📚 KS3 Cambridge Philosophy: Formula & Theorem Quick Reference | KS3 Cambridge 哲学：公式定理速查手册
Welcome to the KS3 Cambridge Philosophy Formulae and Theorems Quick Reference. Philosophy sharpens your thinking about existence, knowledge, values and reasoning. Just as mathematics uses formulae and theorems to capture patterns, this handbook presents essential philosophical principles, argument structures, and logical laws in a condensed, easy‑to‑recall format. Use it to revise key ideas, spot invalid moves, and strengthen your own arguments.
欢迎使用 KS3 剑桥哲学公式定理速查手册。哲学能磨砺你对存在、知识、价值与推理的思考。就像数学用公式和定理来捕捉模式一样，这本手册将核心的哲学原则、论证结构和逻辑规律浓缩成易于记忆的形式。用它来复习关键概念、识别无效推论，并让你自己的论证更有力。
1. Propositional Logic & Truth‑Functional Connectives | 命题逻辑与真值函项连接词
Propositional logic is the foundation of clear reasoning. It replaces whole statements with single letters (P, Q, R) and uses connectives to build compound propositions.
命题逻辑是清晰推理的基础。它用单个字母（P、Q、R）代替整个陈述，并用连接词构造复合命题。
The five core truth‑functional connectives and their Unicode symbols are: Negation (¬), Conjunction (∧), Disjunction (∨), Material Implication (→) and Biconditional (↔).
五个核心真值函项连接词及其 Unicode 符号为：否定（¬）、合取（∧）、析取（∨）、实质蕴含（→）和等值（↔）。
A truth table displays every possible combination of truth values for the atomic sentences and determines the truth value of the compound formula.
真值表展示原子命题所有可能的真值组合，并确定复合公式的真值。
| P | Q | P → Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
The table makes it easy to see that an implication is false only when the antecedent (P) is true and the consequent (Q) is false.
该表让人一目了然地看出：只有当前件（P）为真、后件（Q）为假时，蕴含才为假。
Remember these equivalences which often appear in arguments: P → Q is logically equivalent to ¬P ∨ Q, and ¬(P ∧ Q) is equivalent to ¬P ∨ ¬Q (De Morgan’s Law).
记住论证中常见的等价式：P → Q 逻辑上等价于 ¬P ∨ Q，而 ¬(P ∧ Q) 等价于 ¬P ∨ ¬Q（德摩根律）。
2. Valid Deductive Argument Forms | 有效演绎论证形式
A deductive argument is valid if the truth of its premises guarantees the truth of its conclusion. The following are fundamental valid forms, often treated as “theorems” of natural deduction.
如果一个演绎论证的前提真实能确保结论真实，它就是有效的。下面的基本有效形式常被视为自然演绎的“定理”。
Modus Ponens (Affirming the Antecedent): P → Q, P ⊢ Q
Modus Ponens (肯定前件)：如果 P 则 Q，P 成立，所以 Q 一定成立。
Modus Tollens (Denying the Consequent): P → Q, ¬Q ⊢ ¬P
Modus Tollens (否定后件)：如果 P 则 Q，非 Q 成立，所以非 P 一定成立。
Hypothetical Syllogism: P → Q, Q → R ⊢ P → R
假言三段论：如果 P 则 Q，如果 Q 则 R，所以如果 P 则 R。
Disjunctive Syllogism: P ∨ Q, ¬P ⊢ Q
选言三段论：要么 P 要么 Q，非 P 成立，所以 Q 成立。
Whenever you evaluate an argument, check whether its structure matches one of these valid templates. Any deviation may hide a fallacy.
每当你评估一个论证时，检查其结构是否符合这些有效模板之一。任何偏离都可能隐藏着谬误。
3. Categorical Propositions & Syllogisms | 直言命题与三段论
Categorical logic, developed by Aristotle, deals with propositions about classes of things. The four standard forms are labelled A, E, I, O.
由亚里士多德发展的直言逻辑处理有关事物类别的命题。四种标准形式标记为 A、E、I、O。
- A (Universal Affirmative): All S are P. (全称肯定：所有 S 都是 P。)
- E (Universal Negative): No S are P. (全称否定：没有 S 是 P。)
- I (Particular Affirmative): Some S are P. (特称肯定：有些 S 是 P。)
- O (Particular Negative): Some S are not P. (特称否定：有些 S 不是 P。)
A syllogism consists of two premises and a conclusion, each a categorical proposition. The validity of a syllogism depends on its mood and figure.
