Year 9 AQA Philosophy: Formula & Theorem Quick Reference Handbook | Year 9 AQA 哲学：公式定理速查手册

📚 Year 9 AQA Philosophy: Formula & Theorem Quick Reference Handbook | Year 9 AQA 哲学：公式定理速查手册

In philosophy, we often encounter principles, argument structures, and ethical rules that work like ‘formulas’ in mathematics. Once you understand their patterns, you can analyse and evaluate almost any philosophical problem. This handbook presents the essential ‘formulas and theorems’ for Year 9 AQA Philosophy, covering logic, ethics, epistemology, and critical thinking. Each entry gives you a clear definition, a symbolic formula (where applicable), a worked example, and exam tips to help you revise efficiently.

在哲学中，我们经常会遇到像数学中的“公式”一样起作用的原则、论证结构和伦理规则。一旦理解了它们的模式，你几乎可以分析和评估任何哲学问题。这本手册为你提供了 Year 9 AQA 哲学课程中的核心“公式定理”，涵盖逻辑学、伦理学、认识论和批判性思维。每个条目都给出了清晰的定义、符号化公式（如适用）、实例和备考提示，助你高效复习。

1. The Utilitarian Principle Formula | 功利主义原则公式

Moral Rightness = Σ(Pleasure − Pain) for all affected

Utilitarianism, developed by Jeremy Bentham and John Stuart Mill, claims that the morally right action is the one that produces the greatest happiness for the greatest number. We can express this as a hedonic calculus: the total pleasure produced minus the total pain. If an action maximises net happiness compared to all alternatives, it is the right thing to do. This formula reminds you that every person’s pleasure and pain count equally.

由边沁和密尔发展的功利主义认为，道德上正确的行为是为最大多数人带来最大幸福的行为。我们可以将其表达为快乐计算法：总快乐减去总痛苦。如果一个行为相较于所有其他选择能最大化净幸福，那它就是正确的。这个公式提醒我们，每个人的快乐和痛苦都同等重要。

Example: Should you lie to protect a friend’s feelings? Calculate: lying may cause mild guilt (−2) but prevents major upset to friend (+8). Overall net happiness = +6. Not lying: friend upset (−7), you feel honest (+3). Net = −4. Lying maximises utility.
示例：你应该说谎来保护朋友的感受吗？计算：说谎可能导致轻微内疚（−2），但避免了朋友极大的烦恼（+8），净幸福值为+6。不说谎：朋友难过（−7），你因诚实而欣慰（+3），净值为−4。说谎使功利最大化。

2. Kant’s Categorical Imperative: Universal Law Formula | 康德绝对命令：普遍法则公式

Act only according to that maxim whereby you can at the same time will that it should become a universal law.

Kant’s deontological ethics provides a rule for testing moral duties. A maxim is a personal rule of action. For an action to be morally permissible, you must be able to consistently will that everyone follow the same maxim without contradiction. If universalising the maxim leads to a contradiction in conception or in the will, the action is forbidden. Think of it as a logical consistency test for morality.

康德的义务论伦理学提供了一条检验道德义务的规则。准则是你个人的行为规则。一个行为在道德上是允许的，你必须能够一贯地愿意所有人都遵循同样的准则而不产生矛盾。如果普遍化该准则会导致概念上的矛盾或意志上的矛盾，那么该行为就是被禁止的。这就好比对道德进行逻辑一致性测试。

Example: Lying to get money. Maxim: ‘Whenever I need money, I will make a false promise.’ Universalise: everyone makes false promises when they need money. Then the institution of promising would collapse, and no one would believe promises, making the action impossible. Contradiction → lying is impermissible.
示例：为得钱而说谎。准则：“每当我需要钱时，我会作虚假承诺。” 普遍化：每个人在需要钱时都作虚假承诺。那么承诺制度就会崩溃，无人再相信承诺，行为本身不再可能。产生矛盾 → 说谎是不允许的。

3. Modus Ponens (Affirming the Antecedent) | 肯定前件式

P → Q, P ∴ Q

This is the most fundamental valid argument form in deductive logic. If we know that a conditional statement ‘If P then Q’ is true, and we also know that P is true, we can validly conclude Q. The truth of the premises guarantees the truth of the conclusion. Spotting this pattern helps you quickly assess whether a philosopher’s reasoning is logically sound.

