📚 IGCSE CAIE Philosophy: Quick Reference Handbook of Formulas and Principles | IGCSE CAIE 哲学：公式定理速查手册
Philosophy may not have equations in the usual sense, but it is built on logical forms, ethical principles, and argument structures that can be presented as concise 'formulas'. This handbook organises the core content of the IGCSE CAIE Philosophy syllabus into a set of quick-reference principles, covering logic, epistemology, ethics, and philosophy of religion. Each entry pairs a formal statement with a clear explanation, designed for rapid revision and memorisation. Master these formulas, and you will be well equipped to analyse unseen passages, construct essays, and evaluate philosophical claims under examination conditions.
哲学或许不像数学那样包含大量方程，但它确实建立在逻辑形式、伦理原则和论证结构之上，这些都可以用简洁的“公式”呈现。本手册将IGCSE CAIE哲学课程的核心内容整理为一套速查原则，涵盖逻辑学、认识论、伦理学与宗教哲学。每一条都以形式化陈述配合清晰解释，专为快速复习和背诵设计。掌握这些公式，你将有能力在考试中准确地分析陌生文段、构建论文并评估哲学主张。
1. Modus Ponens (Affirming the Antecedent) | 肯定前件式
If P then Q. P is true, therefore Q is true. This is the most basic valid argument form in propositional logic, guaranteeing that if the premises are true the conclusion must be true.
如果P那么Q。P为真，因此Q为真。这是命题逻辑中最基本的有效论证形式，它保证了如果前提为真，结论必然为真。
P → Q, P ⊢ Q
- Example: If it is raining, the ground is wet. It is raining. Therefore, the ground is wet.
- 例子：如果下雨，地面就会湿。现在下雨。因此，地面湿了。
2. Modus Tollens (Denying the Consequent) | 否定后件式
If P then Q. Q is false, therefore P is false. This valid form works by rejecting the consequent to derive the negation of the antecedent. It is heavily used in scientific falsification and skeptical arguments.
如果P那么Q。Q为假，因此P为假。这一有效形式通过否定后件来推出前件的否定。它在科学证伪和怀疑论论证中被大量使用。
P → Q, ¬Q ⊢ ¬P
- Example: If the patient has malaria, then the blood test will be positive. The blood test is negative. Therefore, the patient does not have malaria.
- 例子：如果病人患有疟疾，那么血液检查会呈阳性。血液检查为阴性。因此，病人没有疟疾。
3. Hypothetical Syllogism (Chain Argument) | 假言三段论（连锁论证）
If P then Q. If Q then R. Therefore, if P then R. This formula links two conditionals to form a longer chain, extending implication through a shared middle term. It is the backbone of multi-step ethical reasoning and causal analysis.
如果P那么Q。如果Q那么R。因此，如果P那么R。这个公式将两个条件句连接成一个更长的链条，通过共同的中项将蕴含关系延伸出去。它是多步骤伦理推理和因果分析的基础。
(P → Q) ∧ (Q → R) ⊢ P → R
- Example: If I study hard, I will pass the exam. If I pass the exam, I will graduate. Therefore, if I study hard, I will graduate.
- 例子：如果我努力学习，我就会通过考试。如果我通过考试，我就毕业。因此，如果我努力学习，我就能毕业。
4. Disjunctive Syllogism | 选言三段论
Either P or Q. Not P, therefore Q. This pattern eliminates one alternative to infer the other. It is valid only when the disjunction is exclusive and both options are genuinely exhaustive in the context of the argument.
要么P，要么Q。非P，因此Q。这种模式通过排除一个选项来推出另一个。只有当析取是排斥性的且两个选项在论证语境中确实穷尽所有可能时才有效。
P ∨ Q, ¬P ⊢ Q
- Example: Either the butler committed the murder or the gardener did. The butler has an airtight alibi. Therefore, the gardener did it.
- 例子：要么是管家杀了人，要么是园丁。管家有确凿的不在场证明。因此，是园丁干的。
5. Reductio ad Absurdum (Proof by Contradiction) | 归谬法
Assume the negation of what you want to prove. Show that this assumption leads to a contradiction. Conclude that the original statement must be true. This method is fundamental in philosophy to demonstrate the incoherence of a rival position.
假设你想要证明的命题的否定成立。证明这一假设会导致矛盾。结论：原命题必然为真。该方法在哲学中被广泛用于证明对立观点的不一致性。
Assume ¬P, derive (Q ∧ ¬Q), ∴ P
- Example: To prove 'A square has four sides', assume a square does not have four sides. This contradicts the definition of a square. So a square must have four sides.
- 例子：要证明“正方形有四条边”，假设正方形没有四条边。这与正方形的定义矛盾。所以正方形必须有四条边。
6. Principle of Non-Contradiction | 不矛盾律
A proposition cannot be both true and false at the same time and in the same respect. This is a first principle of thought without which meaningful discourse collapses. Any argument that violates non-contradiction is inherently invalid.
一个命题不可能在同一时间、同一方面既真又假。这是思维的第一原则，没有它，任何有意义的论述都将崩溃。任何违反不矛盾律的论证本质上都是无效的。
¬(P ∧ ¬P)
- Example: It cannot be the case that the cat is both on the mat and not on the mat at exactly the same moment.
- 例子：猫不可能在同一时刻既在垫子上又不在垫子上。
7. The Tripartite Definition of Knowledge (JTB) | 知识的三元定义（JTB）
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. This 'formula' for propositional knowledge dominated epistemology until Gettier cases challenged its sufficiency. Nevertheless, it remains the starting point for any discussion of knowledge in the IGCSE syllabus.
