📚 Year 13 AQA Philosophy: Formula & Theorem Quick Reference | 13年级AQA哲学：公式定理速查手册
This quick reference handbook converts the major arguments, principles, and conceptual models from the Year 13 AQA Philosophy specification into a set of precise, memorable ‘formulas’ and ‘theorems’. It covers moral philosophy, epistemology, metaphysics of mind, and the philosophy of religion. Each entry is designed to help you analyse and evaluate philosophical reasoning with maximum clarity under exam conditions.
这本速查手册将AQA 13年级哲学课程中的主要论证、原则和概念模型转化为一套精确易记的“公式”与“定理”。内容涵盖道德哲学、认识论、心灵形而上学和宗教哲学。每一个条目都旨在帮助你在考试中以最清晰的思路分析和评估哲学推理。
1. Utilitarian Calculus | 功利主义计算
The principle of utility measures the moral worth of an action by its tendency to produce happiness. Jeremy Bentham’s felicific calculus proposes that we can, in theory, quantify pleasure and pain along several dimensions.
功利原则根据行为产生幸福的倾向来衡量其道德价值。边沁的苦乐计算提出，理论上我们可以从若干维度对快乐与痛苦加以量化。
Happiness = ∑Pleasure − ∑Pain
For each affected individual, calculate the net balance of pleasure over pain. The right action is the one that maximises the total net happiness across all sentient beings.
对每个受影响的个体，计算快乐超出痛苦的净值。正确的行为就是能在所有有感知的存在中最大化总净幸福的行为。
Mill modifies this with a distinction between higher and lower pleasures. The formula remains structurally identical, but the quality of pleasures adds a weighting factor: Utility = ∑(Quality × Quantity of Pleasure) − ∑Pain. Critics argue that interpersonal utility comparison is practically impossible.
密尔通过区分高级快乐和低级快乐修正了这一公式。结构上公式保持不变，但快乐的品质增加了一个权重因子：效用 = ∑(快乐的品质 × 数量) − ∑痛苦。批评者指出，人际效用比较在实践上是不可能的。
2. Categorical Imperative Formulations | 定言令式的公式化表述
Kant’s supreme moral principle is a command of reason that applies universally and unconditionally. Its three main formulations are logically equivalent, and each can be expressed in a quasi‑algorithmic form to test maxims.
康德的最高道德原则是理性的命令，普遍且无条件地适用。它的三个主要表述在逻辑上是等价的，每一个都能以准算法的形式检验准则。
Formula of Universal Law: Act only on that maxim through which you can at the same time will that it should become a universal law.
Test: (1) Formulate your subjective principle of action as a maxim. (2) Universalise it. (3) Check for contradiction in conception or contradiction in will. If a contradiction arises, the action is impermissible.
检验步骤：(1) 将你的主观行动原则表述为一条准则。(2) 将其普遍化。(3) 检查是否存在构想上的矛盾或意愿上的矛盾。若产生矛盾，该行为就是不允许的。
Formula of Humanity: Act so that you use humanity, whether in your own person or in the person of another, always at the same time as an end, never merely as a means.
The structural constraint is: treat rational agency as an unconditional value. Any maxim that treats a rational being merely as a tool for achieving a separate goal violates this existential formula.
结构性的约束是：将理性的能动性视为无条件价值。任何仅仅把理性存在者当作实现其他目标的工具的准则，都违反了这一存在论公式。
3. Virtue as the Mean | 美德即中道
Aristotle’s ethical theory analyses virtue as a disposition located between two opposing vices: excess and deficiency. The moral agent must identify the appropriate response relative to the situation.
亚里士多德的伦理理论将美德分析为一种位于两个对立恶习——过度与不及——之间的品质。道德主体必须针对具体情况确定适当的回应。
Virtue = Mean between Excess and Deficiency
The mean is not an arithmetic midpoint but a contextual balance determined by practical wisdom (phronesis). For instance, courage is the mean between cowardice (deficiency in confidence) and rashness (excess in confidence).
中道不是算术中点，而是由实践智慧（phronesis）确定的、依情境而定的平衡。例如，勇敢是怯懦（自信不足）与鲁莽（自信过度）之间的中道。
In action‑guidance terms, the formula requires: (a) identify the relevant sphere of feeling or action, (b) determine the two extremes, and (c) aim for the intermediate disposition that a person with practical wisdom would choose.
