📚 Quick Reference Handbook of Philosophical Formulas and Theorems | 哲学公式定理速查手册
This handbook translates the core arguments, principles, and thought-experiments of SQA Year 11 Philosophy into formal ‘formulas’ and ‘theorems’. Each entry gives you the canonical logical structure, a concise explanation, and a paired English–Chinese rendering – ideal for on‑the‑spot revision and exam precision.
这本速查手册将 SQA 11 年级哲学的核心论证、原则与思想实验转化为类似“公式”和“定理”的形式。每个条目都提供规范的逻辑结构、简洁说明以及英中双语对照，非常适合考前快速回顾与精准答题。
1. Modus Ponens & Modus Tollens | 肯定前件与否定后件
The two fundamental valid deductive forms. Modus Ponens affirms the antecedent; Modus Tollens denies the consequent. In any argument analysis, always test whether the reasoning fits one of these templates.
这是两种基本的有效演绎形式。肯定前件（Modus Ponens）通过肯定前提中的前件得出结论；否定后件（Modus Tollens）则通过否定后件推断前件为假。分析论证时，首先检查推理是否符合其中一种模板。
Modus Ponens: P → Q, P ⊢ Q
Modus Tollens: P → Q, ¬Q ⊢ ¬P
If it rains, the ground is wet. It rains. Therefore the ground is wet. (M.P.)
如果下雨，地面就会湿。下雨了。所以地面湿了。（肯定前件）
The ground is not wet. Therefore it did not rain. (M.T.)
地面没有湿。所以没有下雨。（否定后件）
2. Hypothetical Syllogism & Disjunctive Syllogism | 假言三段论与选言三段论
These chain arguments and eliminate alternatives. Hypothetical Syllogism links conditionals; Disjunctive Syllogism removes one option to leave the other. Both appear regularly in SQA exam MCQ and essay questions.
这两种推理分别通过链条式条件句和排除选项来得出结论。假言三段论连接多个条件句；选言三段论则否定一个选言支而保留另一个。它们在 SQA 选择题与论述题中频繁出现。
Hyp. Syll.: (P → Q), (Q → R) ⊢ P → R
Disj. Syll.: P ∨ Q, ¬P ⊢ Q
If you study, you pass. If you pass, you graduate. Thus, if you study, you graduate.
如果你学习，就会通过考试。如果你通过，就会毕业。所以，如果你学习，就会毕业。
Either the butler did it or the gardener did it. The butler didn’t. So the gardener did it.
要么是管家干的，要么是园丁干的。管家没有干。所以是园丁干的。
3. Necessary and Sufficient Conditions | 必要条件与充分条件
A condition X is sufficient for Y if X guarantees Y (X → Y). X is necessary for Y if Y cannot occur without X (¬X → ¬Y, equivalently Y → X). Confusing the two is a classic exam trap.
如果 X 能保证 Y 成立，X 就是 Y 的充分条件（X → Y）。如果缺少 X 则 Y 不可能发生，X 就是 Y 的必要条件（¬X → ¬Y，等价于 Y → X）。混淆这两者是经典考试陷阱。
| Symbolic Formula | English | 中文 |
|---|---|---|
| S → E | Being a square is sufficient for having four sides. | 是正方形充分保证有四条边。 |
| ¬O → ¬B | Oxygen is necessary for burning. | 氧气是燃烧的必要条件。 |
Check: “If and only if” (↔) means both sufficient and necessary – the strongest link.
请注意：“当且仅当”（↔）表示既充分又必要——这是最强的逻辑联系。
4. The Problem of Evil – Logical Theorem | 恶的问题 – 逻辑定理
The inconsistent triad is often formalised as a logical contradiction between three propositions. Solving it requires denying at least one premise. This is a core ‘theorem’ of Philosophy of Religion.
“不相容三元组”常被形式化为三个命题之间的逻辑矛盾。要解决它，必须至少否认其中一个前提。这是宗教哲学中的核心“定理”。
(1) God is omnipotent (∀ acts, God can do) ∧ (2) God is omnibenevolent (∀ good, God wills) ∧ (3) Evil exists ⊢ ⊥
If God could prevent evil and wanted to, evil would not exist. But evil exists. So either God is not all‑powerful, not all‑good, or does not exist.
如果上帝能够且愿意阻止恶，恶就不会存在。但恶存在。所以，要么上帝并非全能，要么并非全善，要么不存在。
Augustine’s free‑will defence denies premise (2) is violated by permitting free will. The logical structure remains a must‑know.
