📚 Year 10 Edexcel Philosophy: Formulas & Theorems Quick Reference Guide | Year 10 Edexcel 哲学：公式定理速查手册
Philosophy often feels like a vast sea of arguments and ideas, but many core reasoning structures can be expressed as crisp logical forms, decision matrices, or theorem‑like summaries. This guide compiles the essential 'formulas' you will encounter in Year 10 Edexcel Philosophy: deductive rules, argument templates, ethical calculi, and responses to the problem of evil. Each entry is presented as a clear English statement followed by its Chinese equivalent, enabling fast revision and bilingual mastery.
哲学常让人觉得如烟海般的论证与观念，但许多核心推理结构都可以表达为简洁的逻辑形式、决策矩阵或类似定理的总结。本手册汇编了你在 Year 10 Edexcel 哲学中会接触到的关键“公式”：演绎规则、论证模板、伦理演算以及恶的问题的回应。每一条目都以清晰的英文陈述紧随中文对应呈现，便于快速复习和双语掌握。
1. Logical Operators & Symbols | 逻辑运算符与符号
¬ (Negation): The symbol ¬ means 'not'. If a proposition P is true, then ¬P is false. Negation is used to construct contradictions, such as P ∧ ¬P.
¬（否定）：符号 ¬ 表示“非”。若命题 P 为真，则 ¬P 为假。否定用于构造矛盾，如 P ∧ ¬P。
∧ (Conjunction): The wedge ∧ stands for 'and'. P ∧ Q is true only when both P and Q are true. Example: 'It is raining ∧ I have an umbrella.'
∧（合取）：楔形符号 ∧ 表示“且”。P ∧ Q 仅在 P 与 Q 同为真时为真。例如：“天在下雨 ∧ 我带了伞。”
∨ (Disjunction): The vel ∨ means 'or' in the inclusive sense. P ∨ Q is true if at least one of P, Q is true. In logic, we do not assume 'exclusive or' unless stated.
∨（析取）：符号 ∨ 表示相容性的“或”。P ∨ Q 在 P、Q 至少一个为真时即为真。逻辑学中除非特别说明，一般指相容或。
→ (Conditional): P → Q reads 'if P then Q' antecedent, Q the consequent. The conditional is false only when P is true and Q is false.
→（条件式）：P → Q 读作“如果 P 那么 Q”。P 是前件，Q 是后件。条件句仅在前件真而后件假时才为假。
↔ (Biconditional): P ↔ Q means 'P if and only if Q'. It is true when P and Q share the same truth value. Think of it as (P → Q) ∧ (Q → P).
↔（双条件）：P ↔ Q 意为“P 当且仅当 Q”。当 P 与 Q 真值相同即为真。可视为 (P → Q) ∧ (Q → P)。
∀ (Universal Quantifier): ∀x reads 'for all x' or 'every x'. It asserts a property holds for every member of a domain.
∀（全称量词）：∀x 读作“对所有 x”或“每一个 x”。它断言某个性质对论域中的所有成员都成立。
∃ (Existential Quantifier): ∃x means 'there exists at least one x'. It expresses that some member of the domain satisfies a property.
∃（存在量词）：∃x 表示“至少存在一个 x”。它表达论域中某个成员满足某一性质。
⊢ & ⊨ (Syntactic & Semantic Consequence): Γ ⊢ φ means φ is provable from Γ by rules. At GCSE level, simply know that 'therefore' (∴) indicates a conclusion drawn from premises.
⊢ 与 ⊨（语法后承与语义后承）：Γ ⊢ φ 表示可由 Γ 依规则证明 φ。GCSE 阶段只须了解“因此”(∴) 表示从前提推出结论。
2. Valid Deductive Argument Forms | 有效演绎论证形式
Modus Ponens (Affirming the Antecedent): P → Q, P ⊢ rains, the field is wet. It rains. Therefore, the field is wet. This form is always truth‑preserving.
