First, logic existed before we did. Whether we are around or not, e=mc2. The universe existed before we did, and we did not change it by beginning to think.
The egg came first, evoluntionarily speaking. The kingdom Avia predates the chicken. If you refer to a paradox, then it can be solved by 'thinking outside the box'.
Existence does not require thought, however, logic is everywhere. Causes come before effects, and math is always true.
If you refer to consciousness, then yes, that does require thought and life. Do not confuse logic with being conscious.
-Penguin