N.B. Some of the ideas presented in this essay are not mine. When I throw a disclaimer up front, I'll not stand accused of plagiarism. Bits and pieces of Roy Sorensen's analysis are the most decisive weapons for attacking this "problem". The initial formulation of Moore's paradox was given by (or at least attributed by Wittgenstein to) G. E. Moore - as follows:
Moore: "It's raining but I believe that it is not raining."
Let's look at this paradox from the perspective of its logical-form. It can be cashed-out in both its omissive (1) and commissive (2) versions. On their surfaces - in a "natural language", 1 is not a contradiction, yet 2 is a contradiction.
1. It's raining but it is not the case that I believe that it is raining.
2. It's raining but I believe that it is not raining.
But - after converting them into symbolic logic and inspecting their logical forms, the logical waters become murky.
R = It's raining.
-Br = It is not the case that I believe that it is raining.
B-r = I believe that it is not raining.
3. R ^ -Br
4. R ^ B-r
3 (the omissive version) now appears rather like a contradiction (A ^ -A) - while 4 (the commissive) appears unlike a contradiction. So, 3 (which is 1) appears