📚 Year 12 CCEA Philosophy: Formula & Theorem Quick Reference | CCEA 哲学：公式定理速查手册
Philosophy may not have equations in the mathematical sense, but it is full of logical structures, principled calculations, and conceptual 'formulae' that allow us to evaluate arguments, define ethical actions, and dissect metaphysical claims. This quick reference guide distills key CCEA Year 12 philosophical concepts into memorable 'formulae' and 'theorems' — compact, exam-ready expressions that capture the essence of each idea.
哲学也许没有数学意义上的方程式，但它充满了逻辑结构、原则性计算和概念性“公式”，让我们能够评估论证、界定伦理行为并剖析形而上学主张。这份速查手册将 CCEA 12 年级哲学的关键概念提炼为易于记忆的“公式”和“定理”——简洁、适用于考试的表达，抓住了每个思想的精髓。
1. Deductive Validity & Soundness | 演绎有效性与可靠性
A deductive argument is valid if and only if it is impossible for the premises to be true and the conclusion false. Validity concerns the logical form alone.
一个演绎论证是有效的，当且仅当不可能出现前提为真而结论为假的情况。有效性仅取决于逻辑形式。
Formula:
Valid = (P₁ ∧ P₂ ∧ … ∧ Pₙ) → C is a logical truth
A valid argument with all true premises is sound.
一个有效且所有前提为真的论证是可靠的。
Sound = Valid + True Premises
A sound argument thus guarantees a true conclusion; it is the gold standard of reasoning.
因此，一个可靠的论证保证了结论为真；它是推理的黄金标准。
2. The Tripartite Definition of Knowledge (JTB) | 知识的三元定义（JTB）
Since Plato, knowledge has often been analysed as justified true belief.
自柏拉图以来，知识常被分析为得以辩护的真信念。
K = J + T + B
where K is knowledge, J is justification, B is belief, and T is truth. For a subject S and proposition p:
其中 K 代表知识，J 代表辩护，B 代表信念，T 代表真理。对主体 S 和命题 p：
- S knows that p ↔ S believes p, p is true, and S is justified in believing p.
- S 知道 p ↔ S 相信 p，p 为真，并且 S 有正当理由相信 p。
Gettier cases challenge this by showing that J, T, and B can all be present without knowledge, prompting attempts to add a 'no false lemmas' condition or a fourth criterion.
盖梯尔案例通过表明 J、T、B 可以同时存在却缺乏知识，挑战了这一定义，从而促使人们添加“无误导前提”条件或第四条标准。
3. Descartes' Cogito | 笛卡尔的“我思”
Descartes’ foundational certainty arises from the very act of doubting: 'I think, therefore I am' (Cogito, ergo sum).
笛卡尔的基础确定性来自怀疑行为本身：“我思故我在”。
∃x (Thinking(x) → Existing(x))
Even if an evil demon deceives me, I must exist to be deceived. The cogito resists even the most radical scepticism.
即便有恶魔欺骗我，我也必须存在才能被欺骗。我思甚至抵御了最彻底的怀疑论。
The formula captures the immediate inference: the act of thinking entails the existence of a thinker.
这个公式捕捉了直接推理：思考行为必然包含思考者的存在。
4. Hume's Problem of Induction | 休谟的归纳问题
Inductive reasoning assumes that the future will resemble the past, but this assumption itself cannot be justified without circularity.
归纳推理假定未来与过去相似，但这一假定本身不能在没有循环论证的情况下得到辩护。
Induction: (A₁, A₂, …, Aₙ) all F are G → All F are G
The principle of uniformity of nature (PUN) is needed to justify this move:
需要自然齐一性原理（PUN）来辩护这一步骤：
PUN → Inductive Inference is Reliable
But PUN is itself known only inductively, leading to the famous circularity. Hume's 'theorem' is that induction cannot be rationally justified, only psychologically explained.
但 PUN 本身只能通过归纳而得知，从而导致著名的循环论证。休谟的“定理”是：归纳无法获得理性辩护，只能得到心理学上的解释。
5. The Principle of Utility (Bentham's Felicific Calculus) | 功利原则（边沁的幸福计算）
Classical utilitarianism evaluates actions by the quantity of pleasure or pain they produce.
