📚 Year 9 AQA Philosophy: Formulas & Theorems Quick Reference | Year 9 AQA 哲学：公式定理速查手册
Philosophy is often seen as a subject of open-ended debate, but beneath the surface lies a rigorous structure of logical formulas, argumentative frameworks, and conceptual theorems. This quick-reference handbook distils the essential patterns of reasoning you will encounter in Year 9 AQA Philosophy: the Design Argument, the First Cause Argument, the Problem of Evil, religious experiences, ethical calculation, and logical fallacies. Mastering these templates is like learning the equations of thought — once internalised, you can analyse any philosophical claim with clarity and precision.
哲学常被视为一门开放式辩论的学科，但在表面之下，隐藏着严密的逻辑公式、论证框架和概念定理。这本速查手册提炼了 Year 9 AQA 哲学课程中你会遇到的核心推理模式：设计论证、第一因论证、恶的问题、宗教体验、伦理计算以及逻辑谬误。掌握这些模板就像学会思想的方程——一旦内化，你就能清晰、精确地分析任何哲学主张。
1. The Design Argument (Teleological) Formula | 设计论证（目的论）公式
The Design Argument follows an analogical pattern. It compares natural objects to human-made artefacts and concludes that complex, purposeful structures require an intelligent designer. The standard formulation moves from observed complexity and regularity to the inference of a divine architect.
设计论证遵循一种类比模式。它将自然物体与人造器物进行比较，并得出结论：复杂、有目的的结构必然要求一位智慧的设计者。标准公式从可观察的复杂性与规律性出发，推导出神圣建筑师的存在。
P1: Complex, ordered structures (e.g. a watch) imply a designer.
前提1: 复杂、有序的结构（如一块手表）意味着存在设计者。
P2: The natural world contains vastly more complex, ordered structures (e.g. the eye, ecosystems).
前提2: 自然界包含着远为复杂、有序的结构（如眼睛、生态系统）。
C: Therefore, the natural world implies a vastly greater Designer (God).
结论: 因此，自然界意味着一位远为伟大的设计者（上帝）。
Critics note that analogy relies on a weak comparison: biological organisms replicate and evolve through Darwinian mechanisms, unlike watches. David Hume argued that the universe might be more like a vegetable — self-organising — than a machine.
批评者指出，类比依赖于一种薄弱的比较：生物有机体通过达尔文机制复制和进化，与手表不同。大卫·休谟论证说，宇宙可能更像一株蔬菜——自我组织——而非一台机器。
2. The First Cause Argument (Cosmological) Formula | 第一因论证（宇宙论）公式
The Cosmological Argument does not rely on analogy but on a chain of causation that must, it is claimed, have a starting point. Aquinas formulated this as the argument from motion and causation, insisting that an infinite regress of causes is impossible.
宇宙论论证不依赖于类比，而是依赖于一条因果链，该链据称必须有一个起点。阿奎那将此表述为运动和因果性论证，坚称原因的无限回溯是不可能的。
P1: Everything that exists has a cause.
前提1: 存在的一切事物都有其原因。
P2: The chain of causes cannot stretch back infinitely.
前提2: 因果链不能无限回溯。
C: Therefore, there must be a first, uncaused Cause (God).
结论: 因此，必定存在一个第一因、非被动的因（上帝）。
Bertrand Russell counters: if everything must have a cause, then God must also have a cause. If God is allowed to be uncaused, why not allow the universe itself to be uncaused? This is the ‘fallacy of composition’ — assuming what is true of parts must be true of the whole.
伯特兰·罗素反驳：如果一切都必须有原因，那么上帝也必须有原因。如果允许上帝不被被造成，为什么不直接允许宇宙本身不被被造成？这是“组合谬误”——假设对部分为真的事情必然对整体为真。
3. The Ontological Argument Formula (Anselm) | 本体论论证公式（安瑟伦）
Anselm’s Ontological Argument is unique: it attempts to prove God’s existence from the very definition of God as the greatest conceivable being. It is an a priori argument — based on logic alone, not sensory evidence.
安瑟伦的本体论论证是独一无二的：它试图从上帝作为“可设想的最伟大存在”这一纯粹定义出发来证明上帝的存在。这是一种先验论证——仅基于逻辑，而非感官证据。
P1: God is defined as ‘that than which nothing greater can be conceived’.
前提1: 上帝被定义为“无法设想有比之更伟大的存在”。
P2: Existence in reality is greater than existence merely in the mind.