奥古斯丁的自由意志辩护否认因尊重自由意志而违反前提(2)。其逻辑结构必须掌握。
5. Utilitarian Calculus (Bentham’s Formula) | 功利主义计算公式（边沁公式）
Bentham proposed a hedonic calculus to quantify happiness. Although not a literal numeric formula in the exam, you must treat it as a decision theorem: maximise pleasure over pain across all affected.
边沁提出“快乐计算法”以量化幸福。虽然考试中不是真的让你做数学题，但你必须将其视为一个决策定理：在所有受影响者之间实现快乐减去痛苦的最大化。
Moral Rightness = Σ (Pleasureᵢ − Painᵢ) for all affected individuals i
Intensity, duration, certainty, propinquity, fecundity, purity, and extent – the seven factors. An action is right if it produces the greatest net balance.
强度、持续时间、确定性、远近、继生性、纯度与范围——七大因素。若一个行为产生最大净余额，它就是正当的。
Criticism: Mill refines this into higher and lower pleasures, introducing a quality multiplier Q.
批评：密尔将其提炼为高级快乐与低级快乐，引入了一个品质乘数 Q。
6. Kant’s Categorical Imperative (Formula of Universal Law) | 康德的定言命令（普遍法则公式）
Kant’s ethics behaves like a universalisability test theorem. You run a maxim through it; if it yields a contradiction, the act is morally forbidden. This is arguably the most formulaic moral principle in the SQA syllabus.
康德伦理学如同一个可普遍化测试定理。你将一个准则通过该公式检验；如果产生矛盾，该行为就被道德禁止。这可能是 SQA 大纲中最公式化的道德原则。
Act only on that maxim whereby you can at the same time will that it should become a universal law.
Contradiction in conception: “Make a false promise to get money” → universalised, the institution of promising collapses. Contradiction in will: not helping others cannot be rationally willed in a world where you may need help.
概念中的矛盾：“为了得到钱而做虚假承诺” → 普遍化后承诺制度崩溃。意志中的矛盾：在一个你可能需要帮助的世界里，不愿意帮助他人无法被理性地意愿。
The Formula of Humanity (treat persons as ends) is the complementary expression, often tested together.
人性公式（人是目的而非手段）是其补充表达，二者常一起考查。
7. Aristotle’s Virtue Ethics – The Doctrine of the Mean | 亚里士多德的美德伦理学 – 中庸之道定理
Virtue is a disposition lying in a mean relative to us, determined by reason. For every virtue, there are two vices: excess and deficiency. This can be mapped as a spectrum theorem.
美德是一种相对于我们而言的中道品质，由理性决定。每种美德对应两个恶：过度与不足。这可以表示为一个光谱定理。
| Vice (Deficiency) | Virtue (Mean) | Vice (Excess) | 中文对照 |
|---|---|---|---|
| Cowardice | Courage | Rashness | 怯懦 – 勇敢 – 鲁莽 |
| Stinginess | Generosity | Prodigality | 吝啬 – 慷慨 – 挥霍 |
| Shamelessness | Modesty | Bashfulness | 无耻 – 谦逊 – 羞怯 |
The practical syllogism links virtue to action: Major premise – a virtuous person does X. Minor premise – I am in situation S that calls for X. Conclusion – I do X. This is the action theorem.
实践三段论将美德与行动连接：大前提 – 有德之人做 X。小前提 – 我处于需要 X 的情境 S。结论 – 我做 X。这就是行动定理。
8. Descartes’ Cogito – The First Certainty Theorem | 笛卡尔的我思 – 第一确定性定理
After methodic doubt, Descartes reaches an indubitable formula. The Cogito is not deduced; it is immediately perceived as true whenever uttered or conceived. It serves as the foundational theorem for rationalism.
经过方法论上的怀疑，笛卡尔抵达了一个不可置疑的公式。“我思故我在”不是推导出来的，而是每当说出或想到时就直观地把握为真。它是理性主义的奠基定理。
Cogito, ergo sum / I think, therefore I am.
Even if an evil demon deceives me, I must exist to be deceived. Hence, the self (res cogitans) is known with clarity and distinctness – the criterion of truth.
即使有恶魔欺骗我，我也必须先存在才能被欺骗。因此，自我（思维之物）以清晰分明的方式被认知——这构成了真理的标准。
The ’trademark argument’ for God’s existence (the idea of the infinite) and the Cartesian Circle are the next formulas to master.
上帝存在的“商标论证”（无限观念）与笛卡尔循环是需要掌握的下一个公式。
9. Plato’s Theory of Forms – The Metaphysical Equation | 柏拉图的理型论 – 形上学方程
Plato posits that the visible world (becoming) is a shadow of the intelligible world of Forms (being). The relationship can be expressed as a participation and imitation formula.