这是演绎逻辑中最基本有效的论证形式。如果我们知道一个条件句“如果 P 则 Q”为真，并且还知道 P 为真，我们就可以有效地推出 Q。前提的真保证了结论的真。识别这种模式有助于你快速判断哲学家的推理是否逻辑严谨。

Example: If it is raining (P), the street is wet (Q). It is raining. Therefore, the street is wet.
示例：如果下雨（P），那么街道是湿的（Q）。正在下雨。所以，街道是湿的。
Exam tip: Do not confuse this with the fallacy of affirming the consequent (Q, therefore P), which is invalid.
备考提示：切勿将其与肯定后件的谬误相混淆（由 Q 推出 P），那是无效的。

4. Modus Tollens (Denying the Consequent) | 否定后件式

P → Q, ¬Q ∴ ¬P

Another rock-solid deductive form. If ‘If P then Q’ holds, and Q is false, then P must be false. This pattern is extremely useful in philosophy for eliminating hypotheses or testing predictions. If a theory predicts Q, but we observe not Q, the theory (P) is falsified.

另一种坚如磐石的演绎形式。如果“如果 P 则 Q”成立，并且 Q 为假，那么 P 必然为假。这种模式在哲学中对于排除假说或检验预测非常有用。如果某个理论预测 Q，但我们观察到了非 Q，那么该理论（P）就被证伪了。

Example: If the mind is just the brain (P), then damaging the brain should always erase specific memories (Q). Patient H.M. had brain damage but some memories remained intact (¬Q). Therefore, the mind is not merely the brain (¬P).
示例：如果心灵只是大脑（P），那么损伤大脑应当总是抹去特定记忆（Q）。病人 H.M. 大脑受损，但部分记忆完好（¬Q）。因此，心灵不仅仅是大脑（¬P）。

5. Hypothetical Syllogism | 假言三段论

P → Q, Q → R ∴ P → R

This rule lets you chain conditional statements together. If P implies Q and Q implies R, then we can conclude that P implies R. It embodies pure transitive reasoning and is vital in constructing complex philosophical arguments from simple steps.

这条规则让你能够将条件链连接起来。如果 P 蕴涵 Q，并且 Q 蕴涵 R，那么我们就能推出 P 蕴涵 R。它体现了纯粹的传递性推理，对于从简单步骤构建复杂的哲学论证至关重要。

Example: If I think (P), then I exist (Q). If I exist (Q), then solipsism is false (R). Therefore, if I think, solipsism is false. (This mirrors Descartes’ chain of reasoning.)
示例：如果我在思考（P），那么我存在（Q）。如果我存在（Q），那么唯我论为假（R）。因此，如果我在思考，唯我论为假。（这反映了笛卡尔的推理链。）

6. Disjunctive Syllogism | 选言三段论

P ∨ Q, ¬P ∴ Q

When you are faced with two mutually exclusive options and can eliminate one, the other must be true. This ‘process of elimination’ is a common move in philosophical dilemmas, such as the argument that if something cannot come from nothing, and the universe is not eternal, then the universe must have had an external cause.

