📚 Year 10 CIE Philosophy: Formula & Theorem Quick-Reference Handbook | 十年级CIE哲学：公式定理速查手册
This handbook collects the most important logical argument forms, ethical principles, and critical thinking tools from the CIE Year 10 Philosophy syllabus. Just as mathematicians rely on formulas and theorems, philosophers use these structured patterns to build sound arguments and evaluate claims. Keep this guide handy for quick revision.
本手册汇集了 CIE 十年级哲学大纲中最重要的逻辑论证形式、伦理原则与批判性思维工具。就像数学依赖公式和定理一样，哲学家运用这些结构化的模式来构建合理的论证并评估观点。请随时查阅本指南，以便快速复习。
1. Modus Ponens (Affirming the Antecedent) | 肯定前件式
The most straightforward valid argument form states: if a conditional statement is true and its antecedent holds, the consequent must follow. Modus ponens is the engine of deductive reasoning and appears in countless philosophical proofs.
最直接的有效的论证形式是：如果条件语句为真，且其前件成立，则后件必然成立。肯定前件式是演绎推理的引擎，出现在无数哲学证明之中。
If P then Q, P, ∴ Q
Example: If it is raining, the ground is wet. It is raining. Therefore, the ground is wet. Whenever you recognise this structure, you know the conclusion is inescapable provided the premises are true.
示例：如果下雨，地面就是湿的。正在下雨。因此，地面是湿的。每当你识别出这一结构，只要前提为真，结论就无可避免。
2. Modus Tollens (Denying the Consequent) | 否定后件式
Modus tollens is the companion to modus ponens. It tells us that if a conditional is true and the consequent is false, the antecedent must also be false. This form is essential for testing hypotheses and eliminating false explanations.
否定后件式是肯定前件式的伴随形式。它告诉我们，如果条件语句为真而后件为假，则前件也必为假。这一形式对于检验假设和排除错误解释至关重要。
If P then Q, Not-Q, ∴ Not-P
Example: If the soul is material, it can decay. The soul does not decay (for Plato). Therefore, the soul is not material. This pattern underpins arguments from impossibility and the rejection of theories through counter‑evidence.
示例：如果灵魂是物质的，它就会腐朽。灵魂不会腐朽（柏拉图的观点）。因此，灵魂不是物质的。这一模式支撑着从不可能性出发的论证，以及通过反证来拒斥理论。
3. Hypothetical Syllogism | 假言三段论
This form chains together two conditional statements to produce a new conditional. If one thing leads to another, and that second thing leads to a third, then the first leads directly to the third. It is a tool for linking abstract principles to concrete consequences.
这一形式将两个条件语句串联起来，产生一个新的条件语句。如果一件事导致第二件，而这第二件又导致第三件，那么第一件事就直接导致第三件。它是将抽象原则与具体后果联系起来的工具。
If P then Q, If Q then R, ∴ If P then R
Example: If we have free will, we are morally responsible. If we are morally responsible, punishment can be justified. Therefore, if we have free will, punishment can be justified. Philosophers use this to explore the logical connections between seemingly remote ideas.
示例：如果我们有自由意志，我们就负有道德责任。如果我们负有道德责任，惩罚就可以是正当的。因此，如果我们有自由意志，惩罚就可以是正当的。哲学家运用它来探索看似遥远的观念之间的逻辑联系。
4. Disjunctive Syllogism | 选言三段论
When you face a choice between two options, and you rule one out, the other must be the case – provided the disjunction is exhaustive. This simple argument form is central to decision‑making and to many proofs by elimination in philosophy.
当你面临二选一的情形，并且排除其中一个选项时，另一个就必定成立——前提是这个选言是穷尽的。这一简单的论证形式对决策以及许多哲学中的排除法证明至关重要。
P or Q, Not-P, ∴ Q
Example: Either knowledge comes only from reason, or it comes from experience. Rationalists deny that all knowledge comes from experience. Therefore, knowledge comes only from reason. Notice that the strength of this argument depends on the ‘or’ genuinely covering all possibilities.