前提2: 现实中的存在比仅存在于心灵中的存在更伟大。
C: Therefore, God must exist in reality; otherwise, we could conceive of an even greater being (one that actually exists).
结论: 因此，上帝必然存在于现实中；否则，我们就能设想一个更伟大的存在（一个真实存在的存在）。
Gaunilo’s ‘perfect island’ parody exposes the flaw: one cannot define something into existence. Existence is not a predicate — not a property that can be added to a concept to make it greater, as Kant later rigorously established.
高尼罗的“完美岛屿”戏仿暴露了这一缺陷：你不能通过定义将某物定义进存在。存在不是谓词——不是可以添加到概念中使其更伟大的属性，正如康德后来严谨确立的那样。
4. The Logical Problem of Evil (Deductive Disproof) | 恶的逻辑问题（演绎式反驳）
The Logical Problem of Evil claims that the traditional attributes of God — omnipotence, omnibenevolence, omniscience — are logically incompatible with the existence of evil. If evil exists, the God of classical theism cannot.
恶的逻辑问题声称，上帝的传统属性——全能、全善、全知——与恶的存在在逻辑上不相容。如果恶存在，古典有神论的上帝就不可能存在。
P1: If God is all-powerful, He can prevent evil.
前提1: 如果上帝是全能的，他就能够阻止恶。
P2: If God is all-loving, He would want to prevent evil.
前提2: 如果上帝是全爱的，他就愿意阻止恶。
P3: Evil exists.
前提3: 恶存在。
C: Therefore, an all-powerful and all-loving God does not exist.
结论: 因此，全能全爱的上帝不存在。
This is a modus tollens structure: if P → Q, but not-Q, therefore not-P. The structure converts the theistic premise into a conditional and then negates the consequent through observed evil, thereby negating the antecedent (God’s existence).
这是一个否定后件式结构：如果 P → Q，但非 Q，因此非 P。该结构将有神论前提转化为条件句，然后通过可观察的恶来否定后件，从而否定前件（上帝的存在）。
5. The Free Will Defence (Theorem of Compatible Coexistence) | 自由意志辩护（兼容共存定理）
The Free Will Defence is a response to the logical problem, insisting that God and evil are not logically incompatible because a greater good — free will — requires the possibility of evil. This transforms the attack into a theorem of compatible coexistence.
自由意志辩护是对逻辑问题的回应，坚持认为上帝和恶在逻辑上并非不相容，因为一种更大的善——自由意志——要求恶的可能性。这把攻击转化为一种兼容共存定理。
Genuine free will is incompatible with being determined always to choose good.
真正的自由意志与总是被决定选择善不相容。
A world with free creatures who sometimes choose evil is greater than a world of automated goodness.
一个拥有自由生灵、他们有时选择恶的世界，比一个自动化良善的世界更伟大。
Therefore, God’s allowing the possibility of evil is consistent with His attributes, because free will is a necessary condition for moral good.
因此，上帝允许恶的可能性与他的属性是一致的，因为自由意志是道德善的必要条件。
John Mackie counters: why couldn’t God create free beings who always freely choose good? If free will and guaranteed goodness are logically compatible, the defence collapses. Alvin Plantinga’s version concedes this point but insists the defence works as long as it is logically possible that even an omnipotent God could not actualise such a world, which is sufficient to break the deductive contradiction.
约翰·麦基反驳：为什么上帝不能创造总是自由选择善的自由生灵？如果自由意志和确保的善在逻辑上兼容，辩护就坍塌了。阿尔文·普兰丁格的版本承认这点，但坚称只要在逻辑上可能——即全能的上帝也无法实现这样一个世界——这辩护就足以打破演绎矛盾。
Religious experiences function as a distinct evidentiary basis for belief, operating through the Principle of Credulity and the Principle of Testimony. Richard Swinburne formalises this: we should trust our senses and the sincere reports of others unless we have specific reasons to doubt them.
宗教体验作为信仰的一种独特证据基础，通过“轻信原则”和“见证原则”运作。理查德·斯温伯恩对此进行了形式化：我们应该信任自己的感官和他人的真诚报告，除非我们有特定的理由去怀疑它们。
P1: We generally trust our sensory experiences as veridical unless proven otherwise (Principle of Credulity).
前提1: 我们通常信任自己的感官体验是真实的，除非被证明相反（轻信原则）。
P2: We generally accept the testimony of others unless we have defeaters (Principle of Testimony).
