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