📚 Year 9 CCEA Philosophy: Quick Reference Guide to Formulas and Theorems | Year 9 CCEA 哲学：公式定理速查手册
In philosophy, just as in mathematics or science, certain core principles serve as 'formulas' or 'theorems' that shape rigorous thinking. This quick reference handbook collects the most essential logical laws, argument forms, ethical maxims, and arguments for the existence of God that every Year 9 CCEA Philosophy student should know. Each entry is presented as a clear formula-like statement, unpacked with accessible explanations and examples.
在哲学中，就像在数学或科学中一样，某些核心原则就像“公式”或“定理”，塑造着严谨的思维。这本速查手册汇集了每一位 Year 9 CCEA 哲学学生都应该掌握的最基本的逻辑规律、论证形式、伦理准则和上帝存在论证。每个条目都以清晰的类公式陈述呈现，并配有易于理解的解释和示例。
1. The Law of Identity | 同一律
A = A (or ∀x (x = x))
The Law of Identity states that every entity is identical to itself. In formal terms, for any object x, x is the same as x. This principle guarantees that each thing has a definite nature, making consistent thought and communication possible. If identity were not fixed, the word 'apple' could simultaneously refer to an apple and a non-apple, rendering language useless.
同一律指出，每个实体都与其自身同一。用形式语言来说，对于任意对象 x，x 与其自身相同。这一原则保证了每一事物都具有确定的本性，使前后一致的思维与交流成为可能。如果同一性不固定，“苹果”一词就可能同时指苹果和非苹果，语言就会失效。
2. Law of Non-Contradiction | 矛盾律
¬(P ∧ ¬P) (It is not the case that both P and not-P are true.)
According to the Law of Non-Contradiction, a proposition cannot be both true and false at the same time and in the same respect. For example, the statement 'The door is open' cannot be simultaneously fully open and fully closed. This law underpins all rational debate; without it, any claim could be contradicted immediately and no knowledge could be built.
根据矛盾律，一个命题不能在同一个时间和同一个方面既为真又为假。例如，“门是开着的”这一陈述不可能同时既是完全开的又是完全关的。这条规律是一切理性辩论的基础；没有它，任何主张都可能被立刻反驳，也就无法构建任何知识。
3. Law of Excluded Middle | 排中律
P ∨ ¬P (Either P is true, or its negation is true.)
The Law of Excluded Middle asserts that for any meaningful proposition, either it is true or its negation is true; there is no third option. 'The cat is on the mat' is either true or false—there is no middle state of being half-on-half-off in a logical sense. This principle forces clarity and is essential in classical logic and many philosophical arguments.
排中律断言，对于任何有意义的命题，要么它为真，要么它的否定为真；没有第三种选择。“猫在垫子上”要么为真，要么为假——在逻辑意义上不存在半上半下的中间状态。这一原则迫使人们追求清晰性，是古典逻辑和许多哲学论证的基本前提。
4. Modus Ponens | 肯定前件
If P → Q, and P is true, then Q. (P → Q, P ⊢ Q)
Modus ponens is a valid argument form: if you have a conditional statement 'If P then Q' and you know that P is the case, you can conclude Q. For instance, 'If it rains, the ground will be wet. It rains. Therefore, the ground is wet.' This 'formula' is one of the most fundamental rules of inference used in everyday reasoning and formal logic.
肯定前件是一种有效的论证形式：如果你有一个条件陈述“如果 P 则 Q”，并且你知道 P 成立，就可以得出结论 Q。例如，“如果下雨，地面就会湿。下雨了。因此，地面是湿的。”这个“公式”是日常推理和形式逻辑中最基本的推理规则之一。
5. Modus Tollens | 否定后件
If P → Q, and Q is false, then P is false. (P → Q, ¬Q ⊢ ¬P)
Modus tollens complements modus ponens. If a conditional holds and the consequent (Q) is false, the antecedent (P) must also be false. For example, 'If the alarm works, it will ring at 7am. It did not ring at 7am. Therefore, the alarm does not work.' This pattern is crucial for testing hypotheses and eliminating possibilities.
否定后件是肯定前件的补充。如果条件陈述成立，且后件（Q）为假，那么前件（P）也必然为假。例如，“如果闹钟正常，它会在早上7点响。它没有在7点响。因此，闹钟不正常。”这种模式对于检验假设和排除可能性至关重要。
6. Barbara Syllogism | 芭芭拉三段论
All M are P. All S are M. Therefore, all S are P.
'Barbara' is the medieval mnemonic name for the most basic valid syllogism: a universal affirmative argument. If all members of group M belong to group P, and all members of group S belong to group M, it necessarily follows that all S are P. Example: 'All mammals are warm-blooded. All dogs are mammals. Therefore, all dogs are warm-blooded.' Simplicity and certainty make this a building block of deductive thought.
“芭芭拉”是中世纪为最基本有效三段论所起的助记名：一个全称肯定的论证。如果 M 类的所有成员都属于 P 类，S 类的所有成员都属于 M 类，那么必然得出所有 S 都是 P。例如：“所有哺乳动物都是温血的。所有狗都是哺乳动物。因此，所有狗都是温血的。”简单性和确定性使这一形式成为演绎思维的基石。
7. The Greatest Happiness Principle | 最大幸福原则
The right action is the one that produces the greatest happiness for the greatest number.
