📚 Pre-U CCEA Philosophy Quick Reference: Key Logical Symbols & Arguments | Pre-U CCEA 哲学：公式定理速查手册
Philosophy at Pre-U level under the CCEA specification demands more than fluent essay writing; it requires precise handling of logical notation, argument analysis, and the recurring thought experiments that shape debates in epistemology, ethics, philosophy of mind, and philosophy of religion. This quick-reference guide presents in one place the symbolic ‘formulae’ of deductive and inductive logic, the most cited arguments, and the conceptual tools that examiners expect you to deploy with confidence. Use it as a revision checklist to ensure no logical operator or paradox catches you off guard.
CCEA 考试局 Pre-U 阶段的哲学不仅要求流畅的论文写作，还要求精准运用逻辑符号、分析论证结构，并熟练掌握那些反复塑造知识论、伦理学、心灵哲学和宗教哲学辩论的思想实验。这份速查手册将演绎与归纳逻辑中的“符号公式”、最高频引用的论证以及必备的概念工具汇聚一处，供你自信地运用于考试。把它当作复习清单，确保没有逻辑算子或悖论让你措手不及。
1. Propositional Logic Operators | 命题逻辑算子
The building blocks of formal deductive arguments. Each operator defines how truth values combine; knowing their truth-functional properties is essential for evaluating validity.
形式演绎论证的基本构件。每个算子定义了真值如何组合；理解其真值函项性质是评估有效性的关键。
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| ¬ | Negation | ‘not p’ – reverses truth value | ¬A: It is not the case that A |
| ∧ | Conjunction | ‘p and q’ – true only when both are true | A ∧ B: A and B |
| ∨ | Disjunction | ‘p or q’ – inclusive; false only when both false | A ∨ B: A or B |
| → | Material conditional | ‘if p then q’ – false only when p true and q false | A → B: If A then B |
| ↔ | Biconditional | ‘p if and only if q’ – true when both same | A ↔ B: A iff B |
In examinations, you may be asked to translate English arguments into this notation or to test validity using truth tables. Remember that the material conditional does not capture causal or counterfactual ‘if-then’; it is purely truth-functional.
在考试中，你可能会被要求将英语论证翻译成这种符号，或用真值表检验有效性。请记住，实质条件句并不表达因果或反事实的“如果—那么”关系；它纯粹是真值函项的。
2. Quantificational Logic Symbols | 量化逻辑符号
When propositions involve ‘all’, ‘some’, or ‘none’, quantifiers extend propositional logic. Mastery of these symbols enables formal analysis of syllogisms and existential claims central to ontological arguments.
当命题涉及“所有”、“有些”或“没有”时，量词扩展了命题逻辑。掌握这些符号，就能对三段论和本体论论证中的存在性主张进行形式分析。
| Symbol | Name | Use |
|---|---|---|
| ∀ | Universal quantifier | ‘for all x’ – ∀x(Fx) means every object x has property F |
| ∃ | Existential quantifier | ‘there exists an x such that’ – ∃x(Fx) means at least one x is F |
| ∄ | Negated existential | ‘there does not exist an x’ – equivalent to ∀x(¬Fx) |
Example: Anselm’s famous premise ‘God is that than which nothing greater can be conceived’ can be symbolised to test its logical structure. CCEA expects clarity on the distinction between de dicto and de re necessity, which maps onto the order of quantifiers and modal operators.
示例：安瑟伦的著名前提“上帝是可设想的无与伦比之物”可用符号化来检验其逻辑结构。CCEA 要求你清晰区分“言命题”（de dicto）与“物命题”（de re）必然性，这映射到量词与模态算子的次序上。
3. Modal Logic Operators | 模态逻辑算子
Modal operators add dimensions of necessity and possibility. In CCEA Philosophy, they appear most frequently in discussions of the ontological argument, the problem of evil, and possible-worlds semantics.