前提2: 我们通常接受他人的见证，除非我们有击败证据（见证原则）。
P3: Millions report experiencing a transcendent, divine reality.
前提3: 数以百万计的人报告体验到一种超验的神圣实在。
C: Therefore, it is reasonable to believe that God exists, based on the cumulative weight of testimony.
结论: 因此，基于见证的累积分量，相信上帝存在是合理的。
Objections include the problem of conflicting religious experiences (different faiths report contradictory deities), the naturalistic reduction (neurological triggers can induce transcendent sensations), and the sheer ineffability of the experience making verifiable report impossible.
反对意见包括宗教体验冲突的问题（不同信仰报告互相矛盾的神祇）、自然主义还原（神经触发可以诱导超验感受）以及体验本身的不可言说性使得可验证的报告不可能。
7. Modus Ponens (Affirming the Antecedent) | 肯定前件式
Modus ponens is one of the most fundamental valid inference forms in philosophical logic. If a conditional statement is true and its antecedent is true, the consequent must be true. Recognising this structure helps you deconstruct arguments in any domain.
肯定前件式是哲学逻辑中最基本的有效推理形式之一。如果一个条件陈述为真，并且它的前件为真，那么后件必定为真。识别这一结构有助于你在任何领域解构论证。
General Form: If P then Q. P. Therefore, Q.
一般形式: 如果 P 则 Q。P。因此，Q。
Example: If God exists, objective moral values exist. God exists. Therefore, objective moral values exist.
示例: 如果上帝存在，客观道德价值存在。上帝存在。因此，客观道德价值存在。
Do not confuse this with the fallacy of affirming the consequent: ‘If P then Q. Q. Therefore, P.’ This invalid form traps many casual reasoners into mistaking correlation for logical proof.
不要将此与肯定后件的谬误混淆：“如果 P 则 Q。Q。因此，P。”这种无效形式使许多草率的推理者误将相关性当作逻辑证明。
8. Modus Tollens (Denying the Consequent) | 否定后件式
Modus tollens is the valid deductive move that denies the consequent to prove the negation of the antecedent. This is the logical engine behind the Problem of Evil and many sceptical challenges.
否定后件式是一种有效的演绎动作，通过否定后件来证明前件的否定。这是恶的问题和许多怀疑论挑战背后的逻辑引擎。
General Form: If P then Q. Not-Q. Therefore, not-P.
一般形式: 如果 P 则 Q。非 Q。因此，非 P。
Example: If only material things exist, consciousness is an illusion. Consciousness is not an illusion. Therefore, not only material things exist.
示例: 如果只有物质存在，意识就是幻觉。意识不是幻觉。因此，并非只有物质存在。
In examinations, you can use modus tollens to test the coherence of a claim: identify the conditional, find a clear counterexample to the consequent, and watch the whole edifice fall.
在考试中，你可以使用否定后件式来检验一个主张的连贯性：找出其中的条件句，为后件找到一个明确的反例，整个大厦便会倒塌。
9. The Syllogism (Categorical Reasoning) | 三段论（范畴推理）
A syllogism consists of two premises and a conclusion, connecting three terms (major, minor, middle). Aristotle’s categorical syllogism remains the blueprint for deducing necessary conclusions from universal or particular premises.
三段论由两个前提和一个结论组成，连接三个项（大项、小项、中项）。亚里士多德的范畴三段论仍然是从普遍或特殊前提推导出必然结论的蓝图。
Standard Form:
标准形式:
Major Premise: All M are P. (All humans are mortal.)
大前提: 所有 M 都是 P。（所有人都必死。）
Minor Premise: All S are M. (Socrates is a human.)
小前提: 所有 S 都是 M。（苏格拉底是人。）
Conclusion: Therefore, all S are P. (Socrates is mortal.)
结论: 因此，所有 S 都是 P。（苏格拉底必死。）
Watch for four classic fallacies: the undistributed middle, the illicit major, the illicit minor, and the existential fallacy (assuming existence from universal premises). Mastering the square of opposition helps you spot them instantly.
注意四种经典谬误：中项不周延、大项非法周延、小项非法周延以及存在谬误（从全称前提假设存在）。掌握对立方形图能帮你瞬间识别这些谬误。
10. The Naturalistic Fallacy & The Is-Ought Gap | 自然主义谬误与实然-应然鸿沟
David Hume famously observed that many moral arguments illicitly slide from descriptive statements (‘is’) to prescriptive statements (‘ought’) without justification. G.E. Moore later named the attempt to define ‘good’ in natural terms the ‘Naturalistic Fallacy’.
