📚 AS Edexcel Philosophy: Formulae & Theorems Quick Reference | AS Edexcel 哲学：公式定理速查手册
This quick reference guide distils the essential logical 'formulae' and argument patterns that underpin AS Edexcel Philosophy. Whether you are analysing Descartes' cogito, spotting a formal fallacy, or testing deductive validity with truth tables, these tools will sharpen your philosophical reasoning and help you construct watertight essays. Mastery of these symbolic forms is not an end in itself; it is a means to uncover the hidden structure of arguments in epistemology, philosophy of religion and ethics.
本速查手册浓缩了 AS Edexcel 哲学课程所依托的核心逻辑“公式”与论证模式。无论你在分析笛卡尔的我思、识别形式谬误，还是用真值表检验演绎有效性，这些工具都能磨砺你的哲学推理，并助你构建无懈可击的论文。掌握这些符号形式本身不是目的，而是揭示认识论、宗教哲学与伦理学论证深层结构的手段。
1. Sentential Connectives and Truth Tables | 语句联接词与真值表
Sentential connectives are the building blocks of propositional logic. Each connective is defined by a characteristic truth table that determines the truth value of a compound sentence based on the truth values of its atomic constituents. Below are the five standard connectives with their symbols and truth conditions.
语句联接词是命题逻辑的构件。每个联接词都由一个特征真值表来定义，该表根据原子命题的真值决定复合句的真值。以下是五个标准联接词及其符号与真值条件。
The negation '¬P' reverses the truth value: when P is true, ¬P is false; when P is false, ¬P is true. The conjunction 'P ∧ Q' is true only when both conjuncts are true. The disjunction 'P ∨ Q' is true when at least one disjunct is true (inclusive 'or'). The conditional 'P → Q' ntecedent P is true and the consequent Q is false; otherwise it is true. The biconditional 'P ↔ Q' is true exactly when P and Q share the same truth value.
否定“¬P”翻转真值：当 P 真时 ¬P 假，当 P 假时 ¬P 真。合取“P ∧ Q”仅在两个合取项皆真时为真。析取“P ∨ Q”在至少一个析取项为真时为真（相容的“或”）。条件句“P → Q”仅在前件 P 真而后件 Q 假时为假，其余情况皆真。双条件“P ↔ Q”恰在 P 与 Q 真值相同时为真。
| P | Q | ¬P | P ∧ Q | P ∨ Q | P → Q | P ↔ Q |
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | F |
| F | T | T | F | T | T | F |
| F | F | T | F | F | T | T |
In AS exam questions, you may be asked to compute the truth value of a complex formula given an interpretation, or to recognise that a conditional is not equivalent to its converse. Memorising these truth tables allows you to handle any propositional logic task with confidence.
在 AS 考题中，你可能需要根据给定解释计算复杂公式的真值，或识别条件句与其逆命题不等值。熟记这些真值表能让你自信地应对任何命题逻辑任务。
2. Valid Deductive Argument Forms | 有效演绎论证形式
A deductive argument is valid if and only if it is impossible for the premises to be true and the conclusion false. The following argument forms are unconditionally valid: any argument that instantiates one of these patterns guarantees that the conclusion must be true whenever the premises are true.
一个演绎论证是有效的，当且仅当不可能出现前提真而结论假的情况。下列论证形式是无条件有效的：任何符合这些模式的论证都能保证，只要前提真，结论必然真。
Modus Ponens (MP): P → Q, P ⊢ Q
If P implies Q, and P is given, then Q follows. Example: If it rains, the ground is wet. It rains. Therefore, the ground is wet.
如果 P 蕴含 Q，且给出 P，则 Q 可得。例：如果下雨，地湿。下雨。所以地湿。
Modus Tollens (MT): P → Q, ¬Q ⊢ ¬P
If P implies Q, but Q is false, then P must be false. Example: If the alarm works, the light flashes. The light does not flash. Therefore, the alarm does not work.
如果 P 蕴含 Q，但 Q 为假，则 P 必假。例：如果警报工作，灯会闪。灯没闪。所以警报不工作。
Hypothetical Syllogism (HS): P → Q, Q → R ⊢ P → R
A chain of implications allows the first antecedent to imply the final consequent. Example: If I study, I pass the test. If I pass the test, I graduate. Thus, if I study, I graduate.
