📚 Philosophy Formulas & Theorems Quick Reference | 哲学公式定理速查手册
Sometimes the deepest ideas can be captured in a line of symbols – a 'formula' that helps you remember a whole argument. Philosophy is full of logical moves, ethical calculations, and conceptual equations. This quick reference handbook gathers some of the most famous 'formulas and theorems' from the world of philosophy, written in a way that a Year 8 thinker can decode. Use it to memorise key patterns of reasoning, to compare different approaches, and to see how even the messiest human questions can be expressed with clarity and precision.
有时候，最深刻的思想可以用一行符号来概括——一个能帮你记住整段论证的“公式”。哲学中充满了逻辑推演、伦理计算和概念等式。这本速查手册汇集了哲学世界中最著名的一些“公式与定理”，以八年级学生也能解读的方式呈现。用它来记住推理的关键模式，比较不同的方法，并看看最复杂的人类问题如何能被清晰而精确地表达出来。
1. The Golden Rule Formula | 黄金法则公式
In ethics, nearly every culture has discovered the same basic equation for decent behaviour. The Golden Rule can be written as a straightforward conditional statement. If you would not like X done to you, then do not do X to others. This single instruction generates countless moral duties without needing a long list of rules.
在伦理学中，几乎每种文化都发现了同一个关于得体行为的基本等式。黄金法则可以被写成一个简单的条件陈述：如果某件事 X 你不想别人对你做，那么你就不要对别人做 X。单凭这一指令，就能产生无数道德义务，而不需要长长的规则清单。
¬L(me, X) → ¬Do(me, others, X)
Where 'L(me, X)' means 'I would like X done to me'. The negation in front turns it into 'I would not like it'. The arrow then says: therefore, I must not do it to others. This formula is symmetrical, universal, and requires empathy – it forces you to imagine swapping places with the person affected by your action.
其中“L(me, X)”表示“我愿意别人对我做 X”。前面的否定符号把它变成“我不愿意”，而箭头则表示：因此，我绝不能对别人这样做。这个公式是对称的、普遍的，而且需要同理心——它迫使你想象与他人互换位置，感受自己的行为会带来什么影响。
2. Utilitarian Happiness Calculus | 功利主义的幸福计算
Jeremy Bentham believed that moral choices could be made almost mathematically. He proposed a hedonic calculus – a systematic way to add up pleasure and subtract pain. The basic idea is that the right action is the one that produces the greatest total happiness for everyone affected. Even though Bentham never wrote it as a tidy sum, we can reconstruct the principle as an equation of net utility.
杰里米·边沁相信，道德选择几乎可以用数学方法来做。他提出了一个快乐计算法——一种系统性地累加快乐并减去痛苦的方法。核心理念是：正确的行为就是能为所有受影响者带来最大总幸福的行为。尽管边沁从未把它写成一个整洁的算式，我们可以把这原则重构为一个净效用的等式。
Total Good = Σ(Happiness × Number of people) – Σ(Pain × Number of people)
The sigma (Σ) means 'sum over everyone involved'. Multiply the intensity and duration of happiness by the number of people experiencing it, then do the same for pain. The action with the highest result wins. Critics point out it is very hard to measure happiness in units, and that sometimes a small amount of injustice could be 'justified' if it makes the majority happy – which feels wrong.
希腊字母 Σ 表示“对每个相关人加总”。把幸福的强度和持续时间与体验它的人数相乘，对痛苦也做同样的计算。最后结果最高的行为胜出。批评者指出，幸福很难用单位来衡量，而且有时假如让大多数人开心，少量的不公正也可能被“合理化”——这让人感到不对劲。
3. Descartes' Cogito Theorem | 笛卡尔的“我思”定理
René Descartes wanted to find one absolutely certain truth that could survive even the most radical doubt. He imagined an evil demon deceiving him about everything – the physical world, his own body, even mathematics. After stripping away all beliefs that could be doubted, he arrived at a single, solid foundation: the very act of doubting proved that he existed. This is often called the Cogito argument.
勒内·笛卡尔想找到一个绝对确定的真理，即使经历最彻底的怀疑也能成立。他想象有一个邪恶的恶魔在所有事情上欺骗他——外部世界、他的身体，甚至数学。剥去所有可能被怀疑的信念后，他到达了一个单一的、坚固的根基：怀疑这一行为本身就证明了他的存在。这通常被称作“我思”论证。
Dubito → Sum
(I doubt → I exist)
The theorem is deceptively simple: if I am deceived, I must exist to be deceived; if I doubt that I exist, the doubter must be something that exists. Even an all‑powerful demon cannot make a non‑existent being think. This 'Archimedean point' became the starting point for modern philosophy, though later thinkers questioned whether 'I' can be deduced so quickly – perhaps only 'thinking is happening' is certain.
