陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Lewis Carroll logic puzzles」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

I had another long plane flight recently, so I decided to try making another game, to explore exactly what types of mathematical reasoning might be amenable to gamification. I decided to start with one of the simplest types of logical argument (and one of the few that avoids the disjunction problem mentioned in the previous post ), namely the Aristotelian logic of syllogistic reasoning , most famously exemplified by the classic syllogism:

There is a classic collection of logic puzzles of Lewis Carroll (from his book on symbolic logic), in which he presents a set of premises and asks to derive a conclusion using all of the premises. Here are four examples of such sets:

已知结果和反例

After a certain amount of effort, I was able to gamify the solution process to these sort of puzzles in a Scratch game , although I am not fully satisfied with the results (in part due to the inherent technical limitations of the Scratch software, but also because I have not yet found a smooth user interface for this process). In order to not have to build a natural language parser, I modified Lewis Carroll’s sentences somewhat in order to be machine-readable. Here is a typic

Unfortunately, the gameplay is somewhat clunkier than in the algebra game , basically because one needs three or four clicks and a keyboard press in order to make a move, whereas in the algebra game each click corresponded to one move. This is in part due to Scratch not having an easy way to have drag-and-drop or right-click commands, but even with a fully featured GUI, I am unsure how to make an interface that would make the process of performing a deduction easy; one may ne

证明或构造的主线

There are of course several existing proof assistants one could try to use as a model ( Coq , Isabelle , etc.), but my impression is that the syntax for such assistants would only be easily mastered by someone who already is quite experienced with computer languages as well as proof writing, which would defeat the purpose of the games I have in mind. But perhaps it is possible to create a proof assistant for a very restricted logic (such as one without disjunction) that can b

阅读时建议盯住的点

先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。

值得单独记下的条目

• Major premise: All men are mortal.
• Minor premise: Socrates is a man.
• Conclusion: Socrates is a mortal.
• Nobody is despised who can manage a crocodile;
• Illogical persons are despised.
• My saucepans are the only things that I have that are made of tin;
• I find all your presents very useful;
• None of my saucepans are of the slightest use.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源