📚 Year 8 AQA Philosophy Formula & Theorem Quick Reference Handbook | Year 8 AQA 哲学：公式定理速查手册
Welcome to your Year 8 AQA Philosophy quick reference guide. In mathematics and science, formulas and theorems help you solve problems efficiently. Philosophy works in a similar way. Instead of numbers and symbols, we use structured arguments, key principles, and logical frameworks to navigate complex questions about existence, knowledge, morality, and reality. This handbook collects the essential 'formulas' of philosophy — each one a powerful tool for thinking clearly and constructing sound arguments. Keep this guide handy as you prepare for your assessments; it will help you recall the core logical structures that underpin everything we study this year.
欢迎使用 Year 8 AQA 哲学速查手册。在数学和科学中，公式和定理帮助你高效地解决问题。哲学也以类似的方式运作。我们用结构化的论证、关键原则和逻辑框架，而非数字和符号，来探索关于存在、知识、道德和现实的复杂问题。本手册收集了哲学的核心“公式”——每一个都是清晰思考和构建严谨论证的强大工具。在准备评估时，请将本指南放在手边；它将帮助你回忆起支撑我们今年所学一切的核心逻辑结构。
1. The Greatest Happiness Formula | 最大幸福公式
The Utilitarian principle, advanced by Jeremy Bentham and John Stuart Mill, provides a clear decision-making calculus. To determine the morally right action, we must calculate which option produces the greatest balance of pleasure over pain for everyone affected. Bentham proposed a qualitative measurement called the Hedonic Calculus, which examines seven dimensions of pleasure: intensity, duration, certainty, propinquity, fecundity, purity, and extent. This formula shifts ethical focus from rigid rules to real-world outcomes. The right action is never about personal gain alone — it requires a universal ledger of happiness.
由杰里米·边沁和约翰·斯图尔特·密尔提出的功利主义原则，提供了一个清晰的决策演算。要确定道德上正确的行为，我们必须计算哪个选项能为所有受到影响的人带来最大的快乐相对于痛苦的平衡。边沁提出了一种称为“快乐演算”的定性测量方法，它考察快乐的七个维度：强度、持续时间、确定性、邻近性、丰产性、纯度和范围。这个公式将伦理焦点从僵化的规则转移到了现实世界的结果上。正确的行为从来不是仅仅关乎个人利益——它需要一本关于幸福的普世总账。
or all sentient beings, where the action with the highest net total is morally obligatory.
GHP（最大幸福原则）= 对所有有感知力的生命计算 (快乐 × 时长) - (痛苦 × 范围)，其中净总和最高的行为在道德上具有强制性。
2. The Categorical Imperative Formula | 定言令式公式
Immanuel Kant's ethical framework rejects consequence-based thinking entirely. For Kant, the moral worth of an action lies in its intention and in the principle behind it. The Categorical Imperative is the supreme rational principle that commands absolutely, not hypothetically. It has three main formulations, but the most widely used is the Universal Law Formula. Before acting, you must imagine that your personal rule of conduct, your maxim, becomes a universal law of nature for all rational beings. If you cannot consistently will this to happen without logical contradiction, the action is forbidden.
伊曼努尔·康德的伦理框架完全拒绝基于结果的思维。对康德而言，行为的道德价值在于其意图以及背后的原则。“定言令式”是至高无上的理性原则，它绝对地、而非假设性地发号施令。它有三种主要表述，但最广泛使用的是“普遍法则公式”。在行动之前，你必须想象你的个人行为准则，即你的“格律”，成为所有理性存在者的普遍自然法则。如果你无法在不产生逻辑矛盾的情况下始终如一地意愿这件事发生，那么该行为就是被禁止的。
Act only according to that maxim whereby you can at the same time will that it should become a universal law without contradiction.
