📚 Philosophy Formulas & Theorems Quick Reference | 哲学公式定理速查手册
Philosophy may not have mathematical equations in the traditional sense, but it is rich with logical structures, ethical principles and argument patterns that can be distilled into memorable 'formulas'. This quick reference handbook is designed for Year 9 OCR Philosophy students to grasp core concepts through simplified expressions, diagrams and comparisons. Think of each entry as a mental shortcut for essays, debates and revision. You will find the categorical imperative, utilitarian calculus, fallacy checklists and more, all presented in a clear two‑column format. Use these formulas as tools to deconstruct arguments and build your own reasoned positions.
哲学或许没有传统意义上的数学方程式，但它充满了逻辑结构、伦理原则和论证模式，这些都可以提炼成容易记忆的“公式”。这本速查手册专为 Year 9 OCR 哲学学生编写，通过简化的表达式、图解和对比来掌握核心概念。你可以把每个条目视为写作、辩论和复习时的思维捷径。手册涵盖定言令式、功利计算、谬误清单等内容，全部以清晰的双栏格式呈现。运用这些公式，你将能够拆解他人的论证，并构建自己的理性立场。
1. The Categorical Imperative Formula | 定言令式公式
Kant's central moral law can be stated as a universalisability test. Before you act, ask whether the guiding principle (maxim) behind your action could be willed to become a universal law for all rational beings. If the answer is yes, the action is morally permissible; if the maxim leads to a contradiction when universalised, you must avoid that action.
康德的核心理道德律可以表述为一种可普遍化检验。行动之前，问问自己：你行为背后的准则（格准）是否能够被意愿成为所有理性存在者的普遍法则。如果答案是肯定的，该行为在道德上就是允许的；如果该准则在普遍化时会产生矛盾，你就必须避免那样做。
Formula: Act only on that maxim whereby you can at the same time will that it should become a universal law.
公式：只依据那个你同时愿意它成为一项普遍法则的准则而行动。
This 'formula' turns moral decision‑making into a logical exercise. For example, if you consider lying to get out of trouble, universalising the maxim 'It is permissible to lie whenever it benefits me' would destroy trust in all communication, making the very act of lying pointless. Therefore, lying fails the test.
这个“公式”将道德决策转化为一种逻辑练习。例如，如果你考虑通过撒谎来摆脱麻烦，把“只要对我有利就可以撒谎”这一准则普遍化，就会摧毁所有沟通中的信任，使得撒谎本身毫无意义。因此，撒谎无法通过检验。
2. The Greatest Happiness Principle | 最大幸福原理
Utilitarianism, associated with Bentham and Mill, evaluates actions by their consequences – specifically the amount of pleasure and pain they produce. The 'utility calculus' can be summarised in a simple formula that weighs happiness against suffering for all affected individuals.
以边沁和密尔为代表的功利主义，依据行为的后果——具体而言是其所产生的快乐与痛苦之量——来评价行为。这个“功用计算”可以用一个简单的公式来概括，在所有受影响个体之间权衡幸福与痛苦。
Formula: Net Utility = Total Pleasure – Total Pain
公式：净效用 = 总快乐 – 总痛苦
To apply the formula, you must consider the intensity, duration, certainty and extent of pleasures and pains. The right action is the one that maximises net utility for the greatest number. Critics point out that measuring happiness is difficult, and the formula can justify morally troubling acts if they produce enough pleasure for the majority.
要运用这个公式，你必须考虑快乐与痛苦的强度、持续时间、确定性和范围。正确的行为就是能为最大多数人带来最大净效用的行为。批评者指出，衡量幸福很困难，而且如果某些行为能为多数人带来足够多的快乐，这个公式可能会为道德上有问题的行为辩护。
3. The Socratic Method Loop | 苏格拉底式提问循环
Genuine philosophical inquiry often follows the Socratic method, which can be drawn as a questioning loop. Instead of delivering answers, this technique exposes contradictions and deepens understanding through a structured sequence of questions.
真正的哲学探究往往遵循苏格拉底方法，这个方法可以用一个提问循环来表示。它不是直接给出答案，而是通过一系列结构化的提问来揭示矛盾并加深理解。
Probe Assumptions → Seek Evidence → Consider Alternatives → Examine Implications → Reflect
循环：澄清 → 探查假设 → 寻求证据 → 考虑其他可能性 → 审视含义 → 反思
Start by asking 'What do you mean by X?' (clarify), then 'What must we believe for that statement to be true?' (probe assumptions). Follow with 'Is there any counter‑example?' (seek evidence). This loop helps you test the strength of any claim and is especially useful when analysing ethical dilemmas or the existence of God.
