📚 Year 7 SQA Philosophy: Formula & Theorem Quick Reference Handbook | Year 7 SQA 哲学：公式定理速查手册
Philosophy may not have formulas in the same way as mathematics, but it offers powerful logical tools, ethical principles, and reasoning patterns that act like formulas or theorems for understanding the world. This quick reference handbook compiles key ‘formulas and theorems’ from logic, ethics, politics, and epistemology suitable for Year 7 SQA Philosophy. Each entry provides a memorable pattern, its symbolic form where possible, and a simple example to help you apply it.
哲学或许不像数学那样拥有公式，但它提供了强大的逻辑工具、伦理原则和推理模式，这些可以像公式或定理一样用来理解世界。本速查手册汇编了适合 Year 7 SQA 哲学课程的逻辑、伦理学、政治哲学和认识论中的重要“公式与定理”。每个条目都给出了易于记忆的模式、尽可能的符号形式，以及一个帮助你应用的简单示例。
1. The Principle of Non-Contradiction | 不矛盾律
A fundamental logical principle: nothing can both be and not be at the same time and in the same respect. If a statement and its negation were both true, reasoning would collapse.
基本逻辑原则：同一事物不能在同一方面同时既是又不是。如果一个陈述及其否定同时为真，推理就会崩溃。
Symbolic form:
符号形式：
¬(P ∧ ¬P)
Example: You cannot say ‘It is raining’ and ‘It is not raining’ about exactly the same place and moment without contradiction.
示例：你不能对完全相同的同一地点和同一时刻说“正在下雨”和“没有下雨”而不产生矛盾。
2. Modus Ponens (Affirming the Antecedent) | 肯定前件式
If a conditional statement is true, and its ‘if’ part (antecedent) holds, then its ‘then’ part (consequent) necessarily follows. This is a basic valid argument form.
如果一个条件陈述为真，且其“如果”部分（前件）成立，则其“那么”部分（后件）必然成立。这是一个基本的有效论证形式。
Form:
形式：
P → Q, P ⊢ Q
Example: If it is a dog (P), then it is a mammal (Q). It is a dog (P). Therefore, it is a mammal (Q).
示例：如果它是狗（P），那么它是哺乳动物（Q）。它是狗（P）。因此，它是哺乳动物（Q）。
3. Modus Tollens (Denying the Consequent) | 否定后件式
If the consequent of a true conditional is false, the antecedent must also be false. This allows us to eliminate possibilities.
如果一个真条件句的后件为假，那么其前件必然也为假。这使我们能够排除某些可能性。
Form:
形式：
P → Q, ¬Q ⊢ ¬P
Example: If it is a dog (P), then it is a mammal (Q). It is not a mammal (¬Q). Therefore, it is not a dog (¬P).
示例：如果它是狗（P），那么它是哺乳动物（Q）。它不是哺乳动物（¬Q）。因此，它不是狗（¬P）。
4. The Categorical Imperative Formula | 绝对命令公式
Immanuel Kant’s core ethical principle: Act only according to that maxim whereby you can at the same time will that it should become a universal law. In simpler terms, ask yourself: ‘What if everyone did that?’
康德的伦理学核心原则：只按照你同时能够愿意它成为一条普遍法则的准则去行动。简单来说，就是问自己：“如果每个人都这样做会怎样？”
Formula for testing an action:
检验一个行动的公式：
Maxim → Universalizable? If yes → permissible; If no → impermissible
Example: You consider lying to get out of trouble. If everyone lied whenever it suited them, trust would disappear and lying would become pointless. The maxim fails the test, so lying is impermissible.
示例：你考虑通过说谎来摆脱麻烦。如果每个人都随心所欲地说谎，信任就会消失，说谎本身也就毫无意义了。该准则未能通过检验，因此说谎是不被允许的。
5. Utilitarian Calculus (Greatest Happiness Principle) | 功利主义计算式（最大幸福原则）
Jeremy Bentham and John Stuart Mill argued that the right action is the one that produces the greatest happiness for the greatest number. Happiness = pleasure minus pain.
边沁和密尔认为，正确的行动是为最大多数人带来最大幸福的行动。幸福 = 快乐减去痛苦。
Basic formula:
基本公式：
Utility = Σ(Pleasure) − Σ(Pain)
Example: You have one piece of cake and two friends. Eating it yourself gives you 5 pleasure units but causes each friend 2 pain units (total pain = 4). Sharing gives you 3 pleasure and each friend 3 pleasure (total pleasure = 9). The calculation shows sharing is the better action.
示例：你有一块蛋糕和两个朋友。自己吃掉给你5个快乐单位，但给每个朋友造成2个痛苦单位（总痛苦=4）。分享给你3个快乐，每个朋友3个快乐（总快乐=9）。计算表明，分享是更好的行动。
6. The Social Contract Theorem | 社会契约定理
According to Thomas Hobbes and John Locke, people agree to give up some freedoms and grant power to a government in exchange for security and protection of their rights. This trade-off forms the basis of civil society.
根据霍布斯和洛克的观点，人们同意放弃部分自由并赋予政府权力，以换取安全和对自身权利的保护。这种交换构成了公民社会的基础。
The contract formula:
契约公式：
Natural freedom − Some liberties + Obligation to obey → Protection + Order
Example: You agree not to drive through red lights (giving up a liberty) so that everyone, including you, can travel safely. The state enforces the rule, and order is maintained.
示例：你同意不闯红灯（放弃一项自由），以便所有人包括你自己都能安全出行。国家执行规则，秩序得以维持。
7. Descartes’ Cogito Theorem | 笛卡尔的“我思”定理
René Descartes searched for a foundation of certainty. He concluded that even if he doubted everything, the very act of doubting proved that he existed. This is famously expressed as ‘I think, therefore I am.’
