📚 Formula & Theorem Quick Reference for CAIE Philosophy | CAIE 哲学公式定理速查手册
This quick reference handbook brings together the essential logical forms, ethical principles, epistemological definitions, and classical argument structures that lie at the heart of the Year 13 CAIE Philosophy syllabus. Each entry is presented as a formula or theorem – a pattern you can rely on when analysing arguments, constructing essays, or revising for the exam. Use it to sharpen your philosophical toolkit and to see how abstract ideas are built from precise, repeatable patterns.
这本速查手册汇集了 Year 13 CAIE 哲学课程核心的逻辑形式、伦理学原则、认识论定义以及经典论证结构。每一个条目都以公式或定理的形式呈现——在分析论证、撰写论文或考前复习时，你可以依靠这些模式。用它来打磨你的哲学工具箱，理解抽象理念是如何由精确、可重复的结构构建而成的。
1. Modus Ponens | 肯定前件式
One of the most fundamental valid argument forms. If a conditional statement is true and its antecedent is true, then its consequent must be true. In logical notation, from P → Q and P, infer Q.
这是最基本的有效论证形式之一。如果条件命题为真，且其前件为真，那么后件必然为真。用逻辑符号表示，从 P → Q 和 P，推出 Q。
P → Q, P ⊢ Q
Example: If it is raining, the pavement is wet. It is raining. Therefore, the pavement is wet. The certainty of this inference makes it a cornerstone of deductive reasoning, but note that it does not work backwards – affirming the consequent is a fallacy.
例子：如果下雨，人行道湿。下雨了。所以人行道湿。这种推理的确定性使之成为演绎推理的基石，但请注意它不能反向使用——肯定后件是一种谬误。
2. Modus Tollens | 否定后件式
Another essential valid form of deductive reasoning. If a conditional is true and its consequent is false, then the antecedent must be false. Denying the consequent forces the denial of the antecedent.
另一种基本的有效演绎推理形式。如果条件命题为真，且其后件为假，那么前件必然为假。否定后件迫使否定前件。
P → Q, ¬Q ⊢ ¬P
Example: If the substance is gold, it will dissolve in aqua regia. It did not dissolve. Therefore, the substance is not gold. Modus tollens plays a crucial role in falsification in science and in reductio ad absurdum arguments in philosophy.
例子：如果该物质是黄金，它会溶于王水。它没有溶解。所以该物质不是黄金。否定后件式在科学证伪和哲学归谬论证中起着至关重要的作用。
3. De Morgan’s Laws | 德摩根定律
These equivalences govern the relationship between conjunction, disjunction, and negation. They are indispensable for transforming logical statements and analysing the logical form of ordinary language claims.
这些等价关系支配着合取、析取与否定之间的关系。它们对于转换逻辑命题和分析日常语言主张的逻辑形式不可或缺。
¬(P ∧ Q) ↔ (¬P ∨ ¬Q)
¬(P ∨ Q) ↔ (¬P ∧ ¬Q)
In philosophical argument, De Morgan’s laws are regularly used to expose hidden logical structure. For example, the statement ‘It is not the case that both the universe is designed and the designer is benevolent’ is equivalent to ‘Either the universe is not designed or the designer is not benevolent.’ This allows clear analysis of arguments in the philosophy of religion.
在哲学论证中，德摩根定律经常被用来揭示隐藏的逻辑结构。例如，“宇宙是被设计的且设计者是仁慈的，此说不成立”等价于“要么宇宙不是被设计的，要么设计者不是仁慈的”。这使得宗教哲学中的论证分析变得清晰。
4. The Principle of Utility | 功利原则
At the centre of classical utilitarianism lies a defining formula: the morally right action is the one that produces the greatest happiness for the greatest number. While not a formal logical theorem, it functions as an ethical decision procedure.
古典功利主义的核心是一条定义性公式：道德上正确的行为，就是为最大多数人带来最大幸福的行为。尽管不是形式逻辑定理，但它充当了伦理决策程序。
Right action = max Σ (pleasure – pain) for all sentient beings affected
Bentham’s version expects an actual felicific calculus, measuring intensity, duration, certainty, propinquity, fecundity, purity, and extent. Mill later shifted the emphasis towards quality of pleasures, but the structural formula of maximising aggregate well-being remains central. Critics question whether all values can be reduced to a single utility metric.
