📚 Year 11 WJEC Philosophy: Formula & Theorem Quick Reference Handbook | Year 11 WJEC 哲学：公式定理速查手册
Philosophy may not have mathematical formulas, but it is packed with logical structures, ethical equations, and argument forms that act just like formulas. This quick reference handbook compiles key 'theorems' and 'formulas' from the WJEC Year 11 Philosophy course—covering logic, arguments for God's existence, ethics, and more. Use these simplified patterns to analyse, evaluate, and construct philosophical arguments with precision.
哲学或许没有数学公式，但它充满了逻辑结构、伦理等式和论证形式，就像公式一样。这本速查手册汇总了 WJEC 十一年级哲学课程中的关键‘定理’和‘公式’——涵盖逻辑、上帝存在的论证、伦理等内容。运用这些简化的模式，你可以精确地分析、评价并构建哲学论证。
1. Modus Ponens & Modus Tollens | 肯定前件式与否定后件式
Modus Ponens (affirming the antecedent) is a fundamental valid argument form. It states that if a conditional statement is true and its antecedent is true, then the consequent must be true. This logical 'chain' underlies almost all deductive reasoning.
肯定前件式是一种基本有效论证形式。它断言若一个条件语句为真且其前件为真，则后件必定为真。这一逻辑‘链条’几乎支撑了所有演绎推理。
P → Q, P ∴ Q
Example: If a student revises effectively (P), then they will pass the exam (Q). The student revises effectively (P). Therefore, they will pass the exam (Q).
例子：如果学生有效复习（P），那么他们会通过考试（Q）。该学生有效复习（P）。因此他们会通过考试（Q）。
Modus Tollens (denying the consequent) works by falsification. It argues that if the conditional is true but the consequent is false, the antecedent cannot be true. This pattern is essential for testing hypotheses and eliminating false causes.
否定后件式通过证伪来运作。它论证的是：若条件语句为真而后件为假，则前件不可能为真。此模式对于检验假说、排除错误原因至关重要。
P → Q, ¬Q ∴ ¬P
Example: If it is a genuine Picasso painting (P), it will pass the infrared test (Q). It does not pass the test (¬Q). Therefore, it is not a genuine Picasso (¬P).
例子：如果是真迹毕加索画作（P），它就会通过红外测试（Q）。它未通过测试（¬Q）。因此它并非真迹毕加索（¬P）。
2. Hypothetical Syllogism & Disjunctive Syllogism | 假言三段论与选言三段论
Hypothetical Syllogism chains together conditionals. If P implies Q and Q implies R, we can validly conclude that P implies R. This allows philosophers to build extended lines of reasoning without repeating the intermediate step.
假言三段论将条件语句串成链。如果 P 蕴涵 Q 且 Q 蕴涵 R，我们就可以有效推出 P 蕴涵 R。这让哲学家能建立起延伸的论证线，而无需重复中间步骤。
(P → Q) ∧ (Q → R) ∴ P → R
Example: If you study philosophy (P), you improve your logic (Q). If you improve your logic (Q), you make better arguments (R). Therefore, if you study philosophy, you make better arguments.
例子：如果你学习哲学（P），就能提升逻辑能力（Q）。如果你提升逻辑能力（Q），就能做出更好的论证（R）。因此，如果你学习哲学，你就能做出更好的论证。
Disjunctive Syllogism works by eliminating alternatives. When we have an 'either... or...' statement and one disjunct is known to be false, the other must be true. It is a simple but powerful tool for narrowing down possibilities.
选言三段论通过排除选项来运作。我们有一个‘要么…… 要么……’的陈述，且知道其中一个选言支为假，则另一个必真。这是缩小可能性范围的简单而有力的工具。
P ∨ Q, ¬P ∴ Q
Example: Either the mind is material (P) or it is non-material (Q). The mind is not purely material (¬P). Therefore, it is non-material (Q).
例子：要么心灵是物质的（P），要么它是非物质的（Q）。心灵不纯粹是物质的（¬P）。因此它是非物质的（Q）。
3. Anselm's Ontological Formula | 安瑟伦的本体论公式
Anselm defines God as 'that than which nothing greater can be conceived'. His famous argument turns this definition into a proof of God's existence using a reductio ad absurdum. It can be compressed into a logical formula that highlights the move from thought to reality.
