼罗的“迷失岛”反驳挑战了这一步骤，表明同样的推理可以“证明”任何完美岛屿的存在，暗示该论证从概念到现实之间存在不合理的跳跃。
2. Descartes' Cogito Theorem | 笛卡尔的“我思”定理
Descartes, engaging in hyperbolic doubt, finds that even if an evil demon deceives him about everything, he cannot be deceived about his own existence while he is thinking. The very act of doubting confirms that he exists as a thinking thing. This foundational certainty is expressed in the famous theorem.
笛卡尔在进行极端怀疑时发现，即使有一个恶魔在一切事情上欺骗他，他在思考时也无法对自己存在这一事实被欺骗。怀疑的行为本身确证了他作为一个思维事物的存在。这一基础确定性表达在著名的定理中。
Cogito, ergo sum (I think, therefore I am)
∀x (Doubting(x) → Thinking(x)) ∧ Thinking(x) → Exists(x)
This is often called the Archimedean point of Descartes' epistemology. It establishes a thinking substance (res cogitans) as indubitable and acts as the first premise in his project of reconstructing knowledge through clear and distinct ideas. Critics point out that the cogito only guarantees the existence of a momentary thinking subject, not a persisting self.
这常被称为笛卡尔认识论的阿基米德点。它确立了思维实体（res cogitans）是无可置疑的，并作为他通过清晰分明的观念重建知识工程的第一前提。批评者指出，“我思”只保证了一个当下的思维主体的存在，而非一个持久的自我。
3. The Kalam Cosmological Formula | 卡拉姆宇宙论公式
The Kalam argument is a modern version of the cosmological argument that focuses on the beginning of the universe. It uses two premises to derive a cause for the universe, which is then identified as God.
卡拉姆论证是宇宙论论证的一个现代版本，聚焦于宇宙的起始。它使用两个前提推导出宇宙有一个原因，进而将这个原因认定为上帝。
(1) Everything that begins to exist has a cause.
(2) The universe began to exist.
∴ The universe has a cause (→ God).
Support for (2) often appeals to Big Bang cosmology and the impossibility of an actual infinite series of past events. The formula then applies a conceptual analysis of the cause: it must be uncaused, immaterial, powerful, and personal — attributes of the theistic God. Objections challenge whether the universe as a whole needs to obey the causal principle and whether a personal agent is the only possible explanation.
对前提(2)的支持常诉诸大爆炸宇宙学和真实无限序列的不可能性。然后该公式对原因进行概念分析：它必须是无原因的、非物质的、大能的、有位格的——这正是有神论上帝的属性。反对意见质疑宇宙作为一个整体是否需要遵守因果原则，以及有位格的动因是否是其唯一可能的解释。
4. Paley's Design Inference Rule | 佩利的设计推论规则
William Paley's analogical argument compares the complexity of natural organisms to the complexity of a watch. Just as a watch's complexity and purposeful arrangement imply a watch-maker, so the intricate complexity of the natural world implies a divine designer.
威廉·佩利的类比论证将自然有机体的复杂性比作手表的复杂性。正如手表的复杂性和目的性安排暗示着一位钟表匠，自然世界的精密复杂性也暗示着一位神圣的设计者。
Complex object O exhibits intricacy & purpose.
In all observed cases, such O have a designer.
Natural world N exhibits far greater intricacy & purpose.
∴ N probably has a designer (God).
This is an inference to the best explanation. Hume's criticisms (prior to Darwin) point out that the analogy is weak — the universe is not very like a watch — and that the seeming order could result from other principles. Later, evolutionary theory provided a non-intentional mechanism to explain biological complexity, undermining the inference's key premise.
这是基于最佳解释的推论。休谟的批评（早于达尔文）指出类比是薄弱的——宇宙并不很像一块手表——而且看似秩序可能源于其他原理。后来，进化论提供了一种非意图机制来解释生物复杂性，从而削弱了该推论的关键前提。
5. The Principle of Sufficient Reason (PSR) | 充足理由原则 (PSR)
The Principle of Sufficient Reason states that there must be an explanation for every fact or for why any thing exists and is the way it is rather than otherwise. It is a rationalist axiom employed by Leibniz and others in cosmological arguments.
充足理由原则指出，每一个事实，或者任何事物为何存在且为何是这样而不是那样，都必须有一个解释。这是莱布尼茨等人在宇宙论论证中使用的理性主义公理。
∀p (p is true → ∃q (q explains p))
Applying the PSR to the whole series of contingent facts leads to the requirement for a necessary being that contains the sufficient reason for its own existence, which Leibniz identifies as God. Critics, such as Hume and Russell, reject the principle as dogmatic: they deny that there is any logical necessity for the universe to have a sufficient reason, and simply accept the universe as a brute fact.
