📚 Pre-U CAIE Philosophy: Quick-Reference Handbook of Formulas & Theorems | Pre-U CAIE 哲学：公式定理速查手册
Philosophy is not just a collection of opinions; it relies on rigorous structures of reasoning, precise definitions, and time-tested principles that function much like mathematical formulas. This handbook compiles key logical forms, ethical imperatives, epistemological criteria, and metaphysical shortcuts that every Pre-U CAIE Philosophy student should have at their fingertips. Each entry is presented as a concise statement or 'formula', followed by a brief explanation of its use and significance.
哲学并非仅仅是观点的集合，它依赖于严谨的推理结构、精确的定义以及经过时间检验的原则，这些原则在功能上与数学公式十分相似。本手册汇集了每一位 Pre-U CAIE 哲学学生都应熟记的关键逻辑形式、伦理律令、认识论标准以及形而上学的捷径。每一条都以简洁的陈述或“公式”形式呈现，并附有对其用法和意义的简要说明。
1. Modus Ponens | 肯定前件式
The most fundamental valid argument form in propositional logic. It captures the idea that if a conditional statement is true and its antecedent holds, then the consequent must hold.
命题逻辑中最基本的有效论证形式。它表达了这样一种思想：如果条件陈述为真并且其前件成立，那么后件必然成立。
If P then Q; P; Therefore Q.
如果 P 则 Q；P 成立；因此 Q 成立。
Example: If it rains, the ground is wet. It rains. Therefore the ground is wet.
例子：如果下雨，地面就会湿。下雨了。因此地面是湿的。
Common mistake: Confusing it with the fallacy of affirming the consequent (If P then Q; Q; Therefore P).
常见错误：与肯定后件的谬误（如果 P 则 Q；Q 成立；因此 P 成立）混淆。
2. Modus Tollens | 否定后件式
Another universally valid argument form that works by denying the consequent of a conditional to conclude the denial of the antecedent.
另一种普遍有效的论证形式，它通过否定条件陈述的后件来推出否定前件的结论。
If P then Q; Not Q; Therefore Not P.
如果 P 则 Q；非 Q；因此非 P。
Used extensively in falsificationist reasoning: If the theory is true, we will observe X; we do not observe X; therefore the theory is false.
在证伪主义推理中被广泛使用：如果理论为真，我们会观察到 X；我们没有观察到 X；因此理论为假。
Contrast with the fallacy of denying the antecedent: If P then Q; Not P; Therefore Not Q — this is invalid.
对比否定前件的谬误：如果 P 则 Q；非 P；因此非 Q ——这是无效的。
3. JTB Definition of Knowledge | JTB 知识定义
The classical tripartite analysis of knowledge, widely discussed after Gettier's challenge. Knowledge is traditionally defined as justified true belief.
经典的知识三元分析，在盖梯尔挑战之后被广泛讨论。传统上，知识被定义为经过辩护的真信念。
K = J + T + B (Knowledge = Justified true belief)
知识 = 辩护 + 真 + 信念
S knows that p if and only if: (i) p is true; (ii) S believes that p; (iii) S is justified in believing that p.
S 知道 p 当且仅当：(i) p 为真；(ii) S 相信 p；(iii) S 相信 p 是有辩护的。
Gettier cases show that these three conditions are not sufficient; a fourth condition (e.g., no false lemmas) may be required.
盖梯尔案例表明这三个条件并不充分；可能需要第四个条件（例如，没有错误的引理）。
4. Categorical Imperative (First Formulation) | 绝对命令（第一表述）
Kant's supreme principle of morality, which tests the maxim of an action by considering its universalizability.
康德的最高道德原则，它通过考虑行动准则的可普遍化来检验该准则。
Act only according to that maxim whereby you can at the same time will that it should become a universal law.
只依照你同时愿意它成为一条普遍法则的准则去行动。
This is a test of consistency: if a maxim cannot be universalized without contradiction, the act is morally impermissible.
这是一个一致性检验：如果一个准则无法在没有矛盾的情况下被普遍化，那么该行为在道德上是不被允许的。
Alternative formulation: Treat humanity always as an end in itself, never merely as a means.
另一表述：始终将人性作为目的本身，而绝不仅仅作为手段来对待。
5. The Greatest Happiness Principle | 最大幸福原则
The foundational formula of classical utilitarianism, as articulated by Jeremy Bentham and John Stuart Mill.
古典功利主义的基本公式，由边沁和密尔阐述。
An action is right if and only if it tends to produce the greatest happiness for the greatest number.
一个行为是正当的，当且仅当它倾向于为最大多数人带来最大幸福。
Happiness is defined in terms of pleasure and the absence of pain (hedonistic utilitarianism).