在行动指导方面，这个公式要求：(a) 确定相关的感受或行动领域，(b) 确定两个极端，(c) 以有实践智慧的人所会选择的中庸品质为目标。
4. Descartes’ Clear and Distinct Criterion | 笛卡尔的清楚分明准则
Descartes builds his rationalist epistemology on the foundational certainty of the cogito. From this he derives a general rule: whatever is perceived clearly and distinctly must be true.
笛卡尔将他的理性主义认识论建立在“我思”这一基础确定性之上。由此他得出一个普遍规则：凡是清楚分明地感知到的东西，必定为真。
Clear & Distinct Idea = True
The reliability of this criterion is guaranteed by a non‑deceiving God. The reasoning follows a tight structure: I exist as a thinking thing → I have the idea of God as a perfect being → God’s perfection entails that He is not a deceiver → therefore, my cognitive faculties, when used properly, cannot systematically mislead me.
这个准则的可靠性由一位不欺骗的上帝来保证。推理遵循一个严密的结构：我作为一个思考之物存在 → 我拥有上帝作为完美存在的观念 → 上帝的完美意味着他不是骗子 → 因此，我的认知能力在正确使用时不可能系统地使我出错。
The Cartesian Circle objection threatens this formula: Descartes relies on clear and distinct perception to prove God’s existence, then uses God’s veracity to guarantee clear and distinct perception. This circularity is a key evaluative point for exam essays.
笛卡尔循环的异议威胁着这个公式：笛卡尔依靠清楚分明的感知来证明上帝存在，然后又用上帝的诚实来保证清楚分明的感知。这种循环性是考试论文中的一个关键评估点。
5. The JTB+ Analysis of Knowledge | JTB+ 知识分析公式
The traditional analysis defines knowledge as justified true belief. Gettier’s counter‑examples show that JTB is not sufficient, prompting the search for an additional condition X that blocks the luck involved in Gettier‑style cases.
传统分析将知识定义为得到证成的真信念。盖梯尔的反例表明JTB并不充分，这促使人们寻找一个额外的条件X，用以阻断盖梯尔式案例中的运气因素。
Knowledge = Justified True Belief + X
One prominent candidate for X is the ‘no false lemmas’ condition: the justification must not depend on any false intermediate belief. Thus Knowledge = Justified True Belief that is not inferred from any falsehood.
一个突出的X候选条件是“无错误引理”条件：证成不能依赖于任何错误的中介信念。因此，知识 = 未从任何谬误中推论出的被证成的真信念。
Reliabilism offers a different formula: Knowledge = True Belief produced by a reliable cognitive process. In terms of conditional probability, for a belief B produced by process R, if P(B is true | R produced B) is sufficiently high, the belief counts as knowledge.
可靠主义提供了另一种公式：知识 = 由可靠的认知过程产生的真信念。从条件概率的角度看，对于一个由过程R产生的信念B，如果 P(B为真 | R产生了B) 足够高，则该信念就算作知识。
6. Bayesian Confirmation & the Problem of Induction | 贝叶斯确证与归纳问题
Hume’s problem of induction challenges the rationality of using past regularities to predict future events. A Bayesian framework expresses the degree of belief in a hypothesis H given evidence E, but it still assumes the uniformity of nature as a prior.
休谟的归纳问题挑战了运用过往规律预测未来事件的合理性。贝叶斯框架表达了给定证据E时对假说H的相信程度，但它仍然预设了自然齐一性作为先验。
P(H|E) = P(E|H) × P(H) / P(E)
Where P(H|E) is posterior probability, P(E|H) is likelihood, P(H) is prior probability, and P(E) is the marginal likelihood of the evidence. The formula shows how evidence rationally updates our confidence in a hypothesis.
其中P(H|E)是后验概率，P(E|H)是似然度，P(H)是先验概率，P(E)是证据的边际似然度。该公式表明证据如何理性地更新我们对假说的置信度。
Critics point out that the prior probability P(H) can be arbitrary, and the formula itself does not justify the assumption that future evidence will resemble past evidence in the same likelihood structure.
