📚 Year 10 WJEC Philosophy: Formulae & Theorems Quick Reference Handbook | WJEC 10年级哲学：公式定理速查手册
Welcome to your quick reference companion for Year 10 WJEC Philosophy. Just as mathematics relies on formulae and theorems, philosophical reasoning depends on structured argument patterns, logical rules, and ethical principles that can be applied again and again. This handbook brings together the core 'formulae' of critical thinking and ethical reasoning you need for your course, presented in clear, memorable forms.
欢迎使用这本为 WJEC 10 年级哲学设计的速查手册。正如数学依赖于公式和定理，哲学推理同样依赖于可反复应用的结构化论证模式、逻辑规则和伦理原则。本手册汇集了你课程所需的核心批判性思维和伦理推理“公式”，并以清晰易记的形式呈现。
1. Understanding Arguments | 理解论证
atements where some (premises) are intended to support another (the conclusion). The basic 'formula' + ... → Conclusion. A sound argument must have true premises and valid logical structure.
哲学中的论证是由若干陈述组成的集合，其中一些陈述（前提）旨在支持另一个陈述（结论）。基本“公式”是：前提 1 + 前提 2 + … → 结论。一个可靠的论证必须拥有真实的前提和有效的逻辑结构。
Key symbols worth remembering: ‘∴’ means ‘therefore’ and signals the conclusion. ‘P’, ‘Q’, ‘R’ are often used to represent propositions.
值得记住的关键符号：‘∴’ 表示“所以”，用于引出结论；‘P’、‘Q’、‘R’ 常用来代表命题。
2. Deductive Reasoning: Modus Ponens | 演绎推理：肯定前件
Deductive reasoning aims to provide conclusive support for its conclusion. Modus Ponens (affirming the antecedent) has the form: If P then Q. P is true. Therefore, Q. This pattern is logically flawless when the conditional premise is true.
演绎推理旨在为其结论提供决定性的支持。肯定前件（Modus Ponens）的形式为：如果 P 则 Q。P 为真。所以，Q。当条件前提为真时，这个模式在逻辑上是无懈可击的。
P → Q, P ∴ Q
如果 P 则 Q；P；所以 Q。
Example: If it rains, the ground is wet. It rains. Therefore, the ground is wet. This is a valid deductive formula.
示例：如果下雨，地面会湿。下雨了。所以，地面湿了。这是一个有效的演绎公式。
3. Deductive Reasoning: Modus Tollens | 演绎推理：否定后件
Modus Tollens (denying the consequent) uses the negation of the consequent to conclude the negation of the antecedent. The structure: If P then Q. Not Q. Therefore, not P.
否定后件（Modus Tollens）利用对后件的否定来推出对前件的否定。其结构为：如果 P 则 Q；非 Q；所以，非 P。
P → Q, ¬Q ∴ ¬P
P → Q；¬Q；所以 ¬P。
Example: If the battery is charged, the phone turns on. The phone does not turn on. Therefore, the battery is not charged. This is a vital tool for testing hypotheses.
示例：如果电池有电，手机就会开机。手机没开机。所以，电池没有电。这是检验假说的重要工具。
4. The Syllogism | 三段论
A categorical syllogism connects two statements through a shared term to reach a conclusion. Classic formula: All A are B. All B are C. Therefore, all A are C.
直言三段论通过一个共同项将两个陈述联系起来以推出结论。经典公式：所有 A 是 B；所有 B 是 C；所以，所有 A 是 C。
All A are B. All B are C. ∴ All A are C.
所有 A 是 B；所有 B 是 C；∴ 所有 A 是 C。
Other valid forms include: No A are B. All C are A. Therefore, no C are B. Practise identifying the middle term (B) that disappears in the conclusion.
其他有效形式包括：没有 A 是 B；所有 C 是 A；所以，没有 C 是 B。练习识别在结论中消失的中项（B）。
5. Inductive Reasoning | 归纳推理
Inductive reasoning does not guarantee its conclusion but makes it probable. The core formula: A large sample of X has been observed to have property Y. Therefore, all X probably have Y.
归纳推理不能保证结论必然为真，只能使其具有可能性。核心公式：大量观察到的 X 具有属性 Y；因此，所有 X 很可能都具有 Y。
Observed X₁, X₂ … Xₙ all have Y. ∴ All X likely have Y.
