Those are good questions! Also ones that I would expect from people coming from a mathematics or programming background, where formal language is king :)
The keyword here is semantics - semantic primes are irreducible units of meaning, and as such are used to paraphrase more complex, reducible words into things that cannot be broken down further. They're used as a metalanguage to describe the semantics of words. There's an example given elsewhere on the site I linked that is a good demonstration.
The phrase "someone X is happy (at this time)" can be paraphrased with semantic primes as
someone X thinks like this at this time:
"many good things are happening to me as I want
I can do many things now as I want
this is good"
because of this, this someone feels something good at this time
like someone can feel when they think like this
The people who advance the theory that there are semantic primes would say that the meaning of "happy" can't be broken down any more simply than that, essentially. Semantic primes by definition are universally understood, but can't be defined except by using synonyms. If you've ever seen a circular definition in a dictionary, that's the kind of thing that this theory avoids.
So what does that have to do with your questions? Well, you were right in your guess about your first question - that kind of thing is the domain of syntax :) Natural languages often don't have direct parallels with logical operators - for example, many languages lack words/affixes meaning "or," and have to use other constructions to get that meaning. One way I've seen that done is by creating a negative conditional: something like saying "if it isn't X, it's Y" to mean "X or Y." The way conjunction and disjunction varies crosslinguistically is really interesting - I would recommend reading papers authored or coauthored by Haspelmath to learn more.
As for your second question, I would guess it's because they would argue that they can't actually be defined as a negative - what does "many" really mean? It doesn't mean "some," and it doesn't mean "every." "Many" has a connotation of there being a lot of something, but it depends on the situation - "many ants" is a different amount than "many elephants," right? If there "aren't many ants," it doesn't necessarily follow that "there are few ants." Maybe there's just a normal number of ants.
I don't know a lot about this theory (natural semantic metalanguage), but I think it's useful and interesting as a way to compare meaning crosslinguistically.