Year 12 WJEC Philosophy: Formula and Theorem Quick Reference | Year 12 WJEC 哲学：公式定理速查手册

📚 Year 12 WJEC Philosophy: Formula and Theorem Quick Reference | Year 12 WJEC 哲学：公式定理速查手册

This quick-reference handbook distils the core arguments, logical structures, and conceptual tenets of the Year 12 WJEC Philosophy syllabus into a set of memorable ‘formulae’ and ‘theorems’. Whether you are revisiting the tripartite definition of knowledge, unpacking Anselm’s ontological reasoning, or checking the logical form of modus tollens, the following entries provide a concise yet rigorous retrieval tool for revision.

这本速查手册将 Year 12 WJEC 哲学课程的核心论证、逻辑结构和概念信条提炼为一组易记的“公式”和“定理”。无论你是在重温知识的三元定义、拆解安瑟伦的本体论推理，还是核对否定后件式的逻辑形式，下面的条目都为复习提供了一个简洁而严谨的检索工具。

1. Logic: Valid Argument Forms | 逻辑：有效论证形式

A sound philosophical argument requires propositional logic. The following valid forms appear routinely in philosophical reasoning and can be represented as symbolic ‘formulae’ using P, Q and connectives like → (if…then), ¬ (not) and ∴ (therefore).

可靠的哲学论证需要命题逻辑。以下有效形式在哲学推理中经常出现，可以用 P、Q 以及连接词 →（如果…则）、¬（并非）和 ∴（所以）表示为符号“公式”。

Modus Ponens (MP): (P → Q), P ∴ Q
Modus Tollens (MT): (P → Q), ¬Q ∴ ¬P
Hypothetical Syllogism (HS): (P → Q), (Q → R) ∴ (P → R)
Disjunctive Syllogism (DS): (P ∨ Q), ¬P ∴ Q
Constructive Dilemma: (P → Q) ∧ (R → S), (P ∨ R) ∴ (Q ∨ S)

These forms guarantee that if the premises are true, the conclusion must be true. They are the inferential ‘gears’ of deductive arguments across epistemology, ethics and philosophy of religion.

这些形式保证了如果前提为真，结论必然为真。它们是贯穿知识论、伦理学与宗教哲学演绎论证的推理“齿轮”。

Name	Form	Example
Modus Ponens	If A then B; A ∴ B	If God exists, objective morality exists. God exists. Therefore objective morality exists.
Modus Tollens	If A then B; not B ∴ not A	If the mind is the brain, then after death we cease to exist. We do not cease to exist. Therefore the mind is not the brain.

2. Epistemology: The JTB Definition and the Gettier Problem | 知识论：JTB 定义与葛梯尔问题

The classical tripartite definition of knowledge is often written as a formula: K = JTB, where K stands for knowledge, J for justified, T for true and B for belief. For a subject S to know that p, all three conditions must be met.

经典的命题知识三元定义常写作公式：K = JTB，其中 K 代表知识，J 代表得到辩护的，T 代表真，B 代表信念。对于一个主体 S 知道命题 p，必须同时满足三个条件。

Edmund Gettier (1963) demonstrated that there are cases in which a person has a justified true belief but still lacks knowledge. The ‘Gettier formula’ can be expressed as: J + T + B ≠ K in certain fake-barn or broken-clock scenarios. Thus the JTB condition set is necessary but not sufficient.

埃德蒙·葛梯尔（1963）证明，在某些情形中，一个人拥有得到辩护的真信念却仍然缺乏知识。“葛梯尔公式”可以表示为：在某些假谷仓或坏钟情形中，J + T + B ≠ K。因此 JTB 条件集是知识的必要条件，但不是充分条件。

Standard case: S knows p → J(p) ∧ T(p) ∧ B(p)
Gettier counterexample: J(p) ∧ T(p) ∧ B(p) ∧ ¬K(p)

This drives the post-Gettier project of adding a fourth condition (no defeaters, reliable process, safety, etc.), expressed as K = JTB + X.