将充足理由原则应用于整个偶然事实系列，就会要求存在一个必然存在者，其自身存在的充足理由就在自身之中，莱布尼茨将其等同于上帝。休谟和罗素等批评者拒绝该原则，认为这是一种独断：他们否认宇宙必须有充足理由在逻辑上是必然的，而是将宇宙当作一个无法进一步解释的粗粝事实来接受。
6. Bentham's Felicific Calculus | 边沁的快乐计算法
Jeremy Bentham's act utilitarianism quantifies pleasure and pain to guide moral action. The felicific calculus, or hedonic calculus, is a formula for evaluating the moral worth of an action based on the amount of pleasure it produces. It considers seven dimensions of pleasure and pain.
杰里米·边沁的行为功利主义将快乐与痛苦量化以指导道德行为。快乐计算法，或称享乐计算，是一种基于所产生的快乐量来评估一个行动道德价值的公式。它考量了苦乐的七个维度。
(Intensity + Duration + Certainty + Propinquity + Fecundity + Purity + Extent) of pleasure — pain
Each element: Intensity (how strong), Duration (how long), Certainty (how likely), Propinquity (how soon), Fecundity (chance of being followed by more pleasure), Purity (chance of not being followed by pain), and Extent (number of people affected). This calculus aims to make moral decisions impartial and empirical. Objections include the problem of measuring qualitative differences between pleasures and the demandingness of impartial calculation.
每个要素：强度（多强）、持续时间（多长）、确定性（可能性大小）、邻近性（多快到来）、丰产性（被更多快乐跟随的概率）、纯度（不被痛苦跟随的概率）和范围（受影响的人数）。该计算法旨在使道德决策不偏不倚且基于经验。反对意见包括难以衡量快乐之间的质的差别，以及不偏不倚的计算所带来的严苛性。
7. Kant's Categorical Imperative Formulas | 康德的绝对命令公式
For Kant, moral worth derives from acting according to maxims that can be universalised and that respect rational beings. The Categorical Imperative is the supreme principle of morality, with several formulations that are equivalent.
对康德而言，道德价值源自按照可普遍化且尊重理性存在者的准则去行动。绝对命令是道德的最高原则，它有着几种等价的表述形式。
[Universal Law Formula] Act only on that maxim through which you can at the same time will that it become a universal law.
[Humanity Formula] Act so that you use humanity, whether in your own person or in that of another, always at the same time as an end, never merely as a means.
The universal law formulation generates perfect and imperfect duties by testing for contradictions in conception and in the will. The humanity formula grounds the inviolable dignity of persons. The test of universalisability requires that we check whether a maxim can be willed without contradiction; for example, breaking a promise produces a contradiction because the institution of promising would collapse if everyone did it. This provides a clear decision procedure for ethics.
普遍法则的公式通过检查观念上的矛盾与意愿上的矛盾，产生了完全的义务和不完全的义务。人性公式奠定了人格不可侵犯的尊严。可普遍化测试要求我们检查一个准则是否可以无矛盾地被意愿；例如，违背诺言会产生矛盾，因为如果每个人都这样做，许诺制度就会崩溃。这为伦理学提供了一个清晰的决策程序。
8. The Greatest Happiness Principle | 最大幸福原则
Mill's utilitarian principle defines the right action as that which maximizes general happiness. It is not merely a calculus but a foundation for moral and political theory. Happiness is understood as pleasure and the absence of pain, but Mill introduces qualitative distinctions between higher and lower pleasures.
密尔的功利主义原则将正当行动定义为能够最大化普遍幸福的行为。它不仅仅是一种计算，更是道德和政治理论的基础。幸福被理解为快乐和痛苦的不存在，但密尔引入了高级快乐和低级快乐之间的质的区分。
An action is right iff it promotes the greatest happiness for the greatest number.
Mill expands Bentham's quantitative frame by arguing that competent judges invariably prefer pleasures of the intellect, imagination, and moral sentiments over mere bodily pleasures. This theory is used to ground the harm principle in On Liberty: the only legitimate reason to interfere with an individual's liberty is to prevent harm to others. Criticisms question whether justice, rights, and backward-looking promises can be adequately explained by solely forward-looking consequences.
密尔拓展了边沁的量化框架，主张有资格的评判者总是偏好理智、想象力和道德情操的快乐，而非单纯肉体的快乐。这一理论被用来为《论自由》中的伤害原则奠定基础：干涉个人自由的唯一合法理由是为了防止对他人造成伤害。批评意见质疑正义、权利和追溯性的承诺能否完全由向前看的结果来说明。
9. The Mind-Body Problem Contradiction | 心身问题矛盾式
The mind-body problem arises from the apparent conflict between mental properties and physical properties when we try to conceive of them as unified in a single substance. Substance dualism must explain how a non-physical mind can interact with a physical body.
心身问题产生于当我们试图将心理属性和物理属性设想为统一在一个单一实体中时，它们之间明显的冲突。实体二元论必须解释非物理的心灵如何能与物理的身体发生相互作用。
(1) The mind is non-physical (has no spatial location).
(2) The body is physical (occupies space).
(3) The mind causally affects the body (e.g., my intention moves my arm).
But (1) ∧ (2) ∧ (3) seems impossible under causal closure of the physical.
Princess Elisabeth of Bohemia pressed Descartes on this problem: how can an immaterial soul cause motion in material substances without a mechanical point of contact? This interaction problem remains a central challenge for dualism and leads to alternative positions such as property dualism, epiphenomenalism, and physicalism.
