No, they are not shaky.
I think I answered this a similar question beforesomewhere on MO, but let me try again.
You make a number of assumptions about logic. You are assuming that it is a goal of logic to bootstrap itself and mathematics starting from nothing. You think that a logician is not allowed to use the words "or" and "and" until he has somehow defined them without using them. You assume that "it is logics all the way down", that is to say, that the only way to define a logic is to use another logic. All these assumptions are incorrect.
You want logicans to impart on you a feeling of absolute security in mathematics, starting from nothing. I could probably do that with some cool drugs and a bit of religuous brainwashing. My preferred method is more mundane: teach you reading and writing and basic math, expose you to the wealth of human knowledge, train you in critical thinking, and then have you visit a course in mathematical logic. These are the true foundation of mathematics and logic.
If a logician is not allowed to use "and", "or" and other common words, then perhaps you should explain what he is allowed to use. Nothing? Surely, that is a bit too harsh. If you think about how one might achieve "perfect security" in mathematics, you will of course be reduced to questions about humans and the physical world, which lie outside the scope of mathematics and mathematical logic. All that logicians offer is a a mathematical analysis and reflection upon the nature of mathematics. And that is a very valuable thing to have.