📚 Year 9 SQA Philosophy: Formula & Theorem Quick Reference Handbook | 九年级 SQA 哲学：公式定理速查手册
Philosophy is often seen as a subject of open-ended discussion, but at its core it relies on rigorous structures of reasoning, fundamental principles, and well-tested conceptual ‘formulas’. This quick reference handbook gathers the most essential logical rules, ethical maxims, and epistemological theorems that every Year 9 SQA Philosophy student should commit to memory. Just as mathematics uses equations to model reality, philosophy uses these timeless formulas to clarify thought, evaluate arguments, and build coherent worldviews.
哲学常被视为开放讨论的学科，但其核心依赖于严密的推理结构、基本原则和久经检验的概念“公式”。这本速查手册汇集了每一位九年级 SQA 哲学学生都应牢记的重要逻辑规则、伦理准则和认识论定理。就像数学用方程式模拟现实一样，哲学运用这些经久不衰的公式来澄清思想、评估论证并构建连贯的世界观。
1. Modus Ponens | 肯定前件式
If a conditional statement is accepted, and its antecedent is true, then the consequent must also be true. This is the simplest and most powerful rule of inference.
如果一个条件陈述被接受，且其前件为真，那么后件必然为真。这是最简单且最有力的推理规则。
P → Q, P ⊢ Q
Example: ‘If it rains, the ground gets wet. It rains. Therefore, the ground gets wet.’ The validity of Modus Ponens depends only on logical form, not on the content.
例子：“如果下雨，地面会湿。下雨了。因此，地面湿了。”肯定前件式的有效性仅取决于逻辑形式，而非内容。
2. Modus Tollens | 否定后件式
When the consequent of a conditional is false, the antecedent cannot be true. This rule is essential for falsifying hypotheses.
当一个条件句的后件为假时，其前件不可能为真。这条规则对于证伪假设至关重要。
P → Q, ¬Q ⊢ ¬P
Example: ‘If it is a dog, then it has four legs. This animal does not have four legs. Therefore, it is not a dog.’ Modus Tollens preserves truth and is the backbone of scientific testing.
例子：“如果它是狗，那么它有四条腿。这只动物没有四条腿。因此，它不是狗。”否定后件式保真，是科学检验的支柱。
3. Hypothetical Syllogism | 假言三段论
Chain two conditionals together where the consequent of the first is the antecedent of the second, and you can derive a direct conditional from the first antecedent to the final consequent.
将两个条件句链接起来，若前一个的后件恰为后一个的前件，则可推出从第一个前件到最终后件的直接条件句。
(P → Q) ∧ (Q → R) ⊢ (P → R)
This formula mirrors how we extend reasoning: ‘If I study, I pass the exam. If I pass the exam, I graduate. Therefore, if I study, I graduate.’
这个公式反映我们如何扩展推理：“如果我学习，我就能通过考试。如果我通过考试，我就能毕业。因此，如果我学习，我就能毕业。”
4. Disjunctive Syllogism | 选言三段论
From a disjunction (‘either A or B’) and the negation of one disjunct, the other must be true. This works only when ‘or’ is understood inclusively or when the options are exhaustive.
从一个析取陈述（“要么A要么B”）和其中一个选言支的否定出发，另一个必然为真。只有当“或”被理解为相容的或选项穷尽时才成立。
(P ∨ Q), ¬P ⊢ Q
Example: ‘Either it is Monday or it is Tuesday. It is not Monday. Therefore, it is Tuesday.’ In philosophy, this structure is used to eliminate possibilities.
例子：“要么是周一，要么是周二。不是周一。因此，是周二。”在哲学中，这一结构常用于排除可能性。
5. Law of Non-Contradiction & Excluded Middle | 矛盾律与排中律
Two fundamental logical theorems: a proposition cannot be both true and false in the same respect at the same time (Law of Non-Contradiction: ¬(P ∧ ¬P)). Every meaningful statement is either true or false, with no middle option (Law of Excluded Middle: P ∨ ¬P).
两个基本的逻辑定理：一个命题不能在同一方面同时为真且为假（矛盾律：¬(P ∧ ¬P)）。每一个有意义的陈述非真即假，没有中间状态（排中律：P ∨ ¬P）。
Together they guarantee that logical discourse is possible. Without them, any argument could be both sound and unsound simultaneously, collapsing reason entirely.
它们共同确保了逻辑对话的可能性。没有它们，任何论证都可能既可靠又不可靠，使理性彻底崩溃。
6. Utilitarian Calculus: Greatest Happiness Principle | 功利计算：最大幸福原则
In moral philosophy, Bentham and Mill proposed a formula for ethical decision-making: an action is right if it tends to produce the greatest happiness for the greatest number. The ‘utility’ of an act can be calculated by summing the pleasures and pains it causes across all affected individuals.
在道德哲学中，边沁与密尔提出了一种伦理决策公式：如果一个行为倾向于为最大多数人产生最大幸福，则该行为是正确的。一个行为的“效用”可通过加总它对所有受影响个体带来的快乐与痛苦来计算。
The informal formula is: Total Utility = Sum of Pleasures − Sum of Pains. Mill added that the quality of pleasures matters, not merely quantity, refining the ‘hedonic calculus’.
