📚 CIE IGCSE Philosophy: Logical 'Formulas' & Theorems Quick Reference | CIE IGCSE 哲学：逻辑“公式”与定理速查手册
ical patterns, ethical algorithms, and epistemological formulas that thinkers have used for centuries. This quick reference handbook collects the most essential 'formulas' stemology, and ethics. Treat these structures as tools to sharpen your arguments and deepen your understanding— ne has been test-proven in philosophical debates from ancient Athens to modern exam halls.
哲学或许没有 F = ma 这样的方程式，但它确实拥有久经考验的逻辑模式、伦理算法和认识论公式。这本速查手册收录了 CIE IGCSE 哲学大纲中最核心的“公式”与定理，涵盖逻辑学、认识论和伦理学。把这些结构当作打磨论证、深化理解的工具吧——从古代雅典的辩论场到现代的考场，每一个公式都在哲学审视中反复验证过。
1. The Formula of a Valid Argument | 有效论证的公式
In deductive logic, a valid argument is one where, if all premises are true, the conclusion must be true. The formula for validity does not depend on the actual truth of the premises—only on the structure. Symbolically, we can represent it as: If (Premise₁ ∧ Premise₂ ∧ ... ∧ Premiseₙ) → Conclusion is a logical necessity. A common short form is: True premises guarantee a true conclusion. Note that a valid argument can still have a false conclusion if at least one premise is false; the 'formula' only describes the relationship, not the real-world facts.
在演绎逻辑中，有效论证意味着如果所有前提为真，结论必然为真。有效性的公式不依赖于前提的实际真值，而仅仅取决于结构。我们可以用符号表示为：若（前提₁ ∧ 前提₂ ∧ … ∧ 前提ₙ）为真，则结论必然为真。一个常见的概括是：真前提能保证真结论。请注意，如果至少一个前提为假，有效论证仍然可能得出假结论；这个“公式”仅仅描述的是推理关系，而非事实真伪。
Valid Argument Formula: P₁, P₂ ... Pₙ ⊢ C (where C cannot be false if all P are true)
2. Modus Ponens: The Most Basic Rule | 肯定前件：最基本的推理规则
Modus ponens (Latin for 'method of affirming') is the simplest valid argument form. Its formula is: If P then Q; P is true; therefore, Q is true. This rule is used in everyday reasoning and rigorous proofs alike. For example: If it is raining (P), then the ground is wet (Q). It is raining (P). Therefore, the ground is wet (Q). In symbolic form, it becomes a mechanical check for validity. As long as the conditional premise and the antecedent are accepted, the consequent follows with logical certainty.
肯定前件（modus ponens，拉丁语意为“肯定的方法”）是最简单的有效论证形式。其公式为：若 P 则 Q；P 为真；故 Q 为真。这条规则既用于日常推理，也用于严密证明。例如：如果下雨（P），那么地面是湿的（Q）。天在下雨（P）。因此，地面是湿的（Q）。用符号表示后，它就成了检验有效性的机械步骤。只要承认条件句前提和前件，后件就必然随之成立。
Modus Ponens: P → Q, P, ∴ Q
3. Modus Tollens: Denying the Consequent | 否定后件：反向推理
Modus tollens ('method of denying') is the complementary valid form. Its formula reads: If P then Q; Not Q; therefore, not P. It is particularly powerful in falsifying hypotheses. For instance: If the battery is charged (P), the phone will turn on (Q). The phone does not turn on (¬Q). Therefore, the battery is not charged (¬P). Notice that denying the antecedent (P is false) does not yield a valid conclusion—that is a fallacy. Modus tollens, however, guarantees validity every time.
否定后件（modus tollens，“否定的方法”）是与肯定前件互补的有效形式。其公式为：若 P 则 Q；非 Q；故非 P。它在证伪假设时尤其有力。例如：如果电池有电（P），手机就能开机（Q）。手机无法开机（¬Q）。因此，电池没电（¬P）。请注意，否定前件（P 为假）并不能推出有效结论——那是一种谬误。而否定后件每次都保证有效性。
Modus Tollens: P → Q, ¬Q, ∴ ¬P
4. The JTB Formula for Knowledge | 知识的 JTB 公式
For epistemologists, the classical definition of knowledge works like a formula: Knowledge = Justified True Belief (JTB). According to this model, a subject S knows a proposition p if and only if: (i) p is true; (ii) S believes that p; (iii) S is justified in believing that p. Each element is necessary, and together they are sufficient—at least until Gettier cases challenged the sufficiency. Despite the challenges, JTB remains the starting point for any discussion of knowledge in IGCSE Philosophy, and it helps you unpack what it means to say 'I know'.
