Persuasiveness vs Soundness

The syllogism's premises are not proven, they are assumed. It is really not relevant if Socrates is an alien android. IF the premises are true, the conclusion follows. We go along with it easily, because the premises happen to "seem" true. Tell someone: ALL A is B, C is A, therefore C is B they might not view it is tautological. So I think part of the answer to your question is that verbal / linguistic operations take place more automatically than formal symbolic manipulations in our mind, having been learned at an earlier age. It's also a valid syllogism to say:

All alien androids are mortal, Socrates is an alien android, therefore Socrates is mortal.

You're comparing the soundness of the second argument with the validity of the first. Yes, Euclid and lots of mathematicians have proved that "argument's conclusion completely follows from euclid's axioms" but that just means the argument is valid. Similarly, the second argument's conclusion follows from its premises.

The first premise of the first argument is an assumption. I have no idea how one would show that to be true. E is described by Euclid's axioms because we've decided thats a helpful way to describe it. Usually, the first two premises of the second argument are taken to be assumptions also but we can come up with very good empirical reasons for believing them to be true if we have to. I don't really think either argument is more sound except that question the truth of the first premise of the first argument is probably a category error.

In any case 13 year-old atheists usually screw up the arguments. Do you think a 13 year-old atheist would win a debate with say Alvin Plantinga?

Hmmm...

There is a definite sense in which the second "theorem" is much simpler than the first. To get from "All men are mortal" and "Socrates is a man" to "Therefore Socrates is mortal" requires only a very short proof in first-order predicate logic.

1. ∀x(Man(x)->Mortal(x)) [For all x, if X is a man, then X is mortal] (premise 1)
2. Man(Socrates) ->Mortal(Socrates) [If Socrates is human, then Socrates is mortal] (universal instantiation, applied to 1)
3. Man(Socrates) [Socrates is a man] (premise 2)
4. Mortal(Socrates) [Socrates is mortal] (Modus ponens, applied to 2 and 3)

In most formal proof systems, the proof of the Pythagorean Theorem from the axioms of Euclidean geometry would be much, much longer.

I believe you mean, "The square of the longest side of a right triangle is equal to the sum of the squares of the remaining two sides."

Or alternatively, "The square of the longest side of a triangle is equal to the sum of the squares of the remaining two sides, minus twice the product of those two sides and the cosine of the angle between them."

I'd also link this to the recent marketing rationalism post. Using "dark art" means, an argument can be made as, or more compelling, than the argument would otherwise, all things being equal.

The question there seems to be, is it possible to use those means without negative side effects on the reader, and writer, when we know how easily the human mind can be displaced out of the (fairly artificial) state of rationality ?

Maybe we should cautiously train ourselves into both the defense against, and the use of "dark arts", at the very least to be able to spot them when they're used, and maybe, to know when it can be right to use them. This should follow the lesson in It's okay to be (at least a little) irrational. We are running on a human brain. If we use, or even encounter, the dark arts, we run the risk of warping our own judgement, but if we shy away from them, pretending not to notice, we run the risk of using them without even our realizing it.

We need a safe, error recoverable, way to delve into these. How could we go about doing that ?

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