📚 IGCSE CCEA Philosophy: Quick Reference Handbook of Formulas and Theorems | IGCSE CCEA 哲学：公式定理速查手册
This quick reference handbook collects the key argument structures, logical rules and ethical decision-making formulas that appear across the CCEA Philosophy of Religion and Ethics syllabus. Think of them as ‘theorems’ you can deploy, check for validity and use to analyse classic philosophical texts. The aim is to help you recognise the shape of an argument before you evaluate its content.
这本速查手册汇集了 CCEA 宗教哲学与伦理学大纲中反复出现的关键论证结构、逻辑规则和道德决策公式。你可以把它们看作可以随时调用、检验有效性并用于分析经典哲学文本的‘定理’。目标是在你评价内容之前，先帮你辨认出论证的雏形。
1. Basic Valid Argument Forms | 有效推理基本形式
These deductive forms preserve truth: if the premises are true and the reasoning follows the pattern, the conclusion cannot be false. They underpin almost every philosophical proof you will meet in CCEA, from the cosmological argument to ethical dilemmas.
这些演绎形式保真：若前提为真且推理遵循该模式，结论便不可能为假。它们支撑着你在 CCEA 中遇见的几乎每一个哲学证明，从宇宙论论证到伦理两难。
- Modus Ponens (Affirming the Antecedent): If P then Q; P is given; therefore Q.
- 肯定前件式：如果 P 则 Q；P 成立；因此 Q。
- Modus Tollens (Denying the Consequent): If P then Q; not Q; therefore not P.
- 否定后件式：如果 P 则 Q；非 Q；因此非 P。
- Hypothetical Syllogism: If P then Q; if Q then R; therefore if P then R.
- 假言三段论：如果 P 则 Q；如果 Q 则 R；因此如果 P 则 R。
- Disjunctive Syllogism: P or Q; not P; therefore Q.
- 选言三段论：P 或 Q；非 P；因此 Q。
P → Q, P ∴ Q | P → Q, ¬Q ∴ ¬P
(P → Q) ∧ (Q → R) ∴ (P → R) | P ∨ Q, ¬P ∴ Q
2. Common Fallacies to Detect | 常见谬误侦测
A fallacy is a flawed pattern of reasoning that can appear persuasive. Spotting these ‘invalid formulas’ is as important as recognising valid ones. In CCEA exam answers, identifying a fallacy in a given argument immediately strengthens your evaluation.
谬误是看似有说服力但存在缺陷的推理模式。识别出这些‘无效公式’与认出有效公式同等重要。在 CCEA 考试答案中，指出给定论证里的谬误能立刻提升你的评价质量。
- Affirming the Consequent: P → Q; Q; therefore P. (Matches no valid form.)
- 肯定后件：P → Q；Q；因此 P。（不符合任何有效形式。）
- Denying the Antecedent: P → Q; ¬P; therefore ¬Q.
- 否定前件：P → Q；¬P；因此 ¬Q。
- Begging the Question (Circular Reasoning): The conclusion is assumed within a premise.
- 乞题（循环推理）：结论被事先埋在一个前提里。
- False Dilemma: Only two options are presented when more exist.
- 假两难：只给出两个选项，实际上存在更多可能。
Invalid: P → Q, Q ∴ P | Invalid: P → Q, ¬P ∴ ¬Q
3. Cosmological Argument Formula | 宇宙论论证公式
The cosmological argument seeks to prove God’s existence from the mere fact that the universe exists. Aquinas’ First Three Ways are its CCEA backbone. The core ‘formula’ is a chain of causes that cannot regress infinitely, pointing to a first mover or necessary being.
宇宙论论证试图仅从宇宙存在这一事实推出上帝存在。阿奎那的前三路是 CCEA 的主心骨。核心‘公式’是一个不能无限回溯的因果链，最终指向一个不动的推动者或必然存在。
General Form: Everything that exists has an explanation; the universe exists; therefore the universe has an explanation (God).
一般形式：凡存在的都有解释；宇宙存在；因此宇宙有解释（上帝）。
Aquinas’ Second Way (Causal): We observe efficient causes; nothing can cause itself; an infinite regress of causes is impossible; therefore a first uncaused cause exists.
阿奎那第二路（因果）：我们观察到动力因；无物能自因；无限回溯不可能；因此存在第一非受动因。
∀x (C(x) → ∃y C(y) ∧ y → x), ¬∃x (C(x) ∧ x → x), ¬InfiniteRegress ∴ ∃z (C(z) ∧ ¬∃w (C(w) ∧ w → z))
4. Teleological Argument Formula | 目的论论证公式
The teleological argument reads design-like features in the world as evidence for an intelligent designer. Paley’s watchmaker analogy and the contemporary fine-tuning argument are the standard CCEA versions. The logical formula captures analogy or inference to the best explanation.
