Year 12 CCEA Philosophy: Formula & Theorem Quick-Reference Handbook | CCEA 哲学公式定理速查手册

📚 Year 12 CCEA Philosophy: Formula & Theorem Quick-Reference Handbook | CCEA 哲学公式定理速查手册

This handbook distils the core arguments, principles, and logical structures of the CCEA Year 12 Philosophy syllabus into clearly stated ‘formulae’ and ‘theorems’. By presenting each concept as a formalised statement – often employing logical symbols and step-by-step reasoning – students can quickly recall the essential moves in any essay. From Anselm’s ontological proof to Kant’s categorical imperative, every entry highlights the premise-conclusion form that examiners expect.

本手册将 CCEA 12 年级哲学教学大纲中的核心论证、原则和逻辑结构提炼为清晰的“公式”与“定理”。通过将每个概念表述为使用逻辑符号与分步推理的形式化陈述，学生可以在写作中快速回忆关键环节。从安瑟伦的本体论证明到康德的定言令式，每个条目都突出考纲所要求的前提-结论形式。

1. Anselm’s Ontological ‘Formula’ | 安瑟伦的本体论“公式”

Anselm’s a priori argument defines God as ‘that than which nothing greater can be conceived’ and attempts to derive existence from the very concept. The formula collapses the distinction between existing in the understanding and existing in reality by appealing to the impossibility of a greatest conceivable being lacking real existence.

安瑟伦的先天论证将上帝定义为“不能设想有比之更伟大的存在者”，并试图从概念本身推导出存在。该公式通过指出“最伟大的可设想者缺乏现实存在”这一不可能性，消除了存在于理解与存在于现实之间的区别。

God =df the being than which nothing greater can be conceived (G).
1. G exists in the understanding (Gᵤ).
2. If (Gᵤ ∧ ¬Gᵣ) then a greater being can be conceived (∃x: x > G).
3. By definition, ¬∃x (x > G).
∴ Gᵣ (God exists in reality).

The reductio shows that denying real existence generates a contradiction, so the theist claims necessary existence. Gaunilo’s ‘perfect island’ parody challenges the move from concept to reality, insisting that the logic works only if we already accept the concept’s maximal greatness entails existence.

该归谬法表明，否认现实存在会产生矛盾，因此有神论者主张上帝必然存在。高尼罗的‘完美岛屿’戏仿挑战了从概念到现实的推导，坚持认为只有在已经接受概念的极大伟大蕴含存在时，这种逻辑才有效。

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2. Paley’s Teleological Theorem | 佩利的设计论证定理

William Paley’s analogical argument treats the universe as a machine exhibiting ‘marks of design’, and infers a divine watchmaker. The core theorem compares organized complexity to a watch found on a heath, whose parts are arranged to serve a purpose.

威廉·佩利的类比论证将宇宙视为展现“设计痕迹”的机器，并推断出一位神圣的钟表匠。其核心定理将有机的复杂性比作荒原上拾到的一块怀表，其部件为某个目的而排列。

Theorem (Design Inference):
If an object O displays complex, purpose-ordered parts that work together to produce a function, then O is the product of an intelligent designer.
The universe (and its parts, e.g., the eye) displays such complexity.
∴ The universe has an intelligent designer (God).

Hume’s objections, which pupils must know, expose that the analogy is weak: the universe is not sufficiently like a watch for us to conclude a single, perfect mind. Evolution by natural selection later offered an alternative explanation for apparent design without a designer.

学生们必须了解休谟的反驳，他指出类比是薄弱的：宇宙与怀表的相似程度远不足以推断出唯一、完美的智能。后来的自然选择演化论为没有设计者的外观设计提供了替代解释。

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3. Aquinas’s Cosmological Causation Chain | 阿奎那的宇宙论因果链

Aquinas’s Second Way and the argument from motion depend on the principle that an infinite regress of essentially dependent causes is impossible. The ‘first cause’ theorem shows that any series of causally dependent beings must terminate in a first uncaused cause.

阿奎那的“第二路”以及基于运动的论证依赖于一条原则：本质上依赖的原因不能无限回溯。“第一因”定理表明，任何因果依赖的存在序列都必须终止于一个第一自因。

Cosmological Chain Formula:
1. Some things are in motion / are caused (empirical fact).
2. Whatever is moved is moved by another, and no causal series of essentially ordered causes can proceed to infinity.
3. If every cause depended on a prior cause, an infinite regress would result in no first cause, and thus no intermediate or ultimate effects.
∴ There must exist a first unmoved mover / uncaused cause = God.

The key distinction is between accidentally ordered causes (a grandfather dies but his grandson may live) and essentially ordered causes (a hand moves a stick that moves a stone – remove the hand, the motion stops). Aquinas’s proof relies on essentially ordered series where each member borrows its causal power simultaneously.

