📚 Year 10 SQA Philosophy: Quick Reference Handbook of Formulas and Theorems | SQA 十年级哲学：公式定理速查手册
This handbook gathers the essential logical formulas, rules of inference, and critical concepts that underpin the SQA Year 10 Philosophy course. Use it as a quick reference to sharpen your argument analysis, understand valid reasoning, and avoid common fallacies. Mastering these 'formulas' will give you a solid toolkit for evaluating philosophical arguments with precision.
本手册汇集了支撑 SQA 十年级哲学课程的核心逻辑公式、推理规则与关键概念。你可以将它作为速查工具，用于强化论证分析、理解有效推理并规避常见谬误。掌握这些“公式”将为你精准评判哲学论证提供坚实的工具箱。
1. Argument Form and Structure | 论证的形式与结构
An argument is a set of statements where some (premises) are intended to support another (conclusion). The structural formula is: P₁, P₂, ..., Pₙ ⊢ C, where each P represents a premise and C the conclusion. Recognizing this structure is the first step in logical analysis.
论证是一组陈述，其中一些陈述（前提）旨在支持另一个陈述（结论）。其结构公式为：P₁, P₂, ..., Pₙ ⊢ C，每个 P 代表一个前提，C 代表结论。识别这一结构是逻辑分析的第一步。
A well-formed argument makes the inferential link explicit. In Standard Form, premises are listed above a line and the conclusion below, separated by the symbol ∴ (therefore). For instance: All humans are mortal; Socrates is a human; ∴ Socrates is mortal.
一个结构良好的论证会清晰地表明推论联系。在标准形式中，前提列出在上方，结论在下方，用符号 ∴（所以）隔开。例如：所有人都会死；苏格拉底是人；∴ 苏格拉底会死。
2. Deductive vs Inductive Arguments | 演绎论证与归纳论证
Deductive arguments aim to provide conclusive support: if the premises are true, the conclusion must be true. The relationship is necessity. Inductive arguments aim to provide probable support: if the premises are true, the conclusion is likely to be true. The relationship is probability.
演绎论证旨在提供决定性的支持：如果前提为真，结论必然为真。这种关系体现的是必然性。归纳论证旨在提供概然性的支持：如果前提为真，结论很可能为真。这种关系体现的是或然性。
A quick check: does the conclusion follow necessarily, or is it just a likelihood? Deduction moves from general to specific (often), induction from specific to general. Many philosophical proofs rest on deduction; scientific hypotheses often rely on induction.
快速检验：结论是必然得出的，还是仅仅具有可能性？演绎推理（通常）从一般走向特殊，归纳推理则从特殊走向一般。许多哲学证明依赖于演绎，科学假设则常常依靠归纳。
3. Validity, Soundness, Strength, and Cogency | 有效性、可靠性、强度与说服力
Validity is a property of deductive arguments: an argument is valid if it is impossible for the premises to be true and the conclusion false. Soundness requires both validity and true premises. Formula: Sound = Valid + True Premises.
有效性是演绎论证的一种性质：一个论证有效，当且仅当不可能前提为真而结论为假。可靠性既要求有效，也要求前提真实。公式：可靠 = 有效 + 真实前提。
For inductive arguments, strength measures how well the premises raise the probability of the conclusion. A cogent argument is strong and has true premises. Remember, a valid argument can have false premises; soundness is the gold standard.
对于归纳论证，强度衡量前提在多大程度上提升了结论为真的概率。有说服力的论证既具有强度，又拥有真实的前提。请记住，一个有效的论证可以拥有假前提；可靠性才是黄金标准。
4. Propositional Logic Connectives and Truth Tables | 命题逻辑连接词与真值表
Propositions are combined using logical connectives. The five basic connectives are: ¬ (negation, 'not'), ∧ (conjunction, 'and'), ∨ (disjunction, 'or'), → (conditional, 'if...then'), ↔ (biconditional, 'if and only if'). Their truth-functional definitions are summarised below.
命题借助逻辑连接词进行组合。五种基本连接词为：¬（否定，“并非”）、∧（合取，“且”）、∨（析取，“或”）、→（条件，“如果……则”）和 ↔（双条件，“当且仅当”）。它们的真值函数定义总结如下。
| P | Q | ¬P | P ∧ Q | P ∨ Q | P → Q | P ↔ Q |
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | F |
| F | T | T | F | T | T | F |
| F | F | T | F | F | T | T |
se only when P is true and Q is false. A biconditional is true exactly when both components share the same truth value. These tables are your toolkit for testing argument validity.
关键要义：条件句 P → Q 仅在 P 真而 Q 假时为假。双条件句恰好当两个组成部分真值相同时为真。这些真值表是你检验论证有效性的工具箱。
5. Logical Equivalences and Tautologies | 逻辑等价式与重言式
A tautology is a formula that is always true regardless of the truth values of its components, e.g., P ∨ ¬P (the law of excluded middle). A contradiction is always false, e.g., P ∧ ¬P. Logical equivalences are two formulas that have identical truth tables. Key equivalences (De Morgan's Laws): ¬(P ∧ Q) ≡ ¬P ∨ ¬Q and ¬(P ∨ Q) ≡ ¬P ∧ ¬Q.