模态算子增加了必然性和可能性的维度。在 CCEA 哲学中，它们最常出现在本体论论证、恶的问题以及可能世界语义学的讨论中。
| Symbol | Name | Interpretation |
|---|---|---|
| □ | Necessarily | □p: p is true in all possible worlds |
| ◇ | Possibly | ◇p: p is true in at least one possible world |
The key equivalence in standard S5 modal logic is ◇p ↔ ¬□¬p, and ¬◇p ↔ □¬p. Alvin Plantinga’s modal ontological argument relies on the premise that ‘maximal greatness is possibly instantiated’ (◇∃x Gx), from which, using S5, one can deduce that it is necessarily instantiated. Always specify which system (S4, S5) you are assuming, as the inference differs.
标准 S5 模态逻辑中的关键等价是 ◇p ↔ ¬□¬p，以及 ¬◇p ↔ □¬p。阿尔文·普兰丁格的模态本体论论证依赖于“最大伟大性可能被例示”（◇∃x Gx）这一前提，运用 S5 可由此推出它必然被例示。务必指明你假设的是哪种系统（S4 或 S5），因为推论有所不同。
4. Syllogistic Forms (Aristotelian Logic) | 三段论格式（亚里士多德逻辑）
Before modern quantification, valid argument patterns were categorised by figure and mood. These remain indispensable for analysing arguments about the soul, causality, or the good.
在现代量化逻辑出现之前，有效论证模式按格和式分类。分析关于灵魂、因果关系或善的论证时，这些仍然不可或缺。
The most tested valid moods, each with its medieval name:
| Mood | Form | Example |
|---|---|---|
| Barbara | All M are P; all S are M; therefore all S are P | All humans are mortal; all Athenians are humans; so all Athenians are mortal |
| Celarent | No M is P; all S are M; therefore no S is P | No stone is alive; all statues are stone; so no statue is alive |
| Darii | All M are P; some S is M; therefore some S is P | All virtues are praiseworthy; some habits are virtues; so some habits are praiseworthy |
| Ferio | No M is P; some S is M; therefore some S is not P | No god is evil; some beings are gods; so some beings are not evil |
By testing argument extracts against these patterns, you can swiftly spot whether a syllogism commits the fallacy of the undistributed middle or illicit major. CCEA source booklets often embed such reasoning; effective commentary requires you to name the form.
将论证摘录与这些模式对照检验，你就能迅速发现三段论是否犯了“中词不周延”或“大项不当周延”的谬误。CCEA 的素材册常包含此类推理；有效的评析需要你能说出它的格式。
5. Essential Fallacies | 必知谬误
A formula sheet for philosophy must include the invalid inference patterns that masquerade as cogent reasoning. Spotting and naming these is a core assessment objective.
一份哲学公式表必须收录那些冒充合理推理的无效推论模式。识别并命名它们是一项核心的考核目标。
- Affirming the consequent: p → q; q; therefore p. (Invalid: if it rains the ground is wet; the ground is wet; therefore it rained – ignores sprinklers.)
肯定后件：p → q；q；所以 p。（无效：如果下雨，地面湿；地面湿；所以下雨了——忽略了洒水器。） - Denying the antecedent: p → q; ¬p; therefore ¬q. (Invalid: if it is a dog, it has four legs; it is not a dog; therefore it does not have four legs – a cat may have four legs.)
否定前件：p → q；¬p；所以 ¬q。（无效：如果是狗，它有四条腿；它不是狗；所以它没有四条腿——猫也可能有四条腿。） - Equivocation: Shifting word meanings mid-argument, e.g., ‘light’ as physical vs. ‘light’ as not heavy, often used in critiques of dualism.
偷换概念：在论证中途变换词义，如“light”表物理的光 vs.“light”表不重，常在对二元论的批评中出现。 - Begging the question: Premises assume the conclusion, e.g., ‘God exists because the Bible says so, and the Bible is the word of God.’
循环论证：前提假设了结论，如“上帝存在，因为《圣经》这么说，而《圣经》是上帝的话”。
Identifying such patterns under pressure improves your evaluation marks dramatically.
