📚 Philosophy Formula & Theorem Quick Reference Guide | 哲学公式定理速查手册
Welcome to your Year 8 Philosophy Formula & Theorem Quick Reference Guide! Just as mathematics relies on formulas to solve problems, philosophy uses logical forms and ethical principles to evaluate arguments and decide how to act. This handbook collects the most important reasoning patterns, rules of inference, and conceptual tools you need for the CIE course. Each entry is presented as a clear ‘formula’ you can apply when analysing real-life situations, philosophical thought experiments, or examination questions. Keep it to hand whenever you need to check whether an argument is valid, spot a fallacy, or remind yourself how key ethical theories work.
欢迎使用八年级哲学公式定理速查手册！正如数学依靠公式来解题，哲学借助逻辑形式和伦理原则来评估论证、决定行动。本手册收集了 CIE 课程中最重要的推理模式、推论规则和概念工具。每一个条目都像一条清晰的“公式”，当你在分析现实情境、哲学思想实验或考试题目时，都可以直接套用。无论你是想检查论证是否有效、识别谬误，还是复习核心伦理学理论，这本手册都能随时帮你一把。
1. Modus Ponens (Affirming the Antecedent) | 肯定前件式
Modus Ponens is one of the simplest yet most powerful valid argument forms. It states: if a conditional statement (if P then Q) is true, and the antecedent (P) is true, then the consequent (Q) must also be true. This rule guarantees a true conclusion whenever the premises are true.
肯定前件式是最简单也最有力的有效论证形式之一。它表示：如果一个条件语句（如果 P 则 Q）为真，并且前件 P 为真，那么后件 Q 也必然为真。只要所有前提为真，这条规则就能保证结论为真。
P → Q, P ∴ Q
Example: If it is raining, the ground gets wet. It is raining. Therefore, the ground gets wet.
例子：如果下雨，地面就会湿。现在下雨。所以地面湿了。
This form is always truth-preserving. In philosophical arguments, whenever you see a conditional premise and a statement that matches the ‘if’ part, you can safely draw the ‘then’ part.
该形式永远保真。在哲学论证中，凡是看到一个条件前提和一个与“如果”部分匹配的陈述，你都可以放心地推出“则”部分。
2. Modus Tollens (Denying the Consequent) | 否定后件式
Modus Tollens works in the opposite direction from Modus Ponens. Given that P implies Q, if Q is false, then P must also be false. This rule allows us to eliminate possibilities by showing that a predicted consequence does not occur.
否定后件式与肯定前件式的推理方向相反。已知 P 蕴含 Q，如果 Q 为假，那么 P 也必定为假。此规则让我们能够通过证明预期结果并未发生来排除某个可能性。
P → Q, ¬Q ∴ ¬P
Example: If this animal is a cat, then it has whiskers. This animal does not have whiskers. Therefore, this animal is not a cat.
例子：如果这只动物是猫，那么它有胡须。这只动物没有胡须。所以这只动物不是猫。
Modus Tollens is extremely useful in scientific testing and in philosophical scepticism, where we often ask ‘what would follow if a certain theory were true?’ and then check whether those consequences hold.
否定后件式在科学检验和哲学怀疑论中极有用，我们常会问“如果某个理论为真，会引出什么结果？”然后再检验那些结果是否成立。
3. Hypothetical Syllogism (Chain Argument) | 假言三段论（连锁论证）
Hypothetical syllogism chains two conditional statements together. If P implies Q, and Q implies R, then we can infer that P implies R. It is a fundamental transitive rule of logic.
假言三段论将两个条件语句连接成链。如果 P 蕴含 Q，且 Q 蕴含 R，那么可以推出 P 蕴含 R。这是逻辑里一条基本的传递规则。
P → Q, Q → R ∴ P → R
Example: If I study hard, I will understand the topic. If I understand the topic, I will pass the exam. Therefore, if I study hard, I will pass the exam.
例子：如果我努力学习，我就能理解这个课题。如果我理解这个课题，我就能通过考试。所以，如果我努力学习，我就能通过考试。
This form helps us construct extended arguments and trace long chains of cause and effect without testing each link individually.
这一形式帮助我们构建延伸论证，并追踪长链条的因果关系，而不必单独检验每一个环节。
4. Disjunctive Syllogism (Elimination) | 选言三段论（排除法）
Disjunctive syllogism starts with an ‘or’ statement and the denial of one option. If we know that at least one of P or Q is true, and we learn that P is false, then Q must be true. This rule is the foundation of reasoning by elimination.
选言三段论从一个“或”句出发，并否定其中一个选项。如果我们知道 P 或 Q 至少有一个为真，同时又了解到 P 为假，那么 Q 必然为真。这条规则是排除法推理的基础。
P ∨ Q, ¬P ∴ Q
Example: Either the light is on or the bulb is broken. The light is not on. Therefore, the bulb is broken.
例子：要么灯开着，要么灯泡坏了。灯没有开。所以灯泡坏了。
In philosophy, disjunctive syllogism is often used to narrow down metaphysical or ethical possibilities until only the most plausible option remains.
