📚 Year 8 CCEA Philosophy: Formula & Theorem Quick Reference | 八年级CCEA哲学：公式定理速查手册
In philosophy, the word ‘formula’ does not mean a mathematical equation with numbers, but rather a reliable pattern of thinking, reasoning, and argument. Theorems in this context are tried-and-tested logical structures that help you analyse ideas, spot weaknesses in arguments, and build your own with clarity. This quick-reference handbook collects the most important philosophical formulae and theorems you will encounter in Year 8 CCEA Philosophy, from Socratic questioning to trolley problems, and presents each one as a memorable tool for critical thought.
在哲学中，“公式”一词不指带有数字的数学方程式，而是指一种可靠的思维、推理和论证模式。哲学术语中的定理则是经过检验的逻辑结构，能帮助你分析观点、发现论证中的弱点，并清晰构建自己的论证。这本速查手册汇集了八年级 CCEA 哲学课程中最重要的哲学公式与定理，从苏格拉底提问法到电车难题，把每一个都变成便于记忆的批判性思维工具。
1. The Socratic Formula | 苏格拉底问答公式
The Socratic method is a questioning formula that moves from a claim to deeper understanding through systematic probing. It follows the pattern: Claim → Clarify → Examine Assumptions → Refine → Gain Insight. Socrates believed that genuine knowledge emerges only when we rigorously test what we think we know.
苏格拉底方法是一种提问公式，通过系统追问，从一个主张走向更深刻的理解。它遵循的模式是：主张 → 澄清 → 检验假设 → 修正 → 获得洞见。苏格拉底认为，只有当我们严格审问自以为知道的东西时，真正的知识才会浮现。
In dialogue, you start by asking ‘What do you mean by X?’ Then you question the hidden assumptions behind that meaning. Eventually, the person either discovers a contradiction or is forced to refine their view. The formula is not about winning an argument—it’s about uncovering truth.
在对话中，你首先问“你说的X是什么意思？”，然后质疑其背后隐藏的假设。最终，对方要么发现矛盾，要么被迫修正看法。这个公式不是为了赢得争论，而是为了揭示真理。
Question → Assumption → Contradiction → Revised Understanding
提问 → 假设 → 矛盾 → 修正后的理解
2. Descartes’ Doubt Theorem | 笛卡尔怀疑论定理
René Descartes developed a radical formula for finding certain knowledge: If a belief can be doubted, then it is not absolutely certain knowledge. He imagined a powerful ‘evil demon’ that could deceive all his senses. Whatever could survive this extreme doubt would count as a solid foundation for knowledge.
勒内·笛卡尔提出了一条寻找确定知识的激进公式：如果一个信念可以被怀疑，那它就不是绝对可靠的知识。他设想有一个强大的“恶魔”能欺骗他所有的感官，凡是能在这种极端怀疑下存活下来的，才能算作知识的坚实基础。
The theorem leads step by step: Sense experience breaks down because dreams can mimic reality; mathematical truths might be planted by the deceiver; but the very act of being deceived requires a thinking self. Hence the famous bedrock: Cōgitō, ergō sum—I think, therefore I am.
该定理一步步推导：感官经验不可靠，因为梦境可以模仿现实；数学真理可能是骗子植入的；但“被欺骗”这一行为本身需要一个正在思考的自我。于是得到了那个著名基石：我思故我在。
Doubt Everything → Find One Indubitable Truth → I Think → I Exist
怀疑一切 → 找到一个不可怀疑的真理 → 我思考 → 我存在
3. The Utilitarian Calculus | 功利主义计算
Jeremy Bentham and John Stuart Mill proposed a moral formula that measures the rightness of an action by its consequences. The core ‘equation’ is: Net Good = Total Pleasure − Total Pain. An action is morally right if it produces the greatest happiness for the greatest number of people. This is the Utilitarian Calculus.
边沁和密尔提出了一种道德公式，通过后果来衡量行为对错。核心“方程式”是：净好 = 总快乐 − 总痛苦。如果一项行为能为最大多数人带来最大幸福，它就是道德上正确的，这便是功利主义计算。
When you use this formula, you list everyone affected, estimate the intensity, duration, certainty, and extent of pleasure or pain for each, and then sum them up. The action with the highest net pleasure wins. Critics point out that measuring happiness is difficult and that minority rights might be sacrificed.
当你使用这个公式时，要列出每个受影响者，估算其快乐或痛苦的强度、持续时间、确定性和范围，然后汇总。净快乐值最高的行为胜出。批评者指出，幸福难以度量，且少数人的权利可能被牺牲。
∑Pleasureall affected − ∑Painall affected = Right Action
∑快乐全体受影响者 − ∑痛苦全体受影响者 = 正确行为
4. Kant’s Duty Formulae | 康德义务公式
Immanuel Kant rejected consequence-based morality and instead gave a deontological formula: Act only according to that maxim whereby you can, at the same time, will that it should become a universal law. In other words, before you act, ask: ‘What if everyone did this?’
