📚 Philosophy Formulas & Theorems Quick Reference Handbook | 哲学公式定理速查手册
Welcome to your Year 8 AQA Philosophy Quick Reference Handbook. Philosophy might seem like a subject built only on long texts and open debates, but underneath it lies a powerful set of logical structures, ethical calculus, and conceptual formulae that help us reason clearly. This handbook collects the essential ‘formulas’ and ‘theorems’ you will need to analyse arguments, evaluate ethical theories, and spot fallacies. Each entry provides a concise logical form alongside a practical explanation, helping you see that philosophy, like mathematics, has its own elegant notation.
欢迎使用这本八年级 AQA 哲学速查手册。哲学看起来似乎只是一门由长篇文本和开放辩论构成的学科，但在其底層，隐藏着一套强大的逻辑结构、伦理计算方式和概念公式，帮助我们清晰地推理。本手册汇集了你分析论证、评估伦理理论和识别谬误所需的基本“公式”与“定理”。每个条目都提供了简洁的逻辑形式以及实用解释，让你发现哲学正如数学一样，拥有自身的优雅符号体系。
1. Modus Ponens | 肯定前件式
Modus Ponens, or ‘affirming the antecedent’, is the most direct way to draw a conclusion from a conditional statement. If we accept that P implies Q, and we also accept that P is true, then we must accept that Q is true. This deductive move guarantees truth preservation; if the premises are true, the conclusion cannot be false.
肯定前件式是从条件句得出结论的最直接方式。如果我们承认P蕴含Q，并且我们也承认P为真，那么我们必须承认Q为真。这一演绎步骤能保证真值的传递：如果前提为真，结论就不会为假。
P → Q, P ∴ Q
Example: If it is raining (P), then the ground is wet (Q). It is raining (P). Therefore, the ground is wet (Q). This formula underpins a huge amount of everyday reasoning and scientific prediction. When evaluating someone's argument, always check whether the conditional premise is actually true and whether the antecedent is satisfied.
例如：如果下雨(P)，那么地面湿(Q)。下雨了(P)。所以地面湿(Q)。这一公式支撑着大量日常推理和科学预测。在评估他人论证时，始终要检查条件前提是否确实为真，以及前件是否被满足。
2. Modus Tollens | 否定后件式
Where Modus Ponens works by affirming the antecedent, Modus Tollens works by denying the consequent. If P implies Q, and Q is false, then P must also be false. This is a cornerstone of falsification in science: we use it to rule out hypotheses when their predicted outcomes do not occur.
肯定前件式通过肯定前件来运作，而否定后件式则通过否定后件来运作。如果P蕴含Q，而Q为假，那么P必定也为假。这是科学证伪的基石：我们用它来排除那些预测结果并未出现的假设。
P → Q, ¬Q ∴ ¬P
Example: If the battery is charged (P), the phone turns on (Q). The phone does not turn on (¬Q). So, the battery is not charged (¬P). In philosophy, Modus Tollens helps us test universal claims. A single counterexample can demolish a theory.
例如：如果电池有电(P)，手机就能开机(Q)。手机没有开机(¬Q)。所以，电池没电(¬P)。在哲学中，否定后件式帮助我们检验普遍性主张。一个反例就能摧毁一套理论。
3. Hypothetical Syllogism | 假言三段论
When reasoning chains together multiple conditional statements, we use Hypothetical Syllogism. This rule allows us to infer a direct link between the first condition and the last consequence, streamlining complex causal or logical chains. It reveals that implication is transitive.
当推理将多个条件陈述串联起来时，我们会使用假言三段论。这条规则允许我们推断出第一个条件与最后一个结果之间的直接联系，从而简化复杂的因果链或逻辑链。它揭示了蕴含关系的传递性。
(P → Q) ∧ (Q → R) ∴ (P → R)
Example: If I study consistently (P), I understand the topics (Q). If I understand the topics (Q), I do well in the exam (R). Therefore, if I study consistently (P), I do well in the exam (R). This theorem is essential for building long ethical or metaphysical arguments in philosophy essays.
例如：如果我坚持学习(P)，我就能理解课题(Q)。如果我理解课题(Q)，我就能在考试中表现出色(R)。所以，如果我坚持学习(P)，我就能在考试中表现出色(R)。这一定理对于构建哲学论文中的长篇伦理或形而上学论证至关重要。
4. Disjunctive Syllogism | 选言三段论
When faced with an ‘either… or…’ situation, Disjunctive Syllogism allows us to eliminate one option and thereby conclude the other. This reasoning is valid only when the disjunction is genuinely exhaustive; otherwise, we risk a false dilemma. Used carefully, it forces clarity in decision-making.
当我们面对“要么……要么……”的情形时，选言三段论允许我们排除其中一个选项，从而得出结论可以是另一个选项。这种推理只有在析取项确实穷尽时才有效；否则，我们就可能犯下假两难谬误。谨慎使用时，它能迫使决策变得更加清晰。
(P ∨ Q), ¬P ∴ Q
Example: Either the mind is entirely physical (P) or it has a non‑physical aspect (Q). The mind is not entirely physical (¬P). Therefore, it must have a non‑physical aspect (Q). Notice that this formula is only as strong as the initial disjunction. Always ask: are these really the only options?