只依据那个你同时能够意愿其成为一条普遍法则的格律去行动，并且要做到毫无矛盾。
3. The Virtue Mean Theorem | 美德中庸定理
Aristotle's ethical system concentrates not on isolated actions but on the long-term cultivation of a virtuous character. Virtue is defined as a stable disposition to choose the mean between two opposing vices — one of excess and one of deficiency. This mean is not a cold mathematical average; it is relative to the individual and the specific circumstances, as determined by practical wisdom (phronēsis). For example, courage is the mean between the vice of cowardice (deficiency of confidence) and the vice of rashness (excess of confidence). The formula requires extensive life experience and moral perception to apply correctly.
亚里士多德的伦理体系不关注孤立的行为，而是关注长期品德修养的培育。美德被定义为一种稳定的倾向，即在两种对立的恶之间选择中道——一种是过度之恶，另一种是不及之恶。这个中道并非冰冷的数学平均数；它是相对于个人和具体情境而言的，需要由实践智慧来加以判断。例如，勇敢是怯懦之恶（信心不足）和鲁莽之恶（信心过度）之间的中道。该定理的正确应用需要广泛的生活经验和道德洞察力。
Virtue = The disposition to feel and act at the mean (Mesotēs) between the vice of excess and the vice of deficiency, guided by phronēsis.
美德 = 在实践智慧（phronēsis）的引导下，在过度之恶和不及之恶之间感受并行动于中道的稳定倾向。
4. The Materialist Identity Theorem | 唯物主义同一性定理
In philosophy of mind, this theorem states a direct and complete identity between mental states and physical brain states. According to the Mind-Brain Identity Theory, every time you experience pain, love, or a specific thought, that experience is numerically identical to a particular firing pattern of neurons in your brain. Pain is C-fibres firing; the sensation of red is a specific activation in the visual cortex. This is not a causal relationship — the mental state does not 'produce' the brain state or vice versa; they are strictly one and the same thing, just as water is identical to H₂O molecules.
在心灵哲学中，这一定理宣称心理状态和物理大脑状态之间存在直接和完全的同一性。根据心脑同一论，每当你体验疼痛、爱或某种特定想法时，该体验在数值上就等同于你大脑中神经元的特定放电模式。疼痛就是C纤维的放电；红色的感觉就是视觉皮层中一种特定的激活。这不单纯是一种因果关系——心理状态并不“产生”大脑状态，反之亦然；它们严格就是同一个事物，就如同水等同于H₂O分子一样。
∀ Mental State M → ∃ Brain State B: (M ≡ B). The mental is strictly identical to the physical.
对于任何心理状态M，都存在一个大脑状态B，使得M严格同一于B。心理的东西就是物理的东西。
5. The Personal Identity Memory Criterion | 人格同一性记忆判准
John Locke proposed a radical formula for what makes a person the same over time. Personal identity is grounded not in an unchanging soul substance (as Descartes thought) or in the biological continuity of the human animal, but solely in the continuity of consciousness and memory. Person A at time T₁ is identical to Person B at time T₂ if, and only if, B can remember or extend their consciousness back to the experiences and actions of A. This formula appears clean in theory but generates challenging counterexamples: what if a brave general cannot remember stealing apples as a schoolboy? Does the general literally cease to be that boy?
约翰·洛克提出了一个关于是什么让一个人在时间流逝中保持同一的激进公式。人格同一性的根基不在于一个不变的心灵实体（如笛卡尔所想），也不在于人类动物的生物学连续性，而仅仅在于意识和记忆的连续性。在时间点T₁的人A与在时间点T₂的人B是同一的，当且仅当B能够记住或将其意识回溯到A的经历和行为。这一公式在理论上看似简洁，却引发了棘手的反例：如果一位英勇的将军记不起自己作为学童偷苹果的事呢？将军真的就不再是那个男孩了吗？
₂) ↔ Memory continuity (M) extends backwards from T₂ to the actions and experiences of T₁.