这推动了后葛梯尔计划，即添加第四个条件（无否决因子、可靠过程、安全性等），表示为 K = JTB + X。

3. Foundationalism vs Coherentism: Structure of Justification | 基础主义与融贯论：辩护结构

Foundationalism structures justification as a pyramid: basic beliefs (self-justifying or evident) support non-basic beliefs. The formula can be thought of as J(Bₙ) = B₀ (basic belief) + chain of deduction/induction. Coherentism, by contrast, models justification as a web: J(Bᵢ) ↔ coherence with the whole belief-system Σ.

基础主义将辩护构建为金字塔：基本信念（自明的或直接证成的）支撑非基本信念。其公式可视为 J(Bₙ) = B₀ (基本信念) + 演绎/归纳链条。与此相对，融贯论将辩护建模为一张网：J(Bᵢ) ↔ 与整个信念系统 Σ 的融贯性。

For the foundationalist, ∃ a set of basic beliefs B₀ such that ∀B ∈ Σ, B is justified if and only if it is either a member of B₀ or traceable to B₀. For the coherentist, ∀B ∈ Σ, J(B) depends on mutual support relations among all members of Σ.

对基础主义者而言，存在一组基本信念 B₀，使得 对于 Σ 中的任意信念 B，B 得到辩护当且仅当它要么是 B₀ 的成员，要么可追溯至 B₀。对融贯论者，对于 Σ 中的任意信念 B，J(B) 取决于 Σ 所有成员间的相互支持关系。

WJEC students should note the regress argument: if every belief requires justification from another belief, we face infinite regress, circle, or arbitrary stopping. Foundationalism stops the regress at basic beliefs; coherentism accepts the circle as virtuous.

WJEC 学生应注意回溯论证：若每个信念都需要另一个信念来辩护，我们便面临无穷回溯、循环或任意终止。基础主义通过基本信念终止回溯；融贯论则接受良性循环。

4. Cartesian Scepticism: The Evil Demon Argument | 笛卡尔式怀疑论：邪恶精灵论证

Descartes’ sceptical argument can be formulated as a logical reductio: (1) I cannot be certain that there is no evil demon deceiving me about p. (2) If I cannot be certain of that, I cannot know p. ∴ I cannot know p. This yields global scepticism about the external world.

笛卡尔的怀疑论论证可表述为一个逻辑归谬：（1）我不能确定没有一个邪恶精灵在关于 p 方面欺骗我。（2）如果我对此不能确定，我便不知道 p。∴ 我不知道 p。 这导致对于外部世界的全盘怀疑。

The structure of the argument relies on a closure principle for knowledge: If S knows that p, and S knows that p entails q (not deceived), then S knows that q. Denying the consequent through modus tollens forces the sceptical conclusion.

这个论证的结构依赖于知识的闭合原则：如果 S 知道 p，并且 S 知道 p 蕴含 q（未被欺骗），那么 S 知道 q。 经由否定后件，该原则迫使得出怀疑论结论。

Closure Schema: [K(p) ∧ K(p → q)] → K(q)

Responses include denying closure (Nozick), asserting direct realism, or showing that the sceptical hypothesis is self-refuting. The ‘cogito’ (‘I think, therefore I am’) is Descartes’ foundational escape: ¬(I am deceived about my own existence) ∴ sum.

回应策略包括否定闭合原则（诺齐克）、主张直接实在论，或表明怀疑论假设是自相推翻的。“我思故我在”是笛卡尔的基础性突围：¬(我在有关我自身存在的事情上被欺骗) ∴ 我在。

5. Philosophy of Religion: Anselm’s Ontological Argument Formulation | 宗教哲学：安瑟伦的本体论论证公式

Anselm defines God as ‘that than which nothing greater can be conceived’. The ontological argument can be expressed as a reductio ad absurdum:

安瑟伦将上帝定义为“可设想的无与伦比的伟大之物”。本体论论证可表述为一个归谬形式：

(1) God = df the greatest conceivable being.
(2) God exists in the understanding.
(3) Existing in reality is greater than existing in the understanding alone.
(4) If God exists only in the understanding, a greater being can be conceived (one that also exists in reality).
(5) That contradicts (1). Therefore, God must exist in reality.

This is a purely a priori ‘formula’ that attempts to deduce existence from the definition of God. Gaunilo’s ‘perfect island’ parody uses the same logical structure to generate an absurd conclusion, challenging the move from concept to reality.

