Pre-U CIE Philosophy: Essential Formulas & Theorems Quick Reference | Pre-U CIE 哲学：公式定理速查手册

📚 Pre-U CIE Philosophy: Essential Formulas & Theorems Quick Reference | Pre-U CIE 哲学：公式定理速查手册

In Pre-U CIE Philosophy, critical thinking hinges on formal logic structures, key arguments, and principle-based ‘formulae’ that function much like theorems in mathematics. This quick-reference handbook distils the most essential logical rules, argument schemas, ethical calculi, and epistemological models you will encounter across the syllabus. Mastery of these patterns enables you to reconstruct arguments precisely and evaluate them with rigour.

在 Pre-U CIE 哲学中，批判性思维依赖于形式逻辑结构、关键论证以及基于原则的“公式”——它们的功能很像数学中的定理。这本速查手册浓缩了你在整个考纲中会遇到的最重要的逻辑规则、论证图式、伦理计算模型和认识论模型。掌握这些模式，你就能精确地重构论证并严谨地评估它们。

1. Logical Connectives & Truth Tables | 逻辑联结词与真值表

Propositional logic uses five standard connectives: negation (¬), conjunction (∧), disjunction (∨), material implication (→), and biconditional (↔). Their meanings are defined entirely by truth tables. You must be able to translate ordinary language into symbolic form and compute the truth value of compound statements under any interpretation.

命题逻辑使用五种标准联结词：否定（¬）、合取（∧）、析取（∨）、实质蕴涵（→）和双条件（↔）。它们的含义完全由真值表定义。你必须能够将日常语言转译成符号形式，并计算复合陈述在任何解释下的真值。

P	Q	P ∧ Q	P ∨ Q	P → Q	P ↔ Q
T	T	T	T	T	T
T	F	F	T	F	F
F	T	F	T	T	F
F	F	F	F	T	T

The material conditional P → Q is false only when P is true and Q is false; otherwise it is true. This minimal condition often strikes students as counterintuitive, but it is essential for capturing deductive validity. In an argument, we say premises entail the conclusion if and only if there is no row on the truth table where all premises are true and the conclusion is false.

实质蕴涵 P → Q 只有在 P 为真且 Q 为假时才为假，其余情况都为真。这个最低条件常常让初学者觉得反直觉，但它是刻画演绎有效性的关键。在一个论证中，我们说前提蕴含结论，当且仅当真值表中不存在所有前提为真而结论为假的行。

A tautology is a formula that is true under every interpretation, e.g. P ∨ ¬P. A contradiction is false under every interpretation, e.g. P ∧ ¬P. Identifying these helps test consistency and equivalence.

重言式是指在所有解释下都为真的公式，如 P ∨ ¬P。矛盾式是指在所有解释下都为假的公式，如 P ∧ ¬P。识别它们有助于检验一致性和等值关系。

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2. Quantifiers & Predicate Logic | 量词与谓词逻辑

Predicate logic extends propositional logic with quantifiers and predicates. The universal quantifier ∀ reads ‘for all x’, while the existential quantifier ∃ reads ‘there exists an x such that’. These allow us to formalise categorical statements like ‘All humans are mortal’ as ∀x (Hx → Mx).

谓词逻辑通过量词和谓词扩展了命题逻辑。全称量词 ∀ 读作“对所有 x”，存在量词 ∃ 读作“存在一个 x 使得”。这使我们能够将“所有人都会死”这样的直言陈述形式化为 ∀x (Hx → Mx)。

The negation rules for quantifiers are vital: ¬∀x Px is equivalent to ∃x ¬Px, and ¬∃x Px is equivalent to ∀x ¬Px. Mastery of these equivalences prevents errors when translating ‘Not all S are P’ into ∃x (Sx ∧ ¬Px).