一个三段论由两个前提和一个结论组成，每个都是直言命题。三段论的有效性取决于其式（mood）和格（figure）。
| Name | Mood | Pattern |
|---|---|---|
| Barbara | AAA | All M are P. All S are M. ∴ All S are P. |
| Celarent | EAE | No M are P. All S are M. ∴ No S are P. |
| Darii | AII | All M are P. Some S are M. ∴ Some S are P. |
| Ferio | EIO | No M are P. Some S are M. ∴ Some S are not P. |
Learning these patterns helps you spot incomplete or improperly structured syllogisms in everyday reasoning.
学习这些模式能帮助你在日常推理中识别不完整或结构不当的三段论。
4. The Three Laws of Thought | 思维三大定律
Three fundamental principles, often called the Laws of Thought, have been treated as axioms of rational thinking since Aristotle.
自亚里士多德以来，三项基本原则——常被称为思维定律——被视作理性思维的“公理”。
Law of Identity: Each thing is identical with itself. (A = A)
同一律：每个事物都与其自身同一。 (A = A)
Law of Non‑Contradiction: No statement can be both true and false at the same time. ¬(P ∧ ¬P)
矛盾律：没有陈述可以同时为真且为假。 ¬(P ∧ ¬P)
Law of Excluded Middle: Every statement is either true or false; there is no middle truth value. (P ∨ ¬P)
排中律：每一个陈述要么为真，要么为假；不存在中间真值。 (P ∨ ¬P)
These laws underpin coherent dialogue and scientific inquiry. A claim that violates them is usually regarded as logically impossible.
这些定律是连贯对话和科学探究的基础。违反这些定律的主张通常被视为逻辑上不可能。
5. Common Logical Fallacies | 常见逻辑谬误
A fallacy is a pattern of reasoning that is logically invalid yet psychologically persuasive. Recognising fallacies is crucial for critical thinking. Here are some of the most frequent ones, presented as flawed “formulae”.
谬误是一种在逻辑上无效但在心理上具有说服力的推理模式。识别谬误对批判性思维至关重要。以下是一些最常见谬误，以有缺陷的“公式”形式呈现。
Ad Hominem (Against the Person): Instead of attacking the argument, you attack the person making it.
人身攻击：不攻击论点本身，而是攻击提出论点的人。
Fallacy Formula: Person X argues P. Person X is flawed. Therefore, P is false.
谬误公式：某人 X 主张 P。X 有缺陷。因此 P 是假的。
Appeal to Authority (Argumentum ad Verecundiam): Using an authority figure’s opinion in a field outside their expertise as proof.
诉诸权威：将权威人物在其专业领域外的意见当作证据。
Fallacy Formula: Authority A says Q. Therefore, Q is true.
谬误公式：权威 A 说 Q。因此 Q 为真。
Straw Man: Misrepresenting an opponent’s position to make it easier to attack.
稻草人：歪曲对手的立场，使之更容易被攻击。
Fallacy Formula: Person Y holds position R. I distort R into a weaker version S. I refute S. Therefore, R is refuted.
谬误公式：某人 Y 持立场 R。我把 R 歪曲为较弱的版本 S。我驳倒 S。因此 R 被驳倒。
False Dilemma (Either/Or): Presenting only two options when more exist.
假两难：只呈现两种选项，而实际上存在更多选项。
Fallacy Formula: Either A or B must be true. Not A. Therefore, B must be true. (When other possibilities C, D exist.)
谬误公式：要么 A 要么 B 必真。非 A。因此 B 必真。（而存在其他可能性 C、D。）
Detecting these fallacies requires you to focus on the logical structure, not the emotional colouring of the language.
要识别这些谬误，你必须关注逻辑结构，而非语言的情感色彩。
6. The JTB Analysis of Knowledge | 知识的 JTB 分析
For centuries, philosophers have defined knowledge as “justified true belief”. This is known as the JTB analysis, often treated as the classical “formula” for propositional knowledge.
几个世纪以来，哲学家们将知识定义为“得到辩护的真信念”。这就是著名的 JTB 分析，常被视为命题知识的经典“公式”。
Knowledge Formula: S knows that p if and only if (i) p is true; (ii) S believes that p; (iii) S is justified in believing that p.
知识公式：S 知道 p，当且仅当 (i) p 为真；(ii) S 相信 p；(iii) S 有理由相信 p。
In symbolic form, we can write: Ksp ↔ (Tp ∧ Bsp ∧ Jsp). Each component is necessary and together they are thought to be sufficient.