从“你说的 X 是什么意思？”（澄清）开始，然后问“我们必须相信什么，那个陈述才能成立？”（探查假设）。接着追问“有没有反例？”（寻求证据）。这个循环帮助你检验任何主张的强度，在分析伦理困境或上帝存在的论证时尤其有用。
4. The Ontological Argument Structure | 本体论论证结构
Anselm's ontological argument attempts to prove God's existence purely through reasoning about the concept of a perfect being. Its logical skeleton can be represented as a chain of definitions leading to an unavoidable conclusion.
安瑟伦的本体论论证试图纯粹通过对一个完美存在的概念进行推理来证明上帝存在。其逻辑骨架可以表示为一条由定义组成的链条，最终引出不可避免的结论。
| Step | Logical Statement |
|---|---|
| 1 | God is defined as the greatest conceivable being (GCB). |
| 2 | A GCB that exists in reality is greater than one that exists only in the mind. |
| 3 | If God exists only in the mind, we can conceive a greater being (one that also exists in reality). |
| 4 | This contradicts the definition of God as the GCB. |
| 5 | Therefore, God must exist in reality. |
The 'formula' highlights that the argument rests entirely on the definition of the greatest conceivable being. Gaunilo challenged it by applying the same logic to a 'perfect island', suggesting that one cannot define something into existence. The ontological argument remains a powerful example of a priori reasoning.
这个“公式”突显了该论证完全依赖于“所能设想的最伟大存在”的定义。高尼罗通过将同样的逻辑应用到“完美岛屿”上来进行反驳，指出我们无法通过定义让某物实存。本体论论证至今仍然是先天推理的一个经典例证。
5. The Teleological Argument Analogy | 目的论论证类比
The design argument, or teleological argument, draws an analogy between the complexity of the natural world and the complexity of human artefacts. William Paley's watchmaker analogy can be captured in a proportional formula.
设计论证，也称目的论论证，在自然世界的复杂性与人造物品的复杂性之间建立类比。威廉·佩利的钟表匠类比可以用一个比例公式来概括。
Analogy: Watch : Watchmaker :: Universe : Universe Maker
类比：钟表 : 钟表匠 :: 宇宙 : 宇宙的制造者
If you found a watch on a heath, its intricate parts working together for a purpose would lead you to infer a watchmaker. Similarly, the eye, the ecosystem or the laws of physics exhibit apparent design, suggesting a divine designer. David Hume's criticisms, however, point out that the analogy is weak: the universe is more like a living organism than a machine, and even if a designer existed, it need not be the all‑powerful God of classical theism.
如果你在荒野中发现一块钟表，其精密的零件为了一个目的而协同运作，你就会推断出有一位钟表匠。同样，眼睛、生态系统或物理定律都展现出明显的目的性设计，暗示着一位神圣的设计者。然而，休谟的批评指出这个类比是薄弱的：宇宙更像一个活的有机体而非机器，而且即使有设计者存在，也不一定就是古典有神论中全能的神。
6. The Problem of Evil Equation | 恶的问题等式
The logical problem of evil challenges the coexistence of an all‑powerful, all‑loving God with the existence of suffering. It is often presented as an inconsistent triad, which can be expressed as an equation that cannot be resolved without dropping one of the attributes.
恶的逻辑问题对全能、全善的上帝与苦难的存在能否共存提出了挑战。它通常被表述为一个不协调的三元组，可以写成一道方程，若不放弃其中某一属性，就无法求解。
Inconsistent Triad: Omnipotence + Omnibenevolence + Evil Exist → Contradiction
不协调三元组：全能 + 全善 + 恶存在 → 矛盾
If God is omnipotent, He can eliminate evil; if He is omnibenevolent, He wants to eliminate evil. Yet evil exists. Therefore, such a God cannot exist. The free will defence attempts to resolve the equation by arguing that some evil is necessary for greater goods like moral freedom. The evidential version of the problem shifts the focus to the sheer quantity and distribution of pointless suffering in the world.