这里的关键区分是偶然排序的原因（祖父去世，孙子仍可存活）与本质上排序的原因（手推动棍子，棍子推动石头——若移开手，运动立即停止）。阿奎那的证明依赖于每一环节同时借用因果力量的本质序列。

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4. Hume’s Fork | 休谟的叉子

Hume’s Fork is a criterion of meaning and knowledge: all legitimate propositions fall into one of two classes – relations of ideas and matters of fact. This theorem underpins Humean empiricism and his attack on metaphysics.

休谟的叉子是一项意义与知识标准：所有合法的命题都属于两类之一——观念的关系与事实。这一定理支撑着休谟的经验主义以及他对形而上学的攻击。

Hume’s Fork:
∀ proposition p: p is cognitively meaningful →
(p is a relation of ideas, demonstrable a priori, whose denial is contradictory) ∨
(p is a matter of fact, known a posteriori, whose denial is possible).
If p satisfies neither, it is ‘sophistry and illusion’.

When applied to philosophical theology, the Fork demands that claims about God’s existence be empirically verified or logically demonstrated. Since they often fail both, Hume recommends consigning them to the flames. This tool is essential for any CCEA essay evaluating religious language or the limits of reason.

当该叉子应用于哲学神学时，它要求关于上帝存在的断言要么被经验证实，要么被逻辑演绎。由于它们往往无法通过任何一叉，休谟建议将其付之一炬。这一工具对于任何评估宗教语言或理性界限的 CCEA 论文都至关重要。

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5. The Problem of Induction | 归纳问题

Hume argues that the principle of induction – that the future will resemble the past – cannot be rationally justified. The problem forms a central sceptical concern in the Reason and Experience unit.

休谟主张，归纳原则——即未来将与过去相似——无法得到理性辩护。该问题是“理性与经验”单元中的一项核心怀疑论议题。

Inductive Justification Formula:
Let I = ‘The future will resemble the past’.
To justify I, we need either a deductive proof or an inductive argument.
Deduction fails: it is not logically necessary, as denying I implies no contradiction.
Induction fails: any inductive justification of I would itself rely on I, making it circular.
∴ I is rationally unjustified; our reliance on induction is a matter of custom or habit.

This humbles the claims of science and everyday reasoning. Philosophers have responded by rejecting Hume’s demand (Strawson: induction is conceptually basic) or by trying to salvage probability (Ramsay, Bayes). In your exam, always show that you grasp the circularity objection.

这让科学与日常推理的断言变得谦卑。哲学家们或者否定休谟的要求（斯特劳森：归纳是概念上原初的），或者尝试用概率挽救它（拉姆齐、贝叶斯）。考试中一定要展现你把握了循环反驳的能力。

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6. Kant’s Categorical Imperative | 康德的定言令式

Kant’s supreme principle of morality is formulated as a universal law test. The categorical imperative provides a decision procedure that does not depend on consequences but on the logical form of the maxim.

康德的最高道德原则被表述为一项普遍法则测试。定言令式提供了一种不依赖于后果而只取决于准则逻辑形式的决策程序。

Categorical Imperative (Formula of Universal Law):
Act only according to that maxim whereby you can at the same time will that it should become a universal law.
Impermissible: a maxim that yields a contradiction in conception (e.g., false promise) or a contradiction in the will (e.g., refusing to help others) when universalised.

The formula treats rational beings as ends in themselves: ‘Act in such a way that you always treat humanity, whether in your own person or in the person of any other, never simply as a means, but always at the same time as an end.’ For a CCEA answer link both formulations to show how Kant derives duties from pure reason.

该程式将理性存在者视为目的本身：“你要如此行动，即无论是你自身中的还是任何他人身上的人性，都永远不只被用作手段，而总是同时被视作目的。”在 CCEA 答案中，联系这两个程式可以展示康德如何从纯粹理性中推导出义务。

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7. The Utility Calculus | 功利主义计算

Classical utilitarianism proposes a hedonic calculus in which the moral rightness of an action is a function of the net happiness produced. Bentham’s formula offers a quantitative method, while Mill introduces qualitative distinctions among pleasures.

古典功利主义提出了一种快乐演算：一个行动的道德正当性是其产生的净幸福的函数。边沁的公式提供了一种定量方法，而密尔引入了快乐质的区分。

Greatest Happiness Principle:
Rightness(A) ∝ Σ (Pᵢ – Lᵢ) over all affected individuals i,
where Pᵢ = pleasure intensity × duration, Lᵢ = pain intensity × duration.
Bentham adds: consider certainty, propinquity, fecundity, purity, and extent.

