Is this logic of Chat GPT correct?

I would not really call Chat GPT “intelligent”.

Chat GPT sometimes fails in simplest logic and mathematics tasks.

It is more like seeing a kind of language model that tries to answer your questions/tasks with the help of stored data and the one you have written, in a way that would possibly also write a person. Chat GPT does not really understand the data, but rather looks like the data in its database. [Also in what contexts the data has occurred and have been queried.]

In the specific case, Chat GPT has probably recognized your “can’t be but not” and realizes that in such cases it is usually apologised for a mistake. Therefore the “voice, that was my mistake.” And possibly Chat GPT avoids the word “simultaneously” in its formulation because of the “not at the same time” you mentioned. Nevertheless, in his database “Bern” is somehow connected to “Capital City of Switzerland” and “Federal City of Switzerland”, so Chat GPT is once again coming to the same assertion as before (just re-formulated).

You should not rely on the correctness of the statements mentioned by Chat GPT. You should always think about yourself if that really fits. [In some areas, such as combining texts, Chat GPT is more suitable due to its being. You have to look less closely. But especially in mathematics and related areas such as logic, physics, etc., one must always look very closely at what Chat GPT actually gives for nonsense.]