肯定前件式 (Modus Ponens):P → Q，P ⊢ Q。若下雨，操场会湿。下雨了。因此，操场湿了。该形式永远保真。
Modus Tollens (Denying the Consequent): P → Q, ¬Q ⊢ ¬ gs at danger. It does not ring. Therefore, there is no danger. Critical for falsification.
否定后件式 (Modus Tollens):P → Q，¬Q ⊢ ¬P。若警报正常，遇险时就会响。它没响。因此，没有险情。此形式的证伪作用很重要。
Hypothetical Syllogism: P → Q, Q → R ⊢ nows, school closes. If school closes, we stay home. Therefore, if it snows, we stay home.
假言三段论：P → Q，Q → R ⊢ P → R。如果下雪，学校停课。如果学校停课，我们待在家。因此，如果下雪，我们就待在家。
Disjunctive Syllogism: P ∨ Q, ¬P ⊢ Q. Either the bus is late or I will walk. The bus is not late. Therefore, I will walk. This eliminates one disjunct.
选言三段论：P ∨ Q，¬P ⊢ Q。要么公车晚点，要么我走路。公车没有晚点。因此，我走路。此式排除一个析取支。
Constructive Dilemma: (P → Q) ∧ (R → S), P ∨ R ⊢ Q ∨ cheat, I get expelled. I either study or cheat. Therefore, I either pass or get expelled.
建构性二难：(P → Q) ∧ (R → S)，P ∨ R ⊢ Q ∨ S。若我学习，则通过。若我作弊，则被开除。我或者学习或者作弊。因此，我或者通过或者被开除。
3. Common Formal Fallacies | 常见形式谬误
Affirming the Consequent: P → Q, Q ⊢ it is a dog. (It could be a cat.) This error confuses necessary and sufficient conditions.
肯定后件：P → Q，Q ⊢ P。若是狗，则有四条腿。它有四条腿。因此，它是狗。（它可能是猫。）此谬误混淆了必要条件与充分条件。
Denying the Antecedent: P → Q, ¬P ⊢ ¬ Therefore, it does not bark. (A seal could bark.) Invalid because Q may be true without P.
否定前件：P → Q，¬P ⊢ ¬Q。若是狗，则会吠。它不是狗。因此，它不会吠。（海豹也会吠。）无效，因为 Q 可以不依赖于 P 而为真。
Equivocation: Using a word in two different senses within an argument. 'A feather is light. What is light cannot be dark. Therefore, a feather cannot be dark.' 'Light' shifts meaning.
一词多义（歧义谬误）：在论证中对一个词使用两种含义。“羽毛很轻（light）。轻的东西不可能是黑暗（dark）。因此，羽毛不可能是黑暗的。”此处的“light”意义偷换。
4. The Design Argument 'Theorem' | 设计论证“定理”
Paley's Watch Formalisation: P1: If an object exhibits complex, purpose‑ t has a designer. P2: The universe (and its parts, like the eye) exhibits complex, purpose‑driven arrangement. C: Therefore, the universe has a designer (God). In shorthand: (Cx ∧ Px) → Dx, Cu ∧ Pu ⊢ Du.
佩利手表论证形式化：前提1：若某物展现出复杂且以目的为导向的秩序，则有其设计者。前提2：宇宙（及其局部，如眼睛）展现出复杂且以目的为导向的秩序。结论：因此，宇宙有一位设计者（上帝）。简写为：(Cx ∧ Px) → Dx，Cu ∧ Pu ⊢ Du。
Hume's Counter via Analogical Weakness: Hume objects that the analogy between a watch and the universe is too weak. A watch is a single machine; the universe is organic and unique. No sound induction.