安瑟伦将上帝定义为‘无法设想比之更伟大的存在’。他著名的论证利用归谬法将这一定义转化为对上帝存在的证明。它可以被压缩成一个逻辑公式，突出从思维到实在的跨越。
God ≡df ¬∃x (x > G) (No conceivable being greater than God)
Step 1: God exists in the understanding (in intellectu). Step 2: Suppose God exists only in the understanding and not in reality (in re). Then we can conceive of a being exactly like God but who also exists in reality—this being would be greater. This contradicts the definition. Therefore, the supposition is false, and God must exist in reality.
步骤1：上帝存在于理智中。步骤2：假设上帝只存在于理智中而不存在于现实中。那么我们就能设想一个与上帝完全相同、但同时存在于现实中的存在者——这个存在者会比上帝更伟大。这就与定义矛盾了。因此假设为假，上帝必然存在于现实中。
G(in intellectu) ∧ ¬G(in re) → Contradiction ∴ G(in re)
4. Paley's Design Argument Formula | 佩利的设计论证公式
William Paley's watchmaker analogy forms the core of the teleological argument. It observes that complex, purposeful arrangements in nature resemble artifacts and therefore require an intelligent designer. The reasoning can be expressed as a conditional proof.
威廉·佩利的手表匠比喻构成了目的论论证的核心。它观察到自然界中复杂、有目的的排列类似于人造物，因此需要一位智慧设计者。这一推理可以表达为一个条件证明。
(Complexity ∧ Purpose) → Design; Design → Designer
Premise 1: If an object exhibits intricate complexity combined with a clear purpose (as a watch does), we infer that it had a designer. Premise 2: The universe (or biological organisms) exhibits even greater complexity and purpose. Conclusion: Therefore, the universe has an intelligent designer—God.
前提1：如果一个物体表现出精细的复杂性并结合明确的目的（如手表），我们就推断它有一位设计者。前提2：宇宙（或生物有机体）表现出更大的复杂性与目的。结论：因此，宇宙拥有一位智慧设计者——上帝。
C ∧ P → D; U exhibits (C ∧ P) ∴ D(God)
5. The Kalam Cosmological Formula | 卡拉姆宇宙论公式
The Kalam argument, popularised by William Lane Craig, relies on two premises: whatever begins to exist has a cause, and the universe began to exist. Its logical skeleton is straightforward, yet it opens a huge metaphysical debate.
由威廉·莱恩·克雷格推广的卡拉姆论证依赖于两个前提：凡开始存在者必有原因，且宇宙开始存在。其逻辑骨架很简单，却开启了一场宏大的形上学辩论。
∀x (Bx → Cx), Bu ∴ Cu
(For any x, if x begins to exist, x has a cause; the universe u begins to exist; therefore, u has a cause.) The cause is then identified as a personal, transcendent creator—God—due to the need for a timeless, spaceless, and immensely powerful first cause.
（对于任一x，若x开始存在，则x有原因；宇宙u开始存在；因此u有原因。）这一原因随后被认定为一位人格化、超越性的创造者——上帝，因为需要一个无时间、无空间且无比强大的第一因。
Cu ∧ (Cause must be God-like) ∴ God exists
6. The Logical Problem of Evil | 恶的逻辑问题公式
The logical problem of evil challenges the coherence of the traditional theistic God. If God is omnipotent, omniscient, and wholly good, then evil should not exist. Yet evil undeniably exists, generating a classic dilemma for believers.
恶的逻辑问题挑战了传统有神论上帝的内在一致性。如果上帝是全能的、全知的且全善的，那么恶就不应存在。但恶又无可否认地存在，这向信者提出了一个经典困境。
(Omnipotent ∧ Omniscient ∧ Omnibenevolent) → ¬Evil
But we observe evil. So the atheologian constructs: (O ∧ K ∧ B) → ¬E, E ∴ ¬(O ∧ K ∧ ise, such as that some evils are logically necessary for greater goods (the free will defence).
但我们观察到恶。于是反神学家构造出：(O ∧ K ∧ B) → ¬E，而 E 为真，所以 ¬(O ∧ K ∧ B)。换言之，至少有一个神圣属性为假，所以这样的上帝不存在。有神论者通常通过加入第四个前提来回应，例如某些恶对更大的善是逻辑上必需的（自由意志辩护）。
(O ∧ K ∧ B) → ¬E, E ∴ ¬(O ∧ K ∧ B) [Theist reply: add necessity of greater goods]
7. Pascal's Wager Decision Matrix | 帕斯卡赌注决策矩阵
Pascal's Wager treats belief in God as a rational bet under uncertainty. Instead of a proof, it offers a decision-theoretic argument: the expected utility of believing in God far outweighs that of not believing. The famous matrix below illustrates the risk-reward structure.
