There are various paradoxes of logic that indicate to a certain degree that it is not entirely certain. The one of these that has gotten the widest coverage is Russel's paradox that demonstrated the naive notion of containment has a flaw, and cannot be arbitrarily combined with both negation and universal quantification. "Does the set of all things that do not have themselves as elements have itself as an element?"
This relatively simple question precipitated a deep concern within the Philosophy of Mathematics, which given our modern take on logic as a part of mathematics, is the same field that would cover the certainty of logic. The primary reactions were basically:
1) Mathematical Platonism: ignore the problem and assume paradoxes are simply so strange in form we remain safe, because no important argument will be nearly that strange. The approaches within this range argue about why this should be the case, about how the paradoxes are or are not 'real enough' and ultimately about different theories of idealism and realism in general.
2) Type Theory: be suspicious of self-reference and referential loops in general and assume the rest of logic is safe. This flows into thinking about recursive function theory, proof theory and category theory, which have their own ways of looking at ordered or simultaneous circular references, and more casual theories like Douglas Hoffstadter's notions of the 'strange loop'.
3) Constructivism or Intuitionism: insist all statements of existence rely upon finite extensions of arguments about finite things (ruling out the idea that infinite or ambiguous things are either true or false.) The different approaches here disagree on what it means for an extension of an argument to be finite.
4) Formalism or Fictionalism: deny that the objects reckoned about are real in any important sense and stick with finite syntactical transformations as ways to preserve truth, excluding the problematical ones as we encounter them. (So in this case, we propose a set of syntactical rules that identify what sets it is safe to discuss. If we really need to discuss more general things, we will add more rules.) The different approaches disagree on whether how we treat the cases we are ruling out -- are they false? are they ambiguous? do we need a hierarchy of models that covers each possible case (a la Woodin's investigations)? or should we just remain open to encountering them as we go along.
There is a lot of discussion about the positive and negative aspects of these theories starting from Russel's era and intensifying through the mid 1970's, when a resurgence of ancient observations about definitions and language resurrected by Quine and the postmodernists reduced the relevance of foundational arguments.
All of these remain interesting theories. (Even the first one.) And the gaps between them are far from resolved.
And the paradox that precipitated all of this is not among the most interesting to many people. These approaches and variants on them may also consider the Barry Paradox, Curry's Paradox, Zeno's paradoxes of continuity and the related Banach-Tarski paradoxes of space and measure, paradoxes of instantaneous ambiguity and symmetry in physics related to the ancient 'ass' and 'sorites' problems, and many others.