柏拉图认为可见世界（生成）是可知的理型世界（存在）的影子。这层关系可表达为分有和模仿的公式。
Particular x is F iff x participates in the Form of F-ness.
For example, a beautiful painting is beautiful because it imitates the Form of Beauty. Knowledge (episteme) is of Forms; opinion (doxa) is of particulars. The Simile of the Divided Line and the Allegory of the Cave are visual proofs of this theorem.
例如，一幅美丽的画是美的，因为它模仿了“美”的理型。知识来源于理型；意见来源于具体事物。“分线比喻”与“洞穴寓言”是这一定理的视觉证明。
The Third Man Argument (self‑predication problem) is the standard objection: a Form must also participate in a further Form, leading to infinite regress. Be ready to write the regress structure.
第三人论证（自谓述难题）是标准的反驳：一个理型自身又必须分有一个更高的理型，导致无穷后退。要能写出后退结构。
10. The Cosmological Argument (Aquinas’ Third Way) – The Necessity Theorem | 宇宙论论证（阿奎那第三路） – 必然性定理
Aquinas’ Third Way moves from contingent beings to a necessary being. The argument’s logical skeleton is a must‑reproduce theorem in the exam.
阿奎那的第三路从偶然存在者推到必然存在者。该论证的逻辑骨架是考试中必须能够再现的定理。
(1) ∀x (Contingent x → at some time x does not exist).
(2) If everything is contingent, at some time nothing existed. (3) If nothing existed, nothing would ever exist. (4) Things exist now. ∴ Not everything is contingent. ∴ ∃ a necessary being (God).
In plain English: contingent things come into and go out of existence. If everything were like that, there would have been a time with nothing – but then nothing could ever begin. Therefore, a being existing of its own necessity is required.
简单来说：偶然事物会生灭。如果一切都偶然，就会有一段空无所有的时间——但这样一来就不可能有什么开始存在。因此必须有一个自身必然的存在者。
The Kalam version (Craig) uses the impossibility of an actual infinite. Equally formulaic: whatever begins to exist has a cause.
卡拉姆版本（克莱格）诉诸实际无限的不可行。同样公式化：一切开始存在的事物都有原因。
11. Fallacies Quick Reference – Invalid Theorem Catalogue | 常见谬误速查 – 无效定理目录
Fallacies are pseudo‑theorems – patterns that look like valid formulas but break a logical rule. You must identify the broken rule to gain full marks.
谬误是一种伪定理——看似有效的公式，实际上违反了某项逻辑规则。你必须指出被违反的规则才能拿满分。
| Fallacy | False Formula | Why Invalid | 中文 |
|---|---|---|---|
| Affirming the Consequent | P → Q, Q ⊢ P | Q may have other causes. | 肯定后件 |
| Denying the Antecedent | P → Q, ¬P ⊢ ¬Q | ¬Q does not necessarily follow. | 否定前件 |
| Ad Hominem | Person A claims P; A is flawed. ∴ ¬P | Attacks person, not proposition. | 人身攻击 |
| Straw Man | Exaggerate opponent’s argument then refute. | Not addressing actual claim. | 稻草人 |
Always map an argument onto its formal structure to detect these. Treat each fallacy as a negative formula you must not use.
始终将论证映射到形式结构上来识别它们。把每个谬误当作一个你不能使用的负面公式。
12. Popper’s Falsification Theorem – Science vs. Pseudo‑Science | 波普尔的证伪定理 – 科学与伪科学
Popper replaces verification with falsification as the demarcation criterion. A theory is scientific only if it is falsifiable – it forbids certain observable states. The logical structure is asymmetric.
波普尔用证伪取代证实作为划界标准。一个理论是科学的，仅当它是可证伪的——即它禁止某些可观察的状况。其逻辑结构是不对称的。
Scientific Law L: ∀t (Under initial conditions C, outcome O occurs) ≡ L ∧ C ⇒ O
Falsification: (C is observed ∧ O is not observed) ⇒ ¬L
In modus tollens form: If theory T is true, evidence E should not appear. E does appear. Therefore T is falsified. This is conclusive, while verification is always provisional.
用否定后件形式表达：若理论 T 为真，则不应出现证据 E。E出现了。因此 T 被证伪。这是决定性的，而证实总是暂时的。
Apply this to the debate around God’s existence: ‘God exists’ is often criticised as unfalsifiable, hence not a scientific claim – but perhaps still meaningful. This is a frequent SQA discussion point.
将这一点用于关于上帝存在的争论：“上帝存在”常被批评为不可证伪，因此不是科学主张——但或许仍有意义。这是 SQA 常考的讨论点。
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