如果上帝全能，他就能消除恶；如果上帝全善，他就愿意消除恶。然而世间确实有恶存在。因此，这样的上帝不可能存在。自由意志辩护试图通过主张某些恶是为了实现更大的善（如道德自由）所必需的，从而解开这个方程式。而恶的明证性问题则将矛头指向世界上无谓苦难的巨大数量和分布方式。
7. The Mind–Body Distinction T‑Chart | 心物区分对照表
Dualism, especially as defended by Descartes, holds that mind and body are two fundamentally different substances. A T‑chart helps memorise the essential dualist formula: mental substance is essentially thinking, non‑extended and private; physical substance is extended, non‑thinking and public.
二元论，尤其是笛卡尔所辩护的那种，认为心灵和身体是两种根本不同的实体。一张 T 形对照表有助于记住二元论的基本公式：心灵实体本质上是能思考、不占空间且私密的；物质实体是广延的、不能思考且公共的。
| Mind (Mental Substance) | Body (Physical Substance) |
|---|---|
| Essence: thinking (cogito) | Essence: extension (occupies space) |
| Indivisible, non‑spatial | Divisible into parts |
| Private, directly accessible only to me | Public, observable by others |
| Subject to free will | Governed by mechanical laws |
The 'formula' of substance dualism generates the interaction problem: how can a non‑physical mind cause changes in a physical body? Descartes speculated that the pineal gland served as a meeting point, but contemporary neuroscience largely rejects this. Still, the distinction raises enduring questions about consciousness and identity.
实体二元论的“公式”引出了交互问题：一个非物理的心灵如何能在物理身体中引起变化？笛卡尔推测松果腺是交汇点，但当代神经科学基本上否定了这一点。尽管如此，这种区分仍引发了关于意识与同一性的永恒追问。
8. The Fallacy Detector | 谬误探测器
Philosophy trains you to spot faulty reasoning. A fallacy is a pattern of argument that seems persuasive but contains a logical error. The 'fallacy detector' formula lists common informal fallacies alongside a simple test to identify them.
哲学训练你识别有缺陷的推理。谬误是一种看似有说服力、实则包含逻辑错误的论证模式。这个“谬误探测器”公式列出了常见的非形式谬误，并附有一个简单的识别检验。
Test: Does the argument attack the person instead of the issue? → Ad Hominem
检验：该论证是否在攻击人而非问题本身？→ 人身攻击
Test: Does it assume a slippery slope without sufficient evidence? → Slippery Slope
检验：它是否在没有充分证据的情况下假定了一个滑坡效应？→ 滑坡谬误
Test: Does it rely on a false either/or choice? → False Dilemma
检验：它是否依赖一个非此即彼的虚假选择？→ 假两难谬误
Practise running arguments through these simple questions. For instance, in a debate about school uniform, shouting 'You're just a stubborn student' is an ad hominem; saying 'If we allow casual wear, next they'll demand to bring pets' is a slippery slope. Recognising fallacies strengthens your own thinking and makes your philosophical writing more rigorous.
试着用这些简单的问题去检验论证。例如，在一场关于校服的辩论中，喊出“你只是个固执的学生”属于人身攻击；说“如果我们允许穿便装，下一步他们就会要求带宠物来学校”则是滑坡谬误。识别谬误可以强化你自己的思考，并使你的哲学写作更加严谨。
9. The Relativism vs Absolutism Spectrum | 相对主义与绝对主义光谱
Moral relativism and absolutism lie at opposite ends of a spectrum. The 'formula' for understanding them is to ask: does the truth of a moral judgment depend on culture, time or individual perspective, or is it universally binding?
道德相对主义和绝对主义处于光谱的两端。理解它们的“公式”是去问：一个道德判断的真假是取决于文化、时代或个人视角，还是无差别地具有普遍约束力？
| Relativism | Absolutism |
|---|---|
| 'X is wrong' means 'X is disapproved of by my society.' | 'X is wrong' means 'X violates an objective moral law.' |
| Moral truths are like table manners – they vary. | Moral truths are like mathematical truths – they hold everywhere. |
| Tolerance of different practices is often prized. | Tolerance is a duty only if it conforms to the absolute moral law. |
A strength of relativism is its promotion of cultural understanding; a weakness is that it leaves no grounds to condemn practices like slavery across all societies. Absolutism offers clear, unwavering guidance but may struggle with moral dilemmas where two absolute rules conflict. Year 9 students can use this spectrum to debate ethical case studies.
