Year 8 CAIE Philosophy: Formula & Theorem Quick Reference Handbook | 八年级CAIE哲学：公式定理速查手册

📚 Year 8 CAIE Philosophy: Formula & Theorem Quick Reference Handbook | 八年级CAIE哲学：公式定理速查手册

Philosophy often feels like a world of abstract thoughts, but many of its most powerful ideas can be expressed as clear principles, rules and logical patterns — almost like ‘formulas’ or ‘theorems’. This quick-reference handbook collects the essential logical forms, ethical tests and argument structures that every Year 8 CAIE Philosophy student should have at their fingertips. Each entry is a tool for sharper thinking, clearer writing and stronger evaluation.

哲学常常让人觉得是一个抽象思维的世界，但许多最有力的思想都可以表达为清晰的原理、规则和逻辑模式——几乎就像“公式”或“定理”。这本速查手册汇集了每个八年级CAIE哲学学生都应熟记的基本逻辑形式、伦理检验和论证结构。每一个条目都是让思维更敏锐、表达更清晰、评价更有力的工具。

1. The Golden Rule of Ethics | 伦理学黄金法则

At the heart of many moral systems lies the principle of reciprocity: treat others as you yourself would wish to be treated. This ‘rule’ functions like a universal ethical formula. If you would not want someone to lie to you, do not lie to them. If you value kindness, extend kindness. The Golden Rule reminds us that ethical reasoning begins by putting ourselves in the place of others. It appears in the teachings of Confucius, Jesus and numerous philosophical traditions. The formula can be written as a simple conditional: if an action towards another would be unacceptable if done to you, then it is morally impermissible.

许多道德体系的核心都是互惠原则：你愿意别人怎样待你，你就应当怎样待人。这条“法则”就像一个普遍的伦理公式。如果你不希望别人对你说谎，你就不要说谎。如果你看重善良，你就应该施以善良。黄金法则提醒我们，伦理推理始于换位思考。它出现在孔子、耶稣以及许多哲学传统中。该公式可以写成一个简单的条件句：如果一个行为发生在你身上是你不能接受的，那么这个行为在道德上就是不允许的。

Golden Rule: Act towards others only as you consent to be treated in the same situation.

2. Categorical Imperative (Universalizability Test) | 绝对命令（可普遍化检验）

Immanuel Kant proposed a stricter rational test for morality: act only according to that maxim which you can at the same time will that it should become a universal law. To evaluate an action, ask, ‘What if everyone did this?’ If universalising the maxim leads to a contradiction or undermines the very practice the action relies on, then the action is morally wrong. For example, making a false promise to borrow money: if everyone made false promises, the institution of promising would collapse, and no one would believe promises anymore. The formula states that permissibility equals universalisability.

康德提出了一种更严格的理性道德检验：只依据你同时愿意它成为普遍法则的准则去行动。评判一个行为时，问自己：“如果人人都这样做，会怎样？”如果将该准则普遍化会导致矛盾，或者会破坏该行为本身所依赖的实践，那么这个行为在道德上就是错误的。例如，为了借钱而作出虚假承诺：如果人人都作虚假承诺，承诺制度就会崩溃，再也没人相信承诺了。这个公式表明，可允许性等于可普遍化性。

Categorical Imperative: Permissible(A) ↔ Maxim A can be willed as a universal law without contradiction.

3. The Hedonic Calculus (Utilitarianism) | 快乐计算（功利主义）

Jeremy Bentham’s utilitarian ethics offers a calculus for measuring pleasure and pain. He suggested seven factors: intensity, duration, certainty, propinquity (nearness), fecundity (chance of leading to more pleasure), purity (freedom from pain) and extent (number of people affected). The moral choice is the one that produces the greatest net pleasure summed over all sentient beings. This ‘formula’ turns ethics into a kind of cost–benefit analysis. When facing a dilemma like the famous trolley problem, a utilitarian would pull the lever to save five lives at the cost of one, because total wellbeing is maximised. The core insight: right action = maximise overall happiness.