蕴含链条使得第一个前件能蕴含最后一个后件。例：如果学习，则通过考试。如果通过考试，则毕业。因此，如果学习，则毕业。
Disjunctive Syllogism (DS): P ∨ Q, ¬P ⊢ Q
When one disjunct is eliminated, the other must hold. Example: Either the battery is flat or the bulb is broken. The battery is not flat. Therefore, the bulb is broken.
当一个析取项被排除时，另一个必须成立。例：要么电池没电，要么灯泡坏了。电池有电。因此灯泡坏了。
Constructive Dilemma (CD): (P → Q) ∧ (R → S), P ∨ R ⊢ Q ∨ S
If either of two antecedents holds, at least one of their consequents follows. This form often appears in ethical arguments where two moral principles lead to different obligations.
如果两个前件至少有一个成立，则它们后件的析取也成立。该形式常见于伦理学论证，两条道德原则导向不同的义务。
3. Formal Fallacies: Invalid Forms | 形式谬误：无效形式
A formal fallacy is an error in the logical structure of an argument that renders it invalid. Two fallacies closely resemble valid forms and are common traps in philosophical reasoning.
形式谬误是论证逻辑结构中的错误，致使论证无效。有两种谬误与有效形式极为相似，是哲学推理中常见的陷阱。
Affirming the Consequent: P → Q, Q ⊢ P
This invalid form assumes that because the consequent is true, the antecedent must be true. Example: If it is a dog, it is an animal. It is an animal. Therefore, it is a dog. (It could be a cat.)
此无效形式假定因为后件真，前件必真。例：如果是狗，则是动物。是动物。所以是狗。（可能是猫。）
Denying the Antecedent: P → Q, ¬P ⊢ ¬Q
Here the argument denies the antecedent and falsely concludes the consequent is false. Example: If it rains, the ground is wet. It does not rain. Therefore, the ground is not wet. (A sprinkler could wet the ground.)
这里否定了前件，却错误地推出后件为假。例：如果下雨，地湿。没下雨。所以地不湿。（洒水器也能弄湿地。）
Recognising these fallacies is vital for evaluating philosophical arguments, particularly in exam questions where you must identify flaws in a given passage.
识别这些谬误对于评估哲学论证至关重要，尤其在考题中需要你指出给定段落里的缺陷时。
4. Testing Validity with Truth Tables | 用真值表检验有效性
A truth table provides a mechanical method for determining whether a deductive argument is valid. You construct a table that exhausts all possible truth-value assignments for the atomic propositions, then check if there is any row where all premises are true and the conclusion false. If no such row exists, the argument is valid; a single counterexample row makes it invalid.
真值表提供了一种判定演绎论证是否有效的机械方法。构建一个穷尽所有原子命题真值指派可能的表格，然后检查是否有哪一行使得所有前提真而结论假。如果不存在这样的行，论证有效；只要有一行反例，论证便无效。
For example, to test Modus Ponens (P → Q, P ⊢ w where P=T, Q=F, premise P → Q is false, so that row is not a counterexample. Thus no row has true premises and false conclusion, confirming validity.
例如，检验肯定前件式 (P → Q, P ⊢ Q)，给 P 和 Q 在四行中指派真值。逐行检查：当两个前提都真（P=T, Q=T 的行），结论为 T；在 P=T, Q=F 的行，前提 P → Q 假，该行不构成反例。因此没有任何一行前提真而结论假，确认有效。
When testing longer arguments, calculate the truth value of each component step by step. In the AS exam, you may be asked to complete a truth table or to use one to justify whether an argument is sound. Remember that soundness also requires all premises to be actually true.
测试较长论证时，逐步计算每个部分的真值。在 AS 考试中，可能要求你完成一个真值表，或用它论证某个论证是否可靠。记住，可靠性还要求所有前提在事实上真实。
5. Quantification and Predicate Logic Basics | 量化与谓词逻辑基础
While AS Edexcel Philosophy concentrates on propositional logic, elementary quantifier notation often illuminates the logical structure of statements about 'all' or 'some'. The universal quantifier '∀x' is read as 'for all x', and the existential quantifier '∃x' is read as 'there exists an x'.