已观察到的 X₁, X₂ … Xₙ 都具有 Y；∴ 所有 X 很可能具有 Y。
The strength of the induction depends on sample size, representativeness, and absence of counter-evidence. This is central to scientific reasoning and design arguments.
归纳的强度取决于样本量、代表性以及没有反例。这是科学推理和设计论证的核心。
6. Analogical Arguments | 类比论证
An analogy compares two things to infer a further similarity. The formula: A and B share properties p, q, r. A has property s. Therefore, B probably also has s.
类比通过比较两个事物来推断进一步的相似性。公式为：A 和 B 共有的属性 p, q, r；A 具有属性 s；因此，B 很可能也具有 s。
A ∽ B (share p,q,r). A has s. ∴ B likely has s.
A 与 B 相似（共有 p, q, r）；A 有 s；∴ B 很可能有 s。
Paley’s watchmaker argument uses analogy: the universe is like a watch, which implies a watchmaker; therefore the universe implies a designer. Critically, the more relevant the shared properties, the stronger the analogy.
佩利的钟表匠论证使用了类比：宇宙像钟表，钟表意味着钟表匠；所以宇宙意味着设计者。关键的是，共享属性越相关，类比越强。
7. Utility Calculations: Bentham’s Hedonic Calculus | 效益计算：边沁的快乐计算
In utilitarian ethics, the right action is the one that maximises pleasure over pain. Bentham proposed the hedonic calculus as a 'formula' to quantify this. It considers seven factors for each pleasure or pain produced by an action.
在功利主义伦理学中，正确的行动是那些能最大化快乐、最小化痛苦的行为。边沁提出了快乐计算法作为一种量化这一过程的“公式”。它考量一个行动产生的每个快乐或痛苦所涉及的七个因素。
| Factor | Meaning |
| Intensity | How strong is the feeling? |
| Duration | How long will it last? |
| Certainty | How likely is it to occur? |
| Propinquity | How soon will it happen? |
| Fecundity | Will it lead to more of the same? |
| Purity | Will it be followed by opposite sensations? |
| Extent | How many people are affected? |
The 'theorem' of utility: an act is right if the total net pleasure is greater than any alternative. This calculative approach was designed to make ethics objective.
功利的“定理”：若某项行动的总净快乐大于任何其他选择，则该行动是正当的。这种计算方法旨在使伦理学客观化。
8. Kant’s Categorical Imperative Formulae | 康德的绝对命令公式
Kant’s ethics centre on the categorical imperative, which comes in several formulations. The most famous is the Formula of Universal Law: 'Act only according to that maxim whereby you can at the same time will that it should become a universal law.'
康德的伦理学核心是绝对命令，它有好几种表述。最著名的是普遍法则公式：“只依据那些你同时也愿意其成为普遍法则的准则去行动。”
In logical form this becomes a test: State your maxim (the rule you intend to follow). Ask: can this maxim be universalised without contradiction? If not, the action is morally impermissible. This is like a logical consistency check.
用逻辑形式表示，这就成为一个测试：陈述你的准则（你打算遵循的规则）。问：这个准则能否一无矛盾地被普遍化？如果不能，该行为在道德上就是不允许的。这类似于逻辑一致性检验。
The Formula of Humanity states: 'Act in such a way that you treat humanity, whether in your own person or in the person of another, always at the same time as an end and never simply as a means.' This provides a non-calculative, respect-based ethical 'constant'.
人性公式指出：“你要如此行动，即无论是你自身中的人性还是他人中的人性，你总是同时将其当作目的，而绝不仅仅当作手段。”这提供了一个非计算性的、基于尊重的伦理“常数”。
9. Argument for God’s Existence: Design Argument Structure | 上帝存在的设计论证结构
The teleological argument (design argument) follows an inductive, analogical structure. Its core formula can be expressed as: The universe exhibits complexity, order, and purpose (like a watch). Such features in an artefact imply an intelligent designer. Therefore, the universe likely has an intelligent designer (God).
目的论论证（设计论证）遵循归纳、类比的结构。其核心公式可表示为：宇宙展现出复杂性、秩序和目的性（像钟表一样）。人造物中的这些特征意味着有智慧的设计者。因此，宇宙很可能有一位智慧的设计者（上帝）。
Universe has property C (complex order). Artifacts with C → designer. ∴ Universe → designer.