边沁的功利主义伦理学提供了一套衡量快乐与痛苦的计算方法。他提出了七个因素：强度、持续时间、确定性、远近性（时间或空间上的接近程度）、生育力（带来更多快乐的可能性）、纯度（不夹杂痛苦）以及范围（受影响的人数）。道德上正确的选择，就是对所有有感知力的生命体产生最大净快乐总额的选择。这个“公式”把伦理学变成了一种成本收益分析。面对电车难题这样的困境时，功利主义者会拉动拉杆牺牲一人拯救五人，因为总体福祉得到了最大化。其核心洞见是：正确行为 = 最大化总体幸福。

Hedonic Calculus: Greatest good = maximize Σ (Pleasure – Pain) over all affected.

4. Modus Ponens: Affirming the Antecedent | 肯定前件式：肯定前件

Modus ponens is one of the most basic valid forms of deductive reasoning. If you have a conditional statement ‘If P then Q’ and you know that P is true, then you can logically conclude Q. For instance: If it is raining, the pavement is wet. It is raining. Therefore the pavement is wet. The reasoning is airtight; if the premises are true, the conclusion must be true. This formula appears everywhere from mathematics to everyday planning. Be careful not to confuse it with the fallacy of affirming the consequent (P → Q, Q, therefore P), which is invalid. Only modus ponens guarantees the conclusion.

肯定前件式是最基本的有效演绎推理形式之一。如果你有一个条件句“如果P则Q”，并且知道P为真，那么你在逻辑上就可以断定Q为真。例如：如果下雨，人行道就会湿。下雨了。所以人行道湿了。这种推理是严密的；如果前提为真，结论必然为真。这个公式从数学到日常计划中随处可见。注意不要把它与肯定后件的谬误混淆（P → Q，Q为真，所以P为真），后者是无效的。只有肯定前件式能保证结论成立。

Modus Ponens: P → Q, P ∴ Q

5. Modus Tollens: Denying the Consequent | 否定后件式：否定后件

Modus tollens is the mirror image of modus ponens and equally valid. Starting again from ‘If P then Q’, if Q is false, then P must be false. Example: If a creature is a bird, then it has feathers. This creature has no feathers. Therefore it is not a bird. This form of argument is powerful for eliminating hypotheses. In science and philosophy, modus tollens is used constantly: if a theory predicts something that does not happen, the theory is falsified. The formula reminds us that a single counterexample can overturn a general claim.

否定后件式是肯定前件式的镜像，同样有效。从“如果P则Q”出发，如果Q为假，则P必为假。例如：如果一个生物是鸟，那么它有羽毛。这个生物没有羽毛。所以它不是鸟。这种论证形式在排除假说时非常有力。在科学和哲学中，否定后件式被不断使用：如果一个理论预测的事情没有发生，该理论就被证伪了。这个公式提醒我们，一个反例就足以推翻一个普遍论断。

Modus Tollens: P → Q, ¬Q ∴ ¬P

6. Reductio ad Absurdum (Proof by Contradiction) | 归谬法（反证法）

Reductio ad absurdum is a strategy that starts by assuming the opposite of what you want to prove. You then reason correctly from that assumption until you reach a contradiction — a statement that cannot possibly be true, like ‘1 = 0’ or ‘the same object is both square and circular’. Since a valid argument cannot lead from true premises to a contradiction, the original assumption must be false, which means its negation is true. To prove there is no largest integer, assume there is a largest integer N; but N+1 is larger, contradicting the assumption. Therefore no largest integer exists. This technique is central to mathematical proof and philosophical argument.

归谬法是一种从假设你想要证明的命题的反面出发的策略。然后你从那个假设出发进行正确的推理，直到推出一个矛盾——一个不可能为真的陈述，比如“1=0”或者“同一个物体既是方的又是圆的”。因为一个有效的论证不可能从真前提推出矛盾，所以最初的假设必定为假，从而它的否定为真。要证明不存在最大整数，先假设存在最大整数N；但N+1更大，与假设矛盾。因此不存在最大整数。这一技巧是数学证明和哲学论证的核心。

Reductio: Assume ¬P, derive contradiction (Q ∧ ¬Q), conclude P.

7. Ockham’s Razor: The Principle of Parsimony | 奥卡姆剃刀：简约原则

Ockham’s razor is a heuristic for choosing between competing explanations. It states that entities should not be multiplied beyond necessity. In practice, if two theories explain the same data equally well, the simpler one — the one making fewer assumptions — is to be preferred. The razor does not prove the simpler theory is true, but it is a guide to avoid unnecessary complexity. If you hear

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