边沁的版本期望一种实际的快乐计算，衡量强度、持续时间、确定性、临近性、丰产性、纯度和范围。密尔后来将重点转向快乐的质，但最大化总体福祉的结构公式仍是核心。批评者质疑所有价值是否都能还原为单一的效用尺度。
5. The Categorical Imperative (Formula of Universal Law) | 绝对命令（普遍法则公式）
Kant’s supreme principle of morality is expressed in several formulations. The first, the Formula of Universal Law, instructs us to act only on maxims that we could simultaneously will to become universal laws without contradiction.
康德的最高道德原则以若干表达式呈现。第一个，普遍法则公式，指示我们只按照你同时能够愿意其成为一项普遍法则的准则去行动，且不会产生矛盾。
Act only according to that maxim whereby you can at the same time will that it should become a universal law.
The test involves two kinds of contradiction: a contradiction in conception (the maxim cannot even be thought as universal law without logical impossibility) and a contradiction in the will (a rational agent could not consistently will such a world). This formula provides a formal procedure for testing the moral permissibility of actions, independent of consequences.
检验涉及两种矛盾：概念矛盾（准则甚至无法被无矛盾地设想为普遍法则）和意志矛盾（一个理性行动者无法一贯地意愿这样一个世界）。该公式提供了一种独立于后果的检验行为道德可允许性的形式程序。
6. The JTB Analysis of Knowledge | 知识的JTB分析
Since Plato’s Theaetetus, knowledge has traditionally been analysed as justified true belief. This tripartite definition can be treated as an epistemological formula: S knows that P if and only if P is true, S believes that P, and S is justified in believing that P.
自柏拉图的《泰阿泰德篇》以来，知识一直被分析为得到辩护的真信念。这个三元定义可以视作一个认识论公式：S 知道 P，当且仅当 P 为真，S 相信 P，且 S 有正当理由相信 P。
K(s, P) ↔ [True(P) ∧ Believes(s, P) ∧ Justified(s, P)]
Gettier’s counterexamples showed that these three conditions are not jointly sufficient. Standard cases involve a justified false belief that luckily leads to a truth. The formula has since been refined with additional conditions, such as a ‘no false lemmas’ clause or reliable causal connection, but the JTB structure remains the starting point for all post-Gettier theories.
盖梯尔的反例表明这三个条件并非联合充分。标准情形涉及一个得到辩护的假信念碰巧导致一个真信念。此后公式被加上额外条件加以修正，例如“无假前提”条款或可靠的因果联系，但JTB结构仍是所有后盖梯尔理论的出发点。
7. The Ontological Argument (Anselm’s Formula) | 本体论论证（安瑟伦公式）
Anselm defines God as ‘that than which nothing greater can be conceived.’ The formula proceeds by showing that such a being must exist in reality, because existing in reality is greater than existing only in the understanding.
安瑟伦将上帝定义为“无与伦比者”。该公式通过证明这样一个存在者必然在现实中存在而推进，因为存在于现实中比仅仅存在于理智中更伟大。
God = df the being than which no greater can be conceived.
If that being exists only in the mind, we could conceive a greater being – one that also exists in reality. Therefore, God must exist in reality.
Gaunilo parodied this formula using the concept of a perfect island, and Kant later objected that existence is not a real predicate. Despite these criticisms, the ontological argument formula continues to be studied because it reveals deep connections between concepts, existence, and logical necessity.
高尼罗用完美之岛的概念戏仿了这一公式，康德随后反对存在不是实在谓词。尽管有这些批评，本体论论证公式仍被持续研究，因为它揭示了概念、存在与逻辑必然性之间的深层联系。
8. The Logical Problem of Evil | 恶的逻辑问题
The logical problem of evil challenges the consistency of the theistic set of claims. If God is omnipotent, wholly benevolent, and omniscient, then evil should not exist. But evil does exist. This inconsistency is often presented as a formal argument formula.
恶的逻辑问题挑战了有神论一系列主张的一致性。如果上帝全知、全能、全善，那么恶就不应存在。但恶确实存在。这种不一致通常以形式论证公式呈现。
(Omnipotent ∧ Omnibenevolent ∧ Omniscient) → ¬Exist(Evil)
Exist(Evil)
Therefore, ¬(Omnipotent ∧ Omnibenevolent ∧ Omniscient)
The formula forces the theist to re-examine at least one of the premises: perhaps God’s power is logically limited, or evil is a necessary by‑product of greater goods, or our understanding of ‘benevolence’ needs refining. The free will defence attempts to break the connection between divine attributes and the absence of evil.
