📚 Year 10 CCEA Philosophy: Formula and Theorem Quick Reference Handbook | CCEA 哲学公式定理速查手册
Philosophy may not have mathematical equations, but it has its own ‘formulas’ and ‘theorems’ – logical patterns, argument structures, and ethical principles that can be applied like a toolkit. This quick reference handbook distils the core logical and argumentative forms you will encounter in the Year 10 CCEA Philosophy course, presented with clear English definitions and matching Chinese explanations. Use it to sharpen your reasoning and to recognise classic moves in philosophical debate.
哲学或许没有数学方程式，但它有自己的“公式”和“定理”——即逻辑模式、论证结构和伦理原则，可以像工具箱一样运用。本速查手册提炼了 Year 10 CCEA 哲学课程中最核心的逻辑形式和论证框架，以清晰的英文定义配以中文解释呈现。用它来磨砺你的推理能力，并识别哲学辩论文中的经典套路。
1. Logical Operators: The Building Blocks | 逻辑运算符：基本构件
Logical operators connect propositions to form complex statements. The main ones you need to know are: conjunction (∧, ‘and’), disjunction (∨, ‘or’), negation (¬, ‘not’), conditional (→, ‘if… then’), and biconditional (↔, ‘if and only if’). In philosophical writing these are often expressed in ordinary language; recognising their formal structure helps in analysing arguments.
逻辑运算符把命题连接起来构成复杂陈述。你需要掌握的主要有：合取（∧，“且”）、析取（∨，“或”）、否定（¬，“非”）、条件（→，“如果……那么”）和双条件（↔，“当且仅当”）。在哲学写作中，它们通常用日常语言表达；识别其形式结构有助于分析论证。
| Symbol | English Name | Meaning | 中文名称 |
|---|---|---|---|
| ∧ | Conjunction | Both are true | 合取（且） |
| ∨ | Disjunction | At least one is true | 析取（或） |
| ¬ | Negation | Opposite truth value | 否定（非） |
| → | Conditional | If P then Q | 条件（如果……则） |
| ↔ | Biconditional | P if and only if Q | 双条件（当且仅当） |
2. Modus Ponens: Affirming the Antecedent | 肯定前件式
Modus ponens is one of the most common valid argument forms. Its structure: If P, then Q. P is true. Therefore, Q is true. Example: If it rains, the ground gets wet. It is raining. Therefore, the ground gets wet. This form is deductively valid – if the premises are true, the conclusion must be true.
肯定前件式是最常见的有效论证形式之一。其结构为：如果 P，则 Q；P 为真；因此 Q 为真。例如：如果下雨，地面就会湿。正在下雨。所以地面会湿。这种形式是演绎有效的——如果前提为真，结论必然为真。
General formula: P → Q, P ⊢ Q
一般公式：P → Q, P ⊢ Q
3. Modus Tollens: Denying the Consequent | 否定后件式
Modus tollens is another valid form. Structure: If P, then Q. Not Q. Therefore, not P. Example: If the butler did it, he would have been at the scene. He was not at the scene. Therefore, the butler did not do it. This form is crucial for testing hypotheses – if a predicted consequence fails to occur, the hypothesis is falsified.
否定后件式是另一种有效形式。结构：如果 P 则 Q；非 Q；因此非 P。例如：如果是管家干的，他当时就在现场。他不在现场。所以不是管家干的。这一形式对检验假说至关重要——如果预测的结果没有出现，假说就被证伪。
P → Q, ¬Q ⊢ ¬P
4. Hypothetical Syllogism: Chain Argument | 假言三段论：连锁论证
When conditionals chain together, a hypothetical syllogism draws a valid conclusion: If P then Q. If Q then R. Therefore, if P then R. This form is used extensively in cosmological arguments: If the universe began to exist, it has a cause. If it has a cause, that cause is God. Therefore, if the universe began to exist, God exists. (Note: the truth of the premises must still be examined.)