目的论论证把世界中类似设计的特征读作存在理智设计者的证据。佩利的钟表类比和当代的微调论证是 CCEA 的标配版本。逻辑公式捕捉的是类比或最佳解释推理。
Paley’s Analogy: A watch exhibits complexity and purpose, which implies a watchmaker; the universe exhibits similar complexity and purpose; therefore, by analogy, the universe has a designer.
佩利的类比：钟表展现复杂性与目的，暗示有钟表匠；宇宙展现类似的复杂性与目的；因此，通过类比，宇宙有一位设计者。
Fine-Tuning version: The constants of physics are precisely set for life; this is either by chance, necessity or design; chance and necessity are highly improbable; therefore design is the best explanation.
微调版本：物理常数被精确设定为生命容许；这要么出于偶然、必然或设计；偶然与必然极不可能；因此设计为最佳解释。
P(A|D) ≫ P(A|C) → D is best explanation | Analogy: O₁ shares P and Q; O₂ shares P; so O₂ likely shares Q.
5. Ontological Argument Formula | 本体论论证公式
Anselm’s ontological argument is a deductive, a priori proof that attempts to derive God’s existence from the very concept of ‘that than which nothing greater can be conceived’. CCEA candidates must understand it as a definitional tautology rather than an empirical claim.
安瑟伦的本体论论证是一个演绎的、先天的证明，试图从‘无法设想比之更伟大者’的概念本身推出上帝存在。CCEA 考生须将其理解为定义上的同义反复而非经验断言。
Anselm’s Formula: God is defined as the greatest conceivable being. A being that exists in reality is greater than one that exists only in the mind. Therefore, if God existed only in the mind, we could conceive of a greater being (one existing in reality), which contradicts the definition. Hence God must exist in reality.
安瑟伦公式：上帝被定义为可设想的至大者。存在于现实比仅存在于心灵更伟大。因此，若上帝仅存在于心灵，我们便可设想一位更伟大者（存在于现实），这与定义相悖。故上帝必存在于现实。
Def: G = that than which nothing greater can be conceived. Assume ¬∃x (x = G). Then ∃y (y > G). Contradiction. ∴ ∃x (x = G).
6. The Logical Problem of Evil as a Formula | 恶的逻辑悖论公式
J.L. Mackie’s inconsistent triad is a direct challenge to theistic coherence. CCEA students treat it as a propositional contradiction: if God is all-powerful and all-loving, then evil would not exist; evil does exist; therefore, such a God does not exist.
J.L. 麦基的不一致三元组是对有神论自洽性的直接挑战。CCEA 学生将其视为一个命题矛盾：若上帝全能全善，则恶不会存在；恶存在；因此，这样的上帝不存在。
Mackie’s Triad Formula:
麦基的三元组公式：
∀x (Omnipotent(x) ∧ Omnibenevolent(x) → ¬∃y Evil(y)), ∃y Evil(y) ∴ ¬∃x (Omnipotent(x) ∧ Omnibenevolent(x))
The theist must either deny the premise that omnipotence and benevolence entail no evil, or deny the reality of evil, or qualify the attributes. The formula forces an explicit response.
有神论者要么否认全能全善蕴涵无恶这一前提，要么否认恶的实在性，要么限制属性。该公式强制给出明确回应。
7. Pascal’s Wager Formula | 帕斯卡赌注公式
Pascal’s Wager is not an argument for God’s existence but a prudential ‘formula’ for belief. It applies decision theory: weigh infinite gain against finite loss. CCEA students often evaluate it under pragmatic justification.
帕斯卡的赌注不是关于上帝存在的论证，而是一个关于信仰的审慎‘公式’。它运用决策论：权衡无限的收益与有限的损失。CCEA 学生常在实用合理性下评价它。
Pascal’s Decision Matrix: If you believe in God and God exists, infinite reward; if God does not exist, negligible loss. If you do not believe and God exists, infinite punishment; if God does not exist, negligible gain. The rational bet is to believe.
帕斯卡决策矩阵：如果你信上帝且上帝存在，无限奖赏；若上帝不存在，微末损失。如果你不信且上帝存在，无限惩罚；若上帝不存在，微末收益。理性赌注是选择信。
Expected Utility(Believe) = ∞ × p + (−f) × (1−p) = ∞ EU(¬Believe) = −∞ × p + g × (1−p) = −∞
8. Utilitarian Calculus Formula | 功利主义计算公式
Jeremy Bentham’s hedonic calculus is a quantitative formula for moral decision-making in CCEA ethics. It sums pleasures and pains according to intensity, duration, certainty and extent. The formula represents a consequentialist ‘theorem’.