量词的否定规则至关重要：¬∀x Px 等价于 ∃x ¬Px，而 ¬∃x Px 等价于 ∀x ¬Px。掌握这些等值关系可以避免在翻译“并非所有 S 都是 P”时出错，它应该形式化为 ∃x (Sx ∧ ¬Px)。

In syllogistic reasoning, the square of opposition captures logical relations between A (universal affirmative), E (universal negative), I (particular affirmative) and O (particular negative) propositions. For the Pre-U exam, you should be able to test validity using Venn diagrams or by translating into predicate logic and checking for countermodels.

在直言三段论推理中，对当方阵刻画了 A（全称肯定）、E（全称否定）、I（特称肯定）和 O（特称否定）命题之间的逻辑关系。对于 Pre-U 考试，你应该能够运用维恩图检验有效性，或者将论证翻译成谓词逻辑并检查是否存在反模型。

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3. Modus Ponens & Modus Tollens | 肯定前件与否定后件

Modus ponens (MP) is the fundamental deductive rule: from P → Q and P, infer Q. Symbolically:

肯定前件（MP）是最基本的演绎规则：从 P → Q 和 P 推出 Q。符号表示为：

P → Q, P ∴ Q

Modus tollens (MT) is its valid counterpart: from P → Q and ¬Q, infer ¬P. It denies the consequent to derive the negation of the antecedent.

否定后件（MT）是与之对应的有效式：从 P → Q 和 ¬Q 推出 ¬P。它通过否定后件来得出前件的否定。

P → Q, ¬Q ∴ ¬P

Beware of the formal fallacies: affirming the consequent (P → Q, Q ∴ P) and denying the antecedent (P → Q, ¬P ∴ ¬Q). These are invalid patterns that frequently appear in everyday reasoning and must be recognised in critical analysis.

小心形式谬误：肯定后件（P → Q, Q ∴ P）和否定前件（P → Q, ¬P ∴ ¬Q）。这些无效模式经常出现在日常推理中，在批判性分析时必须识别出来。

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4. Hypothetical Syllogism & Disjunctive Syllogism | 假言三段论与选言三段论

Hypothetical syllogism (HS) chains conditionals: from P → Q and Q → R, we can infer P → R. This rule is indispensable when reconstructing long chains of reasoning, such as in cosmological arguments where a series of dependencies leads to a first cause.

假言三段论（HS）将条件句连锁起来：从 P → Q 和 Q → R，我们可以推出 P → R。这条规则在重构长串推理时不可或缺，比如在宇宙论论证中，一系列依赖关系导致第一因。

P → Q, Q → R ∴ P → R

Disjunctive syllogism (DS) works with ‘or’: from P ∨ Q and ¬P, infer Q. The rule relies on the exclusive sense that at least one disjunct must be true, so eliminating one forces the other.

选言三段论（DS）处理“或”：从 P ∨ Q 和 ¬P 推出 Q。该规则依赖于析取句“至少有一真”的含义，因此排除一个肢命题就迫使另一个为真。

P ∨ Q, ¬P ∴ Q

In philosophical contexts, these simple forms are the building blocks of more complex arguments, including dilemmas. A constructive dilemma, for instance, combines disjunction and conditionals: (P → Q) ∧ (R → S), P ∨ R ∴ Q ∨ S. Recognising such patterns speeds up evaluation under timed conditions.

在哲学语境中，这些简单形式是更复杂论证——包括二维推论——的建构模块。例如，建设性二维推论结合了析取与条件句：(P → Q) ∧ (R → S), P ∨ R ∴ Q ∨ S。识别这些模式可以加快限时条件下的评估速度。

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5. Reductio ad Absurdum | 归谬法

Reductio ad absurdum (RAA) is a powerful proof strategy: to demonstrate that a proposition P is true, assume its negation ¬P, derive a contradiction (Q ∧ ¬Q), and then conclude P. This method is the engine behind many classical philosophical arguments, from Zeno’s paradoxes to theistic proofs.