非正式的公式是：总效用 = 快乐总和 − 痛苦总和。密尔补充说，快乐的质量也很重要，而不仅仅是数量，从而改良了“快乐计算”。
7. Kant’s Categorical Imperative Formula | 康德的定言命令公式
Kant’s supreme principle of morality can be expressed as a formula for testing maxims: Act only according to that maxim whereby you can at the same time will that it should become a universal law.
康德的最高道德原则可被表述为检验准则的公式：只依据那些你同时愿意它成为普遍法则的准则去行动。
This is not a mathematical equation but a test of consistency. If universalizing a maxim leads to a contradiction in conception or in the will, the action is impermissible. It demands that we treat humanity, whether in our own person or in others, always as an end and never merely as a means.
这不是数学方程式，而是一致性检验。若将某准则普遍化会导致构想或意志中的矛盾，则该行为不被允许。它要求我们永远将人性——无论是自身还是他人——作为目的，而绝不仅仅是手段。
8. Descartes’ Cogito | 笛卡尔的“我思故我在”
Descartes’ foundational theorem of knowledge arises from radical doubt: even if an evil demon deceives him about everything, he cannot doubt that he is thinking. Thinking requires a thinker. Hence the formula: I think, therefore I am (Cogito, ergo sum).
笛卡尔的基础认识定理源于彻底怀疑：即使有一个恶魔在一切事物上欺骗他，他也无法怀疑自己在思考。思考需要一个思考者。因此得出公式：我思故我在（Cogito, ergo sum）。
This indubitable truth became the first principle of his philosophy, a self-verifying theorem that no scepticism can overturn. From this point, he attempted to rebuild all knowledge.
这一无可置疑的真理成为他哲学的第一原理，一个任何怀疑论都无法推翻的自我验证定理。由此出发，他尝试重建所有知识。
9. Ockham’s Razor | 奥卡姆剃刀
Ockham’s Razor is a principle of ontological parsimony, often formulated as: Entities should not be multiplied beyond necessity. In simpler terms, the simplest explanation that fits all the facts is to be preferred.
奥卡姆剃刀是一条本体论简约原则，常被表述为：如无必要，勿增实体。简言之，能够解释所有事实的最简单解释应被优先采纳。
It functions as a methodological rule rather than a logical theorem: given two theories that explain the same data equally well, we ought to choose the one with fewer assumptions. It is widely used in science and philosophy to cut away unnecessary hypotheses.
它起到方法论规则而非逻辑定理的作用：如果有两个理论同样好地解释相同的数据，我们应当选择假设更少的那个。它被广泛应用于科学与哲学，用以剔除不必要的假设。
10. JTB Theory of Knowledge | 知识的三元定义
Since Plato, knowledge has been analyzed as a combination of three necessary conditions: Knowledge = Justified True Belief (JTB). A person S knows that P if and only if (i) P is true, (ii) S believes that P, and (iii) S is justified in believing that P.
自柏拉图以来，知识被分析为三个必要条件的组合：知识 = 合理的真信念（JTB）。某人S知道命题P，当且仅当(i) P为真，(ii) S相信P，并且(iii) S有理由相信P。
This formula has been foundational in epistemology, although Gettier cases later showed that JTB may not be sufficient, prompting further refinements. Still, it remains the starting point for any philosophical discussion of knowledge.
这一公式一直是认识论的基础，尽管后来的盖梯尔案例表明JTB或许并不充分，从而引发了进一步改良。但它仍是任何关于知识的哲学讨论的起点。
11. Skeptical Suspension of Judgment | 怀疑论的悬置判断
The ancient Pyrrhonian sceptics offered a formula for mental tranquillity: when confronted with equally strong arguments for and against a proposition, one should suspend judgment (epoché). This suspension is said to lead to ataraxia, or peace of mind.
古代皮浪怀疑论者提供了一条达到心灵宁静的公式：当面对一个命题的正反论证力度相等时，应当悬置判断 (epoché)。据说这种悬置会带来 ataraxia，即内心的平静。
The underlying principle is: If no criterion of truth can be justified without circularity, then we must say neither ‘it is’ nor ‘it is not’, but simply ‘it appears’.
其底层原则是：如果没有一个真理标准能在不循环的情况下得到辩护，那么我们必须既不说“它是”也不说“它不是”，而只说“它看起来是”。
12. The Golden Rule | 黄金法则
Found across cultures and ethical systems, the Golden Rule provides a universal moral formula: Treat others as you would like to be treated.
黄金法则跨越文化与伦理系统，提供了一条普遍的道德公式：你希望别人怎样对待你，你就怎样对待别人。
In its positive form it prescribes active benevolence; in its negative form (‘Do not do to others what you would not want done to yourself’) it prescribes restraint. It is a practical theorem for empathetic ethical reasoning, encouraging moral agents to universalize their actions from a shared standpoint.
其积极形式规定了主动的善意；其消极形式（“己所不欲，勿施于人”）规定了约束。它是移情式伦理推理的实用定理，鼓励道德主体从一个共通的立场出发将自身行为普遍化。
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