📚 SQA Philosophy Quick Reference: Frameworks and Theorems | SQA 哲学速查手册：核心框架与定理
This is not a handbook of mathematical equations. In SQA Philosophy, a 'formula' is a structured argument, an ethical calculus, or a logical schema that transforms premises into justified conclusions. This guide compresses the essential reasoning frameworks you will need for Year 11 RMPS and Philosophy, treating each one as a theorem to be memorised, applied, and critiqued under timed conditions.
这不是一本数学方程手册。在 SQA 哲学中，“公式”是一个结构化论证、一种伦理演算，或一个将前提转化为合理结论的逻辑图式。本指南压缩了 Year 11 RMPS 和哲学课程所需的核心推理框架，将每一个框架视为一个定理，要求你在限时条件下记忆、应用和批判。
1. The Syllogism Theorem | 三段论定理
A syllogism is the fundamental unit of deductive reasoning. It moves from two premises to a necessary conclusion. If the premises are true and the form is valid, the conclusion cannot be false. The classic structure runs: All A are B; All B are C; therefore, All A are C. In the SQA exam, you must test for both validity (logical structure) and soundness (truth of premises).
三段论是演绎推理的基本单位。它从两个前提推出必然结论。如果前提为真且形式有效，结论就不可能是假的。经典结构为：所有 A 是 B；所有 B 是 C；因此，所有 A 是 C。在 SQA 考试中，你必须同时检验有效性（逻辑结构）和可靠性（前提的真实性）。
l humans are mortal. → Minor Premise: Socrates is human. → Conclusion: Socrates is mortal.
大前提：所有人都会死。→ 小前提：苏格拉底是人。→ 结论：苏格拉底会死。
- Validity test: Does the conclusion follow necessarily from the premises? If yes, the argument is valid.
- 有效性检验：结论是否必然从前提中得出？如果是，该论证就是有效的。
- Soundness test: Are the premises actually true? A valid argument with a false premise is unsound.
- 可靠性检验：前提实际上是真的吗？一个有效但前提为假的论证是不可靠的。
- The four figures of the syllogism alter the distribution of the middle term (M-P, S-M, S-P).
- 三段论的四个格改变了中项的分布（M-P, S-M, S-P）。
2. The Utilitarian Calculus | 功利主义演算
Bentham's hedonic calculus is a quantitative formula for moral decision-making. It measures pleasure and pain across seven dimensions to determine which action produces the greatest net utility. Mill later refined this by distinguishing higher intellectual pleasures from lower bodily ones, insisting that quality matters as much as quantity.
边沁的快乐演算是道德决策的量化公式。它从七个维度衡量快乐与痛苦，以确定哪种行动能产生最大的净效用。密尔后来对此进行了修正，区分了高级的精神快乐和低级的身体快乐，坚持认为质量与数量同等重要。
nsity × Duration × Certainty × Propinquity × Fecundity × Purity × Extent) for all affected
净效用 = Σ（强度 × 持续时间 × 确定性 × 临近性 × 衍生性 × 纯度 × 范围）对全体受影响者
- 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 the outcome to occur?
- 确定性：结果发生的可能性有多大？
- Propinquity: How soon will the pleasure or pain arrive?
- 临近性：快乐或痛苦多快会到来？
- Fecundity: Will this pleasure lead to further pleasures?
- 衍生性：这种快乐会带来更多的快乐吗？
- Purity: Will this pleasure be followed by pain of the opposite kind?
- 纯度：这种快乐之后会伴随着相反类型的痛苦吗？
- Extent: How many sentient beings are affected?
- 范围：有多少有感知力的生命受到影响？
Critics point out that utility cannot be quantified like temperature, and that the calculus permits unjust distributions if the total sum is maximised. In an SQA essay, always contrast Bentham's act utilitarianism with Mill's rule utilitarianism.
批评者指出，效用不能像温度那样被量化，而且如果总和最大化，这种演算允许不公正的分配。在 SQA 论文中，始终要将边沁的行为功利主义与密尔的规则功利主义进行对比。
3. The Categorical Imperative Formula | 定言令式公式
Kant's moral law functions as a logical test: act only according to that maxim which you can at the same time will that it should become a universal law. This is not a suggestion but a command of pure practical reason. The formula exposes contradictions in conception (where the maxim becomes self-defeating when universalised) and contradictions in will (where rational beings would not consistently will the maxim, even if conceivable).