当条件语句连锁在一起时，假言三段论能推出有效结论：如果 P 则 Q；如果 Q 则 R；因此如果 P 则 R。这一形式在宇宙论论证中被广泛使用：如果宇宙开始存在，它就有原因。如果它有原因，那个原因就是上帝。因此，如果宇宙开始存在，上帝就存在。（注意：前提的真实性仍需检验。）
(P → Q), (Q → R) ⊢ (P → R)
5. Disjunctive Syllogism: Eliminating Alternatives | 选言三段论：排除选项
From an ‘either… or’ statement, if one option is denied, the other must be true. Structure: Either P or Q. Not P. Therefore, Q. This works only if the disjunction is exclusive. In philosophy, it often appears in arguments about free will: Either determinism is true or we have free will. Determinism is false. Therefore, we have free will. (Again, check the premises.)
从“要么……要么……”的陈述出发，如果否定了其中一个选项，另一个必为真。结构：要么 P 要么 Q；非 P；因此 Q。这只在选言是互斥的情况下有效。在哲学中，它常出现在关于自由意志的论证中：要么决定论为真，要么我们有自由意志。决定论为假。因此，我们有自由意志。（同样，要检查前提。）
(P ∨ Q), ¬P ⊢ Q
6. Reductio ad Absurdum: Proof by Contradiction | 归谬法：矛盾证明
To prove a proposition, assume its negation and show that this leads to a contradiction or absurdity. Therefore the original proposition must be true. This is the logical engine of many ontological arguments and ethical thought experiments. For example, Anselm assumes that God does not exist, but then the greatest conceivable being lacks existence, which seems contradictory – so God must exist.
要证明一个命题，先假设它的否定，然后推出矛盾或荒谬的结果，因此原命题必为真。这是许多本体论论证和伦理思想实验的逻辑引擎。例如，安瑟伦假设上帝不存在，那么最伟大的可设想者就缺少存在性，这似乎是矛盾的——因此上帝必然存在。
Assume ¬P, derive contradiction (Q ∧ ¬Q), conclude P.
假设 ¬P，推导出矛盾（Q ∧ ¬Q），从而断定 P。
7. Aquinas’ First Way: Argument from Motion | 阿奎那的第一路：运动的论证
This ‘formula’ for the existence of God goes as follows: (1) Some things are in motion (change). (2) Whatever is moved is moved by another. (3) This cannot go back to infinity, because then there would be no first mover. (4) Therefore, there must be a first unmoved mover – and this everyone understands to be God. The logical core relies on avoiding an infinite regress, a common strategy in cosmological reasoning.
这个关于上帝存在的“公式”如下：（1）有些事物在运动（变化）。（2）凡被推动者必被他者推动。（3）这不能无限追溯，因为那样就没有第一推动者了。（4）因此，必有一个不被推动的第一推动者——这被大家理解为上帝。其逻辑核心在于避免无限回溯，这是宇宙论推理中的常见策略。
- Infinite regress: an endless chain of causes with no starting point – seen as logically problematic.
- 无限回溯：没有起点的无终止的原因链——在逻辑上被视为有问题。
8. Paley’s Design Argument: Watchmaker Analogy | 佩利的设计论证：钟表匠类比
William Paley’s design argument can be crystallised as an analogy ‘formula’: (1) A watch exhibits complex parts that work together for a purpose. (2) We infer an intelligent watchmaker. (3) The natural world contains far more complex purposeful arrangements (e.g., the human eye). (4) Therefore, by analogy, the natural world also has an intelligent designer – God. The strength of the argument depends on how similar the two cases are.
威廉·佩利的设计论证可以归结为一个类比“公式”：（1）钟表展现出为某一目的而协同运作的复杂部件。（2）我们由此推断存在一位智慧的钟表匠。（3）自然界包含着远更复杂的目的性安排（例如人眼）。（4）因此，通过类比，自然界也有一位智慧的设计者——上帝。论证的力度取决于这两种情形有多相似。
| Watch | 钟表 | Nature | 自然 |
|---|---|---|---|
| Complex parts → design | 复杂部件 → 设计 | Complex organisms → design | 复杂生物体 → 设计 |
| Watchmaker exists | 钟表匠存在 | Divine Designer exists | 神圣设计者存在 |
9. The Problem of Evil: The Inconsistent Triad | 恶的问题：不相容的三元组
The logical problem of evil is often presented as a ‘formula’ that shows an inconsistency in theism. (1) God is all-powerful. (2) God is all-loving. (3) Evil exists. These three statements appear unable to be true at the same time. If God could stop evil and wanted to, evil would not exist. Since evil does exist, either God is not all-powerful, not all-loving, or does not exist. This is the core challenge that theodicies try to solve.