人格P在时间T₁与在时间T₂是同一的，当且仅当记忆连续性（M）从T₂回溯延伸到T₁的行为和经历。
6. The Justified True Belief (JTB) Equation | 确证的真信念方程
Since Plato's Theaetetus, the tripartite definition has served as the standard analytical formula for propositional knowledge — knowing that something is the case. For a subject S to know a proposition P, three necessary and jointly sufficient conditions must be met: P must be actually true (the fact matters); S must believe that P (you cannot know what you do not accept); and S must have adequate justification for believing P (a lucky guess is not knowledge). This formula dominated epistemology until Edmund Gettier's devastating 1963 paper demonstrated that the conditions, even when satisfied, can fail to produce genuine knowledge through a set of clever counterexamples involving justified falsehoods leading to true conclusions.
自柏拉图的《泰阿泰德篇》以来，这个三元定义一直充当命题性知识——即知道某事是如此——的标准分析公式。要使得一个认知主体S知道命题P，必须满足三个必要且联合充分的条件：P必须是实际上为真的（事实很重要）；S必须相信P（你无法知道你不接受的事情）；以及S必须有充分的理由或确证去相信P（幸运的猜测不是知识）。这个公式一直支配着认识论，直到埃德蒙德·盖梯尔在1963年发表了那篇毁灭性的论文，通过一系列涉及导致真实结论的有确证错误信念的巧妙反例，证明了即使这些条件都被满足，也可能无法产生真正的知识。
Kₛ(P) ↔ [P is True] ∧ [S Believes P] ∧ [S is Justified in believing P], plus a fourth, non-devious condition to defeat Gettier cases.
S知道P，当且仅当P为真，并且S相信P，并且S的确证于相信P是有理由的，同时还需附加第四个、非因果偏离的条件以击败盖梯尔案例。
7. The Cosmological First Cause Theorem | 宇宙论第一因定理
Originating with Aristotle and refined by Thomas Aquinas, this a posteriori theorem argues that the existence of the universe demands an ultimate explanation. The argument operates in a chain: every event or object we observe has a cause. If the causal chain regressed infinitely backwards with no beginning, the present moment and the whole chain itself would remain unexplained. Therefore, reasoning backwards, there must exist a First Cause that is itself uncaused — an unmoved mover or a necessary being. Aquinas identified this First Cause with the concept of God. The theorem does not claim to establish all divine attributes, only the logical necessity of a foundational cause that terminates the infinite regress.
该定理起源于亚里士多德，并由托马斯·阿奎那加以完善，这一后验定理认为，宇宙的存在需要有一个终极解释。该论证以一条因果链条进行运作：我们观察到的每一个事件或物体都有一个原因。如果因果链条无限地回溯而没有开端，那么当下的时刻以及整个链条本身就仍然无法得到解释。因此，向后推理，必定存在一个本身不受任何因素驱动的“第一因”——一个不被推动的推动者或必然的存在者。阿奎那将这个第一因等同于上帝的概念。该定理并不试图确立所有的神圣属性，而只是确立一个终结无限回溯的奠基性原因的逻辑必然性。
Given the totality of contingent, dependent beings C, there must exist a Necessary Being N: ∀c ∈ C, ∃N such that N is the uncaused cause of c.
给定所有偶然、依赖他物的事物的总体C，必定存在一个必然的存在者N，使得对于C中的所有c，N是c的不被引起的终极原因。
8. The Design Inference Theorem | 设计推论定理
Known formally as the Teleological Argument, this theorem uses analogical reasoning as its engine. William Paley's famous formulation compares a complex artifact, such as a pocket watch found on a heath, to the natural world. A watch, with its intricate arrangement of parts all cooperating to tell time, compels us to infer a watchmaker. By analogy, the natural world, especially biological organisms like the human eye, exhibits far greater complexity and functional integration. The theorem concludes that the features of order, purpose, and complexity in nature provide strong inductive grounds for inferring the existence of an intelligent designer. David Hume's counter-formula challenges the weakness of analogical reasoning: a universe is too unique a thing to be compared to a pocket watch.