在考试压力下识别这些模式，能显著提升你的评价得分。
6. Key Principles in Epistemology | 知识论核心原则
Epistemology questions often hinge on your mastery of the tripartite definition of knowledge and the Gettier counterexamples. Treat these like equations you must be able to instantiate.
知识论题目常常取决于你对知识三元定义和盖梯尔反例的掌握。把它们当作你必须能够代入具体例子的公式。
The classical formula: Knowledge = Justified True Belief (JTB).
经典公式：知识 = 确证的真信念（JTB）。
Gettier’s challenge shows that JTB is not sufficient. In one scenario, Smith has justified false belief that ‘Jones will get the job’ and that ‘Jones has ten coins’; from this he deduces ‘The man who will get the job has ten coins.’ Unbeknown to Smith, he himself gets the job and also happens to have ten coins. His belief is true and justified, yet intuitively not knowledge. This reveals that JTB is not a sufficient condition.
盖梯尔的挑战表明 JTB 并不充分。在一个情境中，史密斯有理由地相信“琼斯会得到工作”且“琼斯有十个硬币”；由此他推断出“得到工作的人有十个硬币”。不料，史密斯自己得到了工作，且碰巧也有十个硬币。他的信念是真的且有确证，但直觉上并不是知识。这表明 JTB 并非充分条件。
Responses involve strengthening justification (infallibilism, reliabilism) or adding a fourth condition (no false lemmas, causal connection). Remember these as modifications to the ‘knowledge equation’.
回应方案包括加强确证条件（不谬论、可靠论）或增加第四个条件（无假引理、因果联系）。记住它们是“知识等式”的修正版。
7. Ethical Theories – Formulaic Structures | 伦理学理论 – 公式化结构
Utilitarianism, Kantian deontology, and virtue ethics can each be summarised as decision procedures. Quick recall of these ‘formulae’ saves time and ensures precision.
功利主义、康德义务论和美德伦理学都可以归结为决策程序。快速回忆这些“公式”可以节省时间并确保精确度。
Act Utilitarianism (Bentham): Right action = the one that maximises pleasure and minimises pain across all affected. The ‘felicific calculus’ considers intensity, duration, certainty, propinquity, fecundity, purity, and extent.
行为功利主义（边沁）： 正确的行动 = 在一切受影响者中最大化快乐并最小化痛苦的那个行动。“幸福计算法”考虑强度、持续时间、确定性、邻近性、丰产性、纯度和范围。
Rule Utilitarianism (Mill): Right action = the one conforming to a rule that, if generally followed, would maximise utility. This moves the calculation to the rule level.
规则功利主义（密尔）： 正确的行动 = 符合一条规则，而该规则若被普遍遵守，将使功利最大化。这把计算移到了规则层面。
Kant’s Categorical Imperative (First Formulation): Act only according to that maxim whereby you can at the same time will that it should become a universal law. Test: universalise the maxim; check for contradiction in conception or contradiction in willing.
康德的绝对命令（第一公式）： 只依照你同时愿意它成为普遍法则的那条准则行动。检验：将准则普遍化；检查是否存在概念矛盾或意愿矛盾。
Virtue Ethics (Aristotle): Right action = what the virtuous person would do. Virtue is a mean relative to us, determined by phronesis (practical wisdom). Not an algorithmic formula, but a pattern of character.
美德伦理学（亚里士多德）： 正确的行动 = 有德之人会做的事。美德是相对于我们而言的中道，由实践智慧（phronesis）决定。不是算法式公式，而是一种品格模式。
In essays, stating the ‘formula’ early and then testing a case study against it demonstrates structured philosophical method.
在论文中，先陈述“公式”，然后对照案例研究进行检验，体现了结构化的哲学方法。
8. Philosophy of Religion – Logical Structures of Standard Arguments | 宗教哲学 – 标准论证的逻辑结构
The teleological (design) argument, the cosmological argument, and the problem of evil all have canonical logical shapes. Presenting them with numbered premises and conclusions mirrors the ‘formula’ approach.