归谬法（RAA）是一种强有力的证明策略：为了证明命题 P 为真，先假设其否定 ¬P，推导出一个矛盾（Q ∧ ¬Q），然后由此得出结论 P。从芝诺悖论到有神论证明，这种方法是许多经典哲学论证的引擎。

Assume ¬P, derive Q ∧ ¬Q, therefore P.

In formal natural deduction, RAA allows you to discharge an assumption once a contradiction is reached. It is frequently used in the Pre-U logic paper to prove theorems that cannot be shown by direct derivation alone. Be careful to cite line numbers and indicate the closure of the assumption scope.

在形式自然演绎中，当达到矛盾后，归谬法允许你消解一个假设。它经常用于 Pre-U 逻辑考卷中，以证明那些无法仅靠直接推导建立的定理。注意标明行号，并标示出假设范围的闭合。

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6. Cartesian Foundationalism: ‘Cogito’ | 笛卡尔基础主义：“我思”

Descartes’ cogito ergo sum (I think, therefore I am) can be expressed as an indubitable foundational truth. In logical form, it is a performative insight: the very act of doubting one’s existence confirms that there is a doubter. Formally: Dx → Ex (If x doubts, x exists). The instantiation for the self yields the certainty of one’s own existence as a thinking thing.

笛卡尔的“我思故我在”可以被表述为一个无可怀疑的基础真理。从逻辑形式看，它是一种践言性洞见：怀疑自己存在这一行为本身就证实了存在着一个怀疑者。形式化为：Dx → Ex（如果 x 怀疑，则 x 存在）。将这一例示应用于自我，就产生了作为思维者的自身存在的确定性。

∀x (Dx → Ex), Da ∴ Ea

However, the cogito is not a formal syllogism in Descartes’ own view; it is a direct intuition. For the exam, you should contrast foundationalism with coherentism and assess whether the cogito succeeds as a basic belief immune to sceptical doubt.

不过，在笛卡尔本人看来，我思并非一个形式三段论，而是一种直接的直观。在考试中，你应该对比基础主义与融贯论，并评估“我思”作为免受怀疑论侵蚀的基本信念是否成功。

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7. Anselm’s Ontological Argument in Logical Form | 安瑟伦本体论论证的逻辑形式

Anselm’s Proslogion argument defines God as ‘that than which nothing greater can be conceived’. The logical structure can be reduced to a reductio: assume God exists only in the understanding. A being that exists in reality is greater. Therefore, we can conceive of something greater than God — a contradiction. Hence, God must exist in reality.

安瑟伦在《宣讲》中的论证将上帝定义为“那无法设想有比之更伟大者”。其逻辑结构可以归结为一个归谬推理：假设上帝仅存在于理智中。存在于实在中的存在者更伟大。因此，我们能够设想一个比上帝更伟大的东西——这就产生了矛盾。所以，上帝必须存在于实在中。

1. God =ᵈᶠ the being than which no greater can be conceived.
2. Assume God exists only in intellectu.
3. We can conceive of a being with all God’s properties plus existence in re (greater).
4. But that would be greater than God — contradiction with 1.
5. ∴ God exists in re.

Gaunilo’s ‘Lost Island’ parody attempts to show the argument’s logical form leads to absurdity when applied to contingent objects. The Pre-U paper often asks you to evaluate whether existence is a real predicate, following Kant’s criticism that ‘being’ is not a perfection.

高尼罗的“迷失岛”戏仿试图表明，当把该论证的逻辑形式应用于偶性对象时，会导致荒谬。Pre-U 试卷经常要求你评价存在是否是一个实在的谓词，这正是沿袭康德“‘是’不是实在的谓词”的批评。

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8. Hume’s Fork & the Problem of Induction | 休谟之叉与归纳问题

Hume’s Fork divides all possible knowledge into two sorts: relations of ideas and matters of fact. Relations of ideas are a priori, necessary, and discoverable by thought alone (e.g. 2 + 2 = 4). Matters of fact are a posteriori, contingent, and known through experience.