该定理正式名称为目的论论证，它使用类比推理作为其动力。威廉·佩利的著名表述将一个复杂的工艺品，比如在荒地上发现的一块怀表，与自然界进行了比较。一块表，其零件之间的精妙安排，都协同合作以报时，这迫使我们推断出一位钟表匠的存在。以此类比，自然界，特别是像人眼这样的生物有机体，展现出了远为复杂的结构和功能上的整合。该定理得出结论，自然界中的秩序、目的和复杂性特征，为推断一位智能设计者的存在提供了强有力的归纳性依据。大卫·休谟的反驳公式则挑战了类比推理的弱点：宇宙是如此独特的一个事物，以至于它无法与一块怀表相提并论。
If Object O exhibits specified complexity SC, then P(Designer | O) > P(Chance | O). Nature contains SC; thus, a Designer is probable.
如果物体O表现出特定复杂性（SC），那么设计者导致O的概率大于偶然性导致O的概率。自然界包含SC；因此，设计者是很有可能的。
9. The Problem of Evil Inconsistent Triad | 恶的问题：不一致三元组
The Problem of Evil presents perhaps the most formidable challenge to classical theism. J.L. Mackie formulated this as a logical proof by exposing an inconsistent triad of propositions that cannot all be true simultaneously. If God is wholly good (omnibenevolent) and also infinitely powerful (omnipotent), He would have both the desire and the capacity to eliminate all suffering. Yet gratuitous evil and suffering manifestly exist in the world. Therefore, at least one of the three propositions must be false. The logical problem holds that the existence of any pointless evil is logically incompatible with the existence of a perfect deity. Theodicies, like those by Augustine or Irenaeus, attempt to resolve the triad by adding a morally sufficient reason that justifies God's permission of evil without rejecting His attributes.
恶的问题或许是对古典有神论最严峻的挑战。J.L.麦基通过揭示一个所有命题无法同时为真的不一致三元组，将其表述为一个逻辑证明。如果上帝是全善的（全爱），并且也是无限大能的（全能），那么祂将同时拥有消除所有苦难的意愿和能力。然而，无端的恶和苦难显著地存在于这个世界。因此，至少这三个命题中必定有一个是假的。这个逻辑问题认为，任何无意义的恶的存在，在逻辑上都与一个完美神祇的存在不相容。各种神义论，比如奥古斯丁或爱任纽的理论，试图通过增加一个道德上充分的理由来化解这个三元组，该理由可为上帝允许恶的存在作辩护，同时又不必否定其属性。
¬(Omnibenevolence ∧ Omnipotence ∧ Existence of Gratuitous Evil). At most two can be consistently maintained.
不可能同时维持全善、全能和无端之恶的存在。这三者中最多只能同时主张其中两个。
10. The Mind-Body Interaction Theorem | 心物交互定理
René Descartes's interactive dualism proposes a radical separation between two fundamental kinds of substance. The essential nature of mind is thinking, non-extended substance (res cogitans). The essential nature of body is extended, non-thinking substance (res extensa). The theorem holds that these two ontologically distinct substances nevertheless causally interact within the human brain. The body affects the mind when nerve endings cause sensations; the mind moves the body when acts of will produce physical motion. Princess Elisabeth of Bohemia's devastating challenge demanded the formula for this interaction: how can a non-spatial, non-physical thought push a physical pineal gland, thereby moving animal spirits? No adequate mechanism was provided, leaving causal interaction as the dualist's central unresolved equation.
勒内·笛卡尔的交互式二元论提出了两种基本实体类型之间的彻底分离。心灵的本质属性是思维，是非延展的实体（res cogitans）。身体的本质属性是广延，是无思维的实体（res extensa）。该定理认为，这两个在本体论上完全不同的实体，却在大脑内部进行着因果交互作用。当神经末梢引起感觉时，身体影响心灵；当意志活动产生物理运动时，心灵移译了身体。波希米亚的伊丽莎白公主提出了一个致命性的挑战，要求给出这种交互作用的公式：一个非空间性的、非物理的思想，如何能够推动物理性的松果腺，并从而移动动物精气？笛卡尔没有提供任何充分的机制，这使得因果交互作用成为了二元论者悬而未决的核心方程。
extended) causally interacts with Physical state Pₑ (extended) at the pineal gland, yet Mₚ ∩ Pₑ = ∅ in spatial and mechanical terms.