尽管 AS Edexcel 哲学专注于命题逻辑，但基本的量词符号往往能阐明关于“所有”或“某些”陈述的逻辑结构。全称量词“∀x”读作“对所有 x”，存在量词“∃x”读作“存在某个 x”。
The formula ∀ Man(Socrates) allows the deduction Mortal(Socrates), a classic predicate-logic instantiation. The negation of a universal statement is an existential statement: ¬∀x P(x) ≡ ∃x ¬P(x). Similarly, ¬∃x P(x) ≡ ∀x ¬P(x).
公式 ∀x (人(x) → 会死(x)) 加上 人(苏格拉底) 可推出 会死(苏格拉底)，这是经典的谓词逻辑实例。全称陈述的否定是存在陈述：¬∀x P(x) ≡ ∃x ¬P(x)。类似地，¬∃x P(x) ≡ ∀x ¬P(x)。
Understanding these basic equivalences helps you analyse philosophical claims such as 'All knowledge is derived from experience' or 'Some moral truths are absolute'. In essay writing, pinning down the quantifier can expose hidden ambiguities.
理解这些基本等值有助于分析诸如“所有知识都来源于经验”或“某些道德真理是绝对的”这类哲学主张。在论文写作中，明确量词可以揭露隐藏的歧义。
6. Informal Fallacies | 非形式谬误
Beyond formal mistakes, informal fallacies arise from defects in content or context rather than from faulty logical form. Edexcel AS Philosophy expects you to detect and name these common reasoning errors in stimulus passages.
除了形式错误，非形式谬误源于内容或语境方面的缺陷，而非错误的逻辑形式。Edexcel AS 哲学要求你能在阅读材料中识别并指出这些常见推理错误。
- Ad Hominem – attacking the person instead of the argument. Example: 'You claim we should lower taxes, but you are rich, so your argument fails.' | 人身攻击 – 攻击人而非论证。例：“你主张应减税，但你有钱，所以你的论证不成立。”
- Straw Man – misrepresenting an argument to make it easier to attack. | 稻草人 – 歪曲对方论点以便攻击。
- False Dilemma – presenting only two options when more exist. | 假两难 – 仅给出两种选项，实际上存在更多。
- Begging the Question – assuming what you are trying to prove. | 乞题 – 将待证结论当作前提。
- Slippery Slope – claiming a small step will inevitably lead to extreme consequences without evidence. | 滑坡 – 声称小小一步必将引发极端后果而无证据。
When analysing a text, explicitly name the fallacy and explain why it undermines the argument. This demonstrates both critical thinking and precise philosophical vocabulary.
在分析文本时，明确说出谬误名称并解释它为何削弱论证。这既能展现批判性思维，又体现了精确的哲学词汇量。
7. Deductive vs Inductive Reasoning | 演绎推理与归纳推理
Deduction aims for certainty: if the premises are true, the conclusion must be true. Induction, by contrast, aims for probability: the premises make the conclusion likely but not guaranteed. In AS philosophy, this distinction underpins debates about the limits of reason and the problem of induction.
演绎追求确定性：如果前提真，结论必然真。归纳则追求可能性：前提使结论很可能为真，但不保证。在 AS 哲学中，这一区分是有关理性限度与归纳问题之争论的基础。
A typical deductive argument has a formal structure like MP or MT; its validity is a matter of form, not content. An inductive argument, such as 'The sun has risen every day in recorded history; therefore the sun will rise tomorrow', rests on the uniformity of nature, a principle that itself cannot be deductively proved without circularity (Hume's problem).
典型的演绎论证具有如 MP 或 MT 的形式结构；其有效性取决于形式而非内容。归纳论证，例如“有史以来太阳每天都升起；因此明天太阳也会升起”，依赖自然齐一性原则，而该原则本身无法在无循环的情况下被演绎证明（休谟问题）。
In your essays, always specify whether an argument is deductive or inductive, because the standards of evaluation differ: validity and soundness for deduction, strength and cogency for induction.
在论文中，总要说明论证是演绎还是归纳，因为评价标准不同：演绎看有效性与可靠性；归纳看强度与说服力。
8. Reductio ad Absurdum and Conditional Proof | 归谬法与条件证明
Two powerful methods for establishing a conclusion are frequently employed in philosophical reasoning and can be formally represented. Understanding their structure lets you reconstruct arguments in exam essays.