这是一个纯粹先天的“公式”，试图从上帝的定义推出其存在。高尼罗的“完美海岛”戏仿采用相同的逻辑结构得出荒谬结论，以此挑战从概念到实在的跨越。

In modal versions (Plantinga), the formula shifts to possible worlds: If maximal greatness is possibly exemplified, then it is exemplified in every possible world; hence in the actual world. The key premise is ◇∃x G(x) → □∃x G(x), under S5 axioms.

在模态版本（普兰丁格）中，公式转向可能世界：若最大伟大性可能存在，则它必然存在于所有可能世界；因而存在于现实世界。 关键前提是在 S5 公理下的 ◇∃x G(x) → □∃x G(x)。

6. Aquinas’ Cosmological Argument (Second Way) Schema | 阿奎那的宇宙论论证（第二路）图式

Aquinas argues from efficient causation. The schema can be formalised: (1) Some things undergo change (are caused). (2) Everything that is caused is caused by another. (3) An infinite regress of essentially ordered causes is impossible. ∴ (4) There must be a first uncaused cause: God.

阿奎那从动力因出发进行论证。图式可以形式化：（1）某些事物经历变化（被导致）。（2）凡是被导致的皆由另一者所导致。（3）一个本质上有序的因果无穷回溯是不可能的。∴（4）必有一个第一的、不被导致的原因：上帝。

The second way relies on the principle of sufficient reason and the rejection of an actual infinite chain. In notation: ∀x (Caused(x) → ∃y Causes(y,x)) ∧ ¬﹤∞(chain) → ∃z ¬Caused(z) ∧ ∀y(z causes y or …). Hume would challenge premise (2) and the impossibility of an infinite regress.

第二路依赖于充足理由原则以及对实无限链条的拒斥。用符号表示：∀x (被导致(x) → ∃y 导致(y,x)) ∧ ¬﹤∞(链条) → ∃z ¬被导致(z) ∧ ∀y(z 导致 y 或…)。休谟会质疑前提（2）以及无穷回溯的不可能性。

The Kalam version sharpens the argument: (1) Everything that begins to exist has a cause. (2) The universe began to exist. ∴ (3) The universe has a cause. This avoids the ’caused by another’ objection by focusing on beginning-to-exist.

卡拉姆版使论证更为锋利：（1）凡开始存在之物皆有原因。（2）宇宙开始存在。∴（3）宇宙有原因。 这通过聚焦“开始存在”规避了“凡被导致的皆由另一者导致”这一质疑。

7. The Teleological Argument: Paley’s Watchmaker Inference | 目的论论证：佩利的钟表匠推理

Paley’s design argument uses an analogical inference: Watch : Watchmaker :: Universe : Universe-Maker. The formal structure can be expressed as an argument by analogy:

佩利的设计论证运用类比推理：手表 : 钟表匠 :: 宇宙 : 宇宙创造者。 其形式结构可以表示为类比论证：

(1) Artifacts with complex interlocking parts exhibiting purpose (e.g., a watch) imply a designer.
(2) The universe exhibits even greater complexity, order, and apparent purpose.
∴ (3) The universe implies an even greater designer — God.

Hume’s objections can be modelled as weakening the analogy: the universe is not sufficiently like a watch in relevant respects (it is an organic, self-sustaining whole), so the inference to a single, perfect God is unwarranted. Darwin’s natural selection provides an alternative ‘blind watchmaker’ mechanism.

休谟的反驳可以建模为削弱类比：宇宙在相关方面并不足够像手表（它是有机的、自我维持的整体），所以推出单一的、完美的上帝的推论不成立。达尔文的自然选择则提供了一个替代性的“盲眼钟表匠”机制。

Post-Hume, the fine-tuning argument shifts to an inference to the best explanation: If the fundamental constants were even slightly different, life would be impossible. The best explanation of this ‘fine-tuning’ is a divine designer. This is an abductive formula.

