Existence exists - There is something, as opposed to nothing. etc.,etc.

A is A/The Law of Identity - Everything that exists has a specific nature. Each entity exists as something in particular and it has characteristics that are a part of what it is. etc.,etc.

Consciousness is the faculty that perceives that which exists. - To be a consciousness, it must be conscious of something external to itself. Only after it is conscious of something external can it identify itself. etc.,etc.

While these sound great and succeed in establishing a basis for something, i don't think that something is logic. All that seems to be established is that we exist -in a universe of many things -and we are able to make the claim that 'we exist'. A fairly vague basis, I think. It is also based on the perception that we know the truth of existence which, until we can no longer classify anything as 'unexplained', is incomplete.

Classical mechanics is well understood too. It too is incomplete, not being complete does not mean it is not well understood. It is true for a certain range of measurements.
As to why, not being able to answer why does not mean a model is incomplete.

Does that mean an incomplete sentence is true even if we understand the given words in a sentence? ( Cheese is ___ it is also __. True or not?) Is a house without a roof, even though we know it is a house, able to be called such when it doesn't fulfill the basis tenants of shelter? Is it also possible that, having no roof, that house may be having a second floor added? (I've seen this done, very neat :P) In which case, our previous understanding and use of knowledge about square footage, building height, volume and living capacity, all change drastically. Is not an incomplete model subject to similar flaws, all of which only serve to prove we may yet be mistaken? These model may be incomplete due our attempt to mold observation to theory instead of fitting theory to observation.

Granted, I'm not a well read man, but it seems that a lot of philosophical ideas are being used in the discussion of logic(truth, perception, existence, intuition, etc.). Perhaps our incomplete models are based on our inablity to properly combine or divide the two ( :P I jest-interesting thought though). I have a high regard for logic, so please don't get me wrong, but true logical thought has always been beyond me. Much to my fortune it seems.