休谟之叉将所有可能的知识分为两类：观念关系与实际事情。观念关系是先天的、必然的，单凭思维即可发现（如2 + 2 = 4）。实际事情是后天的、偶性的，需要通过经验来认识。

The problem of induction arises because we assume the future will resemble the past. That ‘the sun will rise tomorrow’ is not a relation of ideas, nor can it be justified by past experience without circularity, because any appeal to the uniformity of nature itself relies on induction. Formally:

归纳问题之所以产生，是因为我们假定未来会与过去相似。“太阳明天会升起”既不是一个观念关系，也无法不循环地凭借过去经验得到辩护，因为任何诉诸自然齐一性的推理本身就依赖归纳。形式化如下：

P1: All inductive inferences presuppose the Uniformity Principle (UP).
P2: UP cannot be justified deductively (not a relation of ideas).
P3: UP cannot be justified inductively without circularity.
C: ∴ Induction has no rational justification.

This sceptical conclusion challenges the logical empiricists and remains central to philosophy of science. Responses like Popper’s falsificationism or pragmatic justifications are important critical developments you should be ready to discuss.

这一怀疑论结论挑战了逻辑经验主义者，并且仍是科学哲学的核心议题。像波普尔的证伪主义或实用主义辩护等回应，是你需要准备好讨论的重要批判性发展。

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9. Kant’s Categorical Imperative Formulations | 康德绝对命令的公式

Kant’s ethical ‘formula’ for the categorical imperative provides a decision procedure for moral action. The Formula of Universal Law (FUL) states: ‘Act only on that maxim whereby you can at the same time will that it should become a universal law.’ In logical terms, a maxim is morally permissible if it can be consistently universalised without contradiction.

康德的绝对命令“公式”为道德行为提供了一个决策程序。普遍法则公式（FUL）陈述为：“只依照你同时能够愿意它成为一条普遍法则的那个准则去行动。”用逻辑术语说，如果一个准则能够不矛盾地被普遍化，那它就是道德上允许的。

The contradiction may be in conception (a perfect duty, such as not making false promises) or in the will (an imperfect duty, such as developing talents). Symbolically, for a maxim M, the test is: ∀x (Mx is universalisable without contradiction) → permissible(Mx).

矛盾可能出现在构想中（完全义务，如不作虚假承诺），也可能出现在意愿中（不完全义务，如发展才能）。用符号表示，对于一个准则 M，检验是：∀x（Mx 被普遍化且不矛盾）→ permissible(Mx)。

The Formula of Humanity (FH) commands: ‘Act so that you treat humanity, whether in your own person or in that of another, always as an end and never merely as a means.’ This provides a different test of respect for rational agency, often used in bioethical debates in the exam.

人性公式（FH）命令道：“你要如此行动，即无论是你自身中的人性还是他人中的人性，你总是同时当作目的，绝不仅仅当作手段。”这提供了另一种对理性行动者的尊重检验，在考试中常被用于生命伦理讨论。

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10. Utilitarian Calculus (Bentham’s Hedonic Calculus) | 功利主义计算（边沁的快乐计算）

Bentham’s hedonic calculus is a quantitative ‘formula’ for measuring the moral rightness of an action based on the utility (pleasure minus pain) it produces. The calculus considers seven dimensions of a pleasure or pain: Intensity, Duration, Certainty, Propinquity (nearness), Fecundity (chance of further pleasures), Purity (chance of not being followed by pain), and Extent (number affected).

边沁的快乐计算是一种定量“公式”，根据行动所产生的效用（快乐减去痛苦）来衡量其道德正当性。该计算考量快乐或痛苦的七个维度：强度、持续时间、确定性、临近性（远近）、丰产性（产生更多快乐的机会）、纯真性（不被痛苦跟随的机会）以及广延性（受影响的人数）。

Utility(U) = Σ (Intensity × Duration × Certainty × Propinquity × Fecundity × Purity × Extent) for all individuals affected.