重言式是不论其组成部分的真值如何，总是为真的公式，例如 P ∨ ¬P（排中律）。矛盾式则总是为假，例如 P ∧ ¬P。逻辑等价式是指两个具有完全相同的真值表的公式。关键等价式（德摩根定律）：¬(P ∧ Q) ≡ ¬P ∨ ¬Q 和 ¬(P ∨ Q) ≡ ¬P ∧ ¬Q。
Other useful equivalences include P → Q ≡ ¬P ∨ Q and P ↔ Q ≡ (P → Q) ∧ (Q → P). These transformations allow you to rewrite propositions and analyse complex arguments more efficiently.
其他有用的等价式包括 P → Q ≡ ¬P ∨ Q 和 P ↔ Q ≡ (P → Q) ∧ (Q → P)。这些变换让你能够改写命题，从而更高效地分析复杂论证。
6. Rules of Inference | 推理规则
Rules of inference are simple, valid argument forms that serve as building blocks for more complex proofs. Memorise these core rules. Modus Ponens (MP): P → Q, P ⊢ Q. Modus Tollens (MT): P → Q, ¬Q ⊢ ¬P. Hypothetical Syllogism (HS): P → Q, Q → R ⊢ P → R.
推理规则是一些简单而有效的论证形式，可作为构建更复杂证明的积木。请熟记以下核心规则。肯定前件（Modus Ponens, MP）：P → Q, P ⊢ Q。否定后件（Modus Tollens, MT）：P → Q, ¬Q ⊢ ¬P。假言三段论（Hypothetical Syllogism, HS）：P → Q, Q → R ⊢ P → R。
Disjunctive Syllogism (DS): P ∨ Q, ¬P ⊢ Q. Constructive Dilemma (CD): (P → Q) ∧ (R → S), P ∨ R ⊢ Q ∨ S. Simplification (Simp): P ∧ Q ⊢ P. Conjunction (Conj): P, Q ⊢ P ∧ Q. Addition (Add): P ⊢ P ∨ Q. These rules form the backbone of natural deduction.
选言三段论（Disjunctive Syllogism, DS）：P ∨ Q, ¬P ⊢ Q。构造性二难（Constructive Dilemma, CD）：(P → Q) ∧ (R → S), P ∨ R ⊢ Q ∨ S。化简律（Simplification, Simp）：P ∧ Q ⊢ P。合取律（Conjunction, Conj）：P, Q ⊢ P ∧ Q。附加律（Addition, Add）：P ⊢ P ∨ Q。这些规则构成了自然演绎的骨架。
7. Common Valid Argument Forms | 常见有效论证形式
Beyond the rules of inference, several argument templates recur in philosophical writing. Recognising them speeds up evaluation. Modus Ponendo Tollens: ¬(P ∧ Q), P ⊢ ¬Q. Destructive Dilemma: (P → Q) ∧ (R → S), ¬Q ∨ ¬S ⊢ ¬P ∨ ¬R.
除了推理规则之外，还有几种论证模板在哲学写作中反复出现。识别它们可以加快评判速度。否定合取式（Modus Ponendo Tollens）：¬(P ∧ Q), P ⊢ ¬Q。破坏性二难（Destructive Dilemma）：(P → Q) ∧ (R → S), ¬Q ∨ ¬S ⊢ ¬P ∨ ¬R。
Reductio ad Absurdum (RAA): Assume the negation of what you want to prove, derive a contradiction, then conclude the original claim. Form: if assuming ¬P leads to Q ∧ ¬Q, then ⊢ P. This is the powerhouse behind many philosophical arguments.
归谬法（Reductio ad Absurdum, RAA）：先假设你想证明的命题的否定，推导出一个矛盾，然后得出结论。形式为：如果假设 ¬P 导致 Q ∧ ¬Q，那么 ⊢ P。这是许多哲学论证背后的强大引擎。
8. Categorical Logic and the Square of Opposition | 直言逻辑与对当关系
Categorical logic deals with classes of things. There are four standard categorical propositions: A (All S are P), E (No S are P), I (Some S are P), O (Some S are not P). The Aristotelian Square of Opposition maps their logical relations: Contradictories (A & O, E & I) cannot both be true and cannot both be false. Contraries (A & E) cannot both be true but can both be false.
直言逻辑处理事物的类。有四种标准的直言命题：A（所有 S 都是 P）、E（没有 S 是 P）、I（有些 S 是 P）、O（有些 S 不是 P）。亚里士多德对当关系图展示了它们之间的逻辑关系：矛盾关系（A 与 O，E 与 I）既不能同真，也不能同假。反对关系（A 与 E）不能同真，但可以同假。
Subcontraries (I & O) can both be true but not both false. Subalternation (A to I, E to O): truth flows downward, falsity flows upward. Knowing these relationships helps in analysing syllogisms and immediate inferences.