在哲学中，选言三段论常被用来逐一排除形而上学或伦理上的可能性，直到只剩下最合理的那一个选项。
5. Fallacy: Affirming the Consequent | 谬误：肯定后件
Affirming the consequent is a common logical fallacy that looks like Modus Ponens but is invalid. It occurs when someone argues from ‘If P then Q’ and the truth of Q to the truth of P. The conclusion does not necessarily follow, because other causes could produce Q.
肯定后件是一种常见的逻辑谬误，看似肯定前件式但实为无效。它发生在人们从“如果 P 则 Q”和 Q 为真，推出 P 为真的情形中。这个结论并不必然成立，因为其他原因也可能产生 Q。
P → Q, Q ∴ P (INVALID)
Example: If it is a swan, then it is white. This bird is white. Therefore, it is a swan. (It could be a goose or a duck.)
例子：如果是天鹅，那么它是白色的。这只鸟是白色的。所以它是天鹅。（它可能是鹅或鸭子。）
Recognising this fallacy helps you avoid being misled by superficial evidence. Always ask: could something else explain the observed effect?
识别这个谬误能帮你避免被表面证据误导。要永远追问：是否可能有别的因素解释了观察到的结果？
6. Fallacy: Denying the Antecedent | 谬误：否定前件
Denying the antecedent is another invalid argument form. It uses the premises ‘If P then Q’ and ‘P is false’ to conclude that ‘Q is false’. This reasoning fails because Q could still be true for a different reason.
否定前件是另一种无效的论证形式。它用“如果 P 则 Q”和“P 为假”这两个前提，推出“Q 为假”。这一推理失败，因为 Q 仍可能因别的原因而为真。
P → Q, ¬P ∴ ¬Q (INVALID)
Example: If I am in Paris, then I am in France. I am not in Paris. Therefore, I am not in France. (But I could be in Lyon, which is also in France.)
例子：如果我在巴黎，那么我在法国。我不在巴黎。所以我不在法国。（但我可能在里昂，那也在法国。）
This fallacy reminds us that a single sufficient condition is not the only possible route to an outcome. Watch out for it in everyday arguments about rules and exceptions.
这一谬误提醒我们，一个充分条件并不是通往结果唯一可能的途径。在关于规则和例外情况的日常争论中要特别当心。
7. JTB Theory of Knowledge (Justified True Belief) | JTB 知识论（确证的真信念）
For centuries, philosophers defined knowledge as ‘justified true belief’. According to this formula, a person S knows that P if and only if: (1) P is true, (2) S believes that P, and (3) S has good justification for believing that P. This triad acts as a checklist for knowledge.
多个世纪以来，哲学家把知识定义为“确证的真信念”。按照这个公式，一个人 S 知道 P，当且仅当：(1) P 为真，(2) S 相信 P，并且 (3) S 有足够好的理由相信 P。这三个条件就像一份核对知识的清单。
Knowledge = Truth + Belief + Justification
Example: You know it is raining outside if it really is raining, you believe it is raining, and you have justification – such as seeing rain through the window or checking a reliable weather app.
例子：你知道外面在下雨，如果确实在下雨，你相信在下雨，而且你有确证——比如透过窗子看到了雨，或者查看了可靠的天气应用。
However, Edmund Gettier showed cases where all three conditions are met but we still hesitate to call it knowledge (e.g., a stopped clock showing the right time by coincidence). This ‘Gettier problem’ tells us the formula may not be complete.
然而，埃德蒙德·盖梯尔展示了一些例子，其中三个条件都满足，我们却仍不愿称之为知识（比如一只停了的钟碰巧显示了正确时间）。这个“盖梯尔问题”告诉我们，这个公式也许并不完整。
8. Utilitarian Principle (Greatest Happiness) | 功利主义原则（最大幸福）
Utilitarianism offers an ethical formula: an action is right if it produces the greatest happiness for the greatest number. This principle, developed by Jeremy Bentham and John Stuart Mill, treats moral decisions like a happiness calculation. We can think of it as maximising the total of pleasure minus pain.
功利主义提供了一条伦理公式：一个行动若能带来最大多数人的最大幸福，就是正确的。这一由边沁和密尔发展的原则，把道德决策视为一种幸福计算。我们可以把它看成最大化快乐减去痛苦的总和。
Maximise ∑ (Pleasure − Pain)
Application: When deciding whether to tell a white lie, you weigh the immediate comfort it brings against any long-term harm or broken trust, and choose the option with the best overall balance of happiness over suffering.
应用：在决定是否说一个善意的谎言时，你要权衡它带来的即时安慰与它可能造成的长期伤害或信任破裂，并选择总体上幸福大于痛苦的最佳选项。
Utilitarianism is a consequentialist theory: it judges actions by their outcomes. This ‘formula’ is widely used in public policy, economics, and everyday moral reasoning.
