On proving logic

Siris has some interesting posts on a method of representing the four types of proposition (universal affirmative, universal negative, particular affirmative, particular negative) but I found it hard going. Propositions are represented by ‘stacking ‘ one term on top of another. But you wouldn’t really understand this unless it were explained that the stacking relation means ‘every’ or ‘some’ or whatever. I.e. this

Animal
Man

means ‘every man is an animal’. So, what is the advantage? Since you need the quantifier ‘every’ to explain it, why not just use the quantifier itself? Thomas Reid in his essay on Aristotle’s logic considered a similar method used by Aristotle and his followers to demonstrate the validity of the syllogism (the ‘dici de omni’). But he objects that when the axioms of this method are put in plain English, they does not seem to have that degree of evidence which axioms ought to have. He says

It may even be suspected, that an attempt, by any method, to demonstrate that a
syllogism is conclusive, is an impropriety somewhat like that of attempting to
demonstrate an axiom. In a just syllogism, the connection between the premises
and the conclusion is not only real, but immediate; so that no proposition can
come between them to make their connection more apparent. The very intention of
a syllogism is to leave nothing to be supplied that is necessary to a complete
demonstration. Therefore, a man of common understanding, who has a perfect
comprehension of the premises, finds himself under a necessity of admitting the
conclusion, supposing the premises to be true; and the conclusion is connected
with the premises with all the force of intuitive evidence. In a word, an
immediate conclusion is seen in the premises by the light of common sense; and,
where that is wanting, no kind of reasoning will supply its place.