📚 Y8 Edexcel Philosophy: Quick Reference Handbook of Formulas & Theorems | Year 8 Edexcel 哲学：公式定理速查手册
Welcome to your Year 8 Edexcel Philosophy Formula & Theorem Quick Reference Handbook! Philosophy is about asking big questions and reasoning carefully. Just as mathematics uses formulas to solve problems, philosophy uses logical structures, ethical principles, and epistemic criteria to analyse arguments and ideas. This handbook collects key 'formulas' and 'theorems' that appear in your Edexcel Year 8 philosophy studies, covering logic, ethics, epistemology and more. Use this to revise and strengthen your philosophical thinking.
欢迎使用 Year 8 Edexcel 哲学公式定理速查手册！哲学关乎提出宏大问题并进行严谨推理。正如数学运用公式解题，哲学运用逻辑结构、伦理原则和认识标准来分析论证和思想。本手册收集了您 Edexcel Year 8 哲学学习中出现的核心“公式”与“定理”，涵盖逻辑、伦理、知识论等领域。请用它复习并强化您的哲学思维。
1. The Syllogism Formula | 三段论公式
A syllogism is a form of deductive reasoning consisting of two premises and a conclusion. The classic formula, first systematised by Aristotle, guarantees a true conclusion if the premises are true and the logical structure is valid. The basic pattern is: All M are P (major premise), All S are M (minor premise), therefore All S are P (conclusion). This is the Barbara syllogism in medieval logic.
三段论是一种由两个前提和一个结论组成的演绎推理形式。由亚里士多德首次系统化的经典公式，如果前提为真且逻辑结构有效，就能保证结论为真。基本模式为：所有 M 是 P（大前提），所有 S 是 M（小前提），所以所有 S 是 P（结论）。这就是中世纪逻辑中的 Barbara 式三段论。
Formally, we can write: ∀x (M(x) → P(x)), ∀x (S(x) → M(x)) ⊢ ∀ sophers are thinkers. All sceptics are philosophers. Therefore, all sceptics are thinkers.
形式上可写为：∀x (M(x) → P(x)), ∀x (S(x) → M(x)) ⊢ ∀x (S(x) → P(x))。不过对 Year 8 而言，关键是识别有效式。示例：所有哲学家都是思考者。所有怀疑论者都是哲学家。因此，所有怀疑论者都是思考者。
Be careful: if the middle term is not distributed or if a premise is negative, the syllogism may be invalid. Always test with Venn diagrams or counterexamples.
当心中项不周延或前提为否定时，三段论可能无效。始终用文恩图或反例检验。
2. Deductive Validity & Soundness | 演绎有效性与可靠性
ty + true premises. The formula: If premises entail conclusion, argument is valid. If valid and premises are true, it is sound. Example: All cats are mammals. All mammals are animals. So all cats are animals. (valid & sound).
一个演绎论证是有效的，如果其结论从前提中逻辑地得出。有效性关涉逻辑形式而非真值。可靠性 = 有效 + 前提为真。公式：如果前提蕴含结论，论证有效。如果有效且前提为真，则可靠。示例：所有猫都是哺乳动物。所有哺乳动物都是动物。因此所有猫都是动物。（有效且可靠）。
ey deductive form is modus ponens: P → Q, P, therefore Q. Another is modus tollens: P → Q, ¬Q, therefore ¬ structures. Watch out for fallacies like affirming the consequent (P → Q, Q, therefore P).
一个关键的演绎形式是肯定前件式：P → Q, P, 所以 Q。另一个是否定后件式：P → Q, ¬Q, 所以 ¬P。这些都是保真结构。要警惕谬误，例如肯定后件（P → Q, Q, 所以 P）。
3. Inductive Strength & Cogency | 归纳强度与说服力
tion (all swans are white). Inductive strength depends on sample size, representativeness, and lack of counterevidence. A strong argument with true premises is cogent.
与演绎不同，归纳提供或然性支持。归纳公式：观察到的模式（所有观察到的天鹅都是白的）→ 概括（所有天鹅都是白的）。归纳强度取决于样本量、代表性及缺乏反例。前提为真的强论证是有说服力的。
David Hume's problem of induction: we cannot rationally justify the assumption that the future will resemble the past without circular reasoning. All inductive reasoning relies on the uniformity of nature, which itself can only be proved inductively. This is a deep epistemic challenge.