笛卡尔寻求确定性的基础。他得出结论，即使怀疑一切，怀疑行为本身也证明了他的存在。这就是著名的“我思故我在”。
The reasoning chain:
推理链条：
Doubt → Thinking → Existence
Symbolically: If I doubt, then I think. If I think, then I exist. Therefore, I exist.
符号表示：如果我怀疑，那么我思考。如果我思考，那么我存在。因此，我存在。
8. The Golden Mean Formula | 中庸之道公式
Aristotle’s virtue ethics holds that moral virtue is a mean between two vices — one of excess and one of deficiency. Courage, for example, lies between cowardice and recklessness.
亚里士多德的德性伦理学认为，道德德性是两种恶之间的中道——一种是过度，一种是不及。例如，勇敢处于懦弱和鲁莽之间。
The virtue formula:
德性公式：
Virtue = Balancing point between Excess and Deficiency
Example: In a scary situation, running away immediately is deficient (cowardice), while charging in without any thought is excessive (recklessness). The virtuous response is a measured, appropriate middle action (courage).
示例：在可怕的情境中，立即逃跑是不及（懦弱），而不假思索地冲进去是过度（鲁莽）。有德性的回应是适度的、恰当的中间行为（勇敢）。
9. Occam’s Razor | 奥卡姆剃刀原则
A principle of parsimony attributed to William of Ockham: ‘Entities should not be multiplied beyond necessity.’ When faced with competing explanations, the simplest one requiring the fewest assumptions is usually preferred.
归因于奥卡姆的威廉的简约原则：“如无必要，勿增实体。”当面对相互竞争的解释时，通常偏好那个需要最少假设的最简单的解释。
Practical formula:
实用公式：
If two theories explain the same data equally well → choose the simpler one
Example: You hear hoofbeats in the park. You could theorise it is a horse or a zebra. Since no zebras are known in the area, the horse explanation is simpler and more likely under Occam’s Razor.
示例：你听到公园里有蹄声。你可以假定那是一匹马或一匹斑马。由于该地区没有已知的斑马，根据奥卡姆剃刀，马的解释更简单且更可能。
10. The Veil of Ignorance Reasoning | 无知之幕推理
John Rawls proposed a thought experiment: imagine you are behind a ‘veil of ignorance’ where you do not know your own position, talents, or social status. From this original position, you would design fair principles of justice that treat everyone equally.
罗尔斯提出了一个思想实验：想象你处于“无知之幕”背后，不知道自己的地位、才能或社会身份。在这种原初状态下，你会设计出公平对待每一个人的正义原则。
Reasoning pattern:
推理模式：
Original Position (no knowledge of self) → Impartial choice → Fair rules
Example: If you did not know whether you would be born rich or poor, you would design a society with safety nets and equal opportunities, because you might end up in the least advantaged group.
示例：如果你不知道自己生来是富是穷，你会设计一个拥有安全网和机会平等的社会，因为你可能最终处于最不利的群体中。
11. The Hedonic Calculus (Bentham’s Felicific Calculus) | 快乐计算（边沁的幸福计算）
Bentham proposed a more detailed method for measuring pleasure and pain produced by an action. The hedonic calculus considers seven factors to quantify the overall tendency toward happiness.
边沁提出了一种更详细的方法来衡量一个行动所产生的快乐和痛苦。快乐计算考虑了七个因素来量化整体的幸福倾向。
The seven factors table:
七因素表：
| Factor | Meaning | 中文含义 |
|---|---|---|
| Intensity | How strong is the pleasure or pain? | 强度：快乐或痛苦有多强烈？ |
| Duration | How long will it last? | 持续时间：它会持续多久？ |
| Certainty | How likely is it to occur? | 确定性：它发生的可能性有多大？ |
| Propinquity | How soon will it occur? | 临近性：它多久之后会发生？ |
| Fecundity | Will it lead to more pleasure? | 丰饶度：它会带来更多快乐吗？ |
| Purity | Will it be followed by opposite sensations? | 纯度：它会不会伴有相反的感受？ |
| Extent | How many people are affected? | 范围：多少人受到影响？ |
Conceptual formula: The overall tendency = aggregate of all factored pleasures minus factored pains.
概念公式：总体倾向 = 所有因素化快乐的总和减去因素化痛苦的总和。
12. The State of Nature Theorem (Hobbes) | 自然状态定理（霍布斯）
Thomas Hobbes described a hypothetical ‘state of nature’ without government, where individuals have absolute freedom but live in constant fear. He concluded that life in such a state would be ‘solitary, poor, nasty, brutish, and short,’ leading people to form a social contract and establish a sovereign power.
霍布斯描述了一个假设的没有政府的“自然状态”，其中个体拥有绝对自由，但生活在持续的恐惧中。他得出结论，这种状态下的生活将是“孤独、贫穷、卑污、残忍且短寿的”，从而促使人们订立社会契约并建立主权权力。
The chain of reasoning:
推理链条：
No authority → War of all against all → Fear and insecurity → Social contract → Leviathan (ruler)
Example: Imagine a playground with no teachers or rules. Everyone would grab the best equipment, fights would break out, and soon no one could play safely. To escape this, children agree to a monitor and rules — a mini Leviathan.
示例：想象一个没有老师或规则的操场。每个人都会抢最好的器材，打架会爆发，很快谁也安全地玩不了。为了摆脱这种局面，孩子们同意设一个监督员和规则——一个微型的利维坦。
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