休谟基于类比弱点的反驳：休谟反对手表与宇宙之间的类比太弱。手表是单一机器；宇宙是类生命且独一无二的。无法做出可靠的归纳。
5. The First Cause Argument Structure | 第一因论证结构
Aquinas' Second Way (Efficient Cause): no first cause, and thus no subsequent causes. C: Therefore, a first efficient cause (God) must exist. Form: ¬∃(first cause) → ¬∃(any cause), ∃(causes) ⊢ ∃(first cause).
阿奎那的第二路（动力因）：前提1：我们在世界中观察到一系列动力因。前提2：没有事物能是自己的动力因。前提3：若因果链无限回溯，则无第一因，进而无后续原因。结论：因此，必须存在第一动力因（上帝）。形式：¬∃(第一因) → ¬∃(任何因)，∃(诸因) ⊢ ∃(第一因)。
Brute Fact Objection: Could the universe itself simply be a necessary, uncaused 'brute fact'? If ‘nothing comes from nothing’ is not absolute, the first premise is questioned.
蛮实事反驳：宇宙本身能否就是一个必然的、无因的“蛮实”？若“无中不能生有”并非绝对，则第一个前提即被质疑。
6. Problem of Evil Logical Contradiction | 恶的问题逻辑矛盾
Epicurean Paradox Formalised: le to prevent all evil. Evil exists. Therefore, such a God does not exist. In symbols: (OP ∧ OS ∧ OB) → ¬∃E, ∃E ⊢ ¬(OP ∧ OS ∧ OB). This is a valid deductive attack.
伊壁鸠鲁悖论形式化：若上帝全知、全能、全善，则上帝会愿意且能够阻止一切恶。恶存在。因此，这样的上帝不存在。符号化为：(全能 ∧ 全知 ∧ 全善) → ¬∃恶，∃恶 ⊢ ¬(全能 ∧ 全知 ∧ 全善)。这是一个有效的演绎攻击。
The Logical Problem vs. Evidential Problem: Logical: the attributes are inconsistent with ANY evil. Evidential: the amount and distribution of evil make God's existence improbable. Many theodicies focus on the evidential version.
逻辑问题与证据问题：逻辑问题：这些属性与任何恶都不相容。证据问题：恶的分量和分布使得上帝的存在不大可能。许多神正论集中于证据问题。
7. Pascal's Wager Decision Matrix | 帕斯卡赌注决策矩阵
Pascal’s Wager can be analysed as a decision under ignorance, where the expected utility of believing far outweighs not believing, given the potential infinite reward. The matrix below illustrates the payoffs.
帕斯卡的赌注可视为一种无知之下的决策，其相信的期望效用因其潜在的无限奖赏而远超不信。下表的矩阵展示了收益情况。
| Decision \ God | God exists (probability p>0) | God does not exist (1−p) |
|---|---|---|
| Believe in God | +∞ (infinite bliss) | −c (finite cost, e.g. earthly pleasures) |
| Do not believe | −∞ (infinite punishment) or 0 | +c (finite gain) |
Expected Utility Calculation: (−∞) + (1−p)×(+c) ≈ −∞. Since infinite gain dominates, rational choice seems to be belief.
期望效用计算：相信的期望效用 = p×(∞) + (1−p)×(−c) ≈ ∞。不信的期望效用 = p×(−∞) + (1−p)×(+c) ≈ −∞。既然无限收益支配一切，理性选择似乎是相信。
Critique of Many Gods: The matrix assumes only one possible God. If many gods with different criteria exist, the decision matrix expands, and betting on one may still lead to infinite loss.
多神批判：该矩阵仅假定唯一可能的神。若有多位标准不同的神存在，决策矩阵将大幅扩展，押注其一只仍可能招致无限损失。
8. Utilitarian Calculus: Bentham's Felicific Calculus | 功利计算：边沁苦乐计算
Bentham proposed that the moral rightness of an action depends on the quantity of pleasure it produces, computed through seven dimensions. The total utility for an action can be conceptualised as the sum over all affected sentient beings of the product of intensity, duration, etc.