哲学推理中常用两种强有力的方法来确立结论，并可形式化表示。理解其结构能让你在考卷论文中重构论证。
Reductio ad Absurdum (RAA): a contradiction, and thereby conclude the original statement. Formally, to prove P, assume ¬P, derive ¬P → (Q ∧ ¬Q), then by modus tollens or the principle of non-contradiction, ¬P is rejected, so P holds.
归谬法：假定欲证命题的否定，推导出矛盾，从而断定原命题。形式上，为证 P，假设 ¬P，推出 ¬P → (Q ∧ ¬Q)，然后根据否定后件式或不矛盾律，拒斥 ¬P，故 P 成立。
Conditional Proof (CP): discharge the assumption to conclude P → Q. This technique mirrors how we often argue for 'if... then...' claims.
条件证明：要证一个条件句 P → Q，临时假设 P，在该假设下推出 Q，然后解除假设，得到 P → Q。这一技巧反映了我们论证“如果……那么……”主张的常用方式。
Both methods are essential for tackling questions on the existence of God, where reductio is used to show that denying a perfect being leads to contradiction, or conditional proof is employed in ontological arguments.
这两种方法对于处理有关上帝存在的题目至关重要，其中归谬法被用来表明否认一个完满存在会导致矛盾，条件证明则被用于本体论论证。
9. Key Philosophical Arguments in Logical Form | 重要哲学论证的逻辑形式
Translating canonical arguments into symbolic logic reveals their inferential skeleton and helps you evaluate their force. Below are three landmark arguments from the AS syllabus, rendered in propositional form.
将经典论证翻译成符号逻辑能展现其推理骨架，并有助于评估其力度。以下是从 AS 大纲中选取的三个标志性论证，以命题形式呈现。
Descartes' Cogito:
1. If I am thinking, then I exist. (T → E)
2. I am thinking. (T)
3. Therefore, I exist. (E) – Valid by Modus Ponens.
笛卡尔的我思：
1. 如果我正在思考，则我存在。（思 → 在）
2. 我正思考。（思）
3. 所以我存在。（在）——由肯定前件式得出，有效。
Anselm's Ontological Argument (simplified):
1. God is the being than which no greater can be conceived. (G)
n the understanding, a greater being could be conceived (one that exists in reality). (U → ¬G)
3. But this leads to a contradiction; therefore, God must exist in reality. (G) – Often reconstructed as a reductio.
安瑟伦的本体论论证（简化）：
1. 上帝是无法设想比之更伟大的存在者。（G）
2. 如果上帝仅在理智中存在，则一个更伟大者能被设想（即在现实中存在者）。（U → ¬G）
3. 但这导向矛盾；因此上帝必在现实中存在。（G）——常被重构为归谬论证。
Hume's Problem of Induction:
1. All inductive inferences presuppose the Uniformity Principle (UP).
2. UP cannot be proved deductively (no contradiction in its falsehood) nor inductively (circular).
3. Therefore, induction lacks rational justification. – A disjunctive syllogism once the possible justifications are exhausted.
休谟的归纳问题：
1. 一切归纳推理都预设自然齐一性原则（UP）。
2. UP 既不能演绎证明（其假不蕴含矛盾），也不能归纳证明（循环论证）。
3. 因此，归纳缺乏理性辩护。——穷尽可能的辩护后，运用了析取三段论。
10. A Quick Reference Glossary | 术语速查
Keep this glossary of core logical and philosophical terms at your fingertips when revising for the AS exam. Each concept is paired with its symbol or formula where applicable.
在准备 AS 考试复习时，将以下核心逻辑与哲学术语表置于手边。每个概念在适当处附有符号或公式。
| Term | Symbol/Formula | Meaning |
|---|---|---|
| Validity | Impossible for premises true & conclusion false | 论证形式有效 |
| Soundness | Valid + all premises actually true | 有效且前提真实 |
| Tautology | P ∨ ¬P | 永真式 |
| Contradiction | P ∧ ¬P | 永假式 |
| Contingency | Neither tautology nor contradiction | 偶真式 |
| Conditional | P → Q | 条件句 |
| Necessary condition | No P without Q; ¬Q → ¬P | 必要条件 |
| Sufficient condition | P guarantees Q; P → Q | 充分条件 |
| De Morgan's Laws | ¬(P ∧ Q) ≡ ¬P ∨ ¬Q; ¬(P ∨ Q) ≡ ¬P ∧ ¬Q | 德摩根律 |
Memorising these terms and their symbolic correlates will save you time in the exam
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