边沁的苦乐计算是 CCEA 伦理学中一种量化的道德决策方式。它按强度、持续时间、确定性和广度加总快乐与痛苦。该公式代表了一种后果论‘定理’。
Bentham’s Hedonic Formula: For any action, compute net happiness = (sum of pleasures) − (sum of pains) across all affected individuals.
边沁的快乐公式：对于任何行为，计算净快乐 =（所有受影响个体的快乐之和）−（痛苦之和）。
Net U(A) = Σᵢ (Iₚ × Dₚ × Cₚ − Iₚₐ × Dₚₐ × Cₚₐ) for each sentient i
Mill’s higher/lower pleasure refinement adds a qualitative weight, but the underlying sum structure remains the same ‘formula’.
密尔关于高级/低级快乐的修正加入了质的权重，但底层的加总结构仍是同一条‘公式’。
9. Categorical Imperative Formula | 定言令式公式
Kant’s categorical imperative operates as a rational consistency test. The first formulation is a universalisability formula: act only on that maxim which you can at the same time will to become a universal law.
康德的定言令式起着理性一致性测试的作用。第一表述就是一条可普遍化公式：只依据你同时能够意愿其成为普遍法则的准则去行动。
Universal Law Formula: If maxim M cannot be conceived or willed without contradiction as a universal law, then acting on M is forbidden.
普遍法则公式：若准则 M 无法被设想或意愿为普遍法则而无矛盾，则依据 M 行动是被禁止的。
∀M (◇ U(M) → Obligatory(Act on M)) ∧ (¬◇ U(M) → Forbidden)
The second formulation, treating humanity as an end in itself, is not a formal calculus but functions as a moral boundary condition.
第二表述，即人是目的而非手段，虽非形式算式，但起着道德边界条件的功能。
10. Cultural Relativism Formula | 文化相对主义判断公式
Cultural relativism claims that moral truth is determined by cultural norms. In CCEA papers you often need to evaluate whether ‘X is right in culture A and wrong in culture B’ collapses into contradiction.
文化相对主义声称，道德真理由文化规范决定。在 CCEA 考卷中，你常常需要评价‘行为X在文化A中是对的，在文化B中是错的’是否沦为矛盾。
Relativist Claim: ‘It is morally right for members of culture C to perform action A’ is true if C’s norms approve A. This can be formalised as a two-place predicate.
相对主义断言：‘对文化C的成员而言做行为A是道德上正确的’，当且仅当C的规范认可A。这可以表述为一个二元谓词公式。
Right(A, C) = true iff Norms(C) ⊢ A ∴ Right(A, C₁) ∧ ¬Right(A, C₂) is consistent.
Critics point out that this formula forbids cross-cultural moral criticism, which many find unacceptable.
批评者指出，这条公式禁止了跨文化的道德批评，而这是许多人无法接受的。
11. Free Will and Determinism Formulas | 自由意志与决定论公式
The free will debate can be captured by a set of conditional claims. Hard determinism states: if determinism is true, then free will does not exist. Compatibilism (soft determinism) redefines freedom as acting according to one’s desires without external coercion.
自由意志争论可用一组条件式来捕捉。硬决定论声称：若决定论为真，则自由意志不存在。相容论（软决定论）将自由重新定义为按照自身欲望行动且不受外在强制。
Hard Determinism Formula: D → ¬FW. D is true (scientific determinism). ∴ ¬FW.
硬决定论公式：D → ¬FW。D为真（科学决定论）。故 ¬FW。
Compatibilist Formula: FW := Act is caused by internal psychological states without compulsion. D ∧ Internal Causation is then compatible with FW.
相容论公式：FW := 行动由内在心理状态引起且无强制。于是 D ∧ 内在因果与 FW 相容。
Hard: (D → ¬FW) ∧ D ∴ ¬FW Compatibilist: FW ⇔ (InternalCausation ∧ ¬Coercion)
12. Religious Experience Argument Structure | 宗教体验论证结构
Arguments from religious experience often follow a ‘principle of credulity’ formula: unless we have a defeater, we should trust what appears to be an experience of God. Richard Swinburne’s principle is central in CCEA.
宗教体验论证常遵循‘轻信原则’公式：除非我们有否决因子，否则应相信看似对上帝的体验。斯温伯恩的原则在 CCEA 中居于核心。
Swinburne’s Formula: If a person sincerely reports an experience of God, and no evidence undermines its reliability, then it is reasonable to believe that God was present.
斯温伯恩公式：若一人真诚地报告了上帝体验，且没有证据削弱其可靠性，则相信上帝临在便是合理的。
Report(G) ∧ ¬Defeater → Reasonable(Belief(G)).
The formula reverses the burden of proof, placing it on the sceptic to show the experience is unreliable.
这公式逆转了举证责任，将其放在怀疑者身上，即必须证明该体验不可靠。
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