批评者指出，先验概率P(H)可能是武断的，并且该公式本身并未证成那个假定——未来的证据在同样的似然结构中将与过去的证据相似。
7. Functionalism: The Input‑Output Formula | 功能主义：输入‑输出公式
Functionalism identifies mental states by their causal roles in mediating between sensory inputs, other mental states, and behavioural outputs. Pain, for example, is not defined by its physical realisation but by its functional profile.
功能主义通过心理状态在感觉输入、其他心理状态和行为输出之间的因果作用来定义它们。例如，疼痛不是由其物理实现来界定的，而是由其功能特性定义的。
Mental State M = f(Input, Other Mental States, Output)
The formula captures the three‑place relation: given a set of sensory stimuli I and a set of current mental states S, M is the state that tends to produce behavioural output O and transitions to further states S’. The inverted qualia argument challenges whether this purely relational character captures the subjective feel of experience.
这个公式捕捉了三者关系：给定一组感觉刺激I和一组当前的心理状态S，M就是倾向于产生行为输出O并过渡到后续状态S’的那种状态。颠倒感受质的论证挑战了这种纯粹关系特性是否能捕捉经验的主观感受。
Machine functionalism extends this to computational processing: Mind as Software, Brain as Hardware. Multiple‑realisability is guaranteed because the same functional organisation can run on different physical substrates, but problems like the knowledge argument (Mary’s room) suggest that facts about experience escape the formula.
机器功能主义将此扩展到计算加工：心灵如同软件，大脑如同硬件。多重可实现性是有保障的，因为同样的功能组织可以运行在不同的物理基质上，但是像知识论证（玛丽的黑白房间）这样的问题表明，关于经验的事实逃脱了这个公式的捕捉。
8. Ontological Argument Formalisation | 本体论论证的形式化
Anselm’s ontological argument defines God as “that than which nothing greater can be conceived.” The logical form of the argument can be reduced to the claim that existence is a perfection, and a being that exists in reality is greater than one that exists only in the understanding.
安瑟伦的本体论论证将上帝定义为“无法设想有比之更伟大的存在者”。该论证的逻辑形式可以归结为：存在是一种完美，而在现实中存在的存在者要比仅存在于理解中的存在者更伟大。
God = the being with all perfections
Existence is a perfection. ∴ God exists.
Descartes’ version strengthens the link: essence implies existence in the case of a supremely perfect being, just as the essence of a triangle implies that its internal angles sum to 180 degrees. Essence of God ⊃ Existence.
笛卡尔的版本加强了这一联系：在一个绝对完美存在者的情形中，本质包含实存，正如三角形的本质包含其内角和为180度一样。上帝的本质必然蕴含实存。
Gaunilo’s perfect island parody and Kant’s objection that existence is not a real predicate challenge the logical steps. Kant’s formula: ‘Being is obviously not a real predicate; that is, it is not a concept of something that could be added to the concept of a thing.’
高尼罗的完美岛屿类比和康德关于“存在不是实在谓词”的异议挑战了这些逻辑步骤。康德的公式：“存在显然不是一个实在的谓词；也就是说，它不是一个可以加在事物概念之上的某物的概念。”
9. Design Argument: Likelihood Ratio | 设计论证：似然比
The teleological argument infers an intelligent designer from the appearance of order, complexity, and purpose in the universe. The Bayesian or likelihood version formalises it as a comparison between two competing hypotheses: design and chance.
目的论论证从宇宙中秩序、复杂性和目的的表象推论出一位智能设计者。贝叶斯或似然性版本将其形式化为两种对立假说——设计与偶然——之间的比较。
Likelihood Ratio = P(Ordered Universe | Design) / P(Ordered Universe | Chance)
If this ratio is significantly greater than 1, the evidence strongly favours design. Paley’s watchmaker analogy makes the intuition vivid: a watch found on a heath demands a watchmaker. The formula clarifies the inferential structure.
如果这个比值显著大于1，证据就强有力地支持设计假说。佩利的钟表匠类比生动地表达了这一直觉：在荒野上发现的一块表，必然要求有一位钟表匠。这个公式澄清了推理结构。
Hume’s objections in the Dialogues attack both the likelihood assessment and the uniqueness of the designer. The evil in the world can be factored into the likelihood, reducing the ratio. Moreover, the inference to a single, perfect designer goes beyond what the evidence licenses.
