📚 Year 9 Edexcel Philosophy: Quick Reference Handbook of Formulas & Theorems | Year 9 Edexcel 哲学：公式定理速查手册
In philosophy, ‘formulas’ and ‘theorems’ are not the same as in mathematics, but they represent reliable patterns of reasoning, ethical decision-making, and argument structures. This quick reference handbook collects the most essential logical forms, moral principles, and critical challenges you will encounter in Year 9 Edexcel Philosophy.
在哲学中，“公式”和“定理”不同于数学中的公式与定理，但它们代表了可靠的推理模式、伦理决策框架和论证结构。这份速查手册汇集了 Year 9 Edexcel 哲学课程中最核心的逻辑形式、道德原则与批判性挑战。
1. Modus Ponens | 肯定前件式
Modus Ponens is a valid deductive argument form that lies at the heart of logical reasoning. It states that if a conditional statement is true, and its antecedent is affirmed, then the consequent necessarily follows.
肯定前件式是一种有效的演绎论证形式，是逻辑推理的核心。它指出：如果一个条件命题为真，并且它的前件被肯定，那么后件必然为真。
P → Q
P
∴ Q
Example: If a student studies effectively, they will understand the topic. The student studies effectively. Therefore, they will understand the topic.
例子：如果学生有效学习，他们就会理解该主题。学生有效学习。所以，他们会理解该主题。
In philosophical writing, Modus Ponens is used to draw straightforward conclusions from accepted premises. It is the simplest form of valid inference.
在哲学写作中，肯定前件式被用来从已被接受的前提中直接得出结论。这是最简单的一种有效推理形式。
2. Modus Tollens | 否定后件式
Modus Tollens works in the opposite direction: if the consequent of a conditional is false, then the antecedent must also be false. It allows us to refute a claim by disproving its predicted outcome.
否定后件式则反向运作：如果一个条件命题的后件为假，那么前件也必然为假。这种方法可以让我们通过否定预测的结果来反驳一个主张。
P → Q
¬Q
∴ ¬P
Example: If a watch has a designer, it shows intricate purpose. The object does not show intricate purpose. Therefore, it does not have a designer.
例子：如果一块表有设计者，它就会展示出精妙的目的性。该物体没有展示出精妙的目的性。所以，它没有设计者。
This form is crucial for testing hypotheses. Philosophers often use Modus Tollens to challenge theories by showing that the expected consequences do not occur.
这种形式对检验假说至关重要。哲学家常通过展示预期后果并未发生，从而运用否定后件式来挑战理论。
3. Hypothetical Syllogism | 假言三段论
A hypothetical syllogism chains together two conditional statements to form a new conditional. This rule justifies many long chains of reasoning in ethics and metaphysics.
假言三段论将两个条件命题串联起来，从而构成一个新的条件命题。这一规则在伦理学和形而上学中为许多长推理链条提供了依据。
P → Q
Q → R
∴ P → R
Example: If pleasure is the only good, then we should maximise pleasure. If we should maximise pleasure, then we must calculate consequences. Therefore, if pleasure is the only good, we must calculate consequences.
例子：如果快乐是唯一的善，那么我们应该最大化快乐。如果我们应该最大化快乐，那么我们必须计算后果。所以，如果快乐是唯一的善，我们必须计算后果。
In Year 9, this structure often appears when evaluating moral theories. It shows how one commitment logically leads to another.
在 Year 9 的课程中，这种结构常出现在评价道德理论时。它揭示了一个观点如何在逻辑上导致另一个观点。
4. Disjunctive Syllogism | 选言三段论
When we are presented with an ‘either-or’ choice, and we can rule out one option, a disjunctive syllogism forces us to accept the remaining alternative. This helps clarify decisions in moral dilemmas.
当我们面对一个“非此即彼”的选择时，如果我们能排除其中一个选项，选言三段论就会迫使我们接受剩下的那个选项。这有助于在道德困境中澄清决策。
P ∨ Q
¬P
∴ Q
Example: Either the universe is eternal or it had a first cause. It is not eternal. Therefore, it had a first cause.
例子：要么宇宙是永恒的，要么它有一个第一因。宇宙不是永恒的。所以，它有一个第一因。
Philosophers rely on disjunctive syllogisms when they argue that one side of a dilemma is impossible, leaving only one rational path forward.
当哲学家论证某个两难困境的一个分支不可能成立时，他们就会依赖选言三段论，从而只留下一条合理的路径。
5. The Design Argument Analogy | 设计论证的类比公式
The Design Argument, famously advanced by William Paley, uses an analogy as its ‘formula’: complex objects with purpose require an intelligent designer. The universe displays purpose; therefore, the universe has a designer.