康德反对以后果为基础的道德观，转而提出一条义务论公式：要只按照你同时能够愿意它成为一个普遍法则的准则去行动。换句话说，在行动之前，先问：“要是每个人都这样做，那会怎样？”
This is the first formulation of the Categorical Imperative. If the maxim of your action leads to a contradiction when universalised (e.g., everyone making false promises destroys the practice of promising), then the act is forbidden. The formula protects human dignity by treating people as ends in themselves, never merely as means.
这是定言命令的第一种表述。如果你行动的准则在被普遍化后会导致矛盾（例如，如果人人都作虚假承诺，承诺这一行为本身就会被摧毁），那么该行为就是被禁止的。这条公式保护人的尊严，要求把人当作目的本身，而绝不仅仅是手段。
Maxim → Universalise → Contradiction? → If yes, impermissible; If no, permissible.
准则 → 普遍化 → 是否矛盾？→ 是，则不可为；否，则可为。
5. The Logical Fallacy Theorems | 逻辑谬误定理
A fallacy is a broken formula of reasoning; it seems convincing but hides a logical error. Recognising fallacies helps you dismantle weak arguments. Here are three core ‘theorems’ that describe common fallacy structures.
谬误是推理的破损公式，它听起来有说服力，但暗藏逻辑错误。识别谬误能帮你拆穿薄弱的论证。以下是三条描述常见谬误结构的核心“定理”。
| Fallacy | Formula / Structure |
|---|---|
| Straw Man | 稻草人 | Misrepresent opponent’s view → Attack the misrepresentation → Claim victory. |
| Slippery Slope | 滑坡 | A small step → Chain of increasingly dire consequences → Avoid A to prevent Z. |
| Ad Hominem | 人身攻击 | Attack the person’s character → Dismiss the argument. |
Each fallacy theorem can be stopped by demanding evidence for each link in the chain. For the straw man, simply say: ‘That is not my position.’ For the slippery slope, ask: ‘What is the actual probability that step B leads to C?’
每条谬误定理都可以通过要求证据来阻断。对于稻草人，只需说：“那不是我的立场。”对于滑坡，可以问：“B 步骤实际导致 C 的概率是多少？”
6. The Trolley Problem Equation | 电车难题方程式
The trolley problem is a moral thought experiment that functions like an ethical formula. The classic scenario: a runaway trolley heads toward five people tied to the track. You can pull a lever to divert it to a side track with one person. The ‘equation’ compares: Action (sacrifice 1) vs. Inaction (let 5 die).
电车难题是一个道德思想实验，它就像一个道德公式。经典情境：一辆失控电车冲向绑在轨道上的五个人。你可以拉动扳道器，让电车转入侧轨，但会撞死一个人。这个“方程式”对比的是：行动（牺牲 1 人）还是不作为（让 5 人死）。
Utilitarians calculate: saving five lives minus one death equals a net gain of four lives, so pulling the lever is right. Deontologists might argue that deliberately causing someone’s death violates a direct duty, making it wrong. The formula exposes the clash between outcome-based and rule-based ethics.
功利主义者计算：救五条命减一条命，得净增四条命，所以扳道是对的。义务论者则可能主张，故意造成他人死亡违背了直接义务，因此是错的。这个方程式揭示了后果论与规则论伦理的冲突。
Pull Lever: Intend to save 5, foresee 1 death | Do Nothing: Allow 5 deaths, no active killing
扳道：意图救 5 人，预见到 1 死 | 什么都不做：容许 5 死，没有主动杀人
7. The Brain in a Vat Hypothesis | 缸中之脑假说
This sceptical scenario provides a formula for challenging our trust in external reality: If your brain were kept alive in a vat and fed electrical signals identical to normal sensory input, you could never know this were the case. Any experience you have could, in principle, be simulated.
这个怀疑论情境提出了一个挑战我们对外部现实信任的公式：假如你的大脑被放置在缸中维持生命，并由电信号输入与正常感觉一模一样的刺激，你永远不会知道这是真的。原则上，你的任何经验都可能只是模拟。
The theorem reminds us that the world we perceive might be a perfect illusion. As a formula: Perception → Could be simulation → No definitive proof of external world. This idea is used in philosophy of mind and in popular films like The Matrix.
该定理提醒我们，我们感知到的世界可能只是一个完美幻象。公式是：知觉 → 可能为模拟 → 无法确证外部世界的存在。这一理念被用于心灵哲学以及像《黑客帝国》这样的流行电影中。
8. The Ship of Theseus Paradox | 忒修斯之船悖论
Identity over time can be expressed as a formula: If every plank of a ship is gradually replaced, is it still the same ship? This paradox asks what makes something the same individual after all its parts have been swapped. It is a theorem of personal identity and material constitution.
