📚 GCSE CIE Philosophy: Formula & Theorem Quick Reference Guide | GCSE CIE 哲学：公式定理速查手册
This quick reference guide distils the essential logical forms, key arguments and foundational principles into a 'formula and theorem' style handbook. It covers deductive and inductive reasoning, validity, common fallacies, landmark philosophical arguments and ethical theories, all tailored to the CIE GCSE Philosophy syllabus. Each entry is presented as a core "formula" or "theorem" to help you memorise and apply them effectively in your exam.
本速查手册将基本逻辑形式、关键论证及基础原理浓缩为“公式与定理”风格的手册。内容涵盖演绎与归纳推理、有效性、常见谬误、里程碑式的哲学论证和伦理理论，均紧扣 CIE GCSE 哲学考纲。每个条目均以核心“公式”或“定理”的形式呈现，助你高效记忆并在考试中灵活运用。
1. Deductive Argument Forms | 演绎论证形式
Modus Ponens (Affirming the Antecedent): P → Q, P ∴ Q
Modus ponens is a valid deductive form. If a conditional statement 'If P then Q' true. It allows you to detach the conclusion Q from the premise P → Q once P is affirmed.
肯定前件式是一种有效演绎形式。如果条件句“若 P 则 Q”为真，且前件 P 成立，则后件 Q 必然为真。一旦 P 被肯定，就可以从前提 P → Q 中分离出结论 Q。
Modus Tollens (Denying the Consequent): P → Q, ¬Q ∴ ¬P
Modus tollens is another valid form. Given that P implies Q, if Q is observed to be false, then P must also be false. It works by eliminating the antecedent through the falsity of the consequent, and it is a common tool for testing the consequences of a hypothesis.
否定后件式是另一种有效形式。已知 P 蕴含 Q，如果观察到 Q 为假，那么 P 也必然为假。它通过后件的虚假性来消除前件，是检验假说推论结果的常用工具。
Hypothetical Syllogism: P → Q, Q → R ∴ P → R
This form chains two conditional statements. If P implies Q and Q implies R, then P directly implies R. It demonstrates the transitivity of implication, allowing you to shorten a chain of reasoning into a single conditional step.
这一形式将两个条件句串联起来。若 P 蕴含 Q 且 Q 蕴含 R，则 P 直接蕴含 R。它展示了蕴含的传递性，可将一长串推理精简为单个条件步骤。
Disjunctive Syllogism: P ∨ Q, ¬P ∴ Q
When you have a disjunction (an 'either… or' statement) and one of the disjuncts is denied, the other must be true. This form is valid only if the disjunction is inclusive or exclusive, but it remains a simple elimination rule for alternatives.
当你有一个选言命题（非此即彼）并且其中一个选言支被否定时，另一个必然为真。无论选言是相容还是不相容，这一形式都是有效的替代排除规则。
2. Inductive Reasoning and the Problem of Induction | 归纳推理与归纳问题
Inductive reasoning moves from specific observations to general conclusions. It does not guarantee truth; rather, it makes the conclusion probable based on the evidence. Inductive strength depends on sample size, representativeness and the absence of defeating counter-instances.
归纳推理从具体观察出发，走向普遍结论。它不保証结论为真，而是基于证据使结论具有或然性。归纳的强度取决于样本大小、代表性以及是否存在反例。
The problem of induction, famously articulated by David Hume, points out that inductive reasoning relies on the assumption that the future will resemble the past. This assumption cannot be justified without circularity or without appealing to induction itself, leaving a deep sceptical gap about causal inferences and predictions.
大卫·休谟提出的归纳问题深刻指出，归纳推理依赖于“未来将符合过去”这一假设。这一假设无法在不陷入循环或不诉诸归纳本身的情况下得到辩护，从而给因果推论和预测留下了深刻的怀疑论裂缝。
3. Validity and Soundness | 有效性与可靠性
In deductive logic, an argument is valid if and only if it is impossible for the premises to be true and the conclusion false. Validity concerns the structural connection between premises and conclusion, not the actual truth of the statements.
在演绎逻辑中，一个论证是有效的，当且仅当前提为真而结论为假是不可能的。有效性关乎前提与结论之间的结构关联，而非陈述的实际真假。
An argument is sound if it is both valid and all its premises are actually true. Soundness guarantees a true conclusion. A deductive argument can be valid but unsound if one or more premises are false, while an inductive argument can be strong but uncogent.
一个论证是可靠的，当且仅当它既有效且所有前提实际为真。可靠性确保结论为真。一个演绎论证可能有效却不可靠（若前提为假），而一个归纳论证可能很强但不具说服力。
For example, 'All cats are reptiles; Fluffy is a cat; therefore Fluffy is a reptile' is valid in structure but unsound because the first premise is false. Recognising this distinction is essential for critically evaluating philosophical arguments.