目的论（设计）论证、宇宙论论证和恶的问题都有经典的逻辑形态。用编号列出前提和结论，正是“公式”方法的体现。
Paley’s Teleological Argument:
(1) A watch implies a watchmaker because of its complexity and purpose.
(2) The universe exhibits even greater complexity and purpose.
(3) So, the universe implies a universe-maker (God).
佩利的钟表论证：
（1）钟表因其复杂性和目的性而暗示钟表匠存在。
（2）宇宙展现出更大的复杂性和目的性。
（3）所以，宇宙暗示着宇宙的制造者（上帝）。
The Kalam Cosmological Argument:
(1) Everything that begins to exist has a cause.
(2) The universe began to exist.
(3) Therefore, the universe has a cause (which is identified as God).
卡拉姆宇宙论论证：
（1）凡开始存在的东西必有原因。
（2）宇宙开始存在。
（3）所以，宇宙有一个原因（这个原因被认定为上帝）。
Logical Problem of Evil (J. L. Mackie):
(1) If God is all-powerful, He can prevent all evil.
(2) If God is all-good, He wants to prevent all evil.
(3) Evil exists.
(4) Therefore, an all-powerful, all-good God does not exist.
恶的逻辑问题（麦基）：
（1）如果上帝全能，祂能防止所有恶。
（2）如果上帝全善，祂愿意防止所有恶。
（3）恶存在。
（4）所以，全能、全善的上帝不存在。
Reciting these as formal arguments helps you engage with them critically rather than descriptively.
把这些当作形式论证来复述，有助于你批判性地而非描述性地讨论它们。
9. Philosophy of Mind – The Hard Formulae | 心灵哲学 – 难点公式
The mind-body debate revolves around logical identity, supervenience, and the intensional fallacy. Expressed formally, these provide the vocabulary for high-level analysis.
心身之争围绕着逻辑同一性、随附性以及内涵谬误。将它们形式化地表达，能为高层次分析提供词汇。
Type Identity Theory: For every mental state M, there is some physical brain state P such that M = P (numerical identity).
类型同一论： 对每种心理状态 M，存在某种物理脑状态 P，使得 M = P（数值同一）。
Supervenience (token identity): No difference in mental states without a difference in physical states. Formally: ◇(A-properties different & B-properties same) is false, where A are mental, B are physical.
随附性（个例同一）： 心理状态没有差异，物理状态就不会有差异。形式表达：◇（A属性不同而B属性相同）为假，其中A是心理属性，B是物理属性。
Intensional Fallacy: Equating mental states based solely on co-extensionality of terms. Having the concept of water ≠ having the concept of H₂O, even if water = H₂O. Functionalism must respect these intensional contexts.
内涵谬误： 仅根据词项的外延重合来等同心理状态。拥有水的概念 ≠ 拥有 H₂O 的概念，即使水 = H₂O。功能主义必须尊重这些内涵语境。
Students often confuse supervenience with identity; stating their formal difference makes essays stand out.
学生常混淆随附性与同一性；阐明它们的形式区别会让论文脱颖而出。
10. The ‘Is-Ought’ Gap & Naturalistic Fallacy | “实然–应然”隔阂与自然主义谬误
Hume’s observation and Moore’s open-question argument are staples in ethics and meta-ethics. Their logical core can be captured in a few lines.
休谟的观察和摩尔的开放问题论证是伦理学和元伦理学中的基本内容。它们的逻辑核心可以用寥寥数行概括。
Hume’s Guillotine: You cannot validly derive a conclusion containing ‘ought’ from premises containing only ‘is’. Any such argument requires an evaluative premise smuggled in.
休谟断头台： 你无法从仅含有“实然”的前提中有效地推出含有“应然”的结论。任何此类论证都需要一个被悄悄带入的评价性前提。
Moore’s Naturalistic Fallacy: For any natural property N proposed to define ‘good’, it remains an open question whether something that is N is good. Good is a simple, non-natural property that cannot be reduced. Formally: if ‘good = N’, then ‘Are N things good?’ should be a closed question – but it is not.