帕斯卡的赌注把信仰上帝看作不确定性下的理性赌博。它没有提供证明，而是给出了一个决策论论证：相信上帝的期望效用远大于不信。下面这个著名矩阵展示了风险-回报结构。
| God exists | God does not exist | |
| Believe | Infinite gain (eternal life) | Finite loss (some earthly pleasures) |
| Not believe | Infinite loss (eternal suffering) | Finite gain (freedom from religious constraints) |
Expected value of believing: infinite reward with non-zero probability. Expected value of not believing: possible infinite loss. Therefore, the rational choice is to believe (or to take steps to cultivate belief). Critics note the 'many gods' objection: which God? Yet the formula remains a landmark in pragmatic philosophy.
相信的期望值：可能获得无限奖赏，概率非零。不信的期望值：可能遭受无限损失。因此，理性选择是信仰（或采取步骤培育信仰）。批评者指出‘多神’反驳：信哪个神？但这一公式仍作为实用主义哲学的一座里程碑存在。
8. Bentham's Hedonic Calculus | 边沁的苦乐计算法
Jeremy Bentham's Utilitarianism proposed that morality is a matter of measuring pleasure and pain. His hedonic calculus is a systematic formula for evaluating the utility of any action. Although not strictly mathematical, it supplies a quasi-algebraic method for ethical decisions.
杰里米·边沁的功利主义提出，道德就是衡量快乐与痛苦。他的苦乐计算法是一种评价任何行为效用的系统化公式。尽管不严格是数学的，却为伦理决策提供了一种准代数式的方法。
Total Net Utility = Σ (Pleasure – Pain) over all affected individuals
The calculus considers seven dimensions: (1) Intensity, (2) Duration, (3) Certainty, (4) Propinquity (nearness in time), (5) Fecundity (chance of producing further pleasure), (6) Purity (chance of producing pain), and (7) Extent (number of people affected). Each action is scored along these axes, and the action with the highest aggregate utility is the right one.
该计算法考量七个维度：（1）强度，（2）持续时间，（3）确定性，（4）邻近性（时间远近），（5）丰产性（产生更多快乐的机会），（6）纯度（带来痛苦的机会），（7）范围（受影响的人数）。每个行为都沿这些轴被打分，总效用最高的行为就是正当的。
U(action) = Σ[I × D × Ce + Fe – Pu] over Extent
Example: Choosing between giving a small pleasure to many people or a large pleasure to one person would involve computing the total utility for each option. Bentham's formula insists only the sum matters—no special regard for whose pleasure it is.
例子：在给许多人小快乐与给一个人大快乐之间作选择，会涉及计算每个选项的总效用。边沁的公式坚持只重总和——不考虑快乐属于谁。
9. Kant's Categorical Imperative Formula | 康德的绝对命令公式
Immanuel Kant's moral philosophy centres on the categorical imperative, which functions like an ethical equation testing whether a maxim can become a universal law. Its first formulation provides a clear, logical test for moral permissibility.
伊曼努尔·康德的道德哲学以绝对命令为核心，它像一种伦理等式，检测一条格准能否成为普遍法则。其第一表述为道德许可性提供了一个清晰的逻辑检验。
Act only according to that maxim whereby you can at the same time will that it should become a universal law.
To apply the test: (1) Formulate your maxim (e.g., 'I will make false promises when it benefits me'). (2) Universalise it: 'Everyone will make false promises when it benefits them.' (3) Check for contradictions—either a contradiction in conception (the action becomes impossible, as promising would lose meaning) or a contradiction in the will (you could not rationally will such a world). If a contradiction arises, the action is morally forbidden.
应用该检验：（1）表述你的格准（例如，‘只要对我有利，我就会作虚假承诺’）。（2）将其普遍化：‘只要对人人有利，人人都会作虚假承诺’。（3）检查矛盾——要么是构想上的矛盾（该行为变得不可能，因为承诺会失去意义），要么是意志上的矛盾（你不可能理性地意愿这样一个世界）。只要出现矛盾，该行为在道德上就是被禁止的。
Maxim M passes ⟺ Universalising M does not lead to contradiction (conceptual or volitional)
10. Situation Ethics Agape Formula | 情境伦理圣爱公式
Joseph Fletcher's Situation Ethics insists that the only absolute rule is agape—unconditional, self-giving love. All other laws, principles, or rules are relative to this single 'formula'. The right action in any situation is the most loving thing that can be done.