休谟在《自然宗教对话录》中的反对意见同时攻击了似然评估和设计者的独特性。世界上的恶可以被纳入似然计算，从而降低比值。此外，推论出单一的、完美的设计者超出了证据所许可的范围。
10. Logical Problem of Evil: The Inconsistent Triad | 恶的逻辑问题：不一致三元组
The logical problem of evil argues that the existence of an omnipotent, omnibenevolent God is logically incompatible with the existence of evil. The argument is often presented as an inconsistent triad of propositions that cannot all be true.
恶的逻辑问题主张，一位全能、全善的上帝的存在与恶的存在在逻辑上是不相容的。该论证常被呈现为一个命题集的不一致三元组，它们不可能同时为真。
1. God is omnipotent
2. God is omnibenevolent
3. Evil exists
The formal contradiction emerges: if God is all‑powerful and all‑loving, He would have both the ability and the desire to eliminate all evil; yet evil exists. Therefore, such a God does not exist. J. L. Mackie’s formulation adds the premise that a good thing always eliminates evil as far as it can.
形式矛盾随之显现：如果上帝是全能和全爱的，他既有能力也有意愿消除一切恶；但恶仍然存在。因此，这样的上帝不存在。麦凯的表述增加了一个前提：一个好的事物总是会尽其所能地消除恶。
The free‑will defence modifies the triad by arguing that a world with morally significant free will that sometimes leads to evil is a greater good than a world without such freedom. The logical consistency is restored if it is shown that God cannot actualise a world with free creatures who never choose evil.
自由意志辩护通过论证一个拥有道德意义上重要的自由、且这种自由有时导致恶的世界，比一个没有这种自由的世界更善，从而修改了三元组。如果能证明上帝无法实现一个拥有自由但从不选择恶的受造物的世界，逻辑一致性就得以恢复。
11. Libertarian Free Will: Consequence Argument | 自由意志论：后果论证
In the metaphysics of mind and ethics, the consequence argument captures the incompatibilist intuition that if determinism is true, we lack free will and moral responsibility. The argument formalises the idea that we have no power over the past or the laws of nature.
在心灵形而上学和伦理学中，后果论证捕捉了不相容论的直觉：如果决定论为真，我们就缺乏自由意志和道德责任。该论证将我们无法掌控过去和自然律这一观念加以形式化。
No one has power over the facts of the remote past and the laws of nature.
Our present actions are necessary consequences of those facts and laws.
∴ No one has power over their present actions.
Peter van Inwagen’s modal version uses the transfer of powerlessness principle: if no one has power over P, and no one has power over the fact that P entails Q, then no one has power over Q. This deductive structure supports the claim that free will and determinism are incompatible.
彼得·范·因瓦根的模态版本运用了无力性传递原则：如果无人对P有掌控力，也无人对P蕴含Q这个事实有掌控力，那么无人对Q有掌控力。这一演绎结构支持了自由意志与决定论不相容的主张。
12. Property Dualism & the Zombie Argument | 属性二元论与僵尸论证
Property dualism holds that mental properties are non‑physical and irreducible to physical properties, even though they emerge from a physical substrate. The zombie argument demonstrates this through a conceivable scenario.
属性二元论主张，心理属性是非物理的、不能还原为物理属性的，尽管它们是在物理基质上涌现的。僵尸论证通过一个可设想的场景展示了这一点。
Conceivable → Metaphysically Possible → Physicalism is False
The argument runs: (1) A philosophical zombie – a being physically identical to a conscious human but lacking any phenomenal consciousness – is conceivable. (2) If zombies are conceivable, they are metaphysically possible. (3) If zombies are metaphysically possible, conscious states involve something over and above the physical. Therefore, physicalism is false.
论证如下：(1) 哲学僵尸——一个在物理层面与有意识的人类完全相同、却没有任何现象意识的生物——是可设想的。(2) 如果僵尸是可设想的，那么它们在形而上学上是可能的。(3) 如果僵尸在形而上学上是可能的，那么意识状态就包含了超出物理层面的东西。因此，物理主义为假。
Physicalists reply by challenging the move from conceivability to possibility, or by arguing that phenomenal concepts are conceptually distinct from physical concepts but refer to the same property – a strategy akin to the identification of water with H₂O.
物理主义者的回应是挑战从可设想到可能性的过渡，或者主张现象概念虽然在概念上有别于物理概念，但指的是同一个属性——这一策略类似于将水等同于H₂O的识别。
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