📚 Year 10 SQA Philosophy: Quick Reference Handbook of Formulae and Theorems | SQA 十年级哲学：公式定理速查手册
In SQA RMPS and philosophy courses, clarity often comes from treating arguments as precise logical structures, much like mathematical formulae. This handbook collects the foundational ‘formulae’ and ‘theorems’ you need to recognise, reconstruct, and evaluate key philosophical positions—from logic patterns to utilitarian calculation, Kant’s imperatives, and classical proofs for the existence of God. Mastering these forms will sharpen your critical thinking and improve your essay analysis.
在 SQA RMPS 和哲学课程中，清晰的思考往往来自将论证视为精确的逻辑结构，就像数学公式一样。这本手册汇集了你需要识别、重构和评估的关键哲学“公式”与“定理”——从逻辑模式到功利计算、康德律令，再到经典上帝存在证明。掌握这些形式将强化你的批判性思维并提升论文分析能力。
1. What Are Philosophical Formulae? | 什么是哲学公式？
Philosophical formulae are abstract patterns of reasoning that guarantee validity when their structure is followed correctly. Unlike scientific equations, they express relationships between concepts, moral principles, or metaphysical claims. Think of a formula as a template that, when filled with consistent content, yields a persuasive argument.
哲学公式是推理的抽象模式，当结构被正确遵循时就可确保有效性。与科学方程不同，它们表达概念、道德原则或形而上学主张之间的关系。你可以把公式理解为一种模板，填入一致的内容后便能产生有说服力的论证。
2. Modus Ponens (Affirming the Antecedent) | 肯定前件式
If P implies Q, and P is true, then Q must be true. This is the most straightforward valid argument form.
如果 P 蕴含 Q，且 P 为真，则 Q 必然为真。这是最直接的有效论证形式。
P → Q, P ∴ Q
Example: If it is raining (P), the pavement is wet (Q). It is raining. Therefore, the pavement is wet.
例子：如果正在下雨（P），那么人行道是湿的（Q）。正在下雨。因此，人行道是湿的。
3. Modus Tollens (Denying the Consequent) | 否定后件式
If P implies Q, and Q is false, then P must also be false. This is a crucial pattern for testing hypotheses and refuting claims.
如果 P 蕴含 Q，而 Q 为假，则 P 必定为假。这是检验假设和反驳主张的重要模式。
P → Q, ¬Q ∴ ¬P
Example: If the battery is charged (P), the phone turns on (Q). The phone does not turn on. Therefore, the battery is not charged.
例子：如果电池有电（P），手机就会开机（Q）。手机没有开机。因此，电池没有电。
4. Hypothetical Syllogism | 假言三段论
When two conditionals link together so that the consequent of the first is the antecedent of the second, you can derive a direct bridge between the first antecedent and the final consequent.
当两个条件句相连，即第一个的后件是第二个的前件时，你可以推导出从前件直接到最终后件的桥梁。
P → Q, Q → R ∴ P → R
Example: If I study (P), I pass the exam (Q). If I pass the exam (Q), I graduate (R). Therefore, if I study, I graduate.
例子：如果我学习（P），我就通过考试（Q）。如果我通过考试（Q），我就毕业（R）。因此，如果我学习，我就毕业。
5. Disjunctive Syllogism | 选言三段论
When one of two possibilities is eliminated, the other must be the case. This works only when the disjunction is genuinely exhaustive.
当两个可能性之一被排除时，另一个必然成立。但这只有在选言确实穷尽所有可能时才有效。
P ∨ Q, ¬P ∴ Q
Example: The cake is either in the fridge (P) or on the counter (Q). It is not in the fridge. Therefore, it is on the counter.
例子：蛋糕要么在冰箱里（P），要么在台面上（Q）。它不在冰箱里。因此，它在台面上。
6. The Utilitarian Calculus | 功利计算
Classical utilitarianism proposes that the moral rightness of an action depends solely on its consequences for overall happiness. The guiding formula is to maximise net utility.
古典功利主义主张，行为的道德正当性完全取决于它对整体幸福产生的后果，其指导公式就是最大化净效用。
Rightness ⇌ Max ∑(Pleasure – Pain) for all affected
Jeremy Bentham’s hedonic calculus suggests factoring in intensity, duration, certainty, and extent. In exam contexts, reconstruct the formula when evaluating policies or moral dilemmas: an act is right if and only if it produces the greatest balance of happiness over suffering.
杰里米·边沁的苦乐计算建议考虑强度、持续时间、确定性和范围等因素。在考试情境中，评估政策或道德困境时重构该公式：一个行为是正确的，当且仅当它产生了最大化的快乐减去痛苦的余额。
7. Kant’s Categorical Imperative: Universal Law Formula | 康德定言令式：普遍法则公式
Kant’s ethics centres on the idea that moral actions must be universalisable. The first formulation works as a test for maxims.
康德的伦理学以道德行为必须可普遍化为核心。第一公式是对行为准则的检验。
Act only on that maxim whereby you can at the same time will that it should become a universal law.
Operationally, this means: take your proposed rule (maxim), imagine everyone acting on it. If this leads to a contradiction in conception or in will, the act is impermissible. For example, making a false promise fails the test because a world where everyone makes false promises undermines the very practice of promising—this yields a contradiction.