约瑟夫·弗莱彻的情境伦理坚持，唯一的绝对规则就是圣爱——无条件的、自我奉献的爱。所有其他律法、原则或规则都相对于这个单一的‘公式’。在任何情境下，正当行为就是所能做出的最符合爱的事情。
Agape Maximisation: Right action A* = argmax Love(A, S)
Where S is the situation and Love(A, S) measures the extent to which action A promotes selfless, neighbor-focused love in that context. Fletcher's working principles—pragmatism, relativism, positivism, and personalism—all serve this calculation. Crucially, traditional moral rules may be set aside if love demands it.
其中S是情境，Love(A, S)衡量行为A在该情境下促进无私、以邻舍为中心的爱的程度。弗莱彻的四项工作原则——实用主义、相对主义、实证主义与位格主义——全都服务于这一计算。关键的是，如果爱有要求，传统道德规则可以被搁置一旁。
For example, in a famous case, a mother smothers her crying baby to save a group of refugees from being discovered. The act is normally wrong, but the agape formula may judge it loving because it prevents greater suffering and loss of life.
举例来说，在一个著名的案例里，母亲闷死自己哭闹的婴儿，以免一群难民暴露。该行为通常是不对的，但圣爱公式可能判定它是爱的行动，因为它防止了更大的苦难和生命损失。
11. Correspondence, Coherence & Pragmatic Truth Tests | 符合论、融贯论与实用论真理检测
In epistemology and metaphysics, students are expected to compare three major theories of truth, each offering a different 'formula' for what makes a belief true. They serve as rival tests for the truth-aptness of statements.
在认识论和形而上学中，学生需要比较三种主要的真理理论，每种理论都为‘什么使一个信念为真’提供了不同的‘公式’。它们构成了相互竞争的陈述真理符合性检测。
Correspondence: Statement S is true ⟺ S matches a mind-independent fact F.
Coherence: S is true ⟺ S coheres with a comprehensive, consistent system of beliefs. No single belief can be tested in isolation; truth emerges from the web of interlocking ideas.
符合论：一个陈述S为真 ⟺ S与一个不依赖心灵的事实F相符合。融贯论：S为真 ⟺ S与一个全面、一致的信念系统相融贯。没有哪个信念能被孤立检验；真理从交织的理念之网中涌现。
Coherence: S is true ⟺ S ∈ Σ, where Σ is a maximally consistent belief set.
Pragmatic: S is true ⟺ believing S has useful, workable consequences in experience. Truth is what works in the long run and in the widest sense. Each formula has strengths: correspondence captures objective reality, coherence handles abstract systems like mathematics, and pragmatism addresses scientific progress.
实用论：S为真 ⟺ 相信S在经验中具有有用的、可行的后果。真理是在长期、最广泛意义上起作用的东西。每个公式各有长处：符合论捕捉了客观实在，融贯论处理如数学般的抽象系统，而实用论则应对科学进步。
Pragmatic: S is true ⟺ Practical consequences of believing S are satisfactory.
12. Free Will vs Determinism Compatibility Test | 自由意志与决定论相容性测试
The free will debate often boils down to whether human actions are determined by prior causes. Here are the key formulas describing the core positions, including compatibilism's attempt to reconcile freedom with determinism.
自由意志辩论常归结为人类行为是否被先前的原因所决定。以下是描述核心立场的关键公式，包括相容论试图调和自由与决定论的尝试。
Hard Determinism: ∀ event e, e is causally determined by prior states + laws → No free will.
Libertarianism (incompatibilist freedom): Some human actions are not causally determined; agents have genuine alternative possibilities, so free will exists. Its formula asserts that free action requires the ability to do otherwise.
强决定论：所有事件e都被先前状态加法则因果决定 → 没有自由意志。自由意志论（不相容论的自由）：一些人类行为并非被因果决定；行动者具有真实的替代可能性，因此自由意志存在。其公式断言，自由行为要求有能力做出不同的选择。
Free will exists ⟺ ∃ action A such that agent could have done ¬A under identical conditions.
Compatibilism (soft
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