📚 Year 11 CAIE Philosophy: Formula & Theorem Quick Reference Handbook | Year 11 CAIE 哲学：公式定理速查手册
Philosophy is not all essays and open debate; it also contains rigid logical forms, ethical calculi, and argument templates that function much like mathematical formulas. This revision booklet collects the essential patterns, rules, and schematic summaries that every Year 11 CAIE learner should have at their fingertips for quick recall before an exam.
哲学不只是论文和开放式辩论，它还包含严格的逻辑形式、伦理学计算式和论证模板，这些内容在很大程度上就像数学公式一样运作。这本复习手册汇集了每位 Year 11 CAIE 学生都应熟练掌握的核心模式、规则和图示总结，方便考前快速回忆。
1. Deductive Argument Forms | 演绎论证形式
Modus Ponens (Affirming the Antecedent): If P then Q; P is true, therefore Q is true. This is always valid.
肯定前件式：如果 P 则 Q；P 为真，因此 Q 为真。该形式始终有效。
Modus Tollens (Denying the Consequent): If P then Q; Q is false, therefore P is false. Equally valid and powerful.
否定后件式：如果 P 则 Q；Q 为假，因此 P 为假。同样有效且有力。
Hypothetical Syllogism: If P then Q; If Q then R; therefore If P then R. Chains conditionals.
假言三段论：如果 P 则 Q；如果 Q 则 R；因此如果 P 则 R。将条件语句串联起来。
Disjunctive Syllogism: P or Q; Not P; therefore Q. Eliminates one option to secure the other.
选言三段论：P 或 Q；非 P；因此 Q。排除一个选项以确定另一个。
2. Inductive Reasoning Strength | 归纳推理强度
Inductive strength is not validity but degree of support. A strong induction: If the sample is large, random, and representative, the conclusion is likely to be true. Weak induction: small, biased, or unrepresentative sample.
归纳的强度不在于有效性而在于支持程度。强归纳：如果样本足够大、随机且具有代表性，那么结论很可能为真。弱归纳：样本小、有偏差或不具代表性。
Formula for evaluating an inductive generalisation: (Representative Sample + Sufficient Size + Unbiased Selection) → Probable Conclusion. Lacking any factor reduces strength to mere possibility.
评估归纳概括的公式：（代表性样本 + 足够规模 + 无偏选择）→ 可能的结论。缺少任一因素都会使强度降至仅是一种可能性。
3. Ethical Theory Formulas | 伦理学理论公式
Utilitarian Calculus (Bentham): An action is right if it maximises net pleasure over pain for all affected. Net utility = (Intensity × Duration × Certainty × Propinquity × Fecundity × Purity × Extent) – Pain equivalents.
功利主义计算（边沁）：一个行为如果能为所有受影响者最大化净快乐优于痛苦，就是正确的。净效用 =（强度 × 持续时间 × 确定性 × 邻近性 × 衍生性 × 纯度 × 范围）– 对应的痛苦量。
Kant’s Categorical Imperative (First Formulation): Act only according to that maxim whereby you can at the same time will that it should become a universal law. Self-contradiction in conception or in willing renders the maxim impermissible.
康德的绝对命令（第一公式）：只按照你同时能够意愿它成为一条普遍法则的准则去行动。概念上的自相矛盾或意愿上的自相矛盾都会使准则不被允许。
Natural Law (Primary Precepts): Preserve life, reproduce, educate children, live in society, worship God. An action is right if it fulfils these purposes and conforms to reason.
自然法（基本戒律）：保全生命、繁衍、教育子女、在社会中生活、敬拜神。一个行为如果实现这些目的并符合理性，就是正确的。
4. The Problem of Evil Logical Form | 恶的问题逻辑形式
The inconsistent triad: 1. God is omnipotent; 2. God is wholly good; 3. Evil exists. If any two are true, the third appears false. A deductive challenge to theism.
不一致的三元组：1. 神是全能的；2. 神是全善的；3. 恶存在。如果任意两者为真，则第三个似乎为假。这是对有神论的演绎性挑战。
Evidential version of the problem: Given the amount and distribution of pointless suffering, it is highly improbable that an omnipotent, perfectly good God exists. This is an inductive, probabilistic form.
