📚 GCSE Edexcel Philosophy: Quick Reference Handbook of Formulas and Theorems | GCSE Edexcel 哲学：公式定理速查手册
In GCSE Edexcel Philosophy, ‘formulas and theorems’ are not equations with x and y but the core logical structures, ethical decision-making models, and argument patterns that philosophers use to build and criticise ideas. This handbook distils the most important ethical theories, arguments for God’s existence, responses to the problem of evil, and principles of logic into clear, easy-to-memorise formulations. You can think of each entry as a blueprint for constructing or deconstructing philosophical arguments under exam conditions.
在GCSE Edexcel 哲学中，“公式与定理”并非带有x和y的数学方程，而是哲学家用来构建与批判思想的核心逻辑结构、伦理决策模型和论证模式。这本速查手册将最重要的伦理理论、上帝存在论证、对恶的问题的回应以及逻辑原则提炼为清晰、易记的公式。你可以将每个条目视为考试条件下构建或解构哲学论证的蓝图。
1. Utilitarianism: The Hedonic Calculus | 功利主义：快乐计算法则
Jeremy Bentham’s act utilitarianism proposes that the moral worth of an action is determined by its total utility: the sum of pleasure minus pain across all sentient beings. The hedonic calculus is a formula for measuring pleasure and pain according to seven criteria.
杰里米·边沁的行为功利主义提出，一个行为的道德价值取决于其总效用：所有有感知力的生命体的快乐减痛苦之和。快乐计算法是根据七个标准衡量快乐与痛苦的公式。
- Intensity – How strong is the pleasure or pain? | 强度 – 快乐或痛苦有多强烈？
- Duration – How long will it last? | 持续时间 – 会持续多久？
- Certainty – How likely is it to occur? | 确定性 – 发生的可能性有多大？
- Propinquity – How soon will it be experienced? | 临近性 – 多久之后会被体验到？
- Fecundity – Will it lead to further pleasures? | 繁衍性 – 会带来更多快乐吗？
- Purity – Will it be followed by pain of the same kind? | 纯度 – 会伴随同类的痛苦吗？
- Extent – How many people are affected? | 广度 – 影响多少人？
Total Utility = Σ (Intensity × Duration × Certainty × Propinquity × Fecundity × Purity) for each person, summed across extent
The action that yields the greatest net pleasure for the greatest number is the morally right one. In an exam, always apply the formula: identify stakeholders, estimate pleasure and pain scores, and calculate the balance.
产生最大净快乐、覆盖最多人数的行为就是道德上正确的。在考试中，始终运用此公式：识别利益相关方，估算快乐与痛苦得分，计算净平衡。
2. Kantian Ethics: The Categorical Imperative Formula | 康德伦理学：绝对命令公式
Immanuel Kant’s deontological theory hinges on the categorical imperative, a universal moral law derived from reason. There are three main formulations, but the most exam-relevant is the Formula of Universal Law.
伊曼努尔·康德的义务论伦理以绝对命令为核心，这是一种由理性推导出的普遍道德法则。主要有三种表述方式，但考试最相关的是普遍法则公式。
Universal Law Formula: ‘Act only according to that maxim whereby you can at the same time will that it should become a universal law.’
The logical test: 1) Formulate the maxim of your action (e.g., ‘I will lie to get money’). 2) Imagine a world where everyone follows this maxim. 3) Check for a contradiction in conception (the maxim becomes impossible) or a contradiction in the will (you could not rationally will such a world). If a contradiction arises, the action is morally impermissible.
逻辑检验：1) 形成你行为的准则（如“我将通过说谎获取金钱”）。2) 想象一个所有人都遵循此准则的世界。3) 检查是否存在构想中的矛盾（该准则变得不可能）或意志中的矛盾（你无法理性地意愿这样一个世界）。如果出现矛盾，该行为在道德上就是不允许的。
Another crucial formula is the Formula of 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.’ This is a powerful shortcut for evaluating business, medical and personal relationships.
另一重要公式是目的公式：“你要如此行动，即无论是你自身中的人性还是他人身上的人性，你始终将其同时作为目的，而绝不仅仅作为手段。”这是评估商业、医疗和人际关系的强大捷径。
3. Natural Law: Primary Precepts and the Synderesis Rule | 自然法：首要诫命与心灵趋向规则
Thomas Aquinas’s Natural Law theory flows from the synderesis rule: ‘Do good and avoid evil.’ From this, reason deduces five primary precepts that reflect God’s plan for human flourishing. The formula for moral action is to act in accordance with these precepts and never directly against them.
托马斯·阿奎那的自然法理论源自心灵趋向规则：“行善避恶”。由此，理性推断出五条反映上帝对人类繁荣计划的首要诫命。道德行为的公式是：按照这些诫命行动，绝不直接违背它们。
| Primary Precept (首要诫命) | Natural Inclination (自然倾向) |
| Preserve life (保护生命) | Self-preservation (自我保存) |
| Reproduce (繁衍后代) | Sexual and nurturing inclination (性与养育倾向) |
| Educate the young (教育后代) | Transmission of knowledge (知识传递) |
| Live in an ordered society (在社会中有序生活) | Social and political inclination (社会与政治倾向) |
| Worship God (敬拜上帝) | Search for ultimate truth and meaning (追寻终极真理) |
The double effect formula: an action with both a good and a bad effect is permissible only if the good effect is intended and the bad effect is merely foreseen but not intended, the action itself is good or neutral, and there is a proportionate reason for allowing the bad effect.
