📚 Year 13 WJEC Philosophy: Quick-Reference Handbook of Formulae and Theorems | Year 13 WJEC 哲学：公式定理速查手册
In WJEC Philosophy, the term ‘formulae and theorems’ may sound alien, yet the subject is built on repeatable patterns of reasoning, foundational principles and logical structures that function much like formal tools in mathematics. This concise handbook gathers the most essential argument forms, ethical imperatives, epistemological criteria and metaphysical principles you need to recognise, apply and evaluate at Year 13. Each entry is presented as a theorem-like statement, unpacked with its logical skeleton and philosophical relevance, helping you move from memorisation to mastery.
在 WJEC 哲学中，“公式与定理”这个说法听起来或许陌生，但整个学科正是建立在可重复的推理模式、基础性原则和逻辑结构之上，它们的作用很像数学中的形式工具。这份简明手册汇集了你在 Year 13 需要识别、运用和评估的最重要的论证形式、伦理命令、认识论标准和形而上学原则。每一个条目都像一条定理般呈现，拆解其逻辑骨架并阐明哲学意义，帮助你从记忆走向精通。
1. Modus Ponens | 肯定前件式
p → q
p
∴ q
Modus ponens is the most straightforward valid argument form: if a conditional statement is accepted and its antecedent holds, the consequent must follow. In philosophical essays, you will encounter this pattern whenever a thinker reasons from a general principle to a specific conclusion. Recognising it allows you to check whether the conditional premise is actually defended, and whether the antecedent is genuinely established by the evidence offered.
肯定前件式是最直接的有效论证形式：如果一个条件句被接受，并且它的前件成立，那么后件必然推出。在哲学论文中，每当思想家从一条普遍原则推出具体结论时，你都会遇见这个模式。识别它能让你检查条件前提是否真的得到了辩护，以及前件是否确实被给出的证据所确立。
Example: ‘If the mind is a thinking substance (p → q), and the mind is indivisible (p), then the mind is not material (q).’ This structure appears in Descartes’ Meditations.
例子：“如果心灵是思维实体（p → q），并且心灵不可分（p），那么心灵不是物质性的（q）。”这一结构出现在笛卡尔的《沉思录》中。
2. Modus Tollens | 否定后件式
p → q
¬q
∴ ¬p
Modus tollens denies the consequent of a conditional to deny its antecedent. This form is the engine of falsification in science and of reductio arguments in philosophy. When a prediction fails, we take that failure as a reason to reject the hypothesis that implied it. In ethical and epistemological debates, modus tollens often underlies criticism: if a theory entails an implausible consequence, we reject the theory.
否定后件式通过否定条件句的后件来否定前件。这个形式是科学中证伪以及哲学中归谬论证的引擎。当一个预测失败时，我们把该失败作为拒绝蕴含该预测的假说的理由。在伦理学和认识论辩论中，否定后件式常常构成批评的基础：如果一个理论蕴涵一个不可接受的推论，我们就拒绝该理论。
Example: ‘If all knowledge comes from sense experience (p → q), then we would have innate knowledge of nothing (q). But we do seem to have innate logical principles (¬q). Therefore, not all knowledge comes from sense experience (¬p).’
例子：“如果一切知识都来自感官经验（p → q），那么我们不会有任何天赋知识（q）。但我们似乎确实有天赋的逻辑原则（¬q）。因此，并非所有知识都来自感官经验（¬p）。”
3. Hypothetical Syllogism | 假言三段论
p → q
q → r
∴ p → r
Hypothetical syllogism chains conditionals, allowing philosophers to build extended lines of reasoning. When a thinker argues that one thesis logically commits you to another, and that second to a third, they are using this form. It is especially valuable in ethics, where a basic principle (e.g., ‘pleasure is the only intrinsic good’) is linked to practical judgments about complex situations through a series of conditional steps.
假言三段论将条件句链接起来，使哲学家能够构建延伸的推理链条。当一位思想家论证说，一个论题在逻辑上使你承诺另一个论题，而第二个又承诺第三个时，他们就是在使用这个形式。这在伦理学中尤其有价值，在那里一条基本准则（例如“快乐是唯一的内在善”）通过一系列条件步骤与关于复杂情境的实践判断相连。
Example: ‘If pleasure is the only good (p → q), then the right action maximises pleasure (q → r). Therefore, if pleasure is the only good (p), then the right action maximises pleasure (r).’
例子：“如果快乐是唯一的善（p → q），那么正当的行为就是最大化快乐的行为（q → r）。因此，如果快乐是唯一的善（p），那么正当的行为就是最大化快乐的行为（r）。”
4. Disjunctive Syllogism | 选言三段论
p ∨ q
¬p
∴ q
Disjunctive syllogism eliminates one option from an exhaustive disjunction to establish the remaining one. In philosophy, it often appears when a problem is framed as an ‘either/or’ choice, and one horn is shown to be untenable. Care must be taken to ensure the disjunction is genuinely exhaustive—critics frequently challenge a disjunctive syllogism by exposing a hidden third possibility.
