📚 Pre-U Cambridge Philosophy: Formula and Theorem Quick Reference | Pre-U Cambridge 哲学：公式定理速查手册
Philosophy may not be a subject of numerical formulas, but it is rich in formal principles, logical equivalences, ethical maxims, and definitional theorems that function just like formulas in science. This quick reference handbook collects the most important ‘formulas and theorems’ for the Pre-U Cambridge Philosophy syllabus, covering propositional and predicate logic, normative ethical principles, the structure of knowledge, and key arguments in metaphysics. Each entry is presented as a concise statement that captures the logical or conceptual form, followed by a brief pairing of English and Chinese explanations.
哲学或许不是一门充满数字公式的学科，但它拥有形式化原则、逻辑等价式、伦理格言以及定义定理，其功能正如科学中的公式。这本速查手册汇集了Pre-U剑桥哲学课程中最重要的“公式与定理”，涵盖命题逻辑与谓词逻辑、规范伦理原则、知识的结构以及形而上学中的核心论证。每一个条目都以简明陈述呈现其逻辑或概念形态，并配以英中双语解释。
1. Propositional Logic Equivalences | 命题逻辑等价公式
The basic substitution rules of propositional logic allow us to replace one formula with an equivalent one. These equivalences are the building blocks of formal proof in philosophical argumentation.
命题逻辑的基本替换规则使我们能够用一个公式等价地替换另一个。这些等价式是哲学论证中形式证明的基石。
Double Negation: ¬¬P ⇔ P
双重否定律： 非非P等价于P。
De Morgan’s Laws: ¬(P ∧ Q) ⇔ (¬P ∨ ¬Q) ; ¬(P ∨ Q) ⇔ (¬P ∧ ¬Q)
德摩根律： 并非(P与Q)等价于非P或非Q；并非(P或Q)等价于非P且非Q。
Commutativity: (P ∧ Q) ⇔ (Q ∧ P) ; (P ∨ Q) ⇔ (Q ∨ P)
交换律： P与Q 等价于 Q与P；P或Q 等价于 Q或P。
Associativity: (P ∧ (Q ∧ R)) ⇔ ((P ∧ Q) ∧ R) ; (P ∨ (Q ∨ R)) ⇔ ((P ∨ Q) ∨ R)
结合律： P且(Q且R) 等价于 (P且Q)且R；P或(Q或R) 等价于 (P或Q)或R。
Material Implication: (P → Q) ⇔ (¬P ∨ Q)
实质蕴涵定义： 如果P则Q 等价于 非P或Q。
Contraposition: (P → Q) ⇔ (¬Q → ¬P)
换质位换： 如果P则Q 等价于 如果非Q则非P。
2. Propositional Inference Rules | 命题逻辑推理规则
Inference rules are the ‘theorems of deduction’ that allow us to derive conclusions from premises. Mastering these patterns is essential for evaluating validity in philosophical logic.
推理规则是“演绎定理”，使我们能够从前提得出结论。掌握这些模式对评估哲学逻辑中的有效性至关重要。
Modus Ponens: P → Q, P ⊢ Q
肯定前件式： 如果P则Q，P，因此Q。
Modus Tollens: P → Q, ¬Q ⊢ ¬P
否定后件式： 如果P则Q，非Q，因此非P。
Hypothetical Syllogism: P → Q, Q → R ⊢ P → R
假言三段论： 如果P则Q，如果Q则R，因此如果P则R。
Disjunctive Syllogism: P ∨ Q, ¬P ⊢ Q
选言三段论： P或Q，非P，因此Q。
Constructive Dilemma: (P → Q) ∧ (R → S), P ∨ R ⊢ Q ∨ S
构造性二难推理： (如果P则Q)且(如果R则S)，P或R，因此Q或S。
Conjunction: P, Q ⊢ P ∧ Q
合取引入： P，Q，因此P且Q。
Simplification: P ∧ Q ⊢ P
合取消去： P且Q，因此P。
3. Predicate Logic Quantifier Rules | 谓词逻辑量词规则
Predicate logic extends propositional logic with quantifiers and individual variables. The instantiation and generalization rules are the formal core of arguments about ‘all’ and ‘some’.
谓词逻辑通过量词和个体变量扩展了命题逻辑。实例化与概括规则是关于“所有”和“有些”论证的形式核心。
Universal Instantiation (UI): ∀x Fx ⊢ Fa (where ‘a’ is any specific individual)
全称实例化： 所有x是F，因此a是F（a为任意特定个体）。
Universal Generalisation (UG): From a proof of Fa for an arbitrary ‘a’, infer ∀x Fx.