康德的道德律作为一个逻辑检验发挥作用：只依照你同时能够意愿其成为普遍法则的准则去行动。这不是一个建议，而是纯粹实践理性的命令。该公式揭示了概念中的矛盾（准则普遍化后会自我挫败）和意愿中的矛盾（即使可以设想，理性存在者也不会一贯地意愿该准则）。
ep 2: Universalise it. → Step 3: Test for contradiction. If contradiction → impermissible.
步骤 1：表述你的准则。→ 步骤 2：将其普遍化。→ 步骤 3：检验是否有矛盾。若有矛盾 → 不允许。
- Formula of Universal Law: Act as if your maxim were to become a universal law of nature.
- 普遍法则公式：行动时仿佛你的准则要成为自然的普遍法则。
- Formula of Humanity: Treat humanity, in yourself and others, always as an end and never merely as a means.
- 人性公式：永远把人性——无论是对你自己还是对他人——当作目的，而绝不仅仅当作手段。
- Kingdom of Ends: Act as if you were a legislating member in a universal kingdom of ends.
- 目的王国：行动时仿佛你是普遍目的王国中的立法成员。
In the SQA exam, the perfect-imperfect duty distinction is critical. Perfect duties (no exceptions, e.g., do not lie) arise from contradictions in conception. Imperfect duties (allow latitude, e.g., develop talents) arise from contradictions in will.
在 SQA 考试中，完全责任与不完全责任的区分至关重要。完全责任（无例外，例如不说谎）源自概念矛盾。不完全责任（有自由裁量空间，例如发展才能）源自愿愿矛盾。
4. The Euthyphro Dilemma Theorem | 游叙弗伦两难定理
Plato's Euthyphro presents a devastating logical fork for divine command theory. The question is: Is the pious loved by the gods because it is pious, or is it pious because it is loved by the gods? The dilemma forces the theist to choose between an independent moral standard that binds even the gods, or an arbitrary voluntarism where morality reduces to divine whim.
柏拉图的《游叙弗伦篇》为神命论提出了一个毁灭性的逻辑分叉。问题是：虔敬是因为被众神所爱才虔敬，还是因为它虔敬才被众神所爱？这个两难迫使有神论者在两个选项中选择：一个是连众神也受其约束的独立道德标准，另一个是道德沦为神之任性的任意唯意志论。
ood is independent of God → God is not the source of morality. Horn B: Good depends on God's command → morality is arbitrary.
角A：善独立于上帝 → 上帝不是道德的来源。角B：善依赖于上帝的命令 → 道德是任意的。
- Horn A consequence: If goodness exists independently of God, then morality does not require God and divine command theory collapses.
- 角A后果：如果善独立于上帝而存在，那么道德不需要上帝，神命论瓦解。
- Horn B consequence: If God's command makes something good, then goodness is arbitrary—God could have commanded cruelty and it would be 'good'.
- 角B后果：如果上帝的命令使某事成为善，那么善就是任意的——上帝本可以命令残忍，而那也会是“善”。
- Modified divine command theory response: God's commands flow from his essentially loving nature, which is not arbitrary but necessary.
- 修正神命论的回应：上帝的命令源于他本质上是爱的本性，这不是任意的，而是必然的。
5. The Cartesian Method of Doubt | 笛卡尔怀疑方法
Descartes employed hyperbolic doubt as a methodological formula to strip away all uncertain beliefs until he arrived at an indubitable foundation. He applied doubt in stages: first to the senses (optical illusions, dreams), then to mathematics (the evil demon hypothesis), until he reached the Archimedean point: Cogito, ergo sum.
笛卡尔使用极端怀疑作为方法论公式，剥离所有不确定的信念，直到抵达一个无可置疑的基础。他分阶段应用怀疑：首先是感官（视错觉、梦境），然后是数学（恶魔假设），直到他到达阿基米德点：我思故我在。
age 2: Dream argument → Stage 3: Evil demon → Foundation: Cogito (I think, therefore I am)
阶段1：感官欺骗 → 阶段2：梦境论证 → 阶段3：恶魔假设 → 基础：我思故我在
- The Cogito is indubitable because even if an evil demon is deceiving me, I must exist to be deceived.
- 我思故我在是无可置疑的，因为即使有恶魔在欺骗我，我也必须存在才能被欺骗。
- Clear and distinct perception criterion: Whatever I perceive clearly and distinctly must be true (backed by a non-deceiving God).