古典功利主义根据行动产生的快乐或痛苦的数量来评估行动。
Action A is right ↔ A maximizes Σ(Pleasure − Pain) for all affected
Bentham's felicific calculus includes seven dimensions: intensity, duration, certainty, propinquity, fecundity, purity, and extent.
边沁的幸福计算包括七个维度：强度、持续时间、确定性、远近、丰产性、纯度和范围。
For any decision between actions A and B, calculate:
对于行动 A 和 B 之间的任何决策，计算：
U(A) = Σ(i×d×c×p×f×pu×e) − Σ(i'×d'×c'×p'×f'×pu'×e')
Choose the action with the higher net utility. Mill later refined this with qualitative distinctions between higher and lower pleasures.
选择净效用更高的行动。密尔后来用高级快乐与低级快乐的质的区分完善了这一理论。
6. Kant's Categorical Imperative (Formula of Universal Law) | 康德的定言命令（普遍法则公式）
Kant’s moral law is expressed as a test of universalisability, independent of consequences.
康德的道德律被表达为一种对可普遍化的检验，完全独立于后果。
Act only on that maxim whereby you can at the same time will that it should become a universal law.
只按照你同时能够愿意它成为一条普遍法则的准则去行动。
The formula for a contradiction test:
矛盾检验的公式：
If (Maxim universalised → Contradiction in conception or in will) → Act is forbidden
A contradiction in conception (e.g., false promising) makes a maxim inconceivable as universal law; a contradiction in will (e.g., not helping others) thwarts rational agency. The Formula of Humanity is another key formulation: treat humanity always as an end, never merely as a means.
概念上的矛盾（例如虚假承诺）使该准则无法被设想为普遍法则；意志上的矛盾（例如不帮助他人）则阻碍了理性主体能力的实现。人性公式是另一个关键表述：始终将人性视作目的，而绝不仅仅当作手段。
7. The Ontological Argument (Anselm's Logical Form) | 本体论论证（安瑟伦的逻辑形式）
Anselm defines God as 'that than which nothing greater can be conceived' and argues that existence in reality is greater than existence in the understanding alone.
安瑟伦将上帝定义为“无法设想比之更伟大的存在者”，并论证实际存在比仅仅存在于理解中更为伟大。
1. God =df the greatest conceivable being (GCB)
2. GCB exists in the understanding
in understanding, a greater being (one existing in reality) is conceivable → Contradiction
4. Therefore, GCB exists in reality.
Gaunilo’s island parody mirrors the same logical structure to show its apparent absurdity. The modal version by Plantinga strengthens it: if a maximally great being is possible, it exists necessarily in some possible world; if necessarily existent, then it exists in all possible worlds, including the actual one.
高尼罗的岛屿反驳模仿了相同的逻辑结构，以显示其明显荒谬之处。普兰丁格的模态版本将其加强：如果一个极大伟大者是可能的，那么它必然存在于某个可能世界中；如果必然存在，则存在于所有可能世界中，包括现实世界。
8. The Teleological Argument (Paley's Watchmaker Analogy) | 目的论论证（佩利的钟表匠类比）
The argument from design is often presented as an analogical inference.
设计论证通常被表述为一个类比推理。
Watch : Watchmaker :: Universe : Universe-Maker
Paley argues that the complex, functional order of natural objects (like the eye) implies an intelligent designer, just as a watch implies a watchmaker. The formula can be expressed:
佩利主张，自然物体（如眼睛）复杂的功能秩序暗示了一位智能设计者，正如一块表暗示了一位钟表匠。可以表达为：
If Object X exhibits complex adaptive order, then X is produced by intelligent design.
Hume’s criticisms (via Philo) show the weakness of the analogy: the universe is not sufficiently like a watch; the argument commits a category mistake or fails to establish a single, omnipotent designer.
休谟（借费罗之口）的批评揭示了类比的弱点：宇宙并不足以像一块表；该论证犯了范畴错误，或无法确立一位单一、全能的设计者。
9. The Logical Problem of Evil | 恶的逻辑问题
The inconsistent triad presents the strongest form of the problem of evil: the coexistence of God's omnipotence, omnibenevolence, and the existence of evil.