这条定理简单得令人吃惊：如果我被欺骗了，为了被欺骗，我必须存在；如果我怀疑自己是否存在，这个怀疑者必定是某种存在的东西。即便是全能的恶魔也无法让一个非存在者思考。这个“阿基米德点”成了现代哲学的起点，尽管后来的思想家质疑“我”是否能如此迅速地推出——或许只有“思考正在发生”是确定的。
4. Plato's Cave Equation of Appearance vs Reality | 柏拉图的洞穴：表象与现实方程
Plato's Allegory of the Cave is a story about prisoners who mistake shadows on a wall for reality. We can model the relationship between the visible world and the true world of Forms with a simple function. The shadows (S) are a projection of the real objects (R), just as the physical world is a dim copy of the eternal Forms. Education is the process of turning around to see the source.
柏拉图的洞穴寓言讲的是囚徒把墙上的影子误认为现实的故事。我们可以用一个简单的函数来模拟可见世界与真实理念世界之间的关系。影子(S)是真实物体(R)的投影，正如物理世界是永恒理念的一个暗淡的摹本。教育就是转身去看那光源的过程。
S = f(R) where f is imperfect perception
The shadow is a function of the real thing, distorted by the conditions of the cave (firelight, chains, angle). True knowledge is when the soul escapes the cave and sees R directly – the Form itself. In a similar way, a drawn circle is never a perfect circle; it merely participates in the Form of Circularity. This 'formula' reminds us always to ask: am I staring at shadows or seeking the genuine article?
影子是真实物体的一个函数，受到洞穴条件（火光、锁链、角度）的扭曲。真正的知识发生在灵魂逃出洞穴、直接看到 R——即理念本身——时。类似地，画出来的圆永远不是完美的圆；它只是分有了“圆形”的理念。这个“公式”提醒我们永远要问：我是在盯着影子，还是在追寻真实的事物？
5. The Socratic Method Loop | 苏格拉底式对话循环
Socrates didn't deliver lectures; he asked questions that acted like a logical feedback loop. His method can be written as a process that repeats until contradictions in someone's beliefs are exposed. The aim is not to win an argument but to give birth to clearer ideas – what he called maieutics (midwifery). The loop diagram shows how a definition is tested, refuted, refined, and tested again.
苏格拉底不作长篇演讲；他提问的方式像一个逻辑反馈回路。他的方法可以写作一个循环过程，不断重复，直到某人信念中的矛盾被暴露出来。目的不是赢得辩论，而是催生出更清晰的想法——他称之为“精神助产术”。这个循环图展示了一个定义如何被检验、反驳、修正并重新检验。
Ask 'What is X?' ceive definition → Test with counterexample → If contradiction: refine definition → Repeat 🔄
This loop is the engine of critical thinking. For instance, 'What is courage? – Standing firm in battle.' Counterexample: a soldier who stands firm out of ignorance, not bravery. The definition is refined to 'knowledge of what is truly fearsome and what is not.' The loop never ends but leads to deeper understanding. You can use this formula in any discussion to move from shallow opinions to insight.
这个循环是批判性思维的发动机。例如，“什么是勇气？——在战斗中坚守岗位。”反例：一个士兵因无知而非勇敢而坚守。定义被修正为“知道什么是真正可怕的、什么不是”。这个循环永无止境，但会导向更深刻的理解。你可以在任何讨论中运用这个公式，从肤浅的观点走向洞见。
6. Aristotle's Law of Non‑Contradiction | 亚里士多德的非矛盾律
Before you can even begin to reason, you need a guarantee that a statement cannot be both true and false in the same sense at the same time. Aristotle identified this as the most certain of all principles. Without it, words would have no stable meaning, and all communication would break down. It is the foundation of logic, maths, and science.
在你甚至能开始推理之前，你需要一个保证：一个陈述不可能在相同意义上同时为真又为假。亚里士多德认为这是所有原理中最确定的一条。没有它，词语就没有稳定的含义，所有的沟通都会崩溃。它是逻辑、数学和科学的基础。
¬(p ∧ ¬p)
Read as: it is not the case that both p and not‑p. Here 'p' stands for any proposition, such as 'the door is open'. The law says the door cannot be both open and not open in the same respect at the same instant. If someone claims it can, they cannot even state their claim coherently – because 'it can' and 'it cannot' would mean the same thing, destroying all distinctions. This 'theorem' is the shield of rationality.
读作：不可能同时存在 p 与非 p。这里的“p”代表任何命题，比如“门是开着的”。这条定律说，门不可能在同一方面、同一瞬间既开着又没开着。如果有人声称这是可能的，他们连自己的主张都无法连贯地表达——因为“能这样”与“不能这样”将会是同一个意思，消除了所有区别。这则“定理”是理性的盾牌。
7. Hume's Induction Scepticism Formula | 休谟的归纳怀疑公式
David Hume noticed a crack in how we learn from experience. All science and everyday prediction rely on induction: assuming the future will resemble the past. But, Hume asked, what is the logical proof that the future will resemble the past? We cannot use past success as evidence without circularity. His argument reveals that induction is a habit of the mind, not a proven rule of reality.