心理状态Mₚ（非延展的）与物理状态Pₑ（延展的）在松果腺处进行因果交互作用，然而在空间和机械层面上，Mₚ与Pₑ的交集是空集。
11. The Logical Positivist Verification Principle | 逻辑实证主义证实原则
During the early 20th century, the Vienna Circle philosophers sought to draw a sharp boundary between meaningful discourse and nonsense. The Verification Principle served as their precise formula for meaning. A statement is cognitively meaningful only if it is either analytically true by definition (like 'all bachelors are unmarried men') or empirically verifiable through sense experience in principle. All theological, metaphysical, and ethical statements that cannot meet this test were deemed literally meaningless — not false, but cognitively void, merely expressing emotion or nonsense. The formula self-destructed upon examination: the Verification Principle itself is neither analytically true nor empirically verifiable. By its own criterion, the principle is a meaningless metaphysical claim.
在20世纪早期，维也纳学派的哲学家们试图在有意义的话语和无意义之间划出一条清晰的边界。“证实原则”便是他们关于意义性的精确公式。一个陈述在认知上是有意义的，当且仅当它在定义上是分析真（即通过定义而为真，如“所有单身汉都是未婚男子”），或者在原则上是可通过感官经验在实证上被证实的。所有不能通过这一检验的神学、形而上学和伦理陈述，都被视作字面上毫无意义——并非错误，而是在认知上是空洞的，仅仅是在表达情感或无意义的声音。这个公式在接受审查时自我摧毁了：证实原则本身既非分析真，亦非在实证上可被证实。按其自身的标准，这个原则本身就是一个无意义的形而上学断言。
A proposition P has cognitive meaning iff P ∈ {Analytic Truths} ∪ {Statements empirically verifiable in principle}. Otherwise, P = ∅ᵐ.
一个命题P具有认知意义，当且仅当P属于{分析真理}或是{在原则上可被实证证实的陈述}。否则，P在意义上是空洞的。
12. The Critical Thinking Validity Theorem | 批判性思维有效性定理
At the heart of Year 8 philosophy is the ability to distinguish between good and bad arguments. The deductive validity theorem states that an argument is logically valid if, and only if, it is impossible for all its premises to be true while its conclusion is false. Truth concerns the actual world; validity concerns the logical structure. A classic valid form is Modus Ponens: If A then B, A is true, therefore B must be true. An invalid deductive argument commits a formal fallacy, such as Affirming the Consequent: If A then B, B is true, therefore A is true. This formula fails because the consequent B could be true for reasons entirely independent of the antecedent A. Mastery of these structures is the foundational logical literacy required for philosophical success.
Year 8 哲学的核心在于区分好论证与坏论证的能力。演绎有效性定理指出，一个论证在逻辑上是有效的，当且仅当其所有前提为真而结论为假的情况是不可能的。“真理性”关乎现实世界；“有效性”关乎逻辑结构。一个经典的有效形式是“肯定前件”（Modus Ponens）：如果A则B，A为真，因此B必定为真。一个无效的演绎论证则会犯下形式谬误，例如“肯定后件”（Affirming the Consequent）：如果A则B，B为真，因此A为真。这个公式之所以失败，是因为后件B可能基于完全独立于前件A的原因而为真。掌握这些结构是获得哲学成就所需的基本逻辑素养。
Argument A is deductively valid ↔ ¬(Premises all True ∧ Conclusion False). Soundness = Validity + True Premises.
一个论证A是演绎有效的，当且仅当并非（所有前提为真且结论为假）。可靠性 = 有效性 + 前提真实。
Published by TutorHao | Philosophy Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图