用符号可以写作：Ksp ↔ (Tp ∧ Bsp ∧ Jsp)。每个成分都是必要的，合在一起则被认为是充分的。
Gettier cases, however, challenged the sufficiency of JTB by showing situations where someone has a justified true belief but still does not seem to have knowledge. Modern epistemology continues to refine the “formula”.
然而，盖梯尔案例质疑了 JTB 的充分性，它展示出某人拥有得到辩护的真信念却似乎不拥有知识的情形。现代认识论仍在不断修订这一“公式”。
7. Ethical Formulae: Utilitarianism & Deontology | 伦理公式：功利主义与义务论
Ethical theories often provide a decision procedure that can be expressed as a rule or formula. Two major frameworks are utilitarianism and Kantian deontology.
伦理理论通常给出一种决策程序，可用规则或公式来表达。两大框架是功利主义和康德式义务论。
Utilitarian Principle (Greatest Happiness): An action is right if and only if it produces the greatest happiness for the greatest number.
功利原则（最大幸福）：一个行为是正当的，当且仅当它能为最大多数人带来最大幸福。
Utilitarian Formula: Right(A) ↔ Utility(A) ≥ Utility(any alternative action)
功利公式：正当(A) ↔ 效用(A) ≥ 效用(任何一个其他可选行为)
Kant’s Categorical Imperative (Universal Law Formula): “Act only according to that maxim whereby you can at the same time will that it should become a universal law.”
康德的定言命令（普遍法则公式）：“只依据你能够同时意愿它成为一项普遍法则的准则去行动。”
Kantian Formula: An action is morally permissible ↔ its maxim can be consistently universalised without contradiction.
康德式公式：一个行为在道德上是允许的 ↔ 其准则可无矛盾地一贯普遍化。
These formulae help you evaluate moral dilemmas by applying either the calculation of consequences or the test of universalisability.
这些公式通过适用后果计算或普遍化检验，帮助你评估道德困境。
8. Philosophical Heuristics: Ockham’s Razor & Hume’s Law | 哲学启发法：奥卡姆剃刀与休谟法则
Some principles are not formal theorems but powerful heuristics that guide philosophical analysis. Two of the most important are Ockham’s Razor and Hume’s Law.
有些原则并非形式定理，却是指导哲学分析的强大启发法。其中最重要的两条是奥卡姆剃刀和休谟法则。
Ockham’s Razor (Principle of Parsimony): Entities should not be multiplied beyond necessity. Among competing explanations, the one with the fewest assumptions is usually preferable.
奥卡姆剃刀（简约原则）：若无必要，勿增实体。在相互竞争的解释中，假定最少的那一个通常更可取。
Simplified Formula: If theories T₁ and T₂ explain the data equally well, and T₁ posits fewer entities, prefer T₁.
简化公式：如果理论 T₁ 和 T₂ 同等出色地解释数据，且 T₁ 设定的实体较少，则优先选择 T₁。
Hume’s Law (The Is‑Ought Gap): You cannot validly deduce an “ought” statement (a value judgement) solely from “is” statements (factual premises).
休谟法则（实然-应然鸿沟）：你不能仅仅从“是”陈述（事实前提）有效地推出“应当”陈述（价值判断）。
Hume’s Formula: Premises containing only ‘is’ statements do not logically entail an ‘ought’ conclusion. (Is ⊭ Ought)
休谟公式：仅包含“是”陈述的前提并不逻辑蕴含“应当”结论。（Is ⊭ Ought）
When reading moral arguments, always check whether an evaluative conclusion sneaks in without an evaluative premise.
在阅读道德论证时，一定要检查是否在缺少评价性前提的情况下偷偷引入了评价性结论。
9. Theories of Mind and Body | 心身理论
The mind–body problem asks how mental states relate to physical states. Several “equation‑style” formulations capture the core claims of rival theories.
心身问题探讨心理状态如何与物理状态相关联。几种“方程式风格”的表述抓住了不同理论的核心主张。
Identity Theory (Reductive Physicalism): Mental states are numerically identical to brain states.
同一论（还原物理主义）：心理状态与大脑状态在数值上是同一的。
Identity Formula: Mental state M = Brain state B
同一公式：心理状态 M = 大脑状态 B
Substance Dualism (Cartesian): Mind and body are two fundamentally different substances. Mental events are not identical
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