休谟的归纳问题：我们无法在不陷入循环推理的情况下，理性地证明“未来会与过去相似”这一假定。所有归纳推理依赖于自然齐一性，而自然齐一性本身只能通过归纳来证明。这是一个深刻的认识论挑战。
4. Descartes' Cogito as a First Principle | 笛卡尔的“我思”作为第一原理
Rene Descartes' 'Cogito, ergo sum' ('I think, therefore I am' must exist as a thinking thing. Even if an evil demon deceives me, the fact that I am being deceived proves that I exist. This serves as a self-verifying axiom.
勒内·笛卡尔的“我思故我在”是一条抵抗怀疑论的根本定理。公式：我在思考 → 我必定作为一个思考的事物而存在。即使有邪恶魔鬼欺骗我，“我正被欺骗”这一事实本身证明了我的存在。此即一个自证公理。
The cogito is immediate and indubitable. Unlike other beliefs, it cannot be doubted without affirming its truth. Thus it provides a secure starting point for knowledge. Descartes used it to rebuild philosophy on a firm rational foundation.
“我思”是直接且不可怀疑的。与其他信念不同，怀疑它就会同时肯定它的真。因此它为一个可靠的知识起点。笛卡尔借此在坚稳固的理性基础上重建哲学。
5. The Golden Rule & Kant's Categorical Imperative | 黄金法则与康德的绝对命令
Ethical formulas guide moral action. The Golden Rule: 'Treat others as you would like to be treated.' Immanuel Kant's first formulation of the Categorical Imperative: 'Act only according to that maxim whereby you can, at the same time, will that it should become a universal law.' This universalisability test works like a logical formula.
伦理公式指导道德行动。黄金法则：“你愿意别人怎样待你，你就怎样待别人”。康德绝对命令的第一表述：“只依据你能同时愿意其成为普遍法则的准则去行动”。这种可普遍化检验就像一条逻辑公式一样运作。
trust breaks down, making lying pointless. Contradiction in conception → lying is impermissible. The formula reveals contradictions in immoral maxims.
示例：为了得到想要的东西而撒谎。准则：每当我想得到什么，我就可以撒谎。普遍化后：每个人想得到什么就撒谎。如此一来信任崩溃，撒谎变得毫无意义。概念矛盾 → 撒谎不允许。该公式揭示了不道德准则的内在矛盾。
6. Utilitarian Calculus | 功利主义计算
Jeremy Bentham's utilitarianism: the right action is the one that produces the greatest happiness for the greatest number. Formula: Calculate the sum of pleasures minus pains for everyone affected. The hedonic calculus factors include intensity, duration, certainty, propinquity (nearness), fecundity, purity, and extent.
边沁的功利主义：正确的行为就是为最大多数人带来最大幸福的行为。公式：计算所有受影响者的快乐总和减去痛苦总和。苦乐计算因子包括强度、持续时间、确定性、邻近性、富饶性、纯度和范围。
other factors such as duration or fecundity decide. John Stuart Mill later distinguished higher and lower pleasures, refining the calculus.
如果行为A产生10单位快乐和3单位痛苦，总功利 = +7。行为B产生8快乐1痛苦 = +7。当总功利相等时，其他因子如持续时间和富饶性就起决定作用。约翰·斯图亚特·密尔后来区分了高级快乐与低级快乐，使计算更为精细。
7. Aristotle's Doctrine of the Mean | 亚里士多德的中庸之道
e intermediate relative to us, determined by reason. Courage is the mean between rashness (excess) and cowardice (deficiency). Generosity is the mean between wastefulness and stinginess. The mean is not a mathematical average but a flexible target.
亚里士多德认为，美德存在于两个极端之间：过度与不足。公式：美德 = 相对于我们的中道，由理性确定。勇敢是鲁莽（过度）与怯懦（不足）之间的中道。慷慨是挥霍与吝啬之间的中道。中道并非数学平均，而是灵活的目标。
Finding the mean requires practical wisdom (phronesis). It is not the same for everyone; it depends on the situation and the individual. For example, what is courageous for a soldier differs from what is courageous for a civilian. The virtue of temperance is the mean regarding physical pleasures.