后休谟时期，精细调谐论证转向最佳解释推理：若基本常数有细微改变，生命便不可能。对这一“精细调谐”的最佳解释是神圣设计者。 这是一个溯因公式。

8. The Logical Problem of Evil (Inconsistent Triad) | 恶的逻辑问题（不一致三元组）

JL Mackie formulated the logical problem of evil as showing that the set {God is omnipotent, God is wholly good, Evil exists} is logically inconsistent. The deductive argument can be written:

J.L. 马基将恶的逻辑问题表述为：集合 {上帝全能，上帝全善，恶存在} 在逻辑上不一致。其演绎论证可写作：

(1) If God is omnipotent, He can prevent all evil.
(2) If God is wholly good, He wants to prevent all evil.
(3) Evil exists.
∴ (4) God is not both omnipotent and wholly good.

The formula ∀E (Omnipotent → CanPrevent(E)) ∧ (WhollyGood → WantsPrevent(E)) ∧ ∃E Evil(E) → ¬∃God expresses the contradiction. Plantinga’s free will defence adds the premise that even an omnipotent God cannot create a world containing morally free creatures and no moral evil. Thus the triad is not strictly explicit: the existence of free will makes evil logically possible.

公式 ∀E (全能 → 能阻止(E)) ∧ (全善 → 欲阻止(E)) ∧ ∃E 恶(E) → ¬∃上帝 表达了矛盾。普兰丁格的自由意志辩护增加了一个前提，即即便全能的上帝也不能创造包含有道德自由且没有道德恶的世界。这样三元组便不是严格显式的：自由意志的存在使得恶在逻辑上可能。

The evidential problem shifts from logic to probability: Given the amount and distribution of suffering, God’s existence is improbable. Rowe’s fawn suffering case uses an inference to the best explanation formula.

恶的证据性问题从逻辑转向概率：鉴于苦难的数量和分布，上帝的存在是低概率的。 罗维的幼鹿受苦案例采用了最佳解释推理公式。

9. Utilitarianism: The Greatest Happiness Principle Calculus | 功利主义：最大幸福原则计算式

Bentham’s classic utilitarianism can be expressed as a calculus: Action A is right iff A maximises net pleasure over pain for all affected. The felicific calculus considers intensity, duration, certainty, propinquity, fecundity, purity, and extent.

边沁的古典功利主义可以表达为一个计算式：行动 A 是正确的，当且仅当 A 为所有受影响者带来的净快乐超过痛苦最大化。 快乐计算考虑强度、持续时间、确定性、邻近性、衍生性、纯度和范围。

Utility(A) = Σᵢ (Pleasureᵢ × Durationᵢ × Certaintyᵢ) − Σⱼ (Painⱼ × Durationⱼ × Certaintyⱼ)

Mill’s higher and lower pleasures modify the calculus qualitatively: If competent judges prefer pleasure P₁ over P₂, then P₁ is qualitatively higher, regardless of quantity. This introduces a ‘competent judges test’ formula: P₁ >qual P₂ ↔ Judges(J) ∧ Preference(J, P₁, P₂).

密尔的高等与低等快乐对计算式做了质的修正：若合格的评判者偏爱快乐 P₁ 胜于 P₂，则 P₁ 在质上更高，无关量的大小。 这引入一个“合格评判者检验”公式：P₁ >qual P₂ ↔ 评判者(J) ∧ 偏爱(J, P₁, P₂)。

Act utilitarianism applies the principle directly to individual actions; rule utilitarianism applies it to general rules: Follow the rule that, if generally obeyed, would maximise utility. Thus the formula becomes Right(A) ↔ A conforms to a rule R such that R maximises expected utility.

行为功利主义直接将原则用于个体行动；规则功利主义则将其用于普遍规则：遵守那条若被普遍遵循便能最大化功利的规则。 于是公式变为 Right(A) ↔ A 符合规则 R，且 R 最大化期望功利。

10. Kantian Deontology: Categorical Imperative Formulae | 康德义务论：定言命令公式

Kant’s moral law is expressed as a set of formulae testing the maxim of an action. The first formulation, the Formula of Universal Law (FUL): ‘Act only on that maxim through which you can at the same time will that it should become a universal law.’

康德的道德法则表现为一组检验行动准则的公式。第一个公式，普遍法则公式（FUL）：“要只按照你同时能够愿意它成为一个普遍法则的那个准则去行动。”

This generates a contradiction test: Maxim M passes FUL if M can be consistently conceived as a universal law and can be willed without contradiction. The logical structure is: If universalising M leads to a contradiction in conception or will, then acting on M is forbidden.