例如：要么心灵完全是物理的(P)，要么它具有非物理的层面(Q)。心灵并不完全是物理的(¬P)。所以，它必定具有非物理的层面(Q)。请注意，这个公式的有效性只取决于最初的析取项。要始终追问：这些真的就是仅有的选项吗？
5. The JTB Account of Knowledge | 知识的JTB定理
For centuries, philosophers have treated knowledge as ‘justified true belief’ (JTB). Although Edmund Gettier later showed that JTB is not sufficient, the JTB formula remains the foundational theorem you must master. It breaks knowledge into three necessary conditions that can each be tested independently.
几个世纪以来，哲学家们一直将知识视为“经过辩护的真信念”（JTB）。尽管埃德蒙・盖梯尔后来表明 JTB 并不充分，但 JTB 公式依然是你必须掌握的基础定理。它将知识分解为三个必要条件，每个条件都可以独立检验。
Knowledge = Belief + True + Justification
- Belief: S must hold the proposition p to be true.
- True: The proposition p must correspond to reality.
- Justification: S must have adequate reasons or evidence for p.
- 信念：认知主体S必须认为命题p为真。
- 真实：命题p必须符合现实。
- 辩护：S必须有充分的理由或证据支持p。
This ‘formula’ is used to test whether someone really knows something or merely has a lucky guess. When writing about epistemology, always attempt to run a claim through the JTB test.
这个“公式”被用来检验一个人是真正知道某事，还是仅仅做出了幸运的猜测。在写作关于认识论的文章时，始终要尝试用 JTB 检验去审视一个断言。
6. The Utilitarian Calculus | 功利主义计算式
Jeremy Bentham proposed that moral decisions could be approached like a quantitative science. His felicific calculus measures the total utility produced by an action. Though simplified here, this formula captures the maximising core of classical utilitarianism: the right action is the one that yields the greatest net pleasure.
杰里米·边沁提出，道德决策可以像量化科学一样来处理。他的快乐计算法衡量一个行为所产生的总功利。尽管此处有所简化，但这个公式捕捉到了古典功利主义“最大化”的核心：正确的行为就是能带来最大净快乐的那个行为。
Right action = argmax ∑ (Pleasure − Pain)
The formula considers intensity, duration, certainty, and extent of pleasure and pain across all affected individuals. You will often be asked to apply utilitarian reasoning to dilemmas like the Trolley Problem. The calculus pushes you to think impersonally and aggregate well-being.
这一计算式考虑了快乐与痛苦在所有受影响个体身上的强度、持续时间、确定性和范围。你常常会被要求将功利主义推理应用于像电车难题那样的两难困境。这个计算式促使你进行非个人化的思考，并汇总福祉。
7. The Categorical Imperative Formula | 绝对命令公式
Immanuel Kant’s ethics rests on a single, formula‑like principle called the Categorical Imperative. Its first formulation asks us to universalise our maxim. If the maxim can be consistently willed as a universal law without contradiction, the action is morally permissible. This provides a logical test for duty.
伊曼努尔·康德的伦理学建立在一条独特的、类似公式的原则之上，即绝对命令。其第一条表述要求我们将自己的准则普遍化。如果这一准则能够被不自相矛盾地意愿成为一条普遍法则，那么该行为在道德上就是可允许的。这就为义务提供了逻辑检验。
Act only on that maxim through which you can at the same time will that it should become a universal law.
Example: If you consider breaking a promise to escape trouble, ask: can I rationally will that everyone breaks promises whenever convenient? The resulting world would destroy the institution of promising, generating a contradiction. Hence, breaking promises is not permissible. This formula forbids treating others merely as means to an end.
例如：如果你考虑通过违背诺言来摆脱麻烦，请自问：我能否合理地意愿，每个人都在方便时违背诺言？由此产生的世界会毁掉承诺制度本身，导致矛盾。因此，违背诺言是不被允许的。这一公式禁止仅仅将他人当成达成目的的手段。
8. The Golden Rule | 黄金法则
Present in many religious and secular traditions, the Golden Rule offers a straightforward ethical formula based on reciprocity. It turns self‑interest into a test for moral action: if you would not want an action done to you, then you should not do it to others. It is a practical guide for empathy.
黄金法则存在于许多宗教和世俗传统中，它提供了一条基于互惠性的直接伦理公式。它将自利转化为对道德行动的检验：如果某件事你不希望发生在自己身上，那么你也不应该对别人这样做。这是一条实践性的同理心指南。
Do unto others as you would have them do unto you.
While not a formal logical theorem, the Golden Rule functions as a consistency formula: it demands that our actions match our preferences when roles are reversed. In AQA Philosophy, you can contrast this principle with utilitarian calculus and Kantian duty to explore different moral frameworks.