选言三段论从一个穷尽的选言判断中排除一个选项，从而确立剩下的那个。在哲学中，当一个难题被框定为“非此即彼”的选择，并且其中一角被证明站不住脚时，就经常出现这种形式。必须小心确保选言确实是穷尽的——批评者常常通过揭露一个隐藏的第三种可能性来挑战选言三段论。
Example: ‘Either moral properties are natural properties or they are non-natural properties (p ∨ q). They cannot be reduced to natural properties (¬p). Therefore, moral properties are non-natural (q).’
例子：“道德属性要么是自然属性，要么是非自然属性（p ∨ q）。道德属性不能还原为自然属性（¬p）。因此，道德属性是非自然属性（q）。”
5. Categorical Imperative (First Formulation) | 定言命令（第一公式）
Act only according to that maxim whereby you can at the same time will that it should become a universal law.
Kant’s formula of universal law functions as a decision procedure for moral permissibility. To test a maxim, universalise it: imagine a world where everyone follows it. If the maxim leads to a contradiction in conception or a contradiction in the will, the act is forbidden. This formula is not a deductive theorem in logic but a practical principle that underpins deontological ethics, demanding consistency and respect for rational agency.
康德的普遍法则公式充当道德许可性的判定程序。检验一条准则时，将其普遍化：想象一个所有人都遵循它的世界。如果该准则导致概念上的矛盾或意愿上的矛盾，该行为就是被禁止的。这条公式并非逻辑中的演绎定理，而是支撑义务论伦理学的实践原则，要求一致性和对理性能动性的尊重。
Example: Testing the maxim ‘make a false promise to get money’ leads to a contradiction in conception because if everyone made false promises, the institution of promising would collapse, making the promise impossible.
例子：检验准则“做虚假承诺来获得金钱”会导致概念上的矛盾，因为如果人人都做虚假承诺，承诺制度就会崩溃，使得承诺本身变得不可能。
6. Principle of Utility | 功利原则
An action is right if and only if it produces the greatest happiness for the greatest number.
Bentham’s and Mill’s utility principle serves as the fundamental axiom of classical utilitarianism. It can be treated as a moral algorithm: sum the pleasures and pains for all affected, and the action with the highest net positive balance is obligatory. This formula invites quantitative reasoning and has been formalised in contemporary cost-benefit analysis. Critics target its aggregative nature and its demand to foresee consequences.
边沁和密尔的功利原则是古典功利主义的基本公理。它可以被当作一个道德算法：将所有受影响者的快乐与痛苦加总，净正值余额最高的行为就是义务性的。这个公式引向定量推理，并已在当代成本-收益分析中被形式化。批评者针对它的聚合性质及其要求预见后果这一点展开攻击。
Calculation model: U(A) = Σ (Pleasureᵢ − Painᵢ) for all sentient beings i affected by action A.
计算模型：U(A) = Σ (快乐ᵢ − 痛苦ᵢ)，对所有受行为A影响的有感知存在者i求和。
7. Occam’s Razor | 奥卡姆剃刀
Entities are not to be multiplied beyond necessity.
Occam’s Razor is a methodological principle of parsimony: given two explanations of equal explanatory power, the one that posits fewer entities, assumptions or principles is to be preferred. In philosophy of mind, it is deployed against substance dualism—if mental states can be explained purely in terms of physical processes, why add an extra non-physical mind? The razor does not decide truth, but it shifts the burden of proof onto the more extravagant theory.
奥卡姆剃刀是一条关于简洁性的方法论原则：在解释力相等的两种解释中，应当优先选择预设更少实体、假设或原则的那一种。在心灵哲学中，它被用来反对实体二元论——如果心理状态可以纯粹用物理过程来解释，为什么还要额外添加一个非物理的心灵？剃刀并不决定真理，但它将举证责任转移给更奢侈的理论。
Application: Physicalism about consciousness explains the same data with fewer fundamental categories than dualism, so it is rationally preferable unless dualists can show a need for the extra category.
应用：关于意识的物理主义用比二元论更少的基本范畴解释了同样的数据，因此它在理性上更可取，除非二元论者能证明需要那个额外范畴。
8. Cogito Argument | 我思故我在论证
I think, therefore I am. (Cogito, ergo sum.)
Descartes’ cogito is the foundational theorem of his epistemology—a self-verifying truth that resists even the most radical doubt. The act of doubting one’s own existence confirms the existence of the doubter. In argument form: ‘If I am doubting, then I am thinking. If I am thinking, then I exist. I am doubting. Therefore, I exist.’ This provides the indubitable starting point for his rationalist reconstruction of knowledge.
笛卡尔的我思是其认识论的基础定理——一条自我验证的真理，即便最彻底的怀疑也无法动摇。怀疑自身存在这一行为本身就确证了怀疑者的存在。论证形式为：“如果我在怀疑，那么我在思考。如果我在思考，那么我存在。我在怀疑。因此，我存在。”这为他的理性主义知识重构提供了不可置疑的起点。
Logical form: D → T, T → E, D ∴ E (where D = ‘I doubt’, T = ‘I think’, E = ‘I exist’). This is a chain of modus ponens.