该公式迫使有神论者重新审视至少一个前提：也许上帝的能力逻辑上受限，或者恶是更大善的必要副产品，或者我们对“全善”的理解需要修正。自由意志辩护试图打破神性属性与恶之不存在之间的联系。
9. Cartesian Dualism: The Divisibility Argument | 笛卡尔二元论：可分性论证
Descartes offered a compact argument formula for substance dualism. Bodies are divisible, extension‑based substances; minds are indivisible, thinking substances. Since they have different essential properties, they must be distinct substances.
笛卡尔为实体二元论提供了一个紧凑的论证公式。物体是可分的、基于广延的实体；心灵是不可分的、能思维的实体。因为它们拥有不同的本质属性，它们必然是截然不同的实体。
P1: Body is divisible.
P2: Mind is indivisible.
P3: If X and Y have contradictory properties, X ≠ Y.
Therefore, mind ≠ body.
This formula has been criticised for its reliance on the ‘masked man’ fallacy, but it remains indispensable for understanding the mind‑body problem. It highlights the conceptual gap between first‑person subjective experience and third‑person physical description, a gap that many contemporary philosophers still try to bridge.
该公式被批评依赖于“蒙面人”谬误，但它对于理解身心问题仍不可或缺。它突显了第一人称主观经验与第三人称物理描述之间的概念鸿沟，当代许多哲学家仍在试图弥合这一鸿沟。
10. The Fallacy of Affirming the Consequent | 肯定后件谬误
This formal fallacy occurs when an argument mistakenly infers the antecedent from the truth of the consequent in a conditional. It is structurally invalid and appears frequently in everyday reasoning and in poorly constructed philosophical arguments.
这是一种形式谬误，发生在论证从条件命题后件为真错误地推出前件为真时。它在结构上是无效的，常出现于日常推理和构建不良的哲学论证中。
P → Q, Q ⊢ P? INVALID
Example: If I am in Paris, I am in France. I am in France. Therefore, I am in Paris. This fails because the truth of the consequent does not guarantee the truth of the antecedent; there are many ways to be in France. Identifying this pattern is essential for evaluating arguments about design, causation, and mind in the CAIE syllabus.
例子：如果我在巴黎，我就在法国。我在法国。因此我在巴黎。这无效，因为后件为真不能保证前件为真；有很多方式可以身处法国。识别这种模式对于评价CAIE大纲中关于设计、因果和心灵的论证至关重要。
11. Hume’s Principle of Empirical Meaning | 休谟的经验意义原则
Hume proposed a cognitive criterion for meaningful discourse. Ideas that cannot be traced back to sense impressions are, for Hume, empty and should be consigned to the flames. This principle functions as a kind of philosophical theorem for separating legitimate from pseudo‑inquiry.
休谟提出了有意义话语的认知标准。对于休谟而言，无法追溯到感觉印象的观念是空洞的，应该被付之一炬。该原则作为一种哲学定理，用以区分合法探究与伪探究。
If a term or proposition cannot be derived from any impression, it is without cognitive meaning (or is a ‘fiction’).
Applied to metaphysics, the formula eliminates unverifiable notions such as substance, necessary connection, and the self as a permanent entity. The logical positivists later adopted a similar verification principle, making Hume’s formula a precursor to 20th‑century meaning criteria.
将该公式应用于形而上学，便消除了不可验证的概念，例如实体、必然联系以及作为恒久实体的自我。逻辑实证主义者后来采纳了类似的证实原则，使休谟的公式成为20世纪意义标准的先驱。
12. Parfit’s Repugnant Conclusion (Formula) | 帕菲特的可憎结论（公式）
In population ethics, Derek Parfit revealed a disturbing implication of total utilitarian principles. The formula shows that any large population with a very high quality of life can be ‘outweighed’ by a much larger population whose lives are only just worth living.
在人口伦理学中，德里克·帕菲特揭示了总体功利原则的一个令人不安的含义。该公式表明，任何拥有极高生活质量的大量人口，都可能被一个庞大得多但生活仅勉强值得过的人口所“压倒”。
For any population A with high welfare, there exists a much larger population Z where each life has barely positive welfare, such that total utility(Z) > total utility(A).
This repugnant conclusion challenges the intuition that a world of many barely happy lives is morally preferable to a world of few very happy lives. It forces ethical theorists to refine the formula of utility – perhaps by adopting person‑affecting principles, variable value theories, or threshold accounts of well‑being.
这个可憎结论挑战了这样一种直觉：一个由许多勉强幸福的生命组成的世界，在道德上优于一个由少数非常幸福的生命组成的世界。它迫使伦理理论家修正功利公式——或许采用影响人的原则、可变价值理论或福祉门槛解释。
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