全称概括： 如果可证明对任意个体a有Fa，则推出所有x是F。
Existential Instantiation (EI): ∃x Fx ⊢ Fa (where ‘a’ is a new constant not previously used)
存在实例化： 存在x是F，因此a是F（a为先前未使用的新常元）。
Existential Generalisation (EG): Fa ⊢ ∃x Fx
存在概括： a是F，因此存在x是F。
Quantifier Negation Equivalences: ¬∀x Fx ⇔ ∃x ¬Fx ; ¬∃x Fx ⇔ ∀x ¬Fx
量词否定等价： 并非所有x是F 等价于 存在x不是F；不存在x是F 等价于 所有x不是F。
4. Kant’s Categorical Imperative Formulas | 康德绝对命令公式
Kant’s moral philosophy rests on the Categorical Imperative, which he formulated in several equivalent ways. These formulations act as ethical ‘formulas’ for testing the morality of maxims.
康德的道德哲学建立在绝对命令之上，他提供了几种等价的表述。这些表述就像检验行为准则道德性的伦理“公式”。
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.
普遍法则公式： 只按照你同时能够愿意它成为一项普遍法则的那个准则去行动。
Formula of the End in Itself: 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 merely as a means.
目的自身公式： 你的行动，要把你自己人身中的人性和他人人身中的人性，在任何时候都同样看作是目的，永远不能只看作是手段。
Formula of Autonomy: Act so that through your maxims you could be a legislator of universal laws.
自律公式： 行动时，你的准则应能通过你的意志成为普遍的自然法则。
Kingdom of Ends Formula: All maxims must harmonise with a possible kingdom of ends as a kingdom of nature.
目的王国公式： 每个有理性的存在者都应当将自己视为在普遍法则下立法的成员。
5. Utilitarian Calculus Principles | 功利主义计算原则
Classical utilitarianism proposes a ‘felicific calculus’ to measure moral value. The central formula is the maximisation of aggregate happiness.
古典功利主义提出了一套“幸福计算”来衡量道德价值。其核心公式是最大化幸福总量。
The Greatest Happiness Principle: An action is right if and only if it tends to produce the greatest happiness of the greatest number of sentient beings affected by it.
最大幸福原则： 一个行为是正当的，当且仅当它倾向于在所有受影响的众生中产生最大数量的幸福。
Bentham’s Hedonic Calculus Elements: Intensity, Duration, Certainty, Propinquity, Fecundity, Purity, Extent.
边沁的苦乐计算要素： 强度、持续时间、确定性、远近性、丰富性、纯度、广度。
Mill’s Higher and Lower Pleasures: Quality of pleasure must be weighed above mere quantity; intellectual pleasures carry higher utility.
密尔的高等与低等快乐： 快乐的质量必须置于数量之上；理智的快乐具有更高的效用。
Act Utilitarianism Decision Rule: For any act A, A is morally right iff Σ U(A) > Σ U(alternative) for all affected individuals.
行为功利主义决策规则： 对于任何行为A，A在道德上正确，当且仅当对所有受影响个体，A带来的效用总和大于其他选项的效用总和。
6. The JTB Account of Knowledge | 知识的三条件定义（JTB）
Since Plato’s Theaetetus, knowledge has been analysed as justified true belief. This formula became standard in epistemology until Gettier’s counterexamples.
自柏拉图的《泰阿泰德篇》以来，知识一直被分析为得到证成的真信念。这一定义成为认识论的标准公式，直到葛梯尔提出了反例。
JTB Definition: S knows that p if and only if (i) p is true, (ii) S believes that p, and (iii) S is justified in believing that p.
JTB定义： S知道p，当且仅当（i）p为真，（ii）S相信p，且（iii）S相信p是得到证成的。
Gettier Counterexample Structure: A person has a justified true belief that is accidentally true – the justification is not appropriately connected to the truth.
葛梯尔反例结构： 某人拥有一个得到证成的真信念，但这信念只是偶然为真——证成与真相之间缺乏适当的关联。
No False Lemmas Condition: S knows that p iff JTB plus S’s belief is not inferred from any false proposition.
无假前提条件： S知道p，当且仅当JTB成立且S的信念未从任何假命题中推出。
Reliability Condition: S knows that p iff S’s true belief was produced by a reliable cognitive process.
可靠性条件： S知道p，当且仅当S的真信念是由可靠的认知过程产生的。
7. Identity Theory of Mind | 心灵同一论公式
The mind-brain identity theory claims that mental states are strictly identical to physical brain states. This identity thesis is often expressed as a type-type identity formula.
心脑同一论主张心理状态与物理脑状态严格同一。这一同一性论题常被表述为类型-类型同一公式。
Type Identity Formula: For any mental state type M, there is a physical state type P such that M = P.
类型同一公式： 对于任何心理状态类型M，存在一个物理状态类型P，使得M等于P。
Token Identity: Each particular mental event token is identical with some physical event token, without requiring type correlation.
标记同一： 每一个特定的心理事件标记等同于某个物理事件标记，但不要求类型对应。
Leibniz’s Law Application: If mental state = brain state, then everything true of the mental state is true of the brain state and vice versa.