逻辑形式：D → T, T → E, D ∴ E（其中D =“我在怀疑”，T =“我在思考”，E =“我存在”）。这是一系列肯定前件式的链接。
9. Descartes’ Evil Demon Argument | 笛卡尔的恶魔论证
It is possible that an all-powerful demon systematically deceives me about all external-world beliefs.
This sceptical scenario functions as a theorem about fallibility: if we cannot rule out the possibility of systematic deception, then no belief about the external world is immune to doubt. The argument sets an extraordinarily high standard for knowledge—absolute certainty. Descartes introduces it not to endorse scepticism but to clear the ground for the cogito and later the proof of a non-deceiving God, which would guarantee clear and distinct ideas.
这个怀疑论情景起着一条关于可错性定理的作用：如果我们不能排除系统性欺骗的可能性，那么任何关于外部世界的信念就都不免于怀疑。这个论证为知识设定了一个极高的标准——绝对的确定性。笛卡尔引入它不是为了支持怀疑论，而是为“我思”以及随后对非欺骗性上帝（保证清楚分明观念）的证明清理地基。
Structure: If I cannot be certain that no evil demon exists (¬C), then I cannot be certain of any sensory belief (¬S). ¬C, therefore ¬S. The challenge is to break the conditional or deny ¬C.
结构：如果我不能确定不存在邪恶恶魔（¬C），那么我就不能确定任何感官信念（¬S）。¬C，因此¬S。挑战在于打破这个条件句或者否定¬C。
10. Hume’s Fork | 休谟的叉子
All meaningful statements are either relations of ideas or matters of fact.
Hume’s Fork is an epistemological and semantic theorem that divides all objects of human reason into two kinds. Relations of ideas are a priori, necessary, and discovered by thought alone (e.g., mathematics). Matters of fact are a posteriori, contingent, and known through experience. Any purported statement that fits neither category is to be ‘committed to the flames’, for it contains nothing but sophistry and illusion. This principle is the ancestor of logical positivism’s verificationism.
休谟的叉子是一条认识论和语义定理，它将人类理性的全部对象分为两类。观念的关系是先天的、必然的，仅凭思想就能发现（如数学）。事实的事情是后天的、偶然的，通过经验认识。任何声称的陈述，如果不属于这两类，就应当“付之一炬”，因为它只包含诡辩和幻想。这一原则是逻辑实证主义证实说的先驱。
Application: The statement ‘God exists’ is not a relation of ideas (it is not demonstrable a priori) nor a matter of fact (we have no constant experience of God). Under Hume’s Fork, it is cognitively meaningless.
应用：“上帝存在”这个陈述既不是观念的关系（不能先天证明），也不是事实的事情（我们对上帝没有恒常的经验）。在休谟的叉子下，它不具有认知意义。
11. Verification Principle | 证实原则
A statement is cognitively meaningful if and only if it is either analytically true or empirically verifiable.
Developed by the logical positivists, the verification principle sharpens Hume’s Fork into a criterion of meaning. Any claim that cannot be verified by logical analysis or sense experience is literally nonsensical, not false. This principle was wielded against metaphysical, theological and ethical discourse. However, it faced the self-refutation problem: the verification principle itself is neither analytically true nor empirically verifiable, so by its own standard it is meaningless.
由逻辑实证主义者发展的证实原则将休谟的叉子精炼为一个意义标准。任何不能通过逻辑分析或感官经验证实的断言，严格来说都是无意义的，而不是假的。这一原则被用来攻击形而上学、神学和伦理学话语。然而，它面临自我反驳的问题：证实原则本身既不是分析真理，也不能被经验证实，因此按照它自己的标准，它是无意义的。
Formalisation: A sentence S is meaningful iff S is a tautology, or S is verifiable-in-principle by observation. The principle is often rejected for its own lack of verifiability.
形式化：一个句子S具有意义，当且仅当S是一个重言式，或者S原则上可以被观察所证实。这个原则常因其自身缺乏可证实性而被否决。
12. Falsification Principle | 证伪原则
A hypothesis is scientific if and only if it can be falsified by possible observations.
Popper’s falsificationism replaces verification with falsifiability as the demarcation criterion between science and non-science. A theory must make risky predictions that could, in principle, clash with experience. Statements that are compatible with all possible observations—such as ‘all events are determined by fate’—are unfalsifiable and therefore unscientific. This principle is not a logical theorem but a methodological rule that values bold conjectures and rigorous refutations over confirmation.
波普尔的证伪主义用可证伪性取代可证实性，作为科学与非科学的划界标准。一种理论必须做出可能原则上与经验冲突的大胆预测。兼容于一切可能观察的陈述——例如“所有事件都由命运决定”——是不可证伪的，因而也是不科学的。这条原则并非逻辑定理，而是一条方法论规则，它珍视大胆的猜想和严格的证伪甚于证实。
Example: Freudian psychoanalysis is criticised as unfalsifiable because it can reinterpret any apparent counter-evidence in terms of its own framework (e.g., resistance). It thus makes no risky predictions.
例子：弗洛伊德精神分析被批评为不可证伪，因为它能够用自己的框架（如阻抗）重新解释任何明显的反证。它因此没有做出大胆的预测。
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