波西米亚的伊丽莎白公主曾就此问题追问笛卡尔：一个非物质的灵魂如何能在没有机械接触点的情况下引起物质实体的运动？这一交互作用问题至今仍是二元论面临的中心挑战，并引出了其他立场，如属性二元论、副现象论和物理主义。
10. Substance Dualism Interaction Formula | 实体二元论交互作用公式
Descartes' interactionist dualism posits two distinct kinds of substance: res cogitans (thinking substance) and res extensa (extended substance). The interaction occurs in the pineal gland, where the soul sends 'animal spirits' through the nerves to produce bodily motion, and the body transmits sensory information back to the soul.
笛卡尔的交互论二元论设定了两种不同的实体：思维实体 (res cogitans) 和广延实体 (res extensa)。交互作用发生在松果腺，灵魂通过那里发出“动物精气”经过神经以产生身体运动，身体则将感觉信息传回给灵魂。
Mental event M → Pineal gland P → Physical motion B
Physical stimulus S → Pineal gland P → Mental sensation M'
This formula attempts to localise the mind-body nexus, but it faces the pairing problem: even if we accept pineal transmission, we still lack an account of why this particular mental state causes this particular physical effect, given that mental events do not have spatial properties that could link them lawfully to spatial events. The formula thus collapses under the conservation of energy objection and the conceptual gap Gilbert Ryle called the 'ghost in the machine'.
该公式试图为心身连接点定位，但它面临配对问题：即使我们接受松果腺传导，我们仍然无法解释，为何这个特定的心理状态会造成那个特定的物理结果，因为心理事件不具备空间属性，无法合法则地与空间性事件相连。因此，这个公式在能量守恒的反对意见和吉尔伯特·赖尔所称的“机器中的幽灵”概念鸿沟面前瓦解。
11. Behaviorism's Dispositional Formula | 行为主义的倾向公式
Philosophical behaviorism, as advanced by Gilbert Ryle and influenced by logical behaviorism, claims that mental states are nothing over and above behavioral dispositions. To have a belief, desire, or pain is just to be disposed to act in certain ways under certain circumstances.
哲学行为主义，由吉尔伯特·赖尔推进并受逻辑行为主义影响，主张心理状态不外乎行为倾向的总和。拥有一个信念、欲望或疼痛，就是在特定环境下倾向于以特定方式行动。
Mental state M =ₑₓ the disposition to emit behavior B given stimulus S
For example, 'John believes it is raining' translates to a set of dispositional statements: John is disposed to take an umbrella, to close windows, to say 'it is raining' if asked, etc. This dissolves the mind-body problem by eliminating the inner ghost. Ryle's 'category mistake' argument explains that dualism mistakenly treats the mind as another thing needing location. Objections include the problem of qualia — the felt quality of experience seems irreducible to mere disposition — and the perfect actor objection, which imagines someone having all the dispositions without the actual feeling.
例如，“约翰相信天在下雨”可被翻译为一组倾向陈述：约翰倾向于带伞，关窗，如果被问到便会说“在下雨”等。它通过消除内部的幽灵来消解心身问题。赖尔的“范畴错误”论证说明二元论错误地将心灵视作另一个需要位置的物。反对意见包括感质问题——经验的感觉特性似乎不可还原为纯倾向——以及完美演员反驳，设想某人具有所有倾向却没有实际感受。
12. Functionalism's Multiple Realizability Theorem | 功能主义的多重实现定理
Functionalism holds that mental states are defined by their causal roles — their typical causes and effects — rather than by their physical constitution. This means the same mental state can be realized by different physical substrates, just as the same software can run on different hardware. The multiple realizability argument is a core theorem against type identity theory.
功能主义认为，心理状态是由其因果角色——即其典型的原因和结果——来定义的，而不是由其物理构成来定义。这意味着同一心理状态可以由不同的物理基质实现，就如同同一软件可以在不同硬件上运行。多重实现论证是反对类型同一论的核心定理。
Mental state M₁ = role R [I₁ → O₁] (input-output functional character)
ibres firing in humans, silicon chips in robots, or alien biology → M₁ ≠ brain state type.
For example, pain is defined as the state typically caused by bodily damage and leading to withdrawal, moaning, etc. This role could be played by C-fibre firing in humans and by hydraulic states in a Martian. Therefore, pain cannot be identical to a specific type of brain state. The formula supports computational theories of mind and artificial intelligence. Its chief vulnerability is the absent qualia objection: a system could functionally realize pain without any conscious experience, leaving out what it is like to be in that state.
举例来说，疼痛被定义为通常由身体损伤引起并导致退缩、呻吟等行为的状态。这一角色可由人类的C类神经纤维激活来实现，也可由火星人体内的液压状态来实现。因此，疼痛不能等同于某种特定类型的脑状态。该公式支持心灵的计算理论和人工智能。其主要弱点是感质缺席反驳：一个系统可以在功能上实现了疼痛，但没有任何意识体验，从而遗漏了处于那种状态的主观感受质。
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