相对主义的优点是倡导文化理解；弱点是它无法为谴责普遍存在的蓄奴等现象提供依据。绝对主义提供了清晰、毫不动摇的指引，但在两条绝对规则发生冲突的道德困境面前可能会捉襟见肘。Year 9 学生可以利用这个光谱来辩论各种伦理案例。
10. The Social Contract Formula | 社会契约公式
Social contract theory explains the origin of political authority through a hypothetical agreement. The core formula states that individuals consent to surrender some freedoms and obey a common set of rules in exchange for security and the benefits of co‑operation.
社会契约论通过一个假想的协议来解释政治权威的起源。其核心公式是：个人同意放弃一部分自由并遵守一套共同规则，以换取安全和合作带来的好处。
Formula: Authority = (Liberty Surrendered) / (Security & Order Gained)
公式：权威 = （让渡的自由）/ （获得的安全与秩序）
Hobbes imagined a 'state of nature' where life is 'solitary, poor, nasty, brutish and short', so people contract to create an all‑powerful sovereign. Locke's version emphasises natural rights and government by consent, with citizens retaining the right to revolt. This formula helps you analyse contemporary issues such as lockdown laws or privacy versus surveillance.
霍布斯设想了一种“自然状态”，其中人生“孤独、贫困、污秽、野蛮而又短促”，因此人们缔结契约建立一个全能的统治者。洛克的版本则强调天赋权利和基于同意的政府，公民保留反抗权。这个公式有助于你分析当代议题，比如封锁法律或隐私与监控之间的权衡。
11. The Causality Sequence | 因果序列
Aquinas' Second Way, the argument from causation, posits that every event has a cause, leading to an infinite regress unless there is a first uncaused cause. The logical sequence can be written as a chain that must be broken by a necessary being.
阿奎那的“第二路”——因果论证——假定每个事件都有一个原因，从而导致无穷回溯，除非存在一个不被任何事物推动的第一推动者。这个逻辑序列可以写作一条必须由一个必然存在者打断的因果链。
C₃ → C₂ → C₁ → Event E. To avoid infinite regress, there must be an Uncaused Cause (God).
序列：…… C₃ → C₂ → C₁ → 事件 E。为避免无穷回溯，必有一个不被引发的第一因（上帝）。
The sequence works on the premise that a series of dependent causes cannot be infinite. Hume objected that we have no experience of the universe's origin and the concept of a necessary being is beyond our understanding. Nonetheless, the causality sequence remains a classic cosmological formula for many theists.
这个序列建立在一个前提上：一连串依存的因不可能是无限的。休谟反对说，我们对宇宙的起源毫无经验，而“必然存在者”这一概念也超出了我们的理解。尽管如此，因果序列依然是众多有神论者的一个经典宇宙论公式。
12. The Virtue Ethics Mean | 美德伦理中道
Aristotle's virtue ethics does not prescribe rigid rules but instead locates moral excellence at the 'golden mean' between two extremes. Each virtue can be captured in a simple balance formula that names the deficiency and the excess.
亚里士多德的美德伦理不规定僵化的规则，而是将道德卓越定位在两个极端之间的“黄金中道”上。每一种美德都可以用一个简单的平衡公式来概括，并标明不足与过度。
| Sphere of Action | Deficiency | Virtue (Mean) | Excess |
|---|---|---|---|
| Fear and confidence | Cowardice | Courage | Rashness |
| Giving and taking money | Stinginess | Generosity | Prodigality |
| Pleasure | Insensibility | Temperance | Over‑indulgence |
The 'formula' is not a one‑size‑fits‑all dose; the mean varies according to the person and the situation. It requires practical wisdom (phronesis) to discern the appropriate response. This framework shifts moral focus from 'What should I do?' to 'What kind of person should I become?', a question central to character education and the OCR Philosophy curriculum.
这个“公式”并非一成不变的剂量；中道因人、因情境而异。它需要实践智慧（phronesis）来辨别恰当的反应。这一框架将道德焦点从“我应当做什么？”转向“我应当成为什么样的人？”，这是一个品格教育和OCR哲学课程的核心问题。
Published by TutorHao | Philosophy Revision Series | aleveler.com
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