由威廉·佩利提出的设计论证，使用类比作为其“公式”：具有目的性的复杂物体需要一个智慧的设计者。宇宙展示出目的性；因此，宇宙有一个设计者。
Complex, purpose‐driven object → Designer
Universe is complex and purpose‐driven
∴ Universe has a Designer
Paley’s watchmaker example: if you find a watch on a heath, you infer a watchmaker. Similarly, observing the natural world, with its order and complexity, leads us to infer a divine watchmaker.
佩利的钟表匠例子：如果你在荒野上发现一只手表，你会推断有一位钟表匠。同样，观察自然世界的秩序与复杂性，会引导我们推断有一位神圣的钟表匠。
Critics, such as David Hume, challenge the strength of this analogy. They question whether the universe genuinely resembles a watch, noting that we have no experience of universe-creation to compare it with.
休谟等批评者质疑这个类比的强度。他们质疑宇宙是否真的像一块手表，并指出我们没有任何关于宇宙创造的经验可以与之比较。
6. The First Cause Formula | 第一因公式（宇宙论论证）
The Cosmological Argument operates like a chain of causation. Its basic logical formula is: everything that exists has a cause; the universe exists; there cannot be an infinite regress of causes; therefore, there must be a first, uncaused cause.
宇宙论论证的运作方式就像一条因果链。其基本逻辑公式是：一切存在的事物都有原因；宇宙存在；不可能有无穷倒退的原因；因此，必定存在一个第一因，即无原因的原因。
Every effect has a cause
Universe is an effect
Infinite regress is impossible
∴ There is an uncaused First Cause
Thomas Aquinas presented versions of this argument. He believed that a sequence of movers must terminate in an unmoved mover, which people call God. For Year 9, the key is understanding the logic of dependency.
托马斯·阿奎那提出了这个论证的多个版本。他相信，运动者的序列最终必须终止于一个不动的推动者，人们称之为上帝。对 Year 9 而言，关键在于理解这种依赖性的逻辑。
Opponents ask whether the first cause must be God, and why the universe itself could not simply be eternal, breaking the chain.
反对者则问道：第一因是否一定是上帝？为什么宇宙本身不能就是永恒的，从而打破这个因果链条？
7. The Greatest Happiness Principle | 最大幸福原则（功利主义公式）
Utilitarianism, founded by Jeremy Bentham and developed by John Stuart Mill, can be summarised as a moral ‘formula’: an action is right if it produces the greatest happiness for the greatest number of people affected.
由边沁创立并由密尔发展的功利主义，可以概括为一条道德“公式”：如果一项行动能为最多受到影响的人产生最大幸福，那么它就是正确的。
Action A increases net pleasure over pain
Net pleasure of A > net pleasure of alternatives
∴ A is the right action
Bentham’s hedonic calculus attempted to measure pleasure using factors like intensity, duration, certainty, and extent. This turns ethics into a kind of arithmetic of happiness.
边沁的快乐计算法试图用强度、持续时间、确定性、范围等因素来衡量快乐。这就将伦理学变成了一种幸福算术。
However, critics argue that not all pleasures are equal, and that this formula might justify terrible acts if they somehow increased overall pleasure. This is a key discussion point in your Edexcel course.
然而，批评者认为，并非所有快乐都是等同的，并且这个公式可能为可怕的行为辩护，只要它们在某种程度上增加了总体快乐。这是你 Edexcel 课程中的一个关键讨论点。
8. The Categorical Imperative Formula | 定言命令公式（康德伦理学）
Immanuel Kant’s ethics is built around the Categorical Imperative, a rule that works like a logical test. The first formulation: ‘Act only according to that maxim whereby you can at the same time will that it should become a universal law.’
康德的伦理学围绕定言命令展开，这条规则如同一个逻辑检验。第一条表述：“只按照你同时能够愿意其成为一条普遍法则的准则去行动。”
Can my maxim be universalised without contradiction?
If yes → action is permissible
If no → action is forbidden
Another formula is the ‘humanity formula’: treat humanity, whether in your own person or in another, always as an end and never merely as a means. This forbids using people simply as tools.
另一个公式是“人性公式”：永远将人性——无论是对自己还是对他人——作为目的，而绝不仅仅是手段。这禁止将人单纯当作工具使用。
For a Year 9 student, this means asking: would it make sense for everyone to do what I am about to do? If the answer is no, the action fails the moral test, even if it brings good outcomes.
对于 Year 9 学生而言，这意味着问自己：如果每个人都做我即将要做的事，这在逻辑上说得通吗？如果答案是否定的，那么即便这个行为能带来好的结果，它也没有通过道德检验。
9. The Logical Problem of Evil | 恶的逻辑问题
The problem of evil is often presented as a powerful deductive challenge to the existence of an all-powerful, all-loving God. Its logical structure can be captured as a formula that leads to a paradox.