示例：要么知识仅来自理性，要么知识来自经验。理性主义者否认所有知识来自经验。因此，知识仅来自理性。注意，该论证的力度取决于这个“要么”是否真正涵盖了所有可能性。
5. Reductio ad Absurdum | 归谬法
Reductio ad absurdum assumes the opposite of what you want to prove and shows that this assumption leads to a contradiction or absurdity. Because contradictions cannot be true, the original assumption must be false, and the desired conclusion true. It is one of the oldest and most powerful logical techniques.
归谬法先假定你要证明的论题的反面，然后表明这一假定会导致矛盾或荒谬。由于矛盾不可能为真，原假定必定为假，所欲证的结论即为真。这是最古老、最有力的逻辑技术之一。
Assume Not-P, derive contradiction, ∴ P
Example: To prove that ‘the world is not purely illusory’, assume it is purely illusory. If it were illusory, we could not consistently speak of truth. Yet we do speak of truth and falsehood meaningfully. Contradiction. Therefore, the world is not purely illusory.
示例：要证明“世界并非纯粹虚幻”，先假定世界是纯粹虚幻的。如果它是虚幻的，我们就无法一致地谈论真理。然而我们确实有意义地谈论真与假。矛盾。因此，世界并非纯粹虚幻。
6. Principle of Utility (The Greatest Happiness Formula) | 功利原则（最大幸福公式）
Bentham’s utilitarian ethics can be viewed as a decision‑making formula: an act is right if it tends to increase the total happiness and wrong if it tends to decrease it. Happiness is measured by the quantity of pleasure minus pain produced for all affected individuals.
边沁的功利主义伦理学可以被视为一个决策公式：一个行为如果倾向于增加总体幸福，就是对的；如果倾向于减少总体幸福，就是错的。幸福通过所有受影响者所产生的快乐减去痛苦的数量来衡量。
Right Action = Maximise (Sum of [Pleasure – Pain] for all affected)
To apply the formula: list everyone affected by an action, estimate the pleasure and pain each will experience, calculate the net balance, and compare it with alternatives. The act with the highest net positive balance is the morally correct choice.
要应用这个公式：列出所有受行为影响的人，估计每个人将体验的快乐与痛苦，计算净余额，并与替代方案相比较。净正值最大的行为就是道德上正确的选择。
7. Categorical Imperative (Universal Law Formula) | 绝对命令（普遍法则公式）
Kant’s ethics provides a test for the moral worth of an action: act only according to that maxim whereby you can at the same time will that it should become a universal law. This functions like a logical consistency check for moral rules.
康德的伦理学提供了一个检验行为道德价值的方法：只依据那个你同时能够愿意它成为一条普遍法则的准则去行动。这就像是对道德规则进行逻辑一致性的检验。
Permissible Action ⇔ Maxim can be willed as universal law without contradiction
Example: The maxim ‘I will make a false promise to get money’ cannot be universalised, because if everyone made false promises, the institution of promising would collapse, making the promise itself impossible. Thus the act is morally impermissible.
示例：“我将做虚假承诺以获取金钱”这一准则不能被普遍化，因为如果人人都做虚假承诺，承诺制度就会崩溃，导致承诺本身不可能存在。因此该行为在道德上是不可允许的。
8. Golden Mean (Virtue Theorem) | 中庸之道（德性定理）
Aristotle’s virtue ethics offers a practical theorem: every virtue lies at a mean between two vices, one of excess and one of deficiency. The right action is the one that expresses the relevant virtue at the right time, in the right way, and to the right degree.
亚里士多德的德性伦理学提供了一个实践性定理：每种德性都处于两个恶之间的中道，一个是过度，一个是不及。正确的行为是那个在恰当时机、以恰当方式、达到恰当程度表现出相关德性的行为。
Virtue = Intermediate between Excess and Deficiency (relative to the situation)
Example: Courage is the mean between recklessness (excess) and cowardice (deficiency). The courageous person faces danger neither rashly nor fearfully, but in accordance with reason. The mean is not a mathematical average but a flexible balance determined by practical wisdom.