当你面对两个互斥的选项，并且可以排除其中一个时，另一个必定为真。这种“排除法”是哲学困境中的常见手段，例如，如果事物不能无中生有，而宇宙并非永恒，那么宇宙必定有一个外在原因。

Example: Either moral values are objective (P) or they are subjective (Q). But if they are objective, radical disagreement should eventually resolve (¬Q → P). Persistent deep disagreement exists (¬P effectively denied). So, moral values are subjective. (Careful: real philosophers debate this.)
示例：要么道德价值是客观的（P），要么它们是主观的（Q）。但如果是客观的，那么根本分歧最终应当能够解决。现实中持续存在深刻分歧。因此，道德价值是主观的。（注意：真正的哲学家仍在这一问题上争论不休。）

7. The JTB Formula for Knowledge | 知识的三要素公式

S knows that p ↔ (i) p is true, (ii) S believes p, (iii) S is justified in believing p.

Since Plato, knowledge has often been analysed as justified true belief (JTB). To claim someone knows a proposition p, all three conditions must be met. This formula provides the classic structure for defining knowledge, although Gettier problems later showed it might not be sufficient, it remains the essential starting point in the AQA specification.

自柏拉图以来，知识常被分析为有辩护的真信念（JTB）。要断言某人知道命题 p，必须同时满足这三个条件。这个公式为定义知识提供了经典结构，尽管后来的盖梯尔问题表明它或许不充分，但它仍然是 AQA 考纲中的基本出发点。

Example: Anna knows that Paris is the capital of France. (i) It is true; (ii) she believes it; (iii) she has seen reliable maps and been taught so — she is justified.
示例：安娜知道巴黎是法国的首都。（i）这是真的；（ii）她相信这一点；（iii）她看过可靠的地图并被教导过—她是有辩护的。

8. Empiricist Source-of-Knowledge Principle | 经验主义知识来源原则

All substantive knowledge of the world originates in sense experience. (Nihil in intellectu nisi prius in sensu)

Empiricism, championed by Locke and Hume, holds that the mind begins as a ‘blank slate’. Sensory impressions are the raw material of all our concepts and factual knowledge. For any idea to be valid, you must be able to trace it back to its origins in sensation. This principle functions as a verification formula to test the legitimacy of philosophical concepts.

以洛克和休谟为代表的经验主义主张，心灵最初是一块“白板”。感官印象是我们所有概念和事实知识的原材料。一个观念若要有效，你必须能追溯到它在感觉中的起源。这条原则就像一套验证公式，用以检验哲学概念的合法性。

Example: Hume challenges the concept of ‘necessary connection’ in causation. We never sense necessity, only constant conjunction. Since the idea cannot be traced to an impression, it lacks legitimacy.
示例：休谟挑战因果关系中的“必然联系”概念。我们从未感知到必然性，只感知到恒常连结。既然该观念不能追溯至印象，它就缺乏合法性。

9. Rationalist Innate Idea Theorem | 理性主义天赋观念定理

Certain concepts and truths are not derived from experience; they are innate structures of the mind or accessible through pure reason.

Rationalists like Descartes and Plato argue that experience alone cannot account for necessary truths (e.g., mathematics) or concepts like infinity and perfection. These must be innate or discovered via reason. The key test: if a truth is universal and known a priori (independent of experience), then it is a rational truth. This theorem opposes the empiricist formula and drives the rationalist project.

笛卡尔和柏拉图等理性主义者认为，仅凭经验无法解释必然真理（如数学）或无限、完美等概念。这些必定是天赋的或通过纯粹理性发现的。关键检验标准是：如果某个真理是普遍的且先天地（独立于经验）被知道，那么它就是一个理性真理。这条定理与经验主义原则相对立，驱动着理性主义的事业。

Example: Descartes claims the idea of God as a supremely perfect being cannot come from my imperfect sensory experience; it must have been placed in me by something truly perfect.
示例：笛卡尔声称，关于上帝作为最完美存在的观念不可能来自我不完美的感官经验；它必定是由某个真正完美者置于我心中的。

10. Cartesian Method of Doubt & Certainty Formula | 笛卡尔怀疑方法与确定性公式

If a belief can be doubted, even by a hypothetical scenario, it is not certain. Seek a foundation that is indubitable.