尽管并非形式化的逻辑定理，黄金法则却起着一致性公式的作用：它要求我们的行为在角色互换时，能够与我们的偏好相匹配。在 AQA 哲学中，你可以将这一原则与功利主义计算式和康德义务论进行对比，以探究不同的道德框架。
9. Occam’s Razor | 奥卡姆剃刀
Occam’s Razor is a principle of parsimony that helps philosophers and scientists choose between competing explanations. The formula advises that entities should not be multiplied beyond necessity. In practice, when two theories explain the same data equally well, we should prefer the one that makes fewer assumptions.
奥卡姆剃刀是一条简约性原则，帮助哲学家和科学家在相互竞争的解释之间做出选择。这一公式建议，如无必要，勿增实体。在实践中，当两个理论对相同数据做出同样良好的解释时，我们应当青睐假设更少的那个。
Entities are not to be multiplied beyond necessity.
This is not a proof that the simpler theory is always true, but a methodological theorem. In the philosophy of religion, for example, the razor is wielded against arguments that posit a deity when a naturalistic explanation suffices. Use Occam’s Razor to trim away unsupported complexity.
这并不能证明更简单的理论总是正确的，而是一条方法论定理。例如，在宗教哲学中，当自然主义的解释已足够时，剃刀就被用来反对假设神灵存在的论证。运用奥卡姆剃刀来削去那些缺乏支撑的复杂结构。
10. The Paradox of the Heap | 谷堆悖论
The Sorites paradox shows how vague concepts unravel under logical pressure. It takes the form of a chain of Modus Ponens steps that lead to a contradiction. Formally, the paradox demonstrates that small changes accumulate, forcing us to question whether our concepts have sharp boundaries.
谷堆悖论展示了模糊概念如何在逻辑压力下瓦解。它采用一系列肯定前件式的步骤，最终导致矛盾。在形式上，该悖论表明微小的变化会累积起来，迫使我们质疑自己的概念是否具有清晰的边界。
If n grains do not form a heap, then n+1 grains do not form a heap.
1 grain is not a heap. ∴ 10,000 grains are not a heap.
Obviously the conclusion is false, yet each inductive step seems valid. This theorem pushes you to examine how language and logic interact. In your revision, remember that Sorites challenges bivalent (true/false) thinking about many real‑world properties.
结论显然为假，但每个归纳步骤看起来都有效。这一定理促使你审视语言与逻辑是如何互动的。在复习中，要记住谷堆悖论挑战了对许多现实世界属性所持有的二值化（真/假）思维。
11. The Liar Paradox Formula | 说谎者悖论公式
The Liar Paradox is a self‑referential sentence that defeats any attempt to assign it a stable truth value. It is usually written as a single statement: ‘This sentence is false.’ If it is true, then it is false; if it is false, then it is true. This logical loop has inspired deep work on language and set theory.
说谎者悖论是一个自指的句子，任何试图赋予它一个稳定真值的做法都会失败。它通常被写作一条单独的陈述：“本句话是假的”。如果它为真，那么它就是假的；如果它为假，那么它就是真的。这个逻辑循环激发了关于语言和集合论的深入研究。
L: L is false.
(L ↔ ¬L)
In AQA Philosophy, you might encounter the Liar when discussing truth or the limits of logic. The theorem reveals that not every grammatically correct sentence can be coherently evaluated. Treat it as a cautionary formula: self‑reference often breeds inconsistency.
在 AQA 哲学中，你在讨论真理或逻辑的局限时可能遇到说谎者悖论。这一定理揭示出，并非每个语法正确的句子都能得到连贯的评价。把它当作一个警示公式来对待：自指常常孕育出不一致性。
12. The Socratic Formula | 苏格拉底反诘法公式
Socratic inquiry follows a recursive, formulaic pattern designed to clarify concepts and expose weak definitions. Though not a deductive theorem, the Socratic method functions as an algorithm for philosophical progress. It iterates through questioning, definition, counterexample, and refined definition until a stable understanding emerges.
苏格拉底式探究遵循一种递归的、公式化的模式，旨在澄清概念并暴露薄弱的定义。尽管并非演绎性定理，苏格拉底方法却充当了一种推动哲学进展的算法。它循环进行提问、定义、反例和修正定义，直到出现稳定的理解。
Question (Tί ἐστι;) → Definition → Counterexample → Revised Definition → …
Example: What is courage? Courage is standing firm in battle. But a soldier who retreats to save others might be courageous. So courage is knowledge of what is to be feared and what is not. The formula invites us to keep moving towards a universal definition. It is the engine of dialectic.
例如：什么是勇气？勇气就是坚守战场。但一个为了拯救同伴而撤退的士兵也可能是勇敢的。所以，勇气是知晓何事可畏、何事不可畏的知识。这一公式邀请我们不断逼近一个普遍定义。它正是辩证法的引擎。
Published by TutorHao | Philosophy Revision Series | aleveler.com
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