这产生了一个矛盾检验：若准则 M 能被无矛盾地设想为普遍法则且能被意愿，则 M 通过 FUL。 逻辑结构是：若将 M 普遍化导致概念矛盾或意志矛盾，则按 M 行动是被禁止的。

The Formula of Humanity (FH) treats rational beings as ends-in-themselves: ‘Act in such a way that you treat humanity, whether in your own person or in any other, always at the same time as an end, never merely as a means.’ This can be formalised as a respect constraint: ∀A (Action(A) → ¬(uses rational being merely as means)).

人性公式（FH）将理性存在者视为目的自身：“你要如此行动，即无论是你自身中的人性，还是任何他人中的人性，你在任何时候都同时当作目的，决不只当作手段来使用。” 这可以形式化为一个尊重约束：∀A (行动(A) → ¬(仅仅将理性存在者用作手段))。

11. Aristotle’s Virtue Ethics: The Doctrine of the Mean | 亚里士多德美德伦理学：中庸之道

Aristotle’s virtue ethics avoids formulaic rules but contains a structural theorem: Virtue = disposition to choose the mean relative to us, determined by reason. For any sphere of action or feeling, virtue lies between two vices: excess and deficiency.

亚里士多德的美德伦理学回避程式化规则，但内含一个结构定理：美德 = 选择相对于我们而言的中庸的倾向，由理性决定。 在任何行动或感受领域，美德处于两个恶之间：过度与不及。

Virtueⱼ = Mean(Vice_of_excessⱼ, Vice_of_deficiencyⱼ)

For example, courage is the mean between rashness (excess of confidence) and cowardice (deficiency). This is not arithmetic; it is sensitive to the situation and the agent, and is determined by phronesis (practical wisdom).

例如，勇敢是鲁莽（自信的过度）和怯懦（不足）之间的中庸。这不是算术平均；它对情境和行动者敏感，并由实践智慧（phronesis）决定。

The Eudaimonia Theorem states that living according to virtue constitutes the good life: Eudaimonia (flourishing) = activity of the soul in accordance with virtue over a complete life. This provides a teleological framework: the function (ergon) of humans is rational activity; virtue perfects this function.

幸福（Eudaimonia）定理指出，依据美德的生活构成好生活：Eudaimonia（繁荣）= 灵魂合乎美德的贯穿整个生命的活动。 这提供了一个目的论框架：人的功能（ergon）是理性活动；美德完善了这一功能。

12. Free Will and Determinism: The Consequence Argument | 自由意志与决定论：后果论证

The classic consequence argument (van Inwagen) formalises the incompatibilist claim that if determinism is true, no one has free will. Using a modal operator Np meaning ‘p is true and no one has, or ever had, any choice about whether p’:

经典的后果论证（范·因瓦根）将不相容论主张形式化：若决定论为真，则无人拥有自由意志。使用模态算子 Np 表示“p 为真，并且没有人对 p 是否成立有任何选择”：

Rule Alpha: □p → Np
Rule Beta: Np ∧ N(p → q) → Nq

Given determinism (the conjunction of past states and the laws of nature entails every future truth), we can show that No one has a choice about any future action. The argument applies Beta repeatedly: from N(past ∧ laws) and N((past ∧ laws) → future action), derive N(future action).

给定决定论（过去状态与自然律的合取蕴含每一个未来真理），我们可以得出没有任何人能够对任何未来行动有所选择。该论证反复使用 Beta：从 N(过去 ∧ 规律) 和 N((过去 ∧ 规律) → 未来行动)，推出 N(未来行动)。

Compatibilists (Hume, Ayer) redefine freedom as the absence of constraint: Free action = doing what one wants, unhindered, even if determined. This yields the conditional analysis: S could have done otherwise iff S would have done otherwise, had S chosen differently. The formula shifts the concept of ability from categorical to conditional.

相容论者（休谟、艾耶尔）将自由重新定义为不受强制：自由行动 = 不受阻碍地做自己想做的事，即便被决定。 这产生了条件分析：S 本可以做出其他行动，当且仅当若 S 做出不同选择，S 就会采取不同行动。 该公式将“能力”概念从绝对转为条件。

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Year 12 WJEC Philosophy: Formula and Theorem Quick Reference | Year 12 WJEC 哲学：公式定理速查手册