- 清楚分明感知标准：凡我清楚分明感知到的，必为真（由不欺骗的上帝保证）。
- The Cartesian Circle objection: Descartes proves God's existence using clear and distinct ideas, then uses God to guarantee clear and distinct ideas.
- 笛卡尔循环的反对：笛卡尔用清楚分明的观念证明上帝存在，又用上帝来保证清楚分明的观念。
6. Hume's Fork: The Analytic-Synthetic Theorem | 休谟叉子：分析-综合定理
Hume' ions of ideas are a priori, necessary, and their denial entails a contradiction (e.g., 2+3=5). Matters of fact are a posteriori, contingent, and their denial is always possible (e.g., the sun will rise tomorrow).
休谟的叉子是一个分类公式，将一切有意义的命题划分为两个穷尽的范畴：观念关系与事实。观念关系是先天的、必然的，否定它们会带来矛盾（例如 2+3=5）。事实是后天的、偶然的，否定它们总是可能的（例如太阳明天会升起）。
If a proposition is not a relation of ideas AND not a matter of fact, then it is 'sophistry and illusion'.
如果一个命题既不是观念关系也不是事实，那么它就是“诡辩和幻象”。
- Application to causality: The causal connection is not a relation of ideas (denying it is conceivable) and not directly a matter of fact (we never perceive necessary connection, only constant conjunction).
- 应用于因果性：因果联系不是观念关系（否定它是可以设想的），也不是直接的事实（我们从未感知到必然联系，只感知到恒常联结）。
- Application to the self: The self is not a matter of fact (no impression of a unified self), nor a relation of ideas (denying a self is conceivable). Therefore, the self is a bundle of perceptions.
- 应用于自我：自我不是事实（没有统一自我的印象），也不是观念关系（否定自我是可以设想的）。因此，自我是一束知觉。
- Problem of induction: Matters of fact are justified by cause and effect, but our expectation that the future will resemble the past is itself neither a relation of ideas nor a matter of fact. It is custom or habit.
- 归纳问题：事实由因果关系证明，但我们关于未来将与过去相似的预期本身既不是观念关系也不是事实。它是习俗或习惯。
7. Plato's Theory of Forms: The Two-World Formula | 柏拉图理型论：两世界公式
Plato partitions reality into two ontological domains: the visible world of appearances (particulars, change, opinion) and the intelligible world of Forms (universals, permanence, knowledge). Every particular in the visible world participates in or imitates its perfect Form. The Form of the Good occupies the summit, illuminating all other Forms just as the sun illuminates the visible world.
柏拉图将实在划分为两个本体论领域：可见的现象世界（殊相、变化、意见）和可理解的理型世界（共相、永恒、知识）。可见世界中的每一个殊相都分有或模仿其完美的理型。善的理型居于顶峰，照亮所有其他理型，正如太阳照亮可见世界。
Participation relation: Particular x is F if and only if x participates in the Form of F-ness.
分有关系：殊相 x 是 F，当且仅当 x 分有了 F 性的理型。
- The Third Man Argument (infinite regress): If the Form of F is itself F, then a further Form is required to explain what F-ness and the Form of F have in common, and so on.
- 第三人论证（无穷后退）：如果 F 的理型本身是 F，那么就需要另一个理型来解释 F 性和 F 的理型之间的共同点，如此以至无穷。
- The Divided Line maps four cognitive states: eikasia (imaging), pistis (belief), dianoia (mathematical reasoning), noesis (dialectic grasping of Forms).
- 线段比喻映射了四种认知状态：想象、信念、数学推理、对理型的辩证把握。
- Allegory of the Cave illustrates the journey from the shadows of the visible world to the sunlight of the Form of the Good.
- 洞穴寓言说明了从可见世界的阴影到善的理型之阳光的旅程。
8. Aristotle's Four Causes Formula | 亚里士多德四因公式
For Aristotle, to know something fully is to grasp its four causes. This explanatory framework applies to both natural organisms and human artefacts. The four causes are not competitors but complementary aspects of a complete explanation. In an SQA essay, always apply all four to a single object to demonstrate mastery.
