Seamus answer highlights a typical misconception concerning intuitinistic logic:
So perhaps the question should really be: Can you replace classical logic in everyday reasoning with intuitionistic reasoning? Rephrasing: can every everyday use of classical logic be replaced with intuitionistic logic?
And the answer there is: "of course not!". Think about all the times you do reason by using excluded middle. None of these inferences will work in intuitionistic logic.
Because the Gödel–Gentzen translation gives a perfectly reasonable embedding of classical logic into intuitionistic logic, the answer "of course not!" seems questionable. (Note that this translation even uses the intuitionistic consequence relation, which was Gerhard Gentzen's contribution.) I used a similar translation to make sense of non-constructive results using the axiom of choice even before I knew that this translation always works. (I thought of proofs using the axiom of choice as showing that trying to disprove (of falsify) the given existence claim would be futile.)
The answer by "anonymous bro" also shows inawareness of the strength of intuitionistic logic:
I would argue that that some fact p has been established is distinct from saying the (weaker) statement that it has not been established that p has not been established. The first is the assertion that p holds, while the second is something like saying that p has not been ruled out and is still possible.
The correct (weaker) statement from intuitionistic logic would be that it has been established that the falsity of p cannot be established. The first is the assertion that p holds, while the second is something like saying that p can never be ruled out.