对认识论学者而言，经典的“知识”定义可视为一个公式：知识 = 经过辩护的真信念（JTB）。根据这一模型，主体 S 知道命题 p，当且仅当：（i）p 为真；（ii）S 相信 p；（iii）S 有理由相信 p。三个条件各自是必要的，合起来则构成充分条件——至少直到盖梯尔难题挑战其充分性之前是这样。尽管面临挑战，JTB 仍然是 IGCSE 哲学中讨论知识的出发点，帮助你剖析“我知道”这句话的含义。
Knowledge = Justified True Belief (JTB): Truth + Belief + Justification
5. The Greatest Happiness Principle | 最大幸福原则（功利主义公式）
In ethics, utilitarianism offers a decision-making formula: An action is right if it promotes the greatest happiness for the greatest number. Jeremy Bentham's 'felicific calculus' even attempted to quantify pleasure and pain using seven factors: intensity, duration, certainty, propinquity, fecundity, purity, and extent. While modern utilitarians rarely compute happiness mathematically, the underlying formula remains the central test—calculate the net balance of pleasure over pain, and choose the action with the highest surplus. Exam answers that apply this 'formula' to moral dilemmas are always well-structured.
在伦理学中，功利主义提供了一个决策公式：若一个行为能为最大多数人带来最大幸福，则该行为是正确的。边沁的“苦乐计算法”甚至尝试用七个因子（强度、持续时间、确定性、邻近性、继生性、纯度和范围）来量化快乐和痛苦。虽然现代功利主义者极少真的用数学计算幸福，但底层的公式仍然是核心检验标准——计算快乐减去痛苦后的净余额，选择余额最高的行为。凡是用这条“公式”去分析道德困境的考试答案，结构都会非常清晰。
Utilitarian Formula: Right action = maximise Σ (pleasure – pain) for all affected
6. Kant's Categorical Imperative Formula | 康德的绝对命令公式
Immanuel Kant's ethics can also be expressed as a logical test. The first formulation of the categorical imperative demands: Act only according to that maxim whereby you can at the same time will that it should become a universal law. This functions like a contradiction-detection formula. To apply, formulate the principle behind your action, then ask whether a world where everyone followed that principle is logically conceivable and willable. If the maxim leads to a contradiction in conception or in the will, the action is impermissible. This test makes Kantian ethics uniquely formal and appealing to those who love moral 'mathematics'.
康德的伦理学同样可以表达为一种逻辑测试。绝对命令的第一公式要求：只按照你同时愿意它成为普遍法则的那个准则去行动。这就像一个矛盾检测公式。应用时，先提炼行为背后的原则，然后问自己：在一个所有人都遵循该原则的世界里，这在逻辑上是否可设想、是否可意愿？如果该准则导致概念矛盾或意志矛盾，该行为就是不允许的。这一检验方法让康德伦理学具有独特的形式感，也深受喜爱道德“数学”的人的青睐。
Categorical Imperative: Act only on maxims that can be willed as universal laws without contradiction.
7. The Formula of Syllogism | 三段论公式
A categorical syllogism is a three-line deductive formula consisting of two premises and a conclusion, built from categorical statements (All A are B, No A are B, Some A are B, Some A are not B). The classic example is: All humans are mortal; Socrates is a human; therefore, Socrates is mortal. Validity in syllogisms depends on the distribution of terms and obeying rules such as avoiding the fallacy of the undistributed middle. Memorising the six valid syllogistic forms (Barbara, Celarent, etc.) gives you a formulaic toolkit for testing any argument of this type.