例如，“所有的猫都是爬行动物；毛毛是猫；因此毛毛是爬行动物”在结构上有效，但并不可靠，因为第一个前提为假。理解这一区别对于批判性地评估哲学论证至关重要。
4. Common Logical Fallacies | 常见逻辑谬误
Ad hominem: Attacking the person making the argument rather than the argument itself. Example: 'You can't trust her views on climate change because she drives a big car.'
人身攻击：攻击提出论証的人，而不是论証本身。例如：“你不能相信她对气候变化的看法，因为她开大排量车。”
Straw man: Misrepresenting or exaggerating an opponent's position to make it easier to attack. Example: Saying that someone who supports reducing military spending wants to leave the country defenceless.
稻草人谬误：歪曲或夸大对方的立场，以便更容易攻击。例如：说一个支持削减军费的人就是想置国家于无防御之地。
False dilemma: Presenting only two alternatives when more are available. Example: 'Either we cut all environmental regulations, or the economy will collapse.' The reality may offer a spectrum of options.
假两难谬误：只呈现两种选择，而实际上存在更多可能。例如：“要么我们取消所有环境法规，要么经济就会崩溃。”现实可能提供多种备选方案。
Begging the question (circular reasoning): Assuming the conclusion in the premises. Example: 'The Bible is the word of God because the Bible says so.' The argument provides no independent support.
乞题谬误（循环推理）：在前提中假设了结论。例如：“圣经是上帝的话，因为圣经是这么说的。”该论证并未提供独立的支持。
Slippery slope: Arguing that a relatively small first step will inevitably lead to a chain of events ending in a dire outcome, without sufficient justification for the inevitability. Example: 'If we allow students to redo one exam, soon they'll demand to redo every assignment, and standards will collapse.'
滑坡谬误：主张一个相对较小的第一步将不可避免地引发一连串事件，最终导致灾难性后果，却没有充分证明其必然性。例如：“如果允许学生重考一次，很快他们就会要求重做所有作业，标准就会崩溃。”
5. Descartes' Cogito Argument | 笛卡尔的我思论证
Cogito ergo sum: I think, therefore I am (or more precisely, I am, I exist, is necessarily true whenever it is put forward by me or conceived in my mind).
Descartes' foundational 'theorem' emerges from his method of radical doubt. He imagines an evil demon deceiving him about everything. Even if he is deceived about the external world, his own thinking cannot be doubted, because doubting is itself a form of thinking. The very act of doubting confirms the existence of the self as a thinking thing.
笛卡尔的基础“定理”诞生于他的彻底怀疑方法。他设想一个邪恶精灵在一切事情上欺骗他。即使他被外部世界所欺骗，他自己的思考本身却无法被怀疑，因为怀疑本身正是一种思考。怀疑这一行为本身便确认了作为思维实体的自我的存在。
The cogito does not prove the existence of a physical body, but establishes a first certainty: a thinking substance exists. It serves as the Archimedean point from which Descartes reconstructs knowledge, though critics question whether it moves beyond a momentary subjective certainty.
“我思”并未证明物理身体的存在，而是建立了第一个确定性：一个思维实体存在。它充当了阿基米德支点，笛卡尔借此重构知识大厦，尽管批评者质疑它是否超越了瞬间的主观确定性。
6. Hume's Fork and Causality | 休谟的叉子与因果关系
Hume divides all meaningful propositions into two categories: Relations of Ideas (analytic, known by pure reason, such as '2+2=4') and Matters of Fact (synthetic, known through experience, such as 'the sun will rise tomorrow'). This is known as Hume's Fork.
休谟将所有有意义的命题分为两类：观念的关系（分析的，通过纯粹理性获知，如“2+2=4”）和事实的命题（综合的，通过经验获知，如“太阳明天会升起”）。这就是休谟的叉子。
Our belief in causality derives from observing the constant conjunction of two events, but we never perceive a necessary connection. The idea that one event produces another is a habit of the mind, not a rationally demonstrable truth. This sceptical conclusion challenges scientific and everyday reasoning about cause and effect.
我们对因果关系的信念源自反复观察到两个事件的恒常连结，但我们从未感知到必然联系。认为一件事产生另一件事，只是心灵的习惯，而非可被理性证明的真理。这种怀疑论结论挑战了科学及日常生活中的因果推理。
7. Utilitarianism: The Greatest Happiness Principle | 功利主义：最大幸福原则
The Greatest Happiness Principle: An action is right if it tends to promote the greatest happiness of the greatest number.
Jeremy Bentham's classical utilitarianism treats happiness as pleasure and the absence of pain. He proposed a 'felicific calculus' to weigh the intensity, duration, certainty and proximity of pleasures and pains across all affected individuals. The moral agent must act as an impartial spectator, counting each person's happiness equally.