恶的逻辑问题常被呈现为一个展示有神论内部不一致性的“公式”。（1）上帝是全能的。（2）上帝是全善的。（3）恶存在。这三个陈述似乎无法同时为真。如果上帝能阻止恶并且愿意阻止，恶就不会存在。既然恶确实存在，那么上帝要么不全能，要么不全善，要么不存在。这正是神义论试图解决的核心挑战。
Omnipotent ∧ Omnibenevolent ∧ Evil exists → Contradiction?
全能 ∧ 全善 ∧ 恶存在 → 矛盾？
10. Utilitarianism: The Hedonic Calculus | 功利主义：快乐计算法
Jeremy Bentham’s utilitarianism can be understood as a moral ‘formula’: The right action is the one that maximises pleasure and minimises pain for the greatest number. Bentham even proposed a hedonic calculus with seven factors: intensity, duration, certainty, propinquity, fecundity, purity, and extent. While later utilitarians simplified the theory, the basic principle remains: Maximise net happiness.
边沁的功利主义可以理解为一个道德“公式”：正确的行为就是为最大多数人最大化快乐并最小化痛苦。边沁甚至提出了一种包含七个因素的快乐计算法：强度、持续时间、确定性、邻近性、丰饶性、纯粹性和范围。虽然后来的功利主义者简化了该理论，但基本原则仍是：最大化净幸福。
- Greatest Happiness Principle: Act so as to produce the greatest balance of pleasure over pain.
- 最大幸福原则：行动要产生快乐多于痛苦的最大余额。
- Hedonic calculus factors: Intensity, Duration, Certainty, Propinquity, Fecundity, Purity, Extent.
- 快乐计算因素：强度、持续时间、确定性、邻近性、丰饶性、纯粹性、范围。
11. Kant’s Categorical Imperative: Universalisation Formula | 康德的定言命令：普遍化公式
Kant’s moral ‘theorem’ is the Categorical Imperative, especially its first formulation: ‘Act only according to that maxim whereby you can at the same time will that it should become a universal law.’ This works as a test: can you consistently conceive of a world where everyone follows your personal rule? If you cannot, the action is morally impermissible. For instance, making a false promise would be self-defeating if universalised, because the institution of promising would collapse.
康德的道德“定理”是定言命令，尤其是其第一个表述：“只依照你同时能够意愿它成为一条普遍法则的准则去行动。”这就像一个测试：你能否一贯地设想一个所有人都遵守你个人行为准则的世界？如果不能，该行为在道德就是不允许的。例如，做虚假承诺如果普遍化，将是自我挫败的，因为承诺制度会崩溃。
- Maxim: the subjective principle of action (what you intend to do and why).
- 准则：主观的行为原则（你打算做什么以及为何做）。
- Contradiction in conception vs. contradiction in will – two ways a maxim can fail the test.
- 概念矛盾 与 意愿矛盾——准则未能通过检验的两种方式。
12. Virtue Ethics: The Doctrine of the Mean | 德性伦理：中庸之道
Aristotle’s virtue ethics does not give you a strict decision rule but an ‘equilibrium formula’: Virtue is a disposition to choose the mean relative to us, between excess and deficiency. For example, courage is the mean between cowardice (deficiency) and rashness (excess). The moral agent must use practical wisdom (phronesis) to find the right balance in each situation. This is not a mathematical midpoint but a context-sensitive optimum.
亚里士多德的德性伦理没有给出严格的决定规则，而是一种“均衡公式”：德性是一种相对于我们而言在过度与不及之间选择中道的品质。例如，勇敢是怯懦（不及）与鲁莽（过度）之间的中道。道德行动者必须运用实践智慧（phronesis）在各个情境中找到恰当的平衡点。这不是数学上的中点，而是随情境变化的最佳点。
| Vice of Deficiency | 不及之恶 | Virtue (The Mean) | 德性（中道） | Vice of Excess | 过度之恶 |
|---|---|---|---|---|---|
| Cowardice | 怯懦 | Courage | 勇敢 | Rashness | 鲁莽 |
| Stinginess | 吝啬 | Generosity | 慷慨 | Prodigality | 挥霍 |
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