📚 A-Level Philosophy Quick Reference Handbook: Formulas & Theorems | A-Level 哲学速查手册：公式与定理
This handbook collects the core ‘formulas’ and ‘theorems’ that structure arguments across the CIE A-Level Philosophy syllabus. Just as in mathematics, mastering these logical patterns, ethical equations, and epistemic principles enables you to construct and deconstruct philosophical reasoning with precision. Each entry pairs a canonical statement with its standard interpretation, followed by a paired Chinese translation to reinforce understanding.
本手册汇集了 CIE A-Level 哲学考纲中构建论证的核心“公式”与“定理”。如同数学一般，掌握这些逻辑模式、伦理等式和认识论原则，能够帮助你精确地建构和拆解哲学推理。每个条目将经典陈述与其标准解释配对，再附上对应的中文翻译，以加深理解。
1. Modus Ponens (Affirming the Antecedent) | 肯定前件式（MP）
If P then Q. P. Therefore, Q. This is the simplest valid deductive form. In symbols: (P → Q), P ⊢ Q.
如果 P 则 Q。P 成立。因此，Q 成立。这是最简单有效的演绎形式。符号表示为：(P → Q), P ⊢ Q。
2. Modus Tollens (Denying the Consequent) | 否定后件式（MT）
If P then Q. Not-Q. Therefore, not-P. Symbolically: (P → Q), ¬Q ⊢ ¬P. This is the backbone of falsification and many sceptical arguments.
如果 P 则 Q。非 Q。因此，非 P。符号表示为：(P → Q), ¬Q ⊢ ¬P。这是证伪主义以及许多怀疑论论证的支柱。
3. Hypothetical Syllogism | 假言三段论
If P then Q. If Q then R. Therefore, if P then R. (P → Q), (Q → R) ⊢ (P → R). This allows chaining of conditionals, crucial for extended philosophical reasoning.
如果 P 则 Q。如果 Q 则 R。因此，如果 P 则 R。(P → Q), (Q → R) ⊢ (P → R)。这允许条件句的链接，对扩展的哲学推理至关重要。
4. Disjunctive Syllogism | 选言三段论
P or Q. Not-P. Therefore, Q. (P ∨ Q), ¬P ⊢ Q. Used in process-of-elimination arguments, e.g., ‘either the mind is physical or non-physical; it is not physical in the reductive sense; therefore, it is non-physical.’
P 或 Q。非 P。因此，Q。(P ∨ Q), ¬P ⊢ Q。用于排除法论证，例如：“心灵要么是物理的，要么是非物理的；它不是还原意义上的物理的；因此，它是非物理的。”
5. Reductio ad Absurdum (RAA) | 归谬法
Assume the negation of the target proposition. Derive a contradiction (both Q and ¬Q). Conclude the target proposition. This form is central to proofs of necessary truths and in refutations.
假设目标命题的否定。推导出矛盾（即 Q 且 ¬Q）。得出结论：目标命题成立。这种形式在证明必然真理以及反驳中处于核心地位。
6. Principle of Non-Contradiction | 不矛盾律
Not both P and not-P simultaneously in the same respect. ¬(P ∧ ¬P). This is a fundamental axiom of classical logic, defended by Aristotle, and a test for coherence.
在同一方面不能同时为 P 且非 P。¬(P ∧ ¬P)。这是古典逻辑的基本公理，由亚里士多德辩护，也是检验融贯性的标准。
7. Law of Excluded Middle | 排中律
For any proposition P, either P is true or its negation is true. P ∨ ¬P. It underpins bivalent logic, though challenged by intuitionists and in some paradoxes.
对于任何命题 P，要么 P 为真，要么 P 的否定为真。P ∨ ¬P。它是二值逻辑的基础，但受到直觉主义者和某些悖论的挑战。
8. Utilitarian Greatest Happiness Formula | 功利主义最大幸福公式
Action A is right iff A produces the greatest net balance of pleasure over pain for all affected, compared with alternatives. Utility = ∑ (pleasure – pain) for each individual. Symbolically: Right(A) ↔ ∀x, Utility(A) ≥ Utility(B).