While this looks like a proper equation, it is notoriously difficult to operationalise. Mill’s qualitative hedonism later distinguishes higher and lower pleasures, refining the calculus. For Pre-U, you should be able to apply the utilitarian calculus to thought experiments like the Trolley Problem and discuss whether it provides a complete decision procedure.

虽然这看起来像是一个正规的方程，但它很难操作化。后来密尔的质的快乐论区分了高级快乐与低级快乐，从而改进了这一计算。对于 Pre-U，你应该能够将功利主义计算应用于像电车难题这样的思想实验，并讨论它是否提供了一个完备的决策程序。

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11. The Principle of Charity & Informal Fallacies | 宽容原则与非形式谬误

The Principle of Charity is a hermeneutic rule: when interpreting an argument, you should reconstruct it in its strongest, most rational form before criticizing it. Formally, given an opponent’s statements S, select the interpretation I that maximises the logical validity and truth of the premises while remaining faithful to the author’s intent.

宽容原则是一条解释学规则：在解释一个论证时，你应该先将其重构成最强、最理性的形式，然后再加以批评。形式化地说，给定对手的陈述 S，选择一个解释 I，它在忠实于作者意图的前提下，最大化论证的逻辑有效性和前提的真理性。

Violations of charity often lead to straw man fallacies (misrepresenting an argument to refute it easily). Common informal fallacies to memorise for the exam: ad hominem, appeal to emotion, false dilemma, begging the question, equivocation, and slippery slope. Each can be analysed by identifying the hidden premise and testing its truth or relevance.

违反宽容原则常常导致稻草人谬误（曲解一个论证以便轻易反驳它）。考试中需要记住的常见非形式谬误包括：人身攻击、诉诸情感、假二维推论、乞题、一词多义和滑坡论证。每一个都可以通过识别隐含前提并检验其真值或相关性来加以分析。

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12. Gettier Cases & the JTB Analysis | 盖蒂尔案例与JTB分析

The traditional analysis of knowledge defines it as justified true belief (JTB): S knows that p if and only if (i) p is true, (ii) S believes that p, and (iii) S is justified in believing that p. This can be written as a biconditional: Ksp ↔ (p ∧ Bsp ∧ Jsp).

传统知识论把知识定义为经过辩护的真信念（JTB）：S 知道 p，当且仅当 (i) p 为真，(ii) S 相信 p，且 (iii) S 有理由地相信 p。这可以写成一个双条件式：Ksp ↔ (p ∧ Bsp ∧ Jsp)。

Edmund Gettier’s 1963 counterexamples showed that JTB is not sufficient. In a typical Gettier case, S has a justified false belief from which S deduces a true conclusion due to luck. Formally, S’s justification for p is strong but p is false; however, p ∨ q (where q is true by chance) becomes something S justifiably believes and is true — yet it is not knowledge.

埃德蒙·盖蒂尔1963年的反例表明 JTB 并不充分。在典型的盖蒂尔案例中，S 持有一个经过辩护但为假的信念，并由此推导出一个因运气而成真的结论。形式化地说，S 对 p 的辩护很强但 p 为假；然而 p ∨ q（其中 q 碰巧为真）就成了 S 有理由相信且为真的东西——但它并非知识。

1. S has justified false belief p.
2. S competently deduces p ∨ q from p.
3. Unbeknownst to S, q is true.
4. ∴ S has a justified true belief that (p ∨ q) but does not know that (p ∨ q).

The post-Gettier industry offers repairs: adding a ‘no false lemmas’ condition, reliabilism, or causal theories. You should be able to explain why such cases undermine the tripartite definition and assess proposed solutions.

后盖蒂尔的知识论产业提供了各种修补方案：添加“无错误引理”条件、可靠主义或因果理论。你应当能够解释为什么这类案例会动摇三元定义，并评估提出的解决方案。

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