边沁提出，行为的道德正当性取决于它所产生的快乐之量，可通过七个维度计算。一项行为的总功利可被构想为所有受影响的有情众生之快乐强度与持续时间等乘积的总和。
Intensity (I): How strong is the pleasure or pain? A mild enjoyment scores lower than intense joy.
强度 (Intensity):快乐或痛苦的强烈程度。温和的享受得分低于强烈的喜悦。
Duration (D): How long does the sensation last? A fleeting pleasure has less weight than a long‑lasting one.
持续时间 (Duration):感觉持续多久。转瞬即逝的快乐权重低于持久的快乐。
Certainty (C): How likely is the pleasure to occur? Certain pleasures count more than improbable ones.
确定性 (Certainty):快乐发生的可能性。确定的快乐比不大可能发生的快乐计数更高。
Propinquity (P): How near in time is the pleasure? Immediate pleasures are weighted more than distant ones.
迫近性 (Propinquity):快乐在时间上接近程度。即时快乐比遥远未来之乐权重更高。
Fecundity (F): The chance a pleasure is followed by more pleasures; a generative quality.
丰富性 (Fecundity):该快乐之后带来更多快乐的机会；一种衍生品质。
Purity (Py): The chance a pleasure is followed by pain, and vice‑versa. A pure pleasure has little pain mixed in.
纯度 (Purity):该快乐之后跟随痛苦的机会，反之亦然。纯粹的快乐极少掺杂痛苦。
Extent (E): The number of sentient beings affected. The wider the scope, the greater the total utility.
广度 (Extent):受影响的有情众生数目。范围越广，总体功利越大。
Simple formulaic thought: tal utility ≈ Σ (I × D × C × P × F × Py) across all affected, weighted by extent. This isn't a strict arithmetic formula but a heuristic for comparing actions.
简明的公式化思路：总功利 ≈ Σ (I × D × C × P × F × Py) 遍及所有受影响者，再以广度加权。这不是严格的算术公式，而是比较行为的启发式工具。
9. Kant's Categorical Imperative Formulations | 康德绝对命令公式
Formula of Universal Law (FUL): 'Act only according to that maxim whereby you can at the same time will that it should become a universal law.' Test: can the maxim be conceived without contradiction as a universal law? If yes, proceed; if it leads to a contradiction in conception or in will, the act is impermissible.
普遍法则公式 (FUL):“要只按照你同时能够意愿它成为一项普遍法则的那个准则去行动。”检验：此准则是否能无矛盾地被设想为一项普遍法则？若能，则可行；若导致设想或意愿上的矛盾，则该行为不允许。
Formula of Humanity (FH): 'Act in such a way that you treat humanity, whether in your own person or in the person of another, always at the same time as an end and never simply as a means.' This forbids using rational beings solely as instruments.
人性公式 (FH):“你要这样行动，无论是对你自身中的人性还是对其他人中的人性，总是同时当作目的，而绝不仅仅当作手段来使用。”这禁止将理性存在者仅仅用作工具。
Universalisation Test as a Logical Procedure: Step 1: Formulate your maxim (e.g., 'I will make a false promise to get money'). Step 2: Universalise it ('Everyone makes false promises to get money'). Step 3: Check for contradiction. Here, the institution of promising would collapse, making the maxim impossible to conceive – thus impermissible.
普遍化检验作为一种逻辑程序：步骤1：形成你的准则（如，“我要通过虚假承诺来获得金钱”）。步骤2：将其普遍化（“所有人都通过虚假承诺来获得金钱”）。步骤3：检查矛盾。此处，承诺制度将崩溃，使得准则无法被设想——因而不允许。
10. Theodicies: Formal Defences | 神正论：形式辩护
Free Will Defence (Plantinga): A world containing creatures who are significantly free (freely choosing good over evil) is logically more valuable than a world of automata. If God creates free beings, He cannot caus
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