行动 A 是正当的，当且仅当 A 在所有备选方案中，为所有受影响者带来了快乐减去痛苦的最大净余额。功利 = 每个个体的 ∑ (快乐 – 痛苦)。符号表示为：Right(A) ↔ ∀x, Utility(A) ≥ Utility(B)。
9. Kant’s Categorical Imperative: Formula of Universal Law | 康德定言令式：普遍法则公式
Act only according to that maxim whereby you can at the same time will that it should become a universal law. Test: can the maxim be conceived without contradiction as a universal law? Can it be willed without contradiction?
只按照你同时能够意愿它成为一条普遍法则的准则去行动。检验：这条准则能否无矛盾地被思考为一条普遍法则？能否无矛盾地被意愿？
10. Kant’s Formula of Humanity as End in Itself | 康德人性公式
Act so that you treat humanity, whether in your own person or in the person of any other, always at the same time as an end, never merely as a means. This forbids using rational beings purely instrumentally.
如此行动，以使你总是将人性——无论在你自身人格中还是在任何一个他人的人格中——同时作为目的，而绝不仅仅当作手段。这禁止纯粹工具化地使用理性存在者。
11. Descartes’ Cogito (Foundational Theorem) | 笛卡尔的我思（基础定理）
I think, therefore I am (Cogito, ergo sum). Even if I am being deceived, I must exist to be deceived. This indubitable truth serves as the first principle of knowledge.
我思，故我在（Cogito, ergo sum）。即使我正在被欺骗，我也必须存在才能被欺骗。这一不容置疑的真理充当了知识的首要原理。
12. Logical Positivist Verification Principle | 逻辑实证主义的证实原则
The meaning of a proposition is its method of verification. A statement is cognitively meaningful iff it is either analytically true (true by definition) or empirically verifiable. Metaphysical claims are thus meaningless.
一个命题的意义就在于其证实方法。一个陈述在认知上有意义，当且仅当它要么是分析为真（根据定义为真），要么是可经验证实的。因此形而上学的断言是无意义的。
13. The Problem of Evil Logical Formulation | 逻辑恶问题的公式
If God is omnipotent, omniscient, and wholly good, then evil would not exist. Evil exists. Therefore, such a God does not exist. (O ∧ K ∧ B) → ¬E; E; therefore ¬(O ∧ K ∧ B).
如果上帝是全能、全知且全善的，那么恶就不会存在。恶存在。因此，这样的上帝不存在。(O ∧ K ∧ B) → ¬E；E；因此 ¬(O ∧ K ∧ B)。
14. Theodicy as Defeater Formula | 神义论作为消解者公式
A successful theodicy must show that the existence of evil is logically compatible with God’s existence by providing a morally sufficient reason for God to permit evil. If reason R is possible, then (O ∧ K ∧ B) is compatible with E.
一个成功的神义论必须通过为上帝容许恶提供一个道德上充分的理由，来表明恶的存在与上帝的存在在逻辑上是相容的。如果理由 R 是可能的，那么 (O ∧ K ∧ B) 与 E 相容。
15. Aristotle’s Virtue as the Mean Formula | 亚里士多德的中道德性公式
Virtue is a state of character concerned with choice, lying in a mean relative to us, determined by reason. Virtue = the intermediate between excess and deficiency in emotion and action. For example, courage is the mean between cowardice and rashness.
德性是一种与选择相关的品质状态，存在于相对于我们而言的中道之中，由理性决定。德性 = 情感和行动中过度与不足之间的中间状态。例如，勇敢是怯懦与鲁莽之间的中道。
16. Gettier Problem Structure | 盖梯尔问题结构
Justified True Belief (JTB) is not sufficient for knowledge. Counterexample: S has a justified true belief that P, but the belief is true only by luck. Formula: JTB + ¬G → ¬K (where G = no epistemic luck).
得到辩护的真信念（JTB）并不构成知识的充分条件。反例：S 对 P 持有得到辩护的真信念，但该信念只是凭借运气为真。公式：JTB + ¬G → ¬K（G = 没有认知运气）。
17. Reliabilism’s Formula for Knowledge | 可靠主义的知识公式
S knows that P iff S’s true belief that P is produced by a reliable cognitive process. Reliability = the process tends to produce a high proportion of true beliefs.
S 知道 P，当且仅当 S 关于 P 的真信念是由一个可靠的认知过程产生的。可靠性 = 该过程倾向于产生高比例的真信念。
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