恶的存疑版本：考虑到无意义苦难的数量和分布，一位全能且全善的神存在的概率极低。这是一个归纳的、概率性的形式。
5. Ontological Argument Structure | 本体论论证结构
Anselm’s formula: God is ‘that than which nothing greater can be conceived’. If God exists only in the mind, a greater being could be conceived (one existing in reality as well). But that is a contradiction, so God must exist in reality. Often presented as a reductio ad absurdum.
安瑟伦的公式：神是“无法设想有比之更伟大者的存在”。如果神只存在于心灵中，那么可以设想一个更伟大的存在（既存在于心灵也存在于现实中）。但这就产生了矛盾，因此神必然存在于现实中。通常以归谬法的形式呈现。
Descartes’ version: Existence is a perfection; God, as the supremely perfect being, must possess all perfections, therefore existence belongs to God’s essence, just as three angles belong to the essence of a triangle.
笛卡尔的版本：存在是一种完满性；神作为至上完满的存在，必定拥有一切完满性，因此存在属于神的本质，正如三角形内角和属于三角形的本质一样。
6. Cosmological Argument Causal Principle | 宇宙论论证的因果原理
Aquinas’ Second Way (First Cause): Every event has a cause; an infinite regress of causes is impossible; therefore there must be a first uncaused cause, which everyone calls God.
阿奎那的第二路（第一因）：每一事件都有原因；原因的无限回溯是不可能的；因此必定存在一个最初的无因之因，人人都称之为神。
Kalam version: Whatever begins to exist has a cause; the universe began to exist; therefore the universe has a cause. This cause must be uncaused, immaterial, and powerful.
卡拉姆版本：凡是开始存在的事物都有其原因；宇宙开始存在；因此宇宙有一个原因。这个原因必定是无因的、非物质的和强大的。
Leibniz’s Principle of Sufficient Reason: For every fact or truth, there must be a sufficient reason why it is so and not otherwise. The ultimate sufficient reason for the whole series of contingent facts must lie outside the series, in a necessary being.
莱布尼茨的充足理由律：对于每一个事实或真理，必定有一个充足理由说明它为何如此而非另样。整个偶然事实系列的最终充足理由必然处于该系列之外，在一个必然存在者之中。
7. Teleological Argument Probability | 设计论证的概率类比
Paley’s watchmaker analogy: A watch implies a watchmaker because of its complexity and purposefulness. The universe exhibits even greater complexity and adaptation of means to ends; therefore, by analogy, the universe implies a divine designer.
佩利的钟表类比：钟表因其复杂性和目的性而意味着钟表匠的存在。宇宙展现出更大的复杂性和手段对目的的适配；因此，通过类比，宇宙意味着一位神圣的设计者。
Probability form (Swinburne): The existence of orderly, complex, life-permitting laws is highly improbable under the hypothesis of brute chance but much more probable under the hypothesis of a designer. Using Bayes’-style reasoning, the evidence raises the probability of theism.
概率形式（斯温伯恩）：在纯粹偶然的假设下，有序、复杂、允许生命的法则存在的概率极低；而在设计者的假设下，其概率则高得多。利用贝叶斯式推理，这些证据提高了有神论的概率。
8. Logical Fallacies Quick Guide | 逻辑谬误速查
Ad hominem: Attacking the person instead of the argument. ‘You’re wrong because you’re selfish.’
人身攻击：攻击人而非论点。“你错了因为你自私。”
Straw man: Misrepresenting an opponent’s view to make it easier to attack. ‘You say we should reduce spending slightly, so you want to abolish all public services.’
稻草人：歪曲对方的观点以便更容易攻击。“你说应该稍微减少开支，所以你想废除所有公共服务。”
False dilemma: Presenting only two options when more exist. ‘Either you support every war or you hate your country.’
假两难：只给出两种选择，而实际上存在更多选项。“你要么支持每一场战争，要么恨你的国家。”
Begging the question: Assuming the truth of what one is trying to prove. ‘The Bible is the word of God because the Bible says so.’
乞题：将正在试图证明的结论当作前提来使用。“《圣经》是神的话语，因为《圣经》这么说。”
Slippery slope: Claiming a small step will inevitably lead to an extreme outcome without justification. ‘If we allow same-sex marriage, soon people will marry animals.’