Descartes’ method uses radical doubt to demolish all beliefs that are not absolutely certain. The formula: present the sceptical scenario (evil demon, dreaming); if the belief is still possibly false in that scenario, reject it as uncertain. Only what remains immune to all doubt can serve as the foundation of knowledge. This gave us the famous ‘Cogito ergo sum’ as the indubitable remnant.

笛卡尔的方法运用彻底怀疑来拆毁所有并非绝对确定的信念。其公式是：提出怀疑主义情景（恶魔、梦境）；如果某个信念在那情景下仍有可能是假的，就把它当作不确定的予以摒弃。只有对一切怀疑免疫的东西才能作为知识的基础。这便产生了著名的“我思故我在”，作为不可怀疑的剩余物。

Example: Can I be sure I have a body? A dream can simulate bodily sensations. So, having a body is doubtable. But even if a demon is deceiving me, I must exist to be deceived. ‘I think, I exist’ is indubitable.
示例：我能确定我有一个身体吗？梦境可以模拟身体感觉。所以，拥有身体是可怀疑的。但即使有个恶魔在欺骗我，我也必须存在才能被欺骗。“我思，我存在”是不可怀疑的。

11. Hume’s Problem of Induction Formula | 休谟的归纳问题公式

We cannot rationally justify the inference from ‘All observed X are Y’ to ‘All future X will be Y’ without circularity.

The sun has risen every morning; we expect it to rise tomorrow. But why? Hume points out that all inductive reasoning relies on the uniformity of nature. However, any attempt to prove nature is uniform itself relies on induction, which is circular. The formula: inductive argument (past → future) is justified only if nature is uniform. But the proof of uniformity requires induction. Therefore, induction lacks rational justification.

太阳每天早上都升起；我们预期它明天还会升起。但为什么？休谟指出，一切归纳推理都依赖自然的齐一性。然而，任何试图证明自然齐一的尝试本身就依赖归纳，从而陷入循环。其公式是：归纳论证（从过去到未来）成立仅当自然是齐一的。但齐一性的证明又需要归纳。因此，归纳缺乏理性的辩护。

Example: All observed swans were white → all swans are white. But this was proven false by the discovery of black swans. The step from observed to unobserved cannot be logically guaranteed.
示例：所有观察到的天鹅都是白的 → 所有天鹅都是白的。但黑天鹅的发现证明了这个推论为假。从已观察到未观察的这一步在逻辑上无法保证。

12. The Naturalistic Fallacy Warning | 自然主义谬误警示

You cannot logically derive an ‘ought’ (value) from an ‘is’ (fact) without additional evaluative premises.

Coined by G.E. Moore and rooted in Hume’s is-ought gap, this ‘fallacy formula’ warns against trying to define moral goodness purely in terms of natural properties (e.g., pleasure). Just because something is pleasurable (a natural fact) does not mean it is good (a moral value). Whenever you see an argument that jumps from describing how things are to prescribing how they should be, check for this missing step. It is a vital critical thinking tool for ethics essays.

这个由摩尔（G.E. Moore）命名并植根于休谟“是-应”鸿沟的“谬误公式”警示我们，不要试图纯粹用自然属性（如快乐）来定义道德上的善。某物令人快乐（自然事实）并不意味着它就是善的（道德价值）。每当你看到一个从“描述事物如何”跳到“规定应当如何”的论证时，就要留意这个缺失的环节。这是伦理学论文中至关重要的批判性思维工具。

Example: ‘Humans evolved to care for their offspring, therefore we ought to care for our offspring.’ The premise describes a biological fact; the conclusion is a moral prescription. The leap requires justification.
示例：“人类经过演化会照顾后代，因此我们应当照顾后代。”前提描述了一个生物学事实；结论是一个道德规定。这种跳跃需要辩护。

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Year 9 AQA Philosophy: Formula & Theorem Quick Reference Handbook | Year 9 AQA 哲学：公式定理速查手册