莱布尼茨律的应用： 如果心理状态等于脑状态，那么所有对于心理状态为真的描述对于脑状态也为真，反之亦然。
Multiple Realisability Objection: The same mental state can be realised by different physical substrates across species, undermining type identity.
多重可实现性反驳： 同一种心理状态可以在不同物种中由不同的物理基质实现，这削弱了类型同一论。
8. Deductive-Nomological Model of Explanation | 科学解释的演绎-律则模型
Hempel’s covering law model expresses scientific explanation as a deductive argument. The ‘theorem’ of explanation shows how an event is entailed by laws and initial conditions.
亨佩尔的覆盖律模型将科学解释表达为一种演绎论证。解释的“定理”展示了一个事件如何由定律和初始条件所蕴含。
Explanans: L₁, L₂, … Lₙ (General Laws) & C₁, C₂, … Cₙ (Particular Conditions) ⊢ E (Explanandum event)
解释项： 一般定律L₁到Lₙ 与 特定条件C₁到Cₙ 演绎推出 被解释项E。
Conditions for Adequacy: The argument must be deductively valid; the laws must be essential for the derivation; the explanans must be true.
充分条件： 该论证必须在演绎上有效；定律对推导必不可少；解释项必须为真。
Symmetry Thesis: Explanation and prediction have the same logical structure; if the conditions are known before the event, the same argument yields a prediction.
对称性论题： 解释与预测具有相同的逻辑结构；如果条件在事件前已知，同一论证可生成预测。
Inductive-Statistical Variant: When laws are statistical, the conclusion is shown to be probable, not logically necessary.
归纳-统计变体： 当定律为统计性时，结论显示出高度概然性，而不是逻辑必然性。
9. John Rawls’s Principles of Justice | 罗尔斯正义原则公式
Rawls’s theory of justice as fairness produces two lexically ordered principles. These can be formalised as a decision procedure for a just basic structure of society.
罗尔斯的“作为公平的正义”理论产生了两个按词典顺序排列的原则。这些原则可以形式化为一种关于社会基本结构的正义决策程序。
First Principle (Equal Liberty): Each person has an equal right to a fully adequate scheme of equal basic liberties, compatible with a similar scheme for all.
第一原则（平等自由）： 每个人对与所有人所拥有的最广泛平等基本自由体系相容的类似自由体系，都应有一种平等的权利。
Second Principle (Difference Principle): Social and economic inequalities are to satisfy two conditions: (a) they must be attached to offices and positions open to all under conditions of fair equality of opportunity; (b) they must be to the greatest benefit of the least advantaged.
第二原则（差别原则）： 社会和经济的不平等应这样安排：（a）在与公平的机会平等的条件下，所有职务和地位向所有人开放；（b）它们必须最有利于社会中处于最不利地位的人。
Lexical Priority Rule: Principle 1 is strictly prior to Principle 2; liberty can only be restricted for the sake of liberty itself.
词典式优先规则： 第一原则绝对优先于第二原则；自由只能为了自由本身的缘故而受到限制。
Original Position Formula: Principles of justice are those that rational agents would choose behind a veil of ignorance, maximising the minimum (maximin rule).
原初状态公式： 正义原则是那些理性行动者在无知之幕背后会选择的，它们遵循最大最小化规则。
10. Ontological Argument Formalisation | 本体论论证的形式化公式
The ontological argument is often presented as a deductive proof of God’s existence from the concept of God alone. Its ‘formula’ has been refined from Anselm to Plantinga.
本体论论证常常被表述为仅从上帝的概念中演绎出上帝存在的证明。其“公式”从安瑟伦到普兰丁格不断完善。
Anselm’s Formula: God is that than which nothing greater can be conceived. That which exists in reality is greater than that which exists only in the understanding. Therefore, God must exist in reality.
安瑟伦公式： 上帝是那无法设想比之更伟大者的存在者。存在于现实中比只存在于理智中更为伟大。因此，上帝必然存在于现实中。
Descartes’s Perfection Formula: Existence is a perfection; God has all perfections; therefore God exists.
笛卡尔的完美性公式： 存在是一种完美性；上帝拥有一切完美性；因此上帝存在。
Modal Ontological Argument: It is possible that a maximally great being exists. Therefore, a maximally great being exists in some possible world. If a being is maximally great, it exists in every possible world. Therefore, it exists in the actual world.
模态本体论公式： 一个最大伟大实体的存在是可能的。因此，它在某个可能世界中存在。如果某实体是最大伟大的，那它存在于每一个可能世界。因此，它存在于现实世界。
Kant’s Objection: Existence is not a real predicate; it does not add to the concept of a thing. The ontological argument therefore fails.
康德的批评： 存在不是一个实际的谓词；它并不增加事物概念的内容。因此本体论论证失败。
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