滑坡谬误：在没有论证的情况下声称一小步将不可避免地导致极端的后果。“如果我们允许同性婚姻，很快人们就会和动物结婚。”
9. Truth Tables for Connectives | 联结词真值表
Negation (¬): True becomes false, false becomes true. The simplest logical operator.
否定（¬）：真变为假，假变为真。最简单的逻辑算子。
Conjunction (∧): P ∧ Q is true only when both P and Q are true; otherwise false.
合取（∧）：只有当 P 和 Q 都为真时，P ∧ Q 才为真；否则为假。
Disjunction (∨): P ∨ Q is true if at least one of P, Q is true; false only if both are false.
析取（∨）：如果 P 或 Q 中至少一个为真，则 P ∨ Q 为真；只有当两者都为假时才为假。
Conditional (→): P → Q is false only when P is true and Q is false; in all other cases it is true. This often puzzles beginners.
条件（→）：只有当 P 为真而 Q 为假时，P → Q 才为假；在所有其他情况下为真。这常令初学者困惑。
Biconditional (↔): P ↔ Q is true when P and Q have the same truth value; false when they differ.
双条件（↔）：当 P 和 Q 真值相同时，P ↔ Q 为真；不同时为假。
10. Categorical Syllogisms | 范畴三段论
A standard form syllogism contains two premises and a conclusion, each in the form: All/No/Some A are/are not B. Example: All humans are mortal; Socrates is human; therefore Socrates is mortal.
标准形式的三段论包含两个前提和一个结论，每个命题的形式为：所有/没有/有些 A 是/不是 B。示例：所有人都会死；苏格拉底是人；因此苏格拉底会死。
Rules for validity: The middle term must be distributed at least once; no term can be distributed in the conclusion unless it is distributed in a premise; a valid syllogism cannot have two negative premises; if one premise is negative, the conclusion must be negative.
有效性规则：中项必须至少周延一次；任何在结论中周延的项必须在前提中周延；有效的三段论不能有两个否定前提；如果一个前提是否定的，结论也必须是否定的。
11. Key Definitions & Distinctions | 关键定义与区分
Analytic vs Synthetic: An analytic statement is true by virtue of its meaning alone (e.g., ‘All bachelors are unmarried’). A synthetic statement’s truth depends on how the world is (e.g., ‘The cat is on the mat’).
分析 vs 综合：分析陈述仅凭其意义即为真（如“所有单身汉都是未婚的”）。综合陈述的真假取决于世界的实际情况（如“猫在垫子上”）。
A priori vs A posteriori: A priori knowledge is independent of experience (e.g., 2 + 2 = 4). A posteriori knowledge is dependent on experience or empirical evidence (e.g., ‘Water boils at 100 °C at sea level’).
先验 vs 后验：先验知识独立于经验（如 2+2=4）。后验知识依赖于经验或经验性证据（如“水在海平面处 100 °C 沸腾”）。
Necessary vs Contingent: A necessary truth is one that could not have been false (e.g., a triangle has three sides). A contingent truth could have been otherwise (e.g., I am wearing a blue shirt).
必然 vs 偶然：必然真理是不可能为假的（如三角形有三条边）。偶然真理可能不是现在这个样子（如我现在穿着蓝色衬衫）。
12. Exam Command Words & Structure | 考试指令词与结构
‘Explain’ questions: Unpack a concept, theory, or argument in clear steps. No evaluation required yet. Give examples to illustrate. Aim for precise definitions and logical order.
“解释”题：以清晰的步骤阐释一个概念、理论或论证。此时尚不需要评估。举例说明。追求准确的定义和逻辑顺序。
‘Evaluate’ / ‘Discuss’ questions: Weigh strengths and weaknesses, consider counter-arguments, and come to a justified conclusion. Use a balanced structure: exposition of view, objections, replies, and a final judgement with reasons.
“评估”/“讨论”题：权衡优点和缺点，考虑反论证，并得出有理由支撑的结论。使用平衡的结构：阐述观点、提出反对意见、回应以及最终的判断与理由。
Standard essay skeleton: Introduction (define terms, state thesis), main body (series of paragraphs each with Point, Evidence, Explanation, Link back to question), conclusion (summarise and give final verdict).
标准论文骨架：引言（定义术语，陈述论点），主体（一系列段落，每段包含要点、论据、解释和回扣问题），结论（总结并给出最终判断）。
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