幸福被定义为快乐和痛苦的缺失（享乐主义功利主义）。
Mill introduces a qualitative distinction of pleasures: higher (intellectual) and lower (bodily) pleasures.
密尔引入了快乐的质的区分：高级快乐（理智的）和低级快乐（身体的）。
6. Ockham's Razor | 奥卡姆剃刀
A principle of ontological parsimony used in metaphysics and philosophy of science, attributed to William of Ockham.
一条用于形而上学和科学哲学中的本体论简约原则，归功于奥卡姆的威廉。
Entities are not to be multiplied beyond necessity. (Do not posit extra entities without good reason.)
如无必要，勿增实体。（没有充分理由不要假设多余的实体。）
When two theories explain the same data equally well, the simpler one (with fewer assumptions or entities) is to be preferred.
当两个理论同样好地解释了相同的数据时，应当选择更简单的理论（假设或实体更少）。
Applied in debates about the existence of universals, minds, and abstract objects.
应用于关于共相、心灵和抽象对象是否存在的辩论中。
7. Falsifiability Criterion | 证伪性标准
Karl Popper's demarcation criterion between science and non-science (metaphysics, pseudo-science).
卡尔·波普尔提出区分科学与非科学（形而上学、伪科学）的划界标准。
A theory is scientific if and only if it is falsifiable — i.e., if it conflicts with some possible observation statements.
一个理论是科学的，当且仅当它是可证伪的 ——即，它与某些可能的观察陈述相冲突。
Falsifiability is a logical property: a statement like 'All swans are white' is falsifiable because we can conceive of observing a non-white swan.
可证伪性是一种逻辑属性：像“所有天鹅都是白色的”这样的陈述是可证伪的，因为我们可以设想观察到一只非白色的天鹅。
Popper argues that Marxism and psychoanalysis are not scientific because their proponents can always reinterpret evidence to avoid falsification.
波普尔认为，马克思主义和精神分析学不是科学的，因为它们的拥护者总是可以重新解释证据以避免被证伪。
8. The Social Contract Formula | 社会契约公式
A family of political formulas that derive the legitimacy of political authority from the consent of the governed.
一组从被统治者的同意中推导出政治权威合法性的政治公式。
Legitimate authority = hypothetical or actual agreement among free and equal individuals.
合法权威 = 自由而平等的个体之间假设的或实际的同意。
Hobbes: Individuals agree to surrender all rights (except self-preservation) to a sovereign in exchange for security.
霍布斯：个体同意放弃除自我保存之外的所有权利交给主权者，以换取安全。
Locke: Individuals consent to form a government that protects natural rights (life, liberty, property); if it fails, revolution is justified.
洛克：个体同意组建一个保护自然权利（生命、自由、财产）的政府；如果政府失败，革命就是正当的。
Rawls's 'veil of ignorance' is a refined contractarian device for deriving principles of justice.
罗尔斯的“无知之幕”是为推导正义原则而设计的一种精致的契约论工具。
9. The Principle of Non-Contradiction | 矛盾律
One of the three classical laws of thought, famously defended by Aristotle as the firmest of all principles.
经典思维三律之一，亚里士多德著名地为其辩护为最坚固的原则。
It is impossible for the same thing to belong and not to belong to the same thing at the same time and in the same respect.
同一事物不可能在同一时间、同一方面既属于又不属于同一事物。
Formal logical version: ¬(P ∧ ¬P). It is not the case that both P and not-P are true.
形式逻辑版本：并非（P 并且非 P）。P 和非 P 不可能同时为真。
Used to demonstrate the limits of skepticism and to ground rational discourse; denying it leads to trivialism.
用于证明怀疑论的局限并为理性论述奠基；否定它会导致 trivialism（即一切陈述皆为真）。
10. The No False Lemmas Condition | 无错误引理条件
A response to Gettier problems that strengthens the JTB definition by adding a fourth necessary condition.
对盖梯尔问题的一种回应，它通过增加第四个必要条件来加强 JTB 定义。
S knows that p iff: p is true; S believes p; S's belief is justified; and S's justification does not rest on any false propositions (false lemmas).
S 知道 p 当且仅当：p 为真；S 相信 p；S 的信念得到辩护；并且 S 的辩护不依赖任何假命题（错误引理）。
Gettier's original cases involve a justified true belief that is inferred from a false premise. The no-false-lemmas condition rules out such inferences.
盖梯尔最初的案例涉及从一个假前提推出的经过辩护的真信念。无错误引理条件排除了这样的推论。
Criticism: This condition may be too strong; sometimes a person seems to have knowledge even if one of the background beliefs is accidentally false.
批评：这个条件可能太强了；有时一个人似乎拥有知识，即使其中一个背景信念碰巧是假的。
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