历时同一性可以用一个公式来表达：如果一艘船的每块木板都被逐渐替换，它还是原来那艘船吗？这个悖论追问的是，在所有部件都被换掉之后，是什么让一个东西还是同一个个体。这是一条关于人格同一性和物质构成的定理。
There are two main solutions. One says identity lies in the continuous history and shape, even if the matter changes. The other says that if you reassembled the old planks into a new ship, that would be the original. The formula is: Replace Part → Repeat until 100% → Original or Copy?
主要有两种解答。一种认为同一性在于连续的历史和形态，即使物质改变也无妨；另一种认为，如果你把所有旧木板重新组装成一艘船，那才是原船。公式形如：替换部件 → 重复至 100% → 原件还是复制品？
9. The Is-Ought Gap Theorem | 实然-应然鸿沟定理
David Hume identified a logical gap that has become a famous theorem in ethics: You cannot validly derive an ‘ought’ statement (a value judgement) solely from ‘is’ statements (descriptions of facts). In other words, factual premises alone cannot logically force a moral conclusion.
大卫·休谟指出了一个逻辑鸿沟，现已成为伦理学著名定理：你不能仅仅从“是”的陈述（事实描述）中有效地推导出“应该”的陈述（价值判断）。换句话说，仅有事实前提，无法在逻辑上必然得出道德结论。
For example: ‘Lying causes pain’ (is) + ‘Pain is undesirable’ (is) does not automatically give ‘You ought not to lie’ without an implicit value premise such as ‘You ought not to cause what is undesirable’. The formula alerts us to hidden value assumptions.
例如：“说谎导致痛苦”（是）加上“痛苦是不好的”（是）并不能自动得出“你不应该说谎”，除非隐含了一个价值前提，比如“你不应该造成不好的事”。这个公式提醒我们要注意隐藏的价值假设。
Facts (Is) + Hidden Value → Ought; Facts alone → Cannot reach Ought
事实（实然） + 隐藏价值 → 应然；仅有事实 → 无法抵达应然
10. Occam’s Razor Principle | 奥卡姆剃刀原则
Occam’s Razor is a philosophical ‘trimming’ formula: Entities should not be multiplied beyond necessity. In simpler terms, among competing explanations, the one with the fewest assumptions is usually preferred. It is a theorem of simplicity and explanatory power.
奥卡姆剃刀是一条哲学“修剪”公式：如无必要，勿增实体。简单说，在相互竞争的解释中，假设最少的那一个通常更受青睐。它是一条关于简洁性和解释力的定理。
If you hear hoofbeats, think horses, not zebras—unless you are in Africa. The razor does not prove truth, but it guides inquiry by eliminating unnecessary complexity. Apply it like this: Explanation A (5 assumptions) vs. Explanation B (2 assumptions) → Tentatively prefer B.
听见蹄声，先想马，别想斑马——除非你身在非洲。剃刀并不证明真理，但通过剔除不必要的复杂性来引导探究。这样运用：解释 A（5 个假设）与解释 B（2 个假设） → 暂时选择 B。
11. The Euthyphro Dilemma | 欧绪弗洛困境
Plato’s Euthyphro presents a theological formula that questions the foundation of morality: Is something good because God commands it, or does God command it because it is good? Each horn of the dilemma creates a problem for divine command theory.
柏拉图的《欧绪弗洛篇》提出了一个质疑道德基础的神学公式：某事是善的，是因为上帝命令它如此；还是因为某事本身是善的，上帝才命令它？困境的每一端都给神命论带来了难题。
If good because commanded, then morality seems arbitrary—God could command cruelty and it would be good. If commanded because good, then goodness exists independently of God, so God is not the source of morality. The formula forces us to consider where moral standards ultimately lie.
如果善是因为被命令，那么道德就显得任意——上帝若命令残忍，残忍也会成为善。如果因为善而被命令，那么善就独立于上帝而存在，上帝便不是道德的源头。这个公式迫使我们思考道德标准究竟源于何处。
12. The Social Contract Formula | 社会契约公式
Thinkers like Thomas Hobbes, John Locke, and Jean-Jacques Rousseau used a formula to explain political authority: In a hypothetical ‘state of nature’, rational individuals agree to give up some freedoms to a governing authority in exchange for security and order. This agreement is the social contract.
霍布斯、洛克和卢梭等思想家运用一条公式来解释政治权威：在假想的“自然状态”下，理性个体同意放弃部分自由，交给统治权威，换取安全与秩序。这一约定就是社会契约。
The formula can be expressed as: Natural Rights + Unlimited Freedom → Insecurity → Contract → Civil Society with Laws. Different philosophers vary the terms: Hobbes emphasised fear and absolute sovereignty; Locke stressed the protection of life, liberty, and property; Rousseau sought the general will.
该公式可以表示为：自然权利 + 无限自由 → 不安全 → 契约 → 有法律的公民社会。不同哲学家对条款的侧重不同：霍布斯强调恐惧和绝对主权；洛克强调保护生命、自由和财产；卢梭则追求公意。
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