找到中道需要实践智慧（phronesis）。它对每个人而言并不相同，要视情境和个人而定。例如，对士兵而言的勇敢不同于对平民的勇敢。节制的美德则是关于身体快乐的中道。
8. The Tripartite Theory of Knowledge (JTB) | 知识的三分理论（JTB）
Knowledge has been traditionally defined as justified true belief (JTB). Formula: S knows that P if and only if (i) P is true, (ii) S believes that P, and (iii) S is justified in believing that P. This formula captures the necessary and sufficient conditions for propositional knowledge.
知识传统上被定义为得到辩护的真信念（JTB）。公式：S 知道 P，当且仅当 (i) P 为真，(ii) S 相信 P，以及 (iii) S 对相信 P 有辩护。该公式抓住了命题性知识的充分必要条件。
Edmund Gettier's counterexamples show that JTB is not sufficient. Cases of lucky true beliefs challenge the formula. For instance, looking at a broken clock that happens to show the right time yields a justified true belief but not knowledge. Yet JTB remains a useful starting point.
盖梯尔的反例表明 JTB 并非充分条件。碰巧为真的信念案例挑战了该公式。例如，看着一个坏掉的钟恰好显示正确时间，形成了一个得到辩护的真信念，但并不是知识。但 JTB 仍是有用的出发点。
9. Paradoxes in Philosophy | 哲学中的悖论
Paradoxes reveal limits of reasoning. The Liar Paradox: 'This statement is false.' If true, it must be false; if false, it must be true. No stable truth value. Another: Zeno's dichotomy paradox – in going from A to B, you must first cover half the distance, then half of the remainder, ad infinitum, so motion seems impossible. These paradoxes force us to examine assumptions.
悖论揭示了推理的局限。说谎者悖论：“这句话是假的。”若为真，则必为假；若为假，则必为真。没有稳定的真值。另外一个：芝诺的二分法悖论 – 从A去B，必须先走过一半距离，再走过剩下的一半，如此无限，因此运动看似不可能。这些悖论迫使我们审视预设。
The Barber Paradox: In a village, the barber shaves all those and only those who do not shave themselves. Does the barber shave himself? If he does, he doesn't; if he doesn't, he does. This exposes issues in naive set theory. Philosophy uses paradoxes to sharpen logic and conceptual clarity.
理发师悖论：一个村子里，理发师给所有且只给那些不自己刮胡子的人刮胡子。那理发师给自己刮胡子吗？如果他刮，他就不该刮；如果他不刮，他就该刮。这暴露了朴素集合论的问题。哲学用悖论磨砺逻辑与概念的清晰性。
10. The Argument Structure: Premises and Conclusion | 论证结构：前提与结论
Every philosophical argument can be broken down into premises and a conclusion. The formula: Premise 1, Premise 2, ..., therefore Conclusion. In standard form, list premises, draw an inference line, then state the conclusion. Identifying hidden premises (enthymemes) is a key skill.
每一个哲学论证都可以分解为前提和结论。公式：前提1，前提2，...，所以结论。在标准形式中，列出前提，画一条推论线，然后陈述结论。识别隐含前提（省略三段论）是一项关键技能。
Example: 'You shouldn't eat meat because it causes animal suffering.' Hidden premise: Causing unnecessary animal suffering is morally wrong. Once the missing premise is made explicit, the argument's strength can be evaluated. Always ask: What else must be true for the conclusion to follow?
示例：“你不该吃肉，因为它造成动物痛苦。”隐含前提：造成不必要的动物痛苦在道德上是错的。一旦补全了隐含的前提，就能评价论证的强度。始终要问：还需要什么为真，结论才能得出？
Premises can be normative or descriptive. Learning to reconstruct arguments fairly is essential in philosophy. Standard form helps avoid confusion and makes it easier to spot fallacies.
前提可以是规范性的，也可以是描述性的。学会公正地重构论证对哲学至关重要。标准形式有助于避免混淆，更容易发现谬误。
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