Abstract:

Most students studying mathematical logic consider it as a difficult, boring and uninteresting subject. And many teachers of mathematical logic believe that this is a feature of its content, and nothing can be done about it. But this author is sure that this a feature of the mode of presentation, not of the content. Let us remember to this end the words of Blaise Pascal, who said that the subject of mathematics is so serious that it is necessary not to miss the occasions of making it a little entertaining.

Very useful in this respect (with the initial study of mathematical logic) are the problems on Knights and Knaves (Truth-tellers and Liars). Raymond Smullyan very convincingly and vividly demonstrated this in his remarkable book named “What is the name of this book?”

However, it must be said that the array of these problems has one significant flaw, namely, all these problems deal with the logical connectives, while the quantifiers are not touched in them at all. So, that is the goal of our paper to fill in this gap by presenting a sufficiently large number of quantifier-oriented problems about Knights and Knaves (with solutions). In addition, the place, the role, the logical analysis of the solutions of these problems, and the methodology of using them in the process of teaching quantifiers in the course of mathematical logic will be indicated. Thus, the aim of our paper is twofold: to fill the gap in the popular scientific literature, and to use this filling for improving the methodology of teaching mathematical logic (in the part related to the study of quantifiers).

The practice of teaching this material to schoolchildren, to their teachers, and also to students of the Faculties of Mathematics and Law, shows that using of these problems increases the interest in studying mathematical logic and the quality of mastering it.

To give the reader an opportunity to get an idea of the problems to be proposed, let us present here two of these problems. In the first one, a famous legend is outplayed according to which the Cretan philosopher Epimenides once said: "All Cretans are liars!" Having placed Epimenides on the Island of Knights and Knaves (where every Islander is either Knight or Knave, and the first always says the truth, while the second always lies), we received the following problem.

A traveler who has arrived at the Island of Knights and Knaves, asked one of the Islanders (whose name was Epi), whether there are many Knights on the Island. To this end, Epi answered: "All Islanders are Knaves!". It is required to find out who is Epi, a Knight or a Knave? Whether there is at least one Knight on the Island? At least one Knave? Whether they are all Knights? Or all are Knaves?

The other problem is close to the modern conditions. It says that the Island has an economic crisis. As usual, they began to search for the guilty. The President of the Island appealed to the Government. In his address he, in particular, said:
1) "I am guilty";
2) "If at least one of you is not guilty, then I am also not guilty!"

It is required to find out who is the President, a Knight or a Knave? Whether he is guilty of a crisis? How are matters with the guilt of members of the Government?

Keywords:

Teaching mathematical logic, Quantifiers, Knight and Knave problems.