双重效果公式：一个同时产生好效果和坏效果的行为，只有在好效果是被意图的、坏效果仅被预见而非意图、行为本身是好的或中性的、且允许坏效果有相称理由时，才可被允许。
4. Situation Ethics: The Agape Calculus | 境遇伦理：圣爱计算
Joseph Fletcher’s Situation Ethics rejects both legalism and antinomianism, replacing them with the single absolute norm of agape (unconditional Christian love). The working formula is: ‘Love and do what you will,’ but this love is not sentimentalism; it is a calculating, justice-seeking love that always asks what serves the neighbour best.
约瑟夫·弗莱彻的境遇伦理拒绝律法主义与反律法主义，以圣爱（无条件基督教之爱）作为唯一的绝对规范。其运作公式是：“心怀爱心，为所欲为”，但这种爱并非情感主义，而是一种精于计算、寻求正义的爱，始终追问什么最有利于邻舍。
Moral action X is right ⇔ X serves agape in the concrete situation
The four working principles clarify the calculation: pragmatism (it must work), relativism (no fixed rules except love), positivism (love is chosen by faith), and personalism (persons come first, not things). In application, the decision-maker must weigh how each option maximises agape for everyone involved, even if it means breaking conventional moral rules.
四项运作原则厘清此计算：实用主义（必须有效）、相对主义（除爱之外无固定规则）、实证主义（爱因信心而被选择）、人格主义（人为先，物为次）。在应用中，决策者必须权衡每个选项如何为所有相关人员最大化圣爱，即使这意味着打破常规道德规则。
5. The Design Argument: Paley’s Inference Formula | 设计论证：佩利的推论公式
The teleological argument (from telos, purpose) infers a designer from the apparent order and complexity of the universe. William Paley’s watchmaker analogy provides the classic formula.
目的论证（源自 telos，目的）从宇宙表面上的秩序与复杂性推断一位设计者。威廉·佩利的手表匠类比提供了经典公式。
P1: Complex objects with parts arranged for a purpose (like a watch) imply a designer.
P2: The natural world (e.g., the eye) exhibits even greater complexity and purpose than a watch.
C: Therefore, the natural world implies a supremely intelligent Designer (God).
The argument is inductive and analogical. Hume’s criticisms provide logical counter-formulas: the analogy is weak (universe is not like a machine; perhaps more like a vegetable), the universe might be an accidental combination of matter, and even if a designer exists, the argument cannot prove the God of classical theism (tri-omni).
该论证是归纳类比。休谟的批判提供了逻辑上的反公式：类比薄弱（宇宙不像机器，可能更像植物），宇宙可能是物质偶然组合的结果，且即使存在设计者，该论证也无法证明古典有神论的上帝（全善全能全知）。
6. The Cosmological Argument: First Cause Formula | 宇宙论论证：第一因公式
The cosmological argument moves from the existence of the universe to a first cause or necessary being. Aquinas’s Third Way is a dense logical formula.
宇宙论论证从宇宙的存在推导出第一因或必然存在。阿奎那的第三论证是一个严密的逻辑公式。
P1: Contingent beings exist (they come into being and pass away).
P2: If all beings were contingent, at some time nothing would have existed, and then nothing would exist now (because nothing comes from nothing).
P3: Things exist now.
C: Therefore, there must be a necessary being (God) causing and sustaining all contingent beings.
The Kalam version offers a simpler formula: Everything that begins to exist has a cause; the universe began to exist; therefore, the universe has a cause (God). The formula is highly examinable: always state the premise-conclusion structure and be ready to evaluate whether the first cause must be God, or if an infinite regression is impossible.
卡拉姆版本提供了一个更简明的公式：一切开始存在的事物都有其原因；宇宙开始存在；因此，宇宙有其原因（上帝）。此公式常出现在考试中：务必陈述前提-结论结构，并准备好评估第一因是否必须是上帝，或无限倒退是否不可能。
7. The Problem of Evil: Logical Inconsistency Formula | 恶的问题：逻辑矛盾公式
The problem of evil challenges the coherence of classical theism. J.L. Mackie formulated the logical version as an inconsistent triad.
恶的问题挑战古典有神论的一致性。J.L.麦基将逻辑版本表述为不一致的三元组。
God is omnipotent (all-powerful).
God is omnibenevolent (all-good).
Evil exists.
If any two are true, the third must be false, generating the atheist’s formula: Because evil exists, either God is not all-powerful, not all-good, or does not exist. The free will defence is the most celebrated theistic rebuttal, with its own formula: Evil is the necessary price for morally significant free will; a world with free agents who sometimes choose evil is better than a world of automata; therefore, God is justified in permitting evil.