Proposed by Jeremy Bentham and later refined by John Stuart Mill, this is the core 'formula' of utilitarianism. It evaluates actions not by their intentions but by their consequences—the amount of pleasure or happiness they generate minus the pain. When faced with a moral dilemma, you should calculate the net happiness of each option and choose the one with the highest positive balance. Critics warn that this could overlook justice and individual rights.
由边沁提出，后经密尔完善，这是功利主义的核心“公式”。它评价行为不是看意图，而是看后果——它们产生的快乐或幸福减去痛苦之后的余量。面对道德困境时，你应该计算每个选项的净幸福值，并选择正向余额最高的那个。批评者警告，这可能忽视正义和个人权利。
8. The Categorical Imperative: Universal Law Formula | 绝对命令：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.'
Immanuel Kant's supreme principle of morality is a test of consistency. Before acting, you ask: 'Could I rationally wish that everyone in the world followed the rule I am acting on?' If the answer is no—because universalising it would lead to a contradiction or an undesirable world—then the action is morally forbidden. For example, lying is wrong because if everyone lied, trust would collapse and the very practice of lying would become meaningless.
康德的最高道德原则是一种一致性检验。在行动之前，你问自己：“我能否理性地希望世界上每个人都遵循我当前依据的准则？”如果答案是否定的——因为普遍化它会导致矛盾或一个不可欲的世界——那么该行为在道德上便是被禁止的。例如，说谎是错误的，因为如果人人都说谎，信任就会崩溃，说谎的行为本身也变得毫无意义。
9. The Golden Rule | 黄金法则
'Do to others what you would want them to do to you.' / 'Treat others as you would like to be treated.'
The Golden Rule is a cross-cultural ethical theorem found in many religious and philosophical traditions. It asks you to use your own desires as a measure: whatever you wish others would do to you, do likewise to them; whatever you detest suffering, do not inflict on others. It promotes empathy and mutual respect. While simple, it can be debated whether a sadomasochist would be permitted to harm others under this rule—most interpretations assume a shared baseline of human flourishing.
黄金法则是一条跨文化的伦理性定理，存在于许多宗教和哲学传统中。它要求你以自己的愿望为尺度：你希望别人怎样对待你，你就怎样对待别人；你憎恶遭遇什么，就不要施加给别人。它倡导同理心和相互尊重。虽然简单，但可以争论的是，一个施虐受虐狂是否可以根据这一法则去伤害他人——多数诠释假定一个共同的人类福祉底线。
10. Anselm's Ontological Definition | 安瑟伦的本体论定义
God is 'that than which nothing greater can be conceived.'
Saint Anselm of Canterbury proposed a famous a priori argument for God's existence based on this definition. If we understand God as the greatest conceivable being, and if existing in reality is greater than existing only in the mind, then God must exist in reality, otherwise He would not be the greatest conceivable being. The 'formula' turns the concept of God into a logical proof. Gaunilo replied with a parody island—the 'greatest conceivable island' does not pop into existence just by definition.
坎特伯雷的圣安瑟伦基于这一定义提出了一个著名的先天上帝存在论证。如果我们把上帝理解为“无法设想有比之更伟大的存在”，而存在于现实中比仅存在于心灵中更伟大，那么上帝必须存在于现实中，否则祂就不是可设想的最伟大的存在了。这个“公式”将上帝的概念转化为逻辑证明。高尼罗用一个戏仿的岛屿来回应——光凭定义并不能让“可设想的最伟大的岛屿”凭空出现。
11. The First Cause Theorem | 第一因定理
Every event has a cause; the chain cannot go back forever; thus, there must be a first uncaused cause.
Thomas Aquinas formulated the Cosmological Argument as a 'theorem' of cause and effect. We observe that things move and change because other things act on them. If we trace back the chain of movers or causes, an infinite regress is impossible—there must be a Prime Mover or First Cause that itself is not caused by anything else. Aquinas identified this as God. Critics ask why the chain must stop at God rather than the universe being eternal or uncaused.
托马斯·阿奎那将宇宙论论证表述为一个关乎因果的“定理”。我们观察到万物因他物的作用而运动变化。如果我们追溯推动者或原因的链条，无限回溯是不可能的——必定存在一个不被任何别的东西推动或产生的第一推动者或第一因。阿奎那将其确认为上帝。批评者质问道：为什么链条必须止于上帝，而不能是宇宙本身是永恒或无需原因的。
12. Paley's Watchmaker Analogy | 佩利的钟表匠类比
Complex, purpose-serving order implies an intelligent designer.
William Paley's Teleological Argument uses an analogy as its core formula: if you found a watch on a heath, you would infer a watchmaker because of its intricate complexity and clear purpose. The natural world exhibits far greater complexity and apparent design—the human eye, for instance. Therefore, by analogy, the universe must have an intelligent designer, God. David Hume anticipated this argument, noting that analogy is weak when comparing the universe to a machine, and that even if design exists, it does not prove the designer is all-good.
威廉·佩利的目的论论证以一个类比作为核心公式：如果你在荒野中发现一块手表，你会因它精巧的复杂性和明确的目的而推断有一位钟表匠。自然界展示出远为巨大的复杂性和表面上的设计——比如人的眼睛。因此，通过类比，宇宙必定有一位智慧的設計者，即上帝。休谟早就批评过这一论证，指出将宇宙比作机器的类比很弱，而且即使存在设计，也不能证明设计者全善。
Published by TutorHao | Philosophy Revision Series | aleveler.com
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