大卫·休谟注意到了我们从经验中学习的方式存在一道裂缝。所有的科学和日常预测都依赖于归纳法：假设未来会与过去相似。但休谟问道，有什么逻辑证据能证明未来与过去相似？我们无法在不陷入循环的情况下使用过去的成功来作为证据。他的论证揭示出，归纳只是心灵的一种习惯，而不是一条被证明的实在规则。
Observed: (A₁ → B), (A₂ → B) … (Aₙ → B)
Conclusion: All A → B
The observed pattern is that whenever event A happened, B followed. But the jump to 'all A lead to B' depends on the hidden premise: 'Nature is uniform'. And what justifies that premise? Only more induction. Hume doesn't say we should stop using induction – we can't – but he shows it rests on animal faith, not reason. This 'formula' labels a problem that still puzzles philosophers of science.
观察到的模式是每当事件 A 发生，B 随之而来。但是跳跃到“所有 A 都导致 B”依赖于一个隐藏的前提：“自然是一律的”。那什么来证明这个前提呢？只能靠更多的归纳。休谟并不是说我们应该停止使用归纳法——我们做不到——但他指出了它依靠的是动物般的信念，而不是理性。这个“公式”标示出了一个至今仍令科学哲学家的困惑的问题。
8. Kant's Categorical Imperative Test | 康德的定言命令检验
Immanuel Kant wanted a moral rule that would be as tight as a mathematical law. He proposed the Categorical Imperative, which comes in several formulations. The most famous version tells you to act only on a principle that you could will to become a universal law for everyone. It turns ethics into a test of logical consistency rather than a calculation of outcomes.
伊曼努尔·康德想要一条像数学定律一样严谨的道德规则。他提出了定言命令，它有好几个表述方式。其中最著名的版本告诉你：只按照你同时也能够愿意它成为一条普遍法则的准则去行动。这把伦理学转变为一种逻辑一致性的检验，而非对结果的算计。
Act only on maxim M such that M could be willed as universal law: ∀x(Mx) is consistent
For example, take the maxim 'I may make a false promise to get money.' If everyone made false promises, the practice of promising would collapse, and no one would believe your promise – so the maxim defeats itself. The test is not about what you would like to happen, but about what logically can be willed without contradiction. This formula captures the idea that moral duties are those that apply to all rational beings equally.
举个例子，考虑这条准则：“我可以为了得到钱而作出虚假承诺。”如果每个人都作出虚假承诺，承诺这种实践就会崩溃，没有人会再相信你的承诺——因此这条准则会自我挫败。这种检验不在于你希望发生什么，而在于什么事能被允许而没有矛盾。这个公式抓住了这一理念：道德义务就是那些平等适用于所有理性存在者的。
9. The Existentialist Theorem: Existence Precedes Essence | 存在主义定理：存在先于本质
Jean‑Paul Sartre flipped the traditional picture of human nature. For a manufactured object – a knife – its essence (its design, purpose) exists first in the mind of the maker before the object is brought into existence. For humans, Sartre claimed, there is no predefined essence. First we exist, and then through our choices we define who we are. This liberating and terrifying idea can be written as a priority rule.
让‑保罗·萨特颠覆了关于人性的传统图景。对于一个人造物——比如一把刀——它的本质（其设计、用途）首先存在于制造者的脑海中，然后物体才被带入存在。萨特认为，对于人类而言，并没有一个预先定义好的本质。我们首先存在，然后通过自己的选择来定义自己是谁。这个既解放又令人恐惧的想法可以写成一条优先规则。
For human beings: Existence → (Choice → Essence)
¬(Essence → Existence)
The formula states that there is no arrow from a fixed essence to our existence; instead, from our bare existence we generate our essence through action. A person is not born a coward or a hero – they become one by their deeds. This theorem places total responsibility on each individual: you are nothing else but what you make of yourself, and you cannot blame human nature or God for your failings.
这个公式表明，并没有一个从固定本质到我们存在的箭头；相反，我们从赤裸的存在出发，通过行动来生成自己的本质。一个人不是生来就是懦夫或英雄——他是通过自己的行为成为那样的。这一定理把全部责任放在了每一个个体身上：你除了自己所造就的东西之外什么都不是，你也不能将自己的失败归咎于人性或上帝。
10. Occam's Razor: The Principle of Parsimony | 奥卡姆剃刀：简约原则
William of Ockham gave philosophers and scientists a tool for slicing away unnecessary assumptions. His principle is often stated as 'entities should not be multiplied beyond necessity.' In modern terms, it means that when two explanations compete, the one with fewer unproven parts is to be preferred. It is not a proof that the simpler theory is true, but a rule of thumb for keeping inquiry clean.
