Where do the topics covered in Lewis Carroll's 1896 book "Symbolic logic" fit in the modern mathematical curriculum?

Question

Where do the topics covered in Lewis Carroll's 1896 book "Symbolic logic" fit in the modern mathematical curriculum? And what is the modern substitute or notation?

It appears to me that all it covers is set logic, syllogisms, & fallacies(invalid syllogisms), I didn't read the whole thing; Does anyone else, remember it covering anything else? "No, it didn't cover anything else" is a legitimate answer here.

In case there is any discrepancy over what I am talking about; Here is the easy to read html version from project Gutenberg. And Here is the original text version from Google Books.

Answer 1

Here's an Aristotelean syllogism in modern mathematics:

Major premise: All recursively enumerable sets are Diophantine. This is called Matiyasevich's theorem. It was proved in 1970 by Yuri Matiyasevich, building on work done over several decades by Julia Robinson, Martin Davis, and Hilary Putnam. The converse is trivial.

Minor premise: Some recursively enumerable sets are non-recursive. This was an important discovery made in the 1930s. I think Alan Turing, Stephen Kleene, and others may have done it indepedently. I am unsure of the details.

Conclusion: Some Diophantine sets are non-recursive. This laid Hilbert's tenth problem to rest.