Logic, the most powerful tool you have, if used correctly, is absolutely reliable, but only if your starting assumption is also absolutely correct. Because it's very very hard to find an indisputable ground or assumption from which you can build up logical chain of thoughts, logic can and will lead you to wrong conclusions. The only weakness of logic I can see is its dependence on premise. Otherwise logic, if used methodically, is as good as it can get. 

When your premise start from intuition rather than logic per se, then your logic will dependent on your intuition. If you have a correct intuition then your logico-intuition is as good as it can be. Intuition is such a thing that you can not cultivate, but it can only be recognized. Why? Because intuition comes from the pre-conscious region of the kingdom of Freudian/Jungian unconscious. You can be only aware of intuition but you cannot see where and how it is coming from, unless you have some special access to your unconscious. All intuition is already there and there is nothing more you can cultivate or concoct. The only thing you can do or have to do is to recognize the nature of the situation. If everything that happened is buried inside the intuition, if intuition already knows the staff, and lastly if any conclusion you draw is dependent of intuition then what is the point of thinking or using logic? This is the grand bug in the system that you have to deal with. You have a infinity degree of freedom to find out the mystery of this bug. To find the answer of this problem, again you have to go back to your intuition. No matter how dramatic is the suggestion of the intuition, you have to accept it unless you can find a better one. My answer, with my best intuition recognition and logical training, is as following: 

An example of logical power: 1➡4 

1. I have the free will, therefore, free will itself signifies its exist. 

2. Something cannot come out of nothing. (If the scientists claim that we came out of nothing then I would say that "nothing" is still a "thing" that they don't understand). 

3. I am, therefore, Creator was. 

4. Creator was, and I am still here, therefore Creator is. 

This is just an example of logic but it still has some implicit intuition embedded deep inside, hence difficult to see. Things are not as simple as you might think. The real, deep mystery of the existence can be summarized as following: 

You already have the answer of your existence (from Unconscious ➡Intuition) but you are trying to find why and how did this happen [for example, by using reason and logic ➡mathematics ➡science ] with an absolute degree of freedom.