不兼容的三元组提出了恶的问题的最强形式：上帝的全能、全善与恶的存在无法兼容。
¬(Omnipotent God ∧ Omnibenevolent God ∧ Evil exists)
If God is willing but not able to prevent evil, He is not omnipotent; if able but not willing, He is not omnibenevolent; if both, why is there evil? The formula shows the logical impossibility of all three propositions being true simultaneously, challenging the coherence of classical theism.
如果上帝愿意但不能阻止恶，祂便不是全能的；如果能够但不愿意，祂便不是全善的；如果两者兼备，为何存在恶？这个公式表明三个命题同时为真在逻辑上是不可能的，挑战了古典有神论的一致性。
Free-will defence and soul-making theodicies attempt to break the triad by reinterpreting evil as necessary for a greater good.
自由意志辩护和灵魂塑造神义论试图通过将恶解释为更大善的必要条件来消解三元矛盾。
10. Free Will, Determinism, and Compatibilism | 自由意志、决定论与相容论
The classic free will problem can be represented as a dilemma: if determinism is true, our actions are necessary; if not, they are random. Neither seems to allow genuine free will.
经典的自由意志难题可表示为一个两难：如果决定论为真，我们的行动是必然的；如果不为真，行动则是随机的。两者似乎都不容许真正的自由意志。
Compatibilism defines free will differently:
相容论对自由意志进行了不同的定义：
Free will ↔ Action caused by internal volitions, even if those volitions are determined.
An agent acts freely if the action flows from their own desires, beliefs, and character without external coercion. The 'formula' for compatibilist freedom is thus:
如果某一行动源于行动者自身的欲望、信念和性格，且无外部强制，那么该行动便是自由的。因此，相容论自由的“公式”是：
Freedom = Volitional Causation − External Constraint
Incompatibilists insist that free will requires alternative possibilities (the principle of alternate possibilities: PAP), generating a deeper 'theorem' of libertarianism: an action A is free at time t only if the agent could have done otherwise at t.
不相容论者坚持自由意志需要可供取舍的可能性（替代可能性原则：PAP），从而产生一个更深层的意志自由论“定理”：行动 A 在时间 t 是自由的，仅当行动者在时间 t 本可以做其他选择。
11. Aristotle's Doctrine of the Mean | 亚里士多德的中道学说
For Aristotle, moral virtue is a disposition to choose the intermediate between extremes relative to us, determined by reason.
对亚里士多德而言，道德德性是一种选择相对于我们而言的极端之间的中间状态的气质，由理性决定。
Virtue = (Excess + Deficiency) ÷ 2, applied to the right person, right time, right motive
However, this is not a strict mathematical mean; it varies with circumstances. The 'formula' captures the idea that each virtue lies between two vices:
然而，这并不是严格的数学中点；它随环境而变化。这个“公式”抓住了每种德性都处于两种恶性之间的思想：
- Courage = Mean between Cowardice and Rashness | 勇敢 = 怯懦与鲁莽的中道
- Generosity = Mean between Stinginess and Prodigality | 慷慨 = 吝啬与挥霍的中道
- Truthfulness = Mean between Mock-Modesty and Boastfulness | 诚实 = 假谦虚与吹嘘的中道
The wise person (phronimos) discerns the appropriate mean through practical wisdom (phronēsis).
明智者通过实践智慧（phronēsis）来辨别恰当的中道。
12. Ethical Theories at a Glance | 伦理理论一览
A final summary table ties together the formal structure of the main normative theories examined in Year 12.
最后的总览表将 12 年级所考察的主要规范伦理理论的形式结构汇总在一起。
| Theory 理论 | Core 'Formula' 核心“公式” | Key Thinker 关键思想家 |
|---|---|---|
| Utilitarianism 功利主义 | Right action = max happiness for max number | Bentham, Mill 边沁，密尔 |
| Deontology 义务论 | Right action = follows universalisable maxim | Kant 康德 |
| Virtue Ethics 美德伦理学 | Right action = what a virtuous agent would do | Aristotle 亚里士多德 |
| Divine Command 神命论 | Right action = commanded by God | Various 诸多神学家 |
These concise 'formulae' are not meant to replace nuanced argument but to provide a mental peg for organising your revision — a springboard into deeper evaluation and exam-ready application.
这些简洁的“公式”不是为了取代细致的论证，而是为你的复习提供思维挂钩——是深入评估与应试应用的跳板。
Published by TutorHao | Philosophy Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图