操作上，这意味着：拿出你提议的行为准则（格律），设想每个人都按它行动。如果这导致构想上的矛盾或意愿上的矛盾，该行为就是不允许的。例如，作出虚假承诺就通不过检验，因为在一个每个人都作虚假承诺的世界里，承诺这一行为本身就被瓦解了——这就产生了矛盾。
8. Kant’s Formula of Humanity | 康德的人道公式
The second major formulation demands that we never treat persons merely as instruments but always as ends in themselves.
第二个重要定言令式要求我们绝不仅仅将人当作工具，而永远要同时当作目的本身。
Act so that you treat humanity, whether in your own person or in that of another, always as an end and never merely as a means.
This formula can be applied as a check on exploitation: if your action uses a person’s rationality or autonomy only for your own benefit without their genuine consent, it violates the principle. For instance, lying to someone manipulates them as a mere tool, thereby disrespecting their rational agency.
这个公式可以用作检验剥削的准则：如果你的行为利用他人的理性或自主性仅仅为你自己的利益服务，而未经其真正同意，那就违背了该原则。例如，对某人撒谎就是将他们仅仅作为工具来操控，从而不尊重其理性主体性。
9. Descartes’ Cogito Ergo Sum | 笛卡尔“我思故我在”
Descartes’ foundational theorem emerged from radical doubt: even if an evil demon deceives me, I must exist to be deceived. The act of thinking guarantees my existence.
笛卡尔的基础性定理源于彻底怀疑：即便有一个恶魔欺骗我，我也必须存在才能被欺骗。思的活动保证了我在。
I think → I am (Cogito → Sum)
This is not a syllogism but a self‑verifying intuition. Formally, it can be expressed as: for any thinking activity T, the existence of the subject is a necessary condition. Whenever you encounter Cartesian scepticism in the course, remember that the cogito is the first indubitable truth, the ‘Archimedean point’ from which knowledge can be rebuilt.
这不是三段论，而是一种自证的直观。形式地看，它可以表达为：对于任何思维活动 T，主体的存在是必要条件。每当课程中遇到笛卡尔式怀疑，请记住“我思”是第一个不可怀疑的真理，是可以重建知识的“阿基米德点”。
10. Anselm’s Ontological Argument | 安瑟伦的本体论论证
Anselm defines God as ‘that than which nothing greater can be conceived’. The formula turns on the idea that existence in reality is greater than existence in the understanding alone.
安瑟伦将上帝定义为“无以设想其更大者”。这个公式的关键在于，真实存在的伟大性要大于仅仅在理智中存在。
uppose G exists only in the mind. Then a being with all G’s properties plus real existence is greater (contradiction). Therefore, G exists in reality.
The structure is a reductio ad absurdum: assume God does not exist in reality, show this leads to an absurdity (a being greater than the greatest conceivable can be conceived), and conclude that God exists. Critics like Gaunilo parodied this form using a perfect island, so you can evaluate whether existence is a genuine predicate.
该论证的结构是归谬法：假定上帝不存在于现实中，会导致荒谬（可设想比那无以设想其更大者还要大的东西），进而推出上帝存在。高尼罗等批评者以完美的岛屿来戏仿这一形式，因此你可以评估存在是否是真正的谓词。
11. The Logical Problem of Evil | 恶的逻辑问题
The classic challenge to theism renders the attributes of God seemingly incompatible with the existence of evil. It can be formulated as a deductive argument.
经典的对有神论的挑战将上帝的属性与恶的存在显得不相容。这可以表达为一个演绎论证。
exists → no unnecessary evil exists. Unnecessary evil exists. ∴ Such a God does not exist.
Premise: A perfectly good being would eliminate evil as far as it could. An omnipotent being could eliminate all evil. Yet evil exists. Therefore, no such being exists. In SQA answers, distinguish between moral and natural evil, and consider theodicies (e.g., soul‑making) that challenge the premise that no good justifies allowing evil.
前提：一个全善的存在会尽可能地消除恶；全能的存在能够消除一切恶。然而恶存在着。因此，这样的存在不存在。在 SQA 的答案中，要区分道德恶与自然恶，并思考试图挑战“没有善能为容许恶提供理由”这一前提的神正论（如灵魂塑造说）。
12. Pascal’s Wager | 帕斯卡赌注
Pascal presents a pragmatic reason for belief in God framed as a decision‑theoretic wager. Even if God’s existence cannot be proven, wagering on belief is rationally preferable.
帕斯卡提出了一个相信上帝的实用理由，把它框定为一场决策论式的赌注。即使上帝的存在无法被证明，押注相信在理性上更可取。
| God exists | God does not exist | |
|---|---|---|
| Believe | Infinite gain (heaven) | Finite loss (worldly pleasures forgone) |
| Do not believe | Infinite loss (hell) | Finite gain (worldly pleasures retained) |
Expected utility dominates: rational to believe.
The formula is a dominance reasoning: believing offers an infinite reward if God exists and only a finite sacrifice if not; not believing risks infinite loss. Objections point to the many‑gods problem (which God?) and the doxastic voluntarism issue (can we simply choose to believe?). Use this table to illustrate pragmatic arguments in your revision.
这个公式是一种占优推理：相信上帝，如果上帝存在将获得无限回报，即使上帝不存在也只有有限的牺牲；而不信则冒无限损失的风险。反驳指向多神问题（该信哪个神？）和信念自愿主义（我们能随意选择相信吗？）。复习时可以用这个表格来说明实用主义论证。
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