I was initially surprised to read this because when I hear Intuitionism, I hear Intuitionist Logic. But IL doesn't have anything to do with denying objective reality; it can use facts on the way to proof. So I don't really know why Intuitionism is so much more adamant about denying constructive reality, or why it's thought to "give rise" to Intuitionist Logic, which at this point seems like a totally different thing. In other words, it's true that truth != proof, but that doesn't mean truth doesn't exist.
If there is an independent source of truth (external reality), then classical logic makes sense and intuitionistic logic doesn't. But intuitionists say mathematics, unlike the physical would, doesn't have such an independent reality. There is no platonic mathematical reality apart from explicit mathematical construction. Then classical logic is inappropriate and intuitionistic logic has to be used for mathematics.
"Socrates is a male" has a value of True; "All men are mortal" has a value of True; and a conclusion of "Socrates is mortal" that is "True" because a True "and" a True yields a True.
And then an immortal man is discovered, making that first premise "False". And False "and" a True yields a "False".
This causes "Socrates is mortal" to have a truth value of "False", which doesn't make sense unless you consider it "proof" instead of "truth": it used to be proven (by the premises) that Socrates is mortal, but now it is "false that it is proven". It might still be true, but it's not proven.
This is closer to intuitionist logic than classical logic, but it still relies on facts like "Socrates is male".
> "Socrates is a male" has a value of True; "All men are mortal" has a value of True; and a conclusion of "Socrates is mortal" that is "True" because a True "and" a True yields a True.
No, for a logical argument the conclusion must be true if the premises are true. A valid argument (proof) only shows: necessarily, if the premises are true, the conclusion is true.
> And then an immortal man is discovered, making that first premise "False". And False "and" a True yields a "False".
> This causes "Socrates is mortal" to have a truth value of "False"