边沁的古典功利主义将幸福视为快乐和痛苦的缺失。他提出“快乐计算法”来衡量所有受影响个体的快乐与痛苦的强度、持续时间、确定性、远近程度等因素。道德行为者须作为不偏不倚的旁观者，将每个人的幸福一视同仁地纳入计算。
John Stuart Mill refined the theory by distinguishing higher (intellectual, moral) pleasures from lower (bodily) pleasures. He argued that 'it is better to be a human being dissatisfied than a pig satisfied', thereby injecting qualitative considerations into the quantitative calculus.
约翰·斯图尔特·密尔改进了这一理论，区分了高级（理智的、道德的）快乐和低级（肉体的）快乐。他认为“做一个不满足的人胜于做一头满足的猪”，从而在量的计算中注入了质的考量。
8. Kant's Categorical Imperative (Formula of Universal Law) | 康德的定言令式（普遍法则公式）
Formula of Universal Law: Act only on that maxim whereby you can at the same time will that it should become a universal law.
Kant's categorical imperative is an unconditional moral law, derived from pure practical reason. To test an action, you first formulate its maxim (the principle behind it), and then ask whether you can consistently will that everyone should act on that same maxim without contradiction.
康德的定言令式是无条件的道德律，源自纯粹实践理性。要检验一个行为，首先需构建其准则（行为背后的原则），然后自问：你能否始终一贯地意愿所有人依据同一准则行动而不陷入矛盾？
A contradiction arises in two ways: contradiction in conception (the maxim cannot even be thought as a universal law, e.g., making a false promise to get money) and contradiction in willing (you cannot rationally will the maxim to be universal, e.g., refusing to help others while expecting help when in need). Kant also offers the Formula of Humanity: '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.'
矛盾有二：概念矛盾（该准则根本无法被设想为普遍法则，如为谋财而作虚假承诺）和意愿矛盾（你无法理性地意愿该准则普遍化，如在需要帮助时拒绝帮助他人却又期待他人相助）。康德还提出人性公式：“你要如此行动，即无论是对待你自己还是他人的人性，总是将其作为目的，而绝不仅仅作为手段。”
9. The Teleological Argument (Design Argument) | 目的论论证（设计论证）
The design argument, in its classical form, is an inductive argument. William Paley's watchmaker analogy: if you found a watch on a heath, its intricate arrangement of parts for a purpose would compel you to infer an intelligent watchmaker. By analogy, the complexity and order of the universe, especially living organisms, point to an intelligent divine designer.
设计论证的经典形式是一种归纳论证。威廉·帕里的钟表匠类比：如果在一片荒地上发现一块表，其零件为特定目的而精密组合，你会必然推论出存在一位智慧的钟表匠。同理，宇宙的复杂性与秩序，特别是生命体，指向了一位智慧的神圣设计者。
Hume's criticisms, delivered through the character Philo, challenge this analogy: the universe is not sufficiently like a human artefact; the analogy would at best suggest a finite, imperfect designer, perhaps multiple designers; and the presence of evil and disorder undermines the inference to a benevolent, all-powerful God. The argument is also challenged by evolutionary biology, which offers a non-intentional explanation for apparent design.
休谟借斐罗之口提出的批评挑战了这一类比：宇宙并不充分类似于人造物品；类比充其量只会指向一个有限、不完美的设计者，甚至多个设计者；而恶与混乱的存在削弱了推导出一位全能至善上帝的可能。进化生物学也提供了对表面设计现象的非意图论解释，进一步挑战该论证。
10. The Problem of Evil (Logical and Evidential Forms) | 恶的问题（逻辑与证据形式）
Logical Problem: If God is all-powerful, all-knowing, and all-good, then evil would not exist. Evil exists. Therefore, such a God does not exist.
The logical problem of evil claims an inconsistency in the theistic set of beliefs. An all-powerful God could prevent evil, an all-knowing God would know about it, and an all-good God would want to prevent it. The existence of any evil at all seems to contradict the existence of such a being.
逻辑形式的恶的问题指出有神论信仰集合中存在不一致。全能的神有能力阻止恶，全知的神知晓恶的存在，全善的神愿意阻止恶。任何恶的存在看似都与这样一位存在者的存在相矛盾。
The free will defence, championed by Alvin Plantinga, argues that it is logically possible that God could not create a world with morally free creatures who always choose good, because true freedom necessarily includes the possibility of moral evil. Thus, the co-existence of God and moral evil may not be logically contradictory, though the evidential problem of evil points to the sheer amount and distribution of seemingly pointless suffering, which makes God's existence improbable.
阿尔文·普兰丁格所倡导的自由意志辩护主张，上帝不可能创造一个拥有总是选择善的道德自由受造物的世界，而同时不违反逻辑，因为真自由必然包含作恶的可能。因此，上帝与道德恶的并存或许并非逻辑矛盾，但证据形式的恶的问题指出，大量看似毫无意义的苦难及其分布使得上帝的存在不太可能。
Published by TutorHao | Philosophy Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图