如果任意两个为真，第三个必然为假，从而得出无神论公式：因为恶存在，所以上帝要么非全能，要么非全善，要么不存在。自由意志辩护是最著名的有神论回应，有其自身的公式：恶是为拥有道德意义自由意志所必须付出的代价；一个拥有有时选择作恶的自由行动者的世界，比一个机械人的世界更好；因此，上帝允许恶存在是合理的。
Evidential evil introduces a probabilistic formula: The sheer amount and distribution of pointless suffering makes God’s existence unlikely. The soul-making theodicy (Hick) counters with the formula: Adversity enables moral and spiritual growth that a paradise could not provide.
检验性恶引入了概率公式：无谓苦难的巨大数量和分布使得上帝的存在不太可能。灵魂塑造神义论（希克）则以如下公式回应：逆境促发道德和灵性成长，而乐园无法提供此等成长。
8. Religious Experience: The Principle of Credulity Formula | 宗教经验：轻信原则公式
In religious epistemology, Richard Swinburne argues that religious experiences should be taken as prima facie evidence for God’s existence. His principle of credulity is a rational formula: In the absence of defeating conditions, we ought to believe that things are as they appear to be.
在宗教认识论中，理查德·斯温伯恩主张，宗教经验应被视为上帝存在的初步证据。他的轻信原则是一个理性公式：在不存在否决条件的情况下，我们应该相信事物如其所呈现。
If S has an experience of X, and there is no reason to think S is deluded or the experience deceptive, then it is rational for S to believe that X exists.
Applied to theistic experiences: If a person seems to experience God, and we have no specific defeater (e.g., intoxication, known hallucination), then it is rational to believe God is the cause. The formula is strengthened by the principle of testimony: we should believe what others tell us unless we have positive reasons to doubt them. Critics offer a reductive formula: Religious experiences are caused by neurophysiological states, not supernatural reality.
应用于有神论经验：如果一个人似乎体验到了上帝，且我们无特定的否决条件（如醉酒、已知幻觉），那么相信上帝是其原因是理性的。该公式因证言原则而加强：我们应该相信别人告诉我们的，除非我们有正面理由怀疑他们。批评者则提出还原论公式：宗教经验由神经生理状态引起，而非超自然实在。
9. Logical Fallacies: Error Detection Formulas | 逻辑谬误：错误识别公式
Philosophical arguments often go wrong because of logical fallacies. Recognising these patterns is essential for evaluation questions. Each fallacy can be expressed as an invalid argument formula.
哲学论证常因逻辑谬误而出错。识别这些模式对评估题至关重要。每个谬误都可表达为一个无效论证公式。
| Fallacy (谬误) | Formula (公式) |
| Ad hominem (人身攻击) | Person A makes claim X; Person B attacks A’s character; Therefore X is false. (Invalid) |
| Straw man (稻草人) | Misrepresent X as a weaker version Y; Refute Y; Therefore X is refuted. (Invalid) |
| Begging the question (循环论证) | Premises assume the truth of the conclusion. (Circular) |
| False dilemma (非此即彼) | Only two options A or B are presented, when other options exist; Eliminate A; Therefore B. (Invalid if options incomplete) |
| Post hoc ergo propter hoc (假因) | Y happened after X; Therefore X caused Y. (Invalid; correlation is not causation) |
In ethical debates, also watch for the naturalistic fallacy formula: Deriving ‘ought’ from ‘is’ without justification. For instance, ‘sex produces babies; therefore sex should only be for procreation’ moves from a descriptive fact to a moral prescription without a linking principle.
在伦理辩论中，也要警惕自然主义谬误公式：在没有论证的情况下从“是”推出“应当”。例如，“性行为导致生育；因此性只应用于生殖”在缺乏连接原则的情况下从描述性事实跳跃到道德规定。
10. Deductive and Inductive Argument Forms | 演绎与归纳论证形式
GCSE Philosophy requires you to identify and assess the structure of arguments. Two fundamental formulas distinguish deduction from induction.
GCSE 哲学要求你识别并评估论证结构。两个基本公式区分了演绎与归纳。
Deductive Validity: If premises are true, the conclusion must be true.
Modus Ponens: P → Q ; P ; ∴ Q
Example: If God exists, objective moral values exist. God exists. Therefore, objective moral values exist. This is valid in form; the debate shifts to the truth of the premises.
例子：如果上帝存在，客观道德价值就存在。上帝存在。因此，客观道德价值存在。形式上有效；争论转向前提的真假。
Inductive Strength: If premises are true, the conclusion is probable.
Analogical Formula: A has properties p,q,r; B has properties p,q; Therefore B probably has r.
The design argument works like this: Watch has p (complexity), q (purpose), r (designer); eye has p (complexity), q (purpose); therefore eye probably has r (designer). In evaluation, always ask whether the sample is representative, and whether the analogy holds under scrutiny.
设计论证以此运作：手表拥有 p（复杂性）、q（目的）、r（设计者）；眼睛拥有 p（复杂性）、q（目的）；因此眼睛很可能拥有 r（设计者）。在评估中，始终追问样本是否具有代表性，且类比在仔细审视下是否成立。
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