下反对关系（I 与 O）可以同真，但不能同假。差等关系（A 到 I，E 到 O）：真值向下传递，假值向上传递。了解这些关系有助于分析三段论与直接推理。
9. Syllogistic Rules and Fallacies | 三段论规则与谬误
A categorical syllogism consists of two premises and a conclusion, involving exactly three terms, each appearing twice. For validity, it must obey these rules: (1) Distributed middle term at least once. (2) If a term is distributed in the conclusion, it must be distributed in the premise. (3) Avoid two negative premises. (4) A negative premise requires a negative conclusion (and vice versa).
一个直言三段论由两个前提和一个结论组成，恰好涉及三个词项，每个词项出现两次。为确保有效，必须遵守以下规则：（1）中项至少要周延一次。（2）如果某个词项在结论中周延，那么它必须在前提中周延。（3）避免两个否定前提。（4）否定前提要求得出否定的结论（反之亦然）。
Violating these rules leads to formal fallacies: Fallacy of the Undistributed Middle, Illicit Major, Illicit Minor, Fallacy of Exclusive Premises, and Fallacy of Drawing an Affirmative Conclusion from a Negative Premise. A quick mnemonic: 'Mood' and 'Figure' determine validity when checked against these rules.
违反这些规则就会导致形式谬误：中项不周延谬误、大项不当周延、小项不当周延、前提排斥谬误以及由否定前提得出肯定结论谬误。一个速记方法：在对照这些规则时，“式”和“格”决定了有效性。
10. Informal Fallacies Quick Reference | 非形式谬误速查
Informal fallacies are mistakes in reasoning that arise from the content of an argument rather than its form. Key types to know: Ad Hominem (attacking the person), Straw Man (misrepresenting an argument to refute it easily), Begging the Question (assuming what you're trying to prove), False Dilemma (restricting options to two when more exist).
非形式谬误是由论证的内容而非其形式引起的推理错误。需要了解的关键类型有：人身攻击（攻击对方个人）、稻草人（歪曲一个论证以便轻易反驳）、乞题（预设了所要证明的结论）、虚假两难（当存在更多选项时却将选择限制为两个）。
Also watch for: Slippery Slope (claiming a small step inevitably leads to extreme outcomes without justification), Appeal to Ignorance (arguing a claim is true because it hasn't been proven false), and Equivocation (using a word in two different senses). Being able to name a fallacy is the first step to defusing it.
同样需要留意的还有：滑坡谬误（在无充分理由的情况下声称一小步会不可避免地导致极端后果）、诉诸无知（因未被证明为假就声称某一主张为真）以及一词多义（在两种不同意义上使用同一个词语）。能够给谬误命名是消除其隐患的第一步。
11. Truth Table Test for Validity | 真值表法检验有效性
To test a deductive argument for validity using truth tables, write the symbolic form of the premises and conclusion. Construct a truth table with all possible truth value combinations for the component propositions. An argument is valid if and only if there is no row where all premises are true and the conclusion is false.
要使用真值表检验一个演绎论证的有效性，先写出前提和结论的符号形式。构建一个包含各组成命题所有可能真值组合的真值表。当且仅当不存在任何一行所有前提均为真而结论为假时，该论证才是有效的。
This mechanical check is especially useful for complex propositional arguments. If you find a counterexample row (premises true, conclusion false), the argument is invalid. For validity proofs, show that whenever the premises are T, the conclusion is forced to be T.
这种机械化的检验方法对于复杂的命题论证特别有用。如果找到一个反例行（前提真而结论假），那么该论证是无效的。而对于有效性的证明，则需要表明每当前提为真时，结论便不得不为真。
12. Brief Introduction to Predicate Logic Notation | 谓词逻辑符号简介
Predicate logic extends propositional logic by analysing the internal structure of statements. two Quantifiers are introduced: ∀ (universal quantifier, 'for all') and ∃ (existential quantifier, 'there exists'). Example: 'All humans are mortal' becomes ∀x (Hx → Mx). 'Some philosophers are wise' becomes ∃x (Px ∧ Wx).
谓词逻辑通过分析语句的内部结构来扩展命题逻辑。这里引入了两个量词：∀（全称量词，“对于所有”）和 ∃（存在量词，“存在”）。例如：“所有人都会死”符号化为 ∀x (Hx → Mx)。“有些哲学家是智慧的”符号化为 ∃x (Px ∧ Wx)。
Important equivalences for quantifier negation: ¬∀x Px ≡ ∃x ¬Px and ¬∃x Px ≡ ∀x ¬Px. This shows how 'not everyone is happy' (¬∀x Hx) is equivalent to 'someone is not happy' (∃x ¬Hx). These simple rules are frequently tested when you evaluate the logical structure of philosophical claims.
量词否定的重要等价式：¬∀x Px ≡ ∃x ¬Px 和 ¬∃x Px ≡ ∀x ¬Px。这说明“并非所有人都是幸福的”（¬∀x Hx）等价于“有些人不是幸福的”（∃x ¬Hx）。这些简单规则在你评估哲学论断的逻辑结构时常常会被考查。
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