恶的问题通常被表述为对一个全能、全爱的上帝之存在的有力演绎挑战。它的逻辑结构可以用一个导致悖论的公式来概括。
God is omnipotent and omnibenevolent
Such a God would eliminate all evil
Evil exists
∴ An omnipotent, omnibenevolent God does not exist
If God were able to stop suffering and wanted to do so, we should not observe evil. Since we do observe evil, one of the premises must be false. This is the core of the logical problem of evil.
如果上帝能够阻止苦难并且想要这样做，我们就不应该观察到恶。既然我们确实观察到了恶，那么前提中至少有一个是假的。这就是恶的逻辑问题之核心。
Believers often respond with the free will defence: God gave humans free will, which is a great good, but it logically requires the possibility of evil. This turns the argument into a debate about greater goods.
信者常常以自由意志辩护来回应：上帝赐予人类自由意志，这是一项极大的善，但它在逻辑上需要恶的可能性。这就把论证转变成了一场关于更大之善的辩论。
10. The Golden Mean | 中庸之道（美德伦理学公式）
Aristotle’s virtue ethics does not provide a single decision procedure like the previous ‘formulas’. Instead, it offers a pattern: every moral virtue is a mean between two vices – one of excess and one of deficiency.
亚里士多德的美德伦理学并不像前述的“公式”那样提供单一的决策程序，而是提供了一种模式：每一种道德德性都是两种恶之间的中庸——一种是过度，一种是不及。
Virtue = Mean between Excess and Deficiency
Example: Courage is the mean between Recklessness (excess) and Cowardice (deficiency)
To be brave, you must find the right balance, which depends on the situation and the individual. This requires practical wisdom (phronesis) rather than blind application of a rule.
要变得勇敢，你必须找到正确的平衡，这取决于情境与个人。这需要实践智慧（phronesis），而不是盲目地套用规则。
When answering Edexcel questions on Aristotle, you can present the Golden Mean as a flexible ‘theorem’ for character development. It explains why simply following a list of rules is not enough to become a good person.
在回答 Edexcel 中关于亚里士多德的问题时，你可以将中庸之道表述为一种灵活的品格发展“定理”。它解释了为什么仅仅是遵循一份规则清单并不足以成为一个好人。
11. Common Logical Fallacies | 常见逻辑谬误
Fallacies are errors in reasoning that can make an argument invalid or unsound. Recognising their formulaic structure helps you criticise weak philosophical claims. Below is a table of four fallacies frequently discussed in Year 9.
谬误是推理中的错误，这些错误可能使论证无效或不可靠。识别它们的公式化结构有助于你批判那些站不住脚的哲学主张。下表列举了 Year 9 中常讨论的四种谬误。
| Fallacy | Formula / Structure | Example |
|---|---|---|
| Ad Hominem 人身攻击 | Person X has a flaw, therefore X's argument is false. | 'You are too young to understand political philosophy, so your argument about justice is wrong.' |
| False Dilemma 错误二分 | Either P or Q; only two options, when others exist. | 'Either you agree with Kant entirely, or you have no morals.' |
| Begging the Question 循环论证 | The conclusion is assumed in the premises. | 'God exists because the Bible says so, and the Bible is the word of God.' |
| Slippery Slope 滑坡谬误 | One small step inevitably leads to extreme consequences without justification. | 'If we allow voluntary euthanasia, soon no one will want to treat the elderly.' |
Each fallacy has a recognisable pattern. Being able to spot these patterns will improve your critical essay writing and help you assess the strength of arguments in the Edexcel exam.
每种谬误都有一种可识别的模式。能够发现这些模式将提升你的批判性写作能力，并帮助你在 Edexcel 考试中评估论证的力度。
12. Ockham’s Razor | 奥卡姆剃刀
Ockham’s Razor is a principle of parsimony commonly used in philosophy: ‘Entities should not be multiplied beyond necessity.’ It functions like a methodological theorem, advising us to choose the simplest adequate explanation.
奥卡姆剃刀是一条常用于哲学的简约性原则：“如无必要，勿增实体。”它就像一条方法论定理，建议我们选择最简单且适当的解释。
If two theories explain the data equally well, prefer the one with fewer assumptions.
This does not guarantee truth, but it helps eliminate unnecessary flourishes. In the philosophy of religion, Ockham’s Razor is sometimes used to challenge theistic arguments by suggesting we should not posit God if natural causes suffice.
这并不保证真理，但有助于去除不必要的花哨假设。在宗教哲学中，奥卡姆剃刀有时被用来挑战有神论论证，因为它建议，如果自然原因已经足够，我们就不应再假设一个上帝。
Year 9 students can use this principle when constructing arguments: always ask yourself whether you are adding extra, unneeded ideas to explain something that could be understood in a simpler way.
Year 9 学生在构建论证时可以使用这一原则：时刻自问，你是否添加了额外的、不必要的观念来解释那些本可以更简单地被理解的事物。
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