Year 12 SQA Philosophy: Formula & Theorem Quick Reference Handbook | Year 12 SQA 哲学：公式定理速查手册

📚 Year 12 SQA Philosophy: Formula & Theorem Quick Reference Handbook | Year 12 SQA 哲学：公式定理速查手册

Philosophy may seem like a discipline of nuanced essays, but beneath the surface lie rigorous patterns — logical formulas, argument forms, and conceptual "theorems" that provide structure to critical thinking. This quick-reference handbook distils the essential "formulas" and "theorems" you need for SQA Higher Philosophy, from propositional logic to normative ethics. Each entry presents a key idea as a memorable formula, followed by its explanation and application.

哲学看似一门细腻的论文写作学科，但在表面之下隐藏着严密的模式——逻辑公式、论证形式以及概念性的“定理”，它们为批判性思维提供了结构。这本速查手册提炼了你在 SQA Higher Philosophy 中所需的基本“公式”和“定理”，涵盖从命题逻辑到规范伦理学的各个方面。每一个词条将一个核心思想呈现为易记的公式，随后给出解释与应用。

1. Propositional Logic: Truth-Functional Connectives | 命题逻辑：真值函项连接词

In arguments in action, we represent simple statements with letters and combine them using logical connectives. Each connective has a precise truth-table definition, forming the "calculus" of reasoning.

在 “行动中的论证 ”部分，我们用字母表示简单陈述并使用逻辑连接词将它们组合起来。每个连接词都有精确的真值表定义，构成了推理的“演算”。

Key symbols and their truth conditions:

关键符号及其真值条件：

• ¬P (negation): true when P is false.
  中文：¬P (否定)：当 P 为假时为真。
• P ∧ Q (conjunction): true only when both P and Q are true.
  中文：P ∧ Q (合取)：仅当 P 和 Q 都为真时为真。
• P ∨ Q (disjunction): true when at least one disjunct is true (inclusive 'or').
  中文：P ∨ Q (析取)：当至少一个析取项为真时为真（相容“或”）。
• P → Q (conditional): false only when P is true and Q is false; otherwise true.
  中文：P → Q (条件句)：仅当 P 为真且 Q 为假时为假；其余情况为真。
• P ↔ Q (biconditional): true when P and Q have the same truth value.
  中文：P ↔ Q (双条件)：当 P 和 Q 真值相同时为真。

Formula: (P → Q) ∧ ¬Q ⊢ ¬P (Modus Tollens example)

公式：(P → Q) ∧ ¬Q ⊢ ¬P （否定后件推理示例）

Memorising these truth tables allows you to test validity of propositional arguments by constructing full truth tables or identifying counterexamples.

记住这些真值表后，你就能通过构造完整真值表或找出反例来检验命题论证的有效性。

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2. Rules of Inference for Deductive Arguments | 演绎论证的推理规则

Valid deductive arguments follow strict patterns called rules of inference. Knowing these "theorems" of logic helps you both construct and evaluate arguments quickly.

有效的演绎论证遵循称为推理规则的严格模式。了解这些逻辑“定理”有助于你快速构建和评估论证。

Rule (English)	Formal Representation	规则 (中文)
Modus Ponens (MP)	P → Q, P ⊢ Q	肯定前件
Modus Tollens (MT)	P → Q, ¬Q ⊢ ¬P	否定后件
Hypothetical Syllogism (HS)	P → Q, Q → R ⊢ P → R	假言三段论
Disjunctive Syllogism (DS)	P ∨ Q, ¬P ⊢ Q	选言三段论

Each of these is an argument form where, if the premises are true, the conclusion must be true. They are the bedrock of 'deductive validity', a central concept in the SQA unit Arguments in Action.

这些论证形式中，如果前提为真，结论必然为真。它们是“演绎有效性”的基石，是 SQA 课程“行动中的论证”单元的核心概念。

In your exam, you may be asked to identify the logical form of an argument and assess its validity.

考试中，你可能需要识别论证的逻辑形式并评估其有效性。

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3. Logical Equivalences & Replacement Theorems | 逻辑等价与替换定理

Complex statements can often be simplified or rewritten using known equivalences. These "replacement theorems" allow you to transform arguments without changing truth values.

复杂的陈述常常可以用已知的等价关系来简化或重写。这些“替换定理”使你能够在不改变真值的情况下转换论证。

Key equivalences to memorise:

需记住的关键等价关系：

• De Morgan's Laws: ¬(P ∧ Q) ≡ (¬P ∨ ¬Q); ¬(P ∨ Q) ≡ (¬P ∧ ¬Q)
  中文：德摩根律：¬(P ∧ Q) ≡ (¬P ∨ ¬Q)；¬(P ∨ Q) ≡ (¬P ∧ ¬Q)
• Material Implication: (P → Q) ≡ (¬P ∨ Q)
  中文：实质蕴含：(P → Q) ≡ (¬P ∨ Q)
• Double Negation: ¬¬P ≡ P
  中文：双重否定：¬¬P ≡ P
• Contraposition: (P → Q) ≡ (¬Q → ¬P)
  中文：换质位：(P → Q) ≡ (¬Q → ¬P)

These equivalences are essential for constructing reductio ad absurdum proofs and for evaluating the logical structure of philosophical arguments.

这些等价关系对于构建归谬法

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