奥卡姆的威廉给了哲学家和科学家一个用来切除不必要假设的工具。他的原则常被表述为“如无必要，勿增实体”。用现代语言来说，意思是当有两种解释在竞争时，应当选择未经证实的成分较少的那一个。这并不是证明更简单的理论为真，而是一条保持探究方向清晰的法则。
If Theory A and Theory B explain the same data, and A has fewer entities than B, then choose A.
This can be visualised as trimming a hedge: if the bush looks the same without the extra branches, cut them off. In philosophy, if you can explain the mind without adding a separate soul‑substance, you do so. In science, if planets move as well with gravity as with angels pushing them, gravity wins. The razor doesn't guarantee truth, but it stops us adding invisible dragons to our theories unnecessarily.
这可以想象成修剪树篱：如果剪掉多余的枝条后，树篱看上去一样，那就剪掉它们。在哲学中，如果你可以不用额外添加一个灵魂实体就能解释心灵，那你就这么做。在科学中，如果行星用引力就可以很好地运动，而不需要天使来推动，那引力就胜出。这把剃刀并不保证真理，但它能阻止我们不必要地在理论中添加看不见的龙。
11. The Problem of Evil Logical Equation | 恶的难题逻辑等式
The existence of suffering presents a famous challenge to belief in an all‑powerful, all‑knowing, all‑good God. The logical problem of evil can be laid out as a set of propositions that, if all true, appear to contradict each other. This is not a 'formula' to solve the problem, but a way to see the logical structure clearly. Countless solutions have been offered, each trying to show that the set is not contradictory.
苦难的存在向相信一位全能、全知、全善的上帝提出了一个著名的挑战。恶的逻辑难题可以被摆成一组命题，如果它们都为真，看起来就会彼此矛盾。这并不是一个用来解决问题的“公式”，而是一种看清逻辑结构的方式。人们提出过无数种解决方案，每一种都试图表明这组命题其实并不矛盾。
1. God is all‑powerful.
2. God is all‑good.
3. Evil exists.
¬ (1 ∧ 2 ∧ 3) appears contradictory if 'all‑good' means wanting to eliminate evil and 'all‑powerful' means able to do so.
The apparent contradiction forces us to question at least one premise. Some deny that God is all‑powerful (process theology), others deny that evil truly exists (illusionism), and many argue that God allows evil for a greater good (theodicies). The formula is a tool to diagnose exactly where the disagreement lies. Even if you think there is a solution, the structure helps you argue precisely.
这种表面上的矛盾迫使我们至少质疑其中一个前提。有些人否认上帝是全能的（过程神学），有些人否认恶真实存在（幻觉主义），而许多人则论证说上帝允许恶是为了更大的善（神义论）。这个公式是一个工具，能准确地诊断分歧所在。即便你认为这个问题有解，这种结构也能帮助你把论证表达得更精确。
12. Nietzsche's Eternal Recurrence Thought‑Experiment | 尼采的永恒轮回思想实验
Friedrich Nietzsche offered not a theorem of logic but a test of life‑affirmation. He asked: what if a demon told you that you must live your entire life over and over again, every detail exactly the same, for all eternity? Would you curse the demon, or would you see it as the most divine thought? This idea can be framed as a conditional, forcing you to evaluate whether you truly love your life.
弗里德里希·尼采提供的不是一个逻辑定理，而是一个对生命肯定的测试。他问道：假如有一个恶魔告诉你，你必须将你的一生，每一细节都完全相同地，永恒地重复活下去，你会诅咒这个恶魔，还是把这视为最神圣的念头？这个想法可以被框定为一个条件句，迫使你评估你是否真的热爱自己的人生。
If you would despair at the thought of eternal repetition, then you have not yet affirmed your life fully.
Nietzsche uses this as a measuring stick for psychological health and courage. A person who can say 'Yes!' to the eternal recurrence loves life so deeply that they wish every second, even the painful ones, would return infinitely. While not a literal physical theory, this 'formula' works as a compass: it asks you to live in such a way that you would be overjoyed to repeat every moment forever.
尼采以此作为衡量心理健康与勇气的标尺。一个人如果能够对永恒轮回说“是的！”，那就意味着他如此深爱生命，以至于希望每一秒，哪怕痛苦的部分，也能无限地回归。虽然这不是一个物理上的真实理论，但这个“公式”像指南针一样起作用：它要求你以一种你愿意永恒重复每一个瞬间的方式去生活。
Published by TutorHao | Philosophy Revision Series | aleveler.com
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