宇宙具有属性 C（复杂秩序）；具有 C 的人造物 → 设计者；∴ 宇宙 → 设计者。
Key theorem: The argument is strengthened if we accept that the universe is more like a biological organism that self-organises; weakened if natural selection can explain apparent design. The principle of Ockham’s Razor is often used against adding an unnecessary designer.
关键定理：如果我们接受宇宙更像一个能自我组织的生物有机体，论证就被削弱；如果自然选择能解释表面的设计，论证也被削弱。奥卡姆剃刀原则常被用来反对添加不必要的设计者。
10. Argument for God’s Existence: Cosmological Argument Structure | 上帝存在的宇宙论证结构
Cosmological arguments seek a first cause or necessary being. A standard formula: Everything that begins to exist has a cause. The universe began to exist. Therefore, the universe has a cause. This cause is identified as God.
宇宙论证寻找第一因或必然存在。标准公式：一切开始存在的事物都有其产生的原因。宇宙开始存在。所以，宇宙有其原因。这个原因被认定为上帝。
∀x (Begins(x) → Cause(x)). Begins(universe). ∴ Cause(universe).
∀x (开始存在(x) → 有原因(x))；宇宙开始存在；∴ 宇宙有原因。
Another version is the argument from contingency: Contingent beings depend on something else. If everything were contingent, an infinite regress would occur. Therefore, there must exist a necessary being (God) on which all contingents depend. The 'theorem' is the rejection of an actual infinite regress of causes.
另一版本是从偶然性出发的论证：偶然存在物依赖于他物。若万物皆为偶然，就会出现无限回溯。因此，必存在一个必然存在者（上帝），一切偶然存在物都依赖于它。这里的“定理”是对因果无限回溯的否定。
11. Problem of Evil Logical Structure | 恶的问题逻辑结构
The logical problem of evil presents an inconsistency within classical theism. The formula: If God is omnipotent, omniscient, and omnibenevolent, then evil should not exist. Evil exists. Therefore, such a God does not exist.
恶的逻辑问题展示了古典有神论内部的不一致性。公式为：如果上帝全知、全能、全善，那么恶就不应该存在。恶确实存在。所以，这样的上帝不存在。
(Omnipotent ∧ Omniscient ∧ Omnibenevolent) → ¬Evil. Evil. ∴ ¬God (as defined).
(全能 ∧ 全知 ∧ 全善) → ¬恶；恶存在；∴ ¬上帝（如定义那般）。
The evidential version adjusts the probability: Given the vast amount of pointless suffering, it is highly improbable that a theistic God exists. Theodicy attempts to show that God and evil are compatible, often invoking free will defence or soul-making as 'theorems' that maintain consistency.
证据式的恶的问题则调整了概率：鉴于大量无意义的苦难，上帝存在的可能性极低。神正论试图表明上帝与恶是兼容的，常以自由意志辩护或灵魂塑造作为保持一致性的“定理”。
12. Ockham’s Razor & Hume’s Fork | 奥卡姆剃刀与休谟的叉子
Ockham’s Razor is a methodological principle: 'Entities should not be multiplied beyond necessity.' In formulaic terms: when two explanations account for the data equally well, prefer the one with fewer assumptions. It is a principle of simplicity, not a proof.
奥卡姆剃刀是一项方法论原则：“如无必要，勿增实体。”用公式化的语言来说：当两个解释都能同样好地说明数据时，选择假设更少的那个。这是关于简洁的原则，而非证明。
Hume’s Fork divides all knowledge into two types: relations of ideas (analytic truths, such as mathematical statements, known by reason alone) and matters of fact (synthetic truths, known through experience). The 'theorem' associated with it is that any book of metaphysics not containing reasoning about quantity or number, nor experimental reasoning about matters of fact, 'commit it then to the flames'. This challenges many theological claims.
休谟的叉子将所有知识分为两类：观念关系（分析真理，如数学陈述，仅凭理性可知）和事实（综合真理，通过经验可知）。与之相关的“定理”是：任何不包含量或数的抽象推理，也不包含关于事实的实验推理的形而上学书籍，就该“付之一炬”。这对许多神学主张构成了挑战。
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