大卫·休谟著名地观察到，许多道德论证在未经辩护的情况下，从描述性陈述（“实然”）滑向规范性陈述（“应然”）。G.E. 摩尔后来将用自然术语定义“善”的企图命名为“自然主义谬误”。
Hume’s Law: No conclusion containing ‘ought’ can be deduced solely from premises containing ‘is’.
休谟法则: 任何包含“应然”的结论都不能仅从包含“实然”的前提中演绎得出。
Moore’s Open Question Argument: For any natural property N (pleasure, survival), it remains an open question whether N is good. Therefore, good cannot be identical with any natural property.
摩尔的开方问答论证: 对于任何自然属性 N（快乐、生存），“N 是善的吗？”始终是一个开放问题。因此，善不能等同于任何自然属性。
When evaluating moral arguments in your exam, underline every ‘ought’ and trace it back to its ‘is’ premise. If there is no bridging premise, the argument commits the fallacy.
在考试中评估道德论证时，在每个“应然”下划线，追溯其对应的“实然”前提。如果没有桥接前提，该论证就犯下了此谬误。
11. The Utility Principle (Greatest Happiness Formula) | 功利原则（最大幸福公式）
Utilitarianism provides a decision-procedure for ethics using a quasi-mathematical weighting of pleasures and pains. Bentham’s hedonic calculus and Mill’s qualitative hedonism offer two distinct formulations of the same core principle.
功利主义为伦理学提供了一套决策程序，采用准数学的方式权衡快乐和痛苦。边沁的快乐计算法和密尔的质化快乐论是对同一核心原则的两种不同表述。
Principle of Utility: An act is right if it tends to promote the greatest happiness for the greatest number.
功利原则: 如果一种行为倾向于促进最大多数人的最大幸福，则该行为是正确的。
Bentham’s Hedonic Calculus variables: Intensity, Duration, Certainty, Propinquity (nearness), Fecundity, Purity, Extent.
边沁的快乐计算变量: 强度、持续期、确定性、邻近度、生殖力、纯度、范围。
Mill’s Quality Distinction: Higher pleasures (intellect, moral sentiment) outweigh lower pleasures (bodily) even if the quantity is less.
密尔的质量区分: 高级快乐（智力、道德情操）压倒低级快乐（身体愉悦），即使数量较少。
Common objections: the calculation is impractical (time constraints), it leads to counterintuitive conclusions (scapegoating, punishment of the innocent), and it cannot guarantee justice for individuals. The ‘tyranny of the majority’ is a cost function built into the equation.
常见反对意见：计算不切实际（时间限制）、导致反直觉结论（替罪羊、惩罚无辜者）、无法保障个体正义。“多数暴政”是内嵌在等式中的代价函数。
12. The Duty Formula (Kant’s Categorical Imperative) | 义务公式（康德的绝对命令）
Kant’s ethical framework replaces calculation with universalisability. An action is morally permissible only if the maxim behind it can be consistently willed as a universal law without contradiction. This is not advice but a logical test.
康德的伦理框架用可普遍化替代计算。只有当行为背后的格准能够前后一致地作为普遍法则被意愿，且不产生矛盾时，该行为在道德上才是被允许的。这不是建议，而是一种逻辑检验。
First Formulation (Universal Law): Act only according to that maxim whereby you can at the same time will that it should become a universal law.
第一公式（普遍法则）: 只依据那种你同时也愿意它成为普遍法则的格准而行动。
Second Formulation (Humanity): 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.
第二公式（人本身）: 行动时，无论对待自己还是他人的人格中的人性，永远同时当作目的，绝不仅仅当作手段。
Contradiction Test: Can the maxim be conceived as universal? E.g., ‘Make false promises when convenient’ → if universalised, the institution of promising collapses → contradiction in conception.
矛盾检验: 格准能否被设想为普遍？例如，“方便时做虚假承诺” → 如果普遍化，承诺制度崩溃 → 设想中的矛盾。
Critics note that the categorical imperative can yield conflicting duties (lying to a murderer at the door), is too rigid (no exceptions), and may be empty without specifying what ends are truly rational to pursue.
批评者指出，绝对命令可能产生相互冲突的义务（对门外的凶手撒谎）、过于僵硬（没有例外），并且在没有指明哪些目的真正合乎理性的情况下可能空洞无物。
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