Mill modifies the calculus by insisting that ‘it is better to be a human being dissatisfied than a pig satisfied; better to be Socrates dissatisfied than a fool satisfied.’ The competent-judge criterion tries to avoid the objection that utilitarianism is a doctrine ‘worthy of swine’.

密尔修正了这一演算，坚持“做个不满足的人胜过做一只满足的猪；做不满足的苏格拉底胜过做一个满足的傻瓜。”合格判断者标准试图规避“功利主义是适合猪的学说”这一反驳。

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8. The JTB Model of Knowledge and Gettier’s Counter | 知识的 JTB 模型与盖梯尔反例

Since Plato, knowledge was standardly analysed as justified true belief (JTB). Gettier’s 1963 paper provided counterexamples which show that the JTB conditions are not jointly sufficient.

自柏拉图以来，知识通常被分析为有证成的真信念（JTB）。盖梯尔 1963 年的论文提供了反例，表明 JTB 条件在联合起来时并不充分。

JTB Formula:
S knows that p iff
(i) p is true;
(ii) S believes that p;
(iii) S is justified in believing that p.
Gettier’s Theorem: ∃ scenarios where (i), (ii), and (iii) are satisfied, yet S does not know that p.

A standard Gettier case: Smith has strong evidence that Jones will get the job and Jones has ten coins in his pocket; Smith infers that the man who will get the job has ten coins. Unbeknownst to Smith, he himself gets the job and also has ten coins. His belief is true and justified, but it is not knowledge because the justification rests on a false lemma. This forces epistemologists to search for a ‘no-false-lemma’ condition or seek a revision to JTB.

一个标准的盖梯尔案例：史密斯有充分证据表明琼斯会得到这份工作且琼斯口袋里有十个硬币；史密斯推断“得到工作的人口袋里有十个硬币”。史密斯不知道，他自己得到了这份工作，并且碰巧也有十个硬币。他的信念是真且得到证成的，但不是知识，因为证成依赖于一个错误的引理。这迫使认识论者寻找“无错误引理”条件或修正 JTB。

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9. Descartes’ Cogito Theorem | 笛卡尔的“我思”定理

In the Meditations, Descartes discovers an indubitable foundation for knowledge after the exercise of hyperbolic doubt: the very act of doubting confirms one’s own existence. The cogito is often treated as a performative truth rather than a logical derivation.

在《沉思录》中，笛卡尔在实施极端怀疑之后发现了一个无可置疑的知识根基：怀疑这一行为本身就确证了自身的存在。我思常被看作一种执行性真理，而非逻辑推导。

Cogito Formula:
If I am thinking (i.e., doubting, being deceived, willing), then I must exist.
C → E (where C = ‘I think’, E = ‘I exist’).
The evil demon cannot deceive me about my existence while I am thinking, for deception presupposes a thinker.

The ‘thinking thing’ (res cogitans) becomes the first clear and distinct idea, and the criterion of clarity and distinctness serves as a general rule of truth. Critical questions in Year 12 ask whether the cogito proves a substantial self or merely the existence of a stream of thoughts.

“思维之物”（res cogitans）成为第一个清楚分明的观念，而清楚分明的标准则成为真理的一般规则。12 年级的批��性问题会问，我思证明的是一个实体自我，还是仅仅证明了一系列思想流的存在。

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10. The Logical Problem of Evil | 恶的逻辑难题

The logical problem of evil, epitomised by Mackie, charges that the existence of any evil at all is logically incompatible with the existence of a wholly good, omnipotent, and omniscient God. It poses an inconsistent triad for the classical theist.

由麦基所总结的恶的逻辑难题指责说，任何恶的存在都与一个全善、全能、全知的上帝的存在在逻辑上不相容。它为古典有神论者提出了一组不相容的三元命题。

Inconsistent Triad:
1. God is omnipotent (can prevent all evil).
2. God is omnibenevolent (wants to prevent all evil).
3. Evil exists.
If (1) ∧ (2) are true, then ¬(3) must be true. Yet (3) is undeniably true. Therefore, the conjunction (1) ∧ (2) is false.
∴ It is logically impossible for such a God to coexist with evil.

The free-will defence (Plantinga) attempts to show that God might permit moral evil in order to preserve significant free will. Introducing the possibility of a greater good that logically requires the permission of evil shows that the triad is not strictly inconsistent, rescuing the logical coherence of theism.

自由意志辩护（普兰丁格）试图表明，上帝可能允许道德恶以维护重大的自由意志。引入一个逻辑上要求容许恶的更大善的可能性，就可以证明这三元命题并非严格矛盾，从而挽救有神论的逻辑一致。

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Published by TutorHao | Philosophy Revision Series | aleveler.com

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