直言三段论是由两个前提和一个结论构成的三段演绎公式，前提和结论都由直言命题（所有 A 是 B，没有 A 是 B，有的 A 是 B，有的 A 不是 B）组成。经典例子是：所有人都会死；苏格拉底是人；所以苏格拉底会死。三段论的有效性取决于词项的周延性，以及是否遵守诸如“中项必须至少周延一次”等规则。记住六种有效三段论格式（如 Barbara、Celarent 等），就等于拥有了一套公式化的工具包，可以用来检验任何此类推理。
Syllogism: All M are P, All S are M, ∴ All S are P
8. Hume's Fork: The Formula of Meaning | 休谟的叉子：意义的公式
David Hume's fork is a razor-sharp sorting tool: statements that are not 'relations of ideas' (analytic truths) or 'matters of fact' (synthetic truths about the world) are, in his view, nothing but sophistry and illusion. The formula works as a meaning test: Does a proposition express a demonstrable logical relation or a verifiable factual claim? If neither, commit it to the flames. This empiricist criterion directly influenced the later verification principle and remains a powerful device for evaluating philosophical claims, especially in metaphysics and theology.
休谟的叉子是一把锋利的分类工具：在他看来，凡不属于“观念的关系”（分析命题）或“事实”（关于世界的综合命题）的陈述，都只不过是诡辩和幻觉。这个公式充当了意义的检验标准：一个命题是表达可证明的逻辑关系，还是一个可验证的事实陈述？如果两者皆非，就把它扔进火里。这一经验主义判准直接影响了后来的证实原则，至今仍是评估哲学主张（尤其是形而上学和神学主张）的有力武器。
Hume' Fork: Meaningful proposition = Relation of Ideas OR Matter of Fact; otherwise meaningless.
9. Fallacy Formulas: Instant Logical Checks | 谬误公式：即时逻辑检查
Logical fallacies are flawed reasoning patterns that often masquerade as valid formulas. Recognising them by their formulaic structure is a core skill. The table below captures several fallacies, their general forms, and quick tests. Treat each row as an anti-formula—when you spot these structures, you know the argument is invalid regardless of the content.
逻辑谬误是常常伪装成有效公式的错误推理模式。通过识别其公式化的结构来辨认它们是一项核心技能。下表列出了几种常见谬误、它们的一般形式以及快速检验方法。把每一行都当作一个“反公式”——一旦发现这些结构，无论具体内容如何，你都知道该论证是无效的。
| Fallacy | Fallacy Formula | Quick Test |
|---|---|---|
| Ad Hominem | Attack the person, not the argument | Is the premise about the arguer rather than the claim? |
| Straw Man | Misrepresent X as Y, then refute Y | Would the opponent accept this as their actual position? |
| False Dilemma | Only A or B; not A, therefore B (when other options exist) | Are there at least three plausible options? |
| Begging the Question | Premises assume the truth of the conclusion | Would someone who doubts the conclusion accept the premises? |
| Post Hoc | A occurred before B, therefore A caused B | Is correlation confused with causation? |
10. Popper's Formula of Falsification | 波普尔的证伪公式
Karl Popper replaced the verification principle with a demarcation criterion based on falsifiability. The 'formula' is: A hypothesis is scientific if and only if there is a possible observation that would prove it false. In logical form: Hypothesis H is scientific if ∃ an observation O such that O makes ¬H true. For example, 'All swans are white' is falsifiable because one black swan contradicts the theory. This test avoids the limitations of induction and gives you a clear line between science and pseudo-science, which is a frequent discussion point in IGCSE Philosophy exams on the nature of knowledge.
波普尔以可证伪性为基础的分界标准取代了证实原则。其“公式”是：一个假设是科学的，当且仅当存在某种可能的观察能够证明它为假。用逻辑形式表达：假说 H 是科学的，当且仅当存在某个观察 O，使得 O 让 ¬H 为真。例如“所有天鹅都是白的”是可证伪的，因为一只黑天鹅就能反驳该理论。这一检验标准避免了归纳法的局限，并在科学与伪科学之间划出了一条清晰的分界线，这也是 IGCSE 哲学考试中关于知识本质的常见讨论点。
Falsification: A theory is scientific ↔ it can be contradicted by a possible empirical observation.
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