功利主义是一种后果论：它按结果来评判行动。这个“公式”被广泛用于公共政策、经济学和日常道德推理中。
9. Categorical Imperative (Simplified) | 定言命令式（简版）
Immanuel Kant’s Categorical Imperative provides a non-consequentialist ethical formula. In its most famous version: ‘Act only according to that maxim whereby you can at the same time will that it should become a universal law.’ Before acting, you test whether your reason for acting could be followed by everyone without contradiction.
康德的定言命令式提供了一条非后果论的伦理公式。其最著名的版本是：“只按照你同时愿意它成为一条普遍法则的准则去行动。” 在行动前，你要检验你行动的理由是否能在没有矛盾的前提下为每个人所遵循。
Universalizability Test: Can I will that everyone act on this maxim?
Example: If you consider breaking a promise to get out of a difficult situation, ask: can I wish that promise-breaking became a universal law? If everyone broke promises whenever it suited them, the practice of promising would collapse. The maxim fails the test, so the action is wrong.
例子：如果你考虑通过违背承诺来脱困，你就要问：我能希望违背承诺成为一条普遍法则吗？如果每个人只要对自己有利就违背承诺，承诺这种实践本身就会崩溃。这个准则通不过检验，因此这一行动是错误的。
This formula focuses on the intention and the rational consistency of the rule behind an action, not on the consequences.
这一公式聚焦于行动背后的意图和规则的理性一致性，而不看重后果。
10. Cogito, ergo sum (I think, therefore I am) | 我思故我在
Rene Descartes formulated one of philosophy’s most famous ‘theorems’: even if a malicious demon deceives me about everything, I cannot doubt that I am thinking. And if I am thinking, I must exist. Thus, ‘I think, therefore I am’ becomes an indubitable foundation for knowledge.
笛卡尔提出了哲学史上最著名的“定理”之一：即便有一个邪恶的魔鬼在一切事情上欺骗我，我也无法怀疑自己正在思考。而如果我在思考，我就必然存在。于是，“我思故我在”成为知识不可怀疑的基石。
Doubt → Thinking → Existence
Reasoning: Every act of doubt confirms the existence of the doubter. This simple but profound insight shows that at least one truth – my own existence as a thinking being – is certain.
推理：每一次怀疑的行动都证实了怀疑者的存在。这个简单却深刻的洞见表明，至少有一个真理——我作为一个思考者的存在——是确定的。
Descartes’ cogito is a foundationalist formula: from this one secure starting point, he attempts to rebuild all knowledge using clear and distinct ideas.
笛卡尔的“我思”是一条基础主义公式：他试图从这个唯一可靠的起点出发，用清楚分明的观念重建全部知识。
11. Inductive Generalization | 归纳概括原理
Inductive generalization is the process of drawing a universal conclusion from a limited set of observations. The formula: a large number of observed As have property B, and no counterexamples have been found yet; therefore, all As have property B. Unlike deductive rules, induction only makes the conclusion probable, not certain.
归纳概括是从有限观察中得出普遍结论的过程。公式如下：大量观察到的 A 都具有性质 B，目前尚未发现反例；因此，所有 A 都具有性质 B。与演绎规则不同，归纳仅使结论成为可能，而非必然。
All observed A are B ⇒ All A are B (probable)
Example: Every raven anyone has seen so far has been black. Therefore, all ravens are black. (But a white raven discovered tomorrow would disprove the conclusion.)
例子：迄今人们看到的每一只渡鸦都是黑色的。因此，所有渡鸦都是黑色的。（但如果明天发现一只白色渡鸦，就会推翻这个结论。）
In philosophy of science, induction is the engine of scientific laws, but it raises the ‘problem of induction’: how can we justify moving from ‘some’ to ‘all’? This remains an open philosophical puzzle.
在科学哲学中，归纳是科学定律的引擎，但它也引发了“归纳问题”：我们如何为从“有些”到“所有”的跳跃辩护？这仍是一个开放的哲学难题。
12. Validity vs Soundness | 有效性与可靠性
When evaluating arguments, two related but distinct ‘formulas’ matter. An argument is valid if and only if it is impossible for the premises to be true and the conclusion false. An argument is sound if and only if it is valid AND all its premises are actually true. Validity is about logical structure; soundness adds real-world truth.
评估论证时，有两个相关但不同的“公式”很重要。一个论证是有效的，当且仅当不可能出现前提为真而结论为假的情况。一个论证是可靠的，当且仅当它有效而且所有前提实际上都为真。有效性关乎逻辑结构，可靠性则加上真实世界的真。
Soundness = Validity + True Premises
Example of valid but unsound: All planets are made of cheese. Earth is a planet. Therefore, Earth is made of cheese. (The structure is valid, but the first premise is false, so the argument is unsound.)
有效但不可靠的例子：所有行星都是奶酪做的。地球是行星。所以，地球是奶酪做的。（结构有效，但第一个前提为假，因此论证不可靠。）
Whenever you analyse a philosophical argument, check for validity first, then test the truth of the premises. Only when both conditions are satisfied can you accept the conclusion with confidence.
每当你分析一个哲学论证时，先检查有效性，再检验前提的真假。只有两个条件都满足，你才能自信地接受结论。
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