S知道P当且仅当 (i) P为真，(ii) S相信P，并且 (iii) S有理由相信P。这一命题性知识的“公式”主导了认识论，直到盖梯尔问题挑战了它的充分性。不过，它仍然是IGCSE课程中任何关于知识的讨论的起点。
Knowledge = Justified True Belief (JTB)
| Condition | Meaning |
| Truth | The proposition must correspond to reality. |
| Belief | The subject must genuinely accept the proposition. |
| Justification | The subject must have adequate evidence or reasons. |
8. The Principle of Utility (Greatest Happiness Principle) | 功利原则（最大幸福原则）
An action is morally right if it produces the greatest balance of pleasure over pain for all affected individuals. In its simplest Benthamite form, the right action maximises net utility. This consequentialist 'formula' is central to ethical evaluation and often compared with duty-based theories.
如果一个行为能为所有受影响者带来最大的快乐与痛苦之差，那么它就是道德上正确的。在其最简单的边沁形式中，正确的行为就是最大化净功用的行为。这一后果主义的“公式”是伦理评价的核心，经常与义务论进行比较。
Right action = max Σ(happiness − suffering) for all affected
- Bentham's factors: intensity, duration, certainty, propinquity, fecundity, purity, extent.
- 边沁的变量：强度、持续时间、确定性、邻近性、繁衍性、纯洁性、范围。
9. Categorical Imperative (First Formulation) | 定言令式（第一公式）
Act only according to that maxim whereby you can at the same time will that it should become a universal law. This Kantian principle tests the morality of an action by its universalisability without contradiction. If the maxim cannot be consistently conceived as a universal law, the action is impermissible.
只依照你能够同时意愿它成为一条普遍法则的准则去行动。这一康德原则通过检验一个准则能否无矛盾地普遍化来判定行为的道德性。如果该准则无法被一贯地构想为普遍法则，则该行为不被允许。
CI₁: Universal Law Formula
- Step 1: Formulate the maxim (e.g., 'I will make a false promise to get money').
- Step 2: Universalise it ('Everyone makes false promises when in need').
- Step 3: Check for contradiction (the institution of promising would collapse). If contradiction, the act is wrong.
- 步骤1：拟定准则（如“我会为了得到钱而做虚假承诺”）。
- 步骤2：将其普遍化（“每个有需要的人都做虚假承诺”）。
- 步骤3：检查矛盾（承诺制度将崩溃）。如有矛盾，行为就是错误的。
10. The Teleological Argument (Design Formula) | 目的论论证（设计论公式）
The universe exhibits order, complexity, and apparent purpose. Such features are best explained by the existence of an intelligent designer. Therefore, God exists. This analogical argument, from Paley's watchmaker to modern fine-tuning, follows the inference to the best explanation structure, often captured as: order implies design, design implies designer.
宇宙展现出了秩序、复杂性和显见的目的。这些特征的最佳解释是一个有智慧的设计者存在。因此，上帝存在。从佩利的钟表匠到现代精调理论，这一类比论证遵循最佳解释推理结构，常被概括为：秩序意味着设计，设计意味着设计者。
Order + Complexity → Design → Designer (God)
- Paley's watch: a watch found on a heath implies a watchmaker; likewise, the eye implies a creator.
- 佩利的表：在荒野中发现一块表意味着有一位钟表匠；同样，眼睛意味着有一位创造者。
11. The Problem of Evil (Logical Form) | 恶的问题（逻辑形式）
If God exists, then God is omnipotent, omniscient, and omnibenevolent. An omnipotent being can prevent all evil. An omniscient being knows when evil occurs. An omnibenevolent being would want to prevent all evil. Yet evil exists. Therefore, such a God does not exist. This deductive formulation presents the logical problem of evil, which theodicies attempt to resolve by challenging the premises.
如果上帝存在，那么上帝是全能的、全知的、全善的。全能的存在能阻止所有恶。全知的存在知道恶何时发生。全善的存在会想要阻止所有恶。然而恶却存在。因此，这样的上帝不存在。这个演绎形式呈现了恶的逻辑问题，神义论试图通过质疑前提来解决它。
(God exists) → ¬(Evil exists). Evil exists. ∴ God does not exist.
- Main theistic responses: free will defence, soul-making theodicy, evil as privation.
- 主要有神论回应：自由意志辩护、灵魂锻造神义论、恶作为善的匮乏。
12. Descartes' Cogito (Foundational Formula) | 笛卡尔的我思（基础公式）
I can doubt the existence of my body, the external world, and even mathematical truths – but I cannot doubt that I am doubting. The very act of thinking and doubting guarantees my existence as a thinking thing. Cogito ergo sum ('I think, therefore I am') is the indubitable foundational truth from which Descartes attempts to rebuild knowledge.
我可以怀疑我的身体、外部世界甚至数学真理的存在——但我不能怀疑我正在怀疑这一事实。正是思考和怀疑的活动确保了我作为一个思维实体的存在。我思故我在，是笛卡尔试图重建知识体系时所依靠的不可怀疑的基础真理。
Doubt → Thinking → Existence (as a thinking thing)
- Even an evil demon cannot deceive me about my own existence, because deception requires a mind to be deceived.
- 即使有一个邪恶的魔鬼也不能欺骗我关于我自身的存在，因为欺骗需要一个被欺骗的心灵。
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