摩尔的自然主义谬误： 对于任何被提议定义“善”的自然属性 N，“具有 N 的事物是善的吗？”这个问题始终是开放的。善是一种单纯的、非自然的属性，不可还原。形式化表达：如果“善 = N”，那么“凡 N 的东西是善的吗？”应为封闭问题——但它并不是。
Grasping these prevents you from carelessly reducing moral terms in evaluation essays.
掌握这些，可以防止你在评价性论文中轻率地还原道德词语。
11. Paradoxes & Thought Experiments as Argument Templates | 悖论与思想实验作为论证模板
Philosophical paradoxes are not mere puzzles; they are compact argument templates that reveal incoherencies. Familiarity with their structure is like knowing a standard chemical equation.
哲学悖论不仅仅是谜题；它们是揭示不融贯性的紧凑论证模板。熟悉其结构就像掌握一个标准化学方程式。
Zeno’s Arrow: (1) At any instant, the arrow occupies a space equal to itself. (2) What occupies a space equal to itself is stationary. (3) The arrow is always at such instants. Therefore, the arrow never moves. This forces a reconsideration of time, change, and the continuum.
芝诺飞矢悖论：（1）在任何瞬间，箭占据与自身相等的空间。（2）占据与自身相等空间的东西是静止的。（3）箭总处在这样的瞬间。因此，箭从不运动。这迫使人们重新思考时间、变化和连续体。
The Ship of Theseus: If every plank of a ship is gradually replaced, is it the same ship? If the removed planks are reassembled, which of the two is the original? Used to probe criteria of identity over time – essential for personal identity debates.
忒修斯之船： 如果一艘船的所有木板被逐一更换，它还是同一条船吗？如果把换下的木板重新组装起来，两者中哪一件是原来的船？用于探究历时同一性的标准——对人格同一性辩论至关重要。
The Teletransporter (Parfit): You step into a machine that scans and destroys your body while creating an exact copy elsewhere. Do you survive? Parfit uses this to argue that personal identity is not ‘all or nothing’ but a matter of psychological continuity and connectedness.
远程传送机（帕菲特）： 你走进一台机器，机器扫描并摧毁你的身体，同时在别处创造出一个精确复制品。你存活下来了吗？帕菲特用此论证人格同一性并非“全有或全无”，而是一个心理连续性和关联性的事情。
Knowing the numbered steps of these thought experiments allows you to reconstruct them under time pressure and deploy them as evidence.
知道这些思想实验的编号步骤，让你能在时间压力下重构它们，并将它们作为证据加以运用。
12. Quick-reference Glossary of Key Concepts | 关键概念速查表
This final table distils recurrent technical terms into concise ‘formulaic’ definitions. Treat it as a mental index to trigger deeper explanations in your essays.
最后这张表格将反复出现的技术术语浓缩为简洁的“公式化”定义。把它当作心理索引，在论文中触发更深入的解释。
| Term | Quick Definition |
|---|---|
| A priori / a posteriori | Justified independently of / dependent on experience |
| Analytic / synthetic | True by meaning alone / true in virtue of how the world is |
| Necessary / contingent | Could not have been false / could have been false |
| Substance dualism | Mind and body are distinct substances (Descartes) |
| Occasionalism | God is the sole cause of all mental-physical correlations (Malebranche) |
| Epiphenomenalism | Mental states are caused by, but do not cause, physical states |
| Compatibilism | Free will is compatible with determinism (Hume, Ayer) |
| Libertarianism (free will) | Agents have categorical ability to do otherwise |
| Ockham’s razor | Do not multiply entities beyond necessity |
| Verification principle | A statement is meaningful iff empirically verifiable or tautological |
Internalising this glossary ensures you never resort to vague paraphrases when precise terminology is expected. On the day of the examination, a swift mental run-through of these ‘formulae’ will steady your analysis and sharpen your argumentative edge.
内化这张术语表，可以确保在需要精确术语时你从不诉诸模糊的转述。考试当天，在脑海中快速过一遍这些“公式”，会让你的分析沉稳健实，让论证的锋芒更为犀利。
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