示例：勇敢是鲁莽（过度）与怯懦（不及）之间的中道。勇敢的人面对危险既不莽撞也不恐惧，而是合乎理性。这个中道不是数学上的平均值，而是由实践智慧决定的灵活平衡。
9. Slippery Slope Argument Form | 滑坡论证形式
The slippery slope is a prominent argument pattern claiming that a particular first step will set off a chain of events leading to an undesirable outcome. To be logically sound, the causal links must be clearly justified rather than merely asserted.
滑坡论证是一种突出的论证模式，它声称某一特定的第一步将引发一连串事件，最终导致一个不可取的后果。要使它在逻辑上可靠，每一步因果关系都必须得到清晰论证，而不能仅仅被断言。
If A, then likely B; if B, then likely C; C is unacceptable; ∴ we should not do A
Example: If we allow voluntary euthanasia, it will lead to pressure on the terminally ill to end their lives, which will eventually erode respect for all human life. This form requires rigorous empirical and logical support for each step to avoid becoming a fallacy.
示例：如果我们允许自愿安乐死，就会导致对绝症患者施压结束生命，这最终会侵蚀对所有人类生命的尊重。这一形式要求每一步都有严格的经验和逻辑支持，以免沦为谬误。
10. Ockham’s Razor (Principle of Parsimony) | 奥卡姆剃刀（精简原则）
Ockham’s Razor advises that among competing explanations of the same set of facts, the one with the fewest assumptions is to be preferred. It is not a proof but a heuristic that guides philosophical theory‑building and scientific reasoning.
奥卡姆剃刀建议，在对同一组事实的相互竞争的解释中，应当优先选择假定最少的那个。它并不是一项证明，而是一条指导哲学理论构建和科学推理的启发式原则。
Entities should not be multiplied beyond necessity
Example: To explain the regularity of the physical world, we could posit an all‑powerful being constantly intervening, or we could propose simple natural laws. Unless there is evidence requiring the supernatural, Ockham’s Razor favours the natural‑law explanation. Philosophers use it to shave off unnecessary metaphysical commitments.
示例：要解释物理世界的规律性，我们可以假定一个全能的存在不断干预，也可以提出简单的自然法则。除非有证据需要超自然因素，否则奥卡姆剃刀偏爱自然法则的解释。哲学家用它来剃除不必要的形而上学承诺。
11. Argument from Analogy Structure | 类比论证结构
An argument from analogy infers that because two things are similar in certain respects, they are likely similar in another respect. This pattern is common in the design argument for God’s existence and in ethical thought experiments.
类比论证通过指出两个事物在某些方面相似，从而推断它们在另一个方面也相似。这一模式在关于上帝存在的设计论论证以及伦理思想实验中非常常见。
A has properties x, y, z and property w. B has properties x, y, z. ∴ B probably has property w.
Example: A watch has complexity, purpose, and a maker. The universe has complexity and apparent purpose. Therefore, the universe probably has a maker. The strength of such analogies depends on the relevance and number of shared properties, and the absence of significant disanalogies.
示例：手表具有复杂性、目的性和制造者。宇宙具有复杂性和表面上的目的性。因此，宇宙很可能有一个制造者。此类类比的强度取决于共有属性的相关性与数量，以及重大差异的不存在。
12. Burden of Proof Principle | 举证责任原则
In philosophical debate, the burden of proof falls on the person who makes a positive claim, especially one that goes beyond accepted common ground. This principle prevents endless shifting of the requirement to provide evidence and maintains dialectical fairness.
在哲学辩论中，举证责任落在提出肯定性主张的一方，尤其是那些超出公认共识的主张。这一原则防止了不断转移举证要求的局面，并维护了辩证公平。
The party asserting a claim must support it; the sceptic need only point to lack of evidence
Example: If someone claims that there is an undetectable dragon in the garage, the burden is on them to produce evidence. It is not the sceptic’s job to disprove the dragon. This rule is foundational in debates about existence claims, miracles, and extraordinary phenomena.
示例：如果有人声称车库里有条不可探测的龙，那么提供证据的负担就在他们身上。怀疑者无需去证明龙的不存在。这一规则是关于存在性主张、奇迹与超常现象等辩论的基础。
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