对亚里士多德而言，充分认识某物就是把握其四因。这个解释框架既适用于自然有机体，也适用于人造物。四因不是竞争关系，而是完整解释的互补方面。在 SQA 论文中，始终将四因全部应用于单一对象以展示掌握程度。
Complete Explanation = Material Cause + Formal Cause + Efficient Cause + Final Cause
完整解释 = 质料因 + 形式因 + 动力因 + 目的因
| Material Cause: What something is made of (e.g., bronze for a statue). | 质料因：某物由什么构成（例如雕像的青铜）。 |
| Formal Cause: The essence or defining pattern (e.g., the shape of Athena). | 形式因：本质或定义性模式（例如雅典娜的形状）。 |
| Efficient Cause: The primary source of change or rest (e.g., the sculptor). | 动力因：变化或静止的主要来源（例如雕塑家）。 |
| Final Cause: The purpose or telos (e.g., to honour the goddess). | 目的因：目的或 telos（例如荣耀女神）。 |
Aristotle's Unmoved Mover is the ultimate final cause of the cosmos: pure actuality, thinking on thinking, that moves all things by being the object of love and desire, without itself moving.
亚里士多德的不动的推动者是宇宙的终极目的因：纯粹的现实、对思想的思想，它通过作为爱与欲望的对象来推动万物，而自身不运动。
9. The Teleological Argument Formula | 目的论论证公式
The teleological argument (from telos, purpose) infers a designer from the appearance of order, complexity, and purpose in the natural world. Paley's watchmaker analogy provides the classic formulation: if you find a watch on a heath, its intricate mechanism compels you to infer a watchmaker. Similarly, the vastly more complex universe compels us to infer a divine designer.
目的论论证（源自 telos，目的）从自然界中的秩序、复杂性和目的表象推断出设计者。佩利的钟表匠类比提供了经典表述：如果你在荒野上发现一块表，其精细的机制迫使你推断出一位钟表匠。同样地，远比表更复杂的宇宙迫使我们推断出一位神圣设计者。
se are best explained by intelligent design. → Conclusion: An intelligent designer exists.
前提1：宇宙展示出复杂的秩序和目的。→ 前提2：复杂的秩序和目的最好由智能设计来解释。→ 结论：存在一位智能设计者。
- Hume's objections via Philo: The analogy is weak (the universe is not sufficiently like a watch). Even if a designer exists, it may be a committee, an infant deity, or a deceased god. The universe may be a botched first attempt.
- 休谟通过斐罗提出的反对：类比是薄弱的（宇宙与表不够相似）。即使存在设计者，也可能是一个委员会、一个幼年神祇或一个已故之神。宇宙可能是一次拙劣的初次尝试。
- Evolutionary objection (Darwin): Natural selection acts as an automatic, non-conscious designer, producing apparent design through variation and differential survival over deep time.
- 进化论的反对（达尔文）：自然选择作为一个自动的、无意识的设计者，通过变异和在漫长时间中的差别生存来产生表面上的设计。
- Fine-tuning argument: Physical constants (gravitational constant, strong nuclear force) appear fine-tuned for life. Any slight variation would make life impossible.
- 微调论证：物理学常数（引力常数、强核力）似乎为生命进行了微调。任何微小变化都会使生命不可能。
10. The Cosmological Argument: Contingency Formula | 宇宙论论证：偶然性公式
The cosmological argument moves from the existence of contingent beings to the existence of a necessary being. Aquinas's Third Way is the definitive formulation: Some beings are generated and corrupted, so they are possible not to be. If everything were contingent, then at some time nothing existed—but if that were true, nothing would exist now, which is false. Therefore, a necessary being exists.
宇宙论论证从偶然存在物的存在推出必然存在物的存在。阿奎那的第三路是决定性表述：有些存在物有生灭，所以它们可能不存在。如果一切都是偶然的，那么某个时刻什么都不存在——但如果那是真的，现在也就什么都不存在，而这是错误的。因此，存在一个必然存在物。
y anything exists. → A necessary being, whose essence is existence, must ground the chain.
如果某物偶然存在，其不存在是可能的。→ 一个无穷的偶然存在者链条无法解释为什么有东西存在。→ 一个本质即存在的必然存在者必须作为该链条的基础。
- Hume's objection: If we can conceive each individual contingent being not existing, we can conceive the whole set not existing. The universe itself could be the necessary being.
- 休谟的反对：如果我们能设想每个单独的偶然存在者不存在，我们就能设想整个集合不存在。宇宙本身可能就是必然存在者。
- Russell-Brute Fact objection: The universe simply exists as a brute fact that requires no further explanation. The question 'Why is there something rather than nothing?' may be meaningless.
- 罗素-原初事实的反对：宇宙只是作为一个原初事实存在，不需要进一步的解释。“为什么有东西存在而不是空无一物？”这个问题可能是无意义的。
- Kalam formulation: Everything that begins to exist has a cause; the universe began to exist; therefore, the universe has a cause.
- 卡拉姆论证表述：凡开始存在的东西都有一个原因；宇宙开始存在；因此宇宙有一个原因。
11. Hegelian Dialectic Formula | 黑格尔辩证法公式
Hegel's dialectic is a triadic movement of thought and reality: thesis, antithesis, synthesis. The thesis is an initial position that contains internal contradictions. The antithesis negates the thesis. The synthesis negates the negation, preserving what was true in both while resolving the contradiction at a higher level. This is not just a method of thinking—for Hegel, it is the very structure of being itself.
黑格尔的辩证法是思维与实在的三合一运动：正题、反题、合题。正题是一个包含内在矛盾的初始立场。反题否定正题。合题否定了这个否定，同时保留了两者中真实的东西，并在更高层次上解决矛盾。这不仅是一种思维方法——对黑格尔而言，这正是存在本身的结构。
Thesis → Antithesis (negation) → Synthesis (negation of the negation, Aufhebung)
正题 → 反题（否定）→ 合题（否定之否定，扬弃）
- Aufhebung (sublation): The synthesis simultaneously cancels, preserves, and elevates the thesis and antithesis. Example applied to SQA: Authority (thesis) versus Autonomy (antithesis) yields Responsible Freedom (synthesis).
- 扬弃（Aufhebung）：合题同时取消、保留并提升了正题和反题。应用于 SQA 的例子：权威（正题）对自治（反题）产生负责任的自由（合题）。
- Dialectical materialism (Marx): Marx inverts Hegel, applying the dialectic to material conditions—class struggle between bourgeoisie (thesis) and proletariat (antithesis) leads to classless society (synthesis).
- 辩证唯物主义（马克思）：马克思颠倒了黑格尔，将辩证法应用于物质条件——资产阶级（正题）与无产阶级（反题）之间的阶级斗争导致无阶级社会（合题）。
- Critique: Popper argued the dialectic is unfalsifiable and encourages sloppy thinking by treating contradictions as productive rather than as errors to be eliminated.
- 批判：波普尔认为辩证法不可证伪，并通过将矛盾视为生产性的而非需要消除的错误来鼓励草率的思考。
12. The Free Will Defence Formula | 自由意志辩护公式
The logical problem of evil presents a trilemma: if God is omnipotent, omniscient, and omnibenevolent, evil should not exist. Evil exists. Therefore, such a God does not exist. Plantinga's Free Will Defence offers a logical formula to show no contradiction: a world containing free creatures who freely choose good is more valuable than a world of automata. It is logically impossible for God to create free creatures and also causally determine them always to choose good.
恶的逻辑问题呈现一个三难困境：如果上帝是全能的、全知的和全善的，那么恶就不该存在。恶存在。因此，这样的上帝不存在。普兰丁格的自由意志辩护提供了一个逻辑公式来表明不存在矛盾：一个包含自由选择善的自由受造物的世界，比一个全是自动机的世界更有价值。上帝在逻辑上不可能既要创造自由的受造物，又因果性地决定他们总是选择善。
(God would want to eliminate evil AND could eliminate evil). But (evil exists). Therefore ¬(God exists). Plantinga: Add 'free will' as a logically necessary condition for moral good.
如果（上帝存在）→（上帝会想要消除恶并且能消除恶）。但（恶存在）。因此 ¬（上帝存在）。普兰丁格：加入“自由意志”作为道德善的逻辑必要条件。
- Transworld depravity: It is possible that every free creature God could create would sin in at least one possible world; thus, God cannot actualise a world of free creatures who always choose good.
- 跨世界堕落论：上帝可能创造的每一个自由受造物在至少一个可能世界中都会犯罪；因此，上帝无法实现一个自由受造物总是选择善的世界。
- Evidential problem of evil: The free will defence may solve the logical problem but not the evidential one—the sheer quantity and distribution of suffering seems gratuitous.
- 恶的证据性问题：自由意志辩护可能解决了逻辑问题，但没有解决证据性问题——苦难的巨大量级和分布似乎是毫无理由的。
- Irenaean theodicy (soul-making): Suffering is necessary for moral and spiritual development. A world without challenges would produce shallow, immature souls.
- 爱任纽神义论（灵魂塑造）：苦难对于道德和灵性发展是必要的。一个没有挑战的世界会产生肤浅、不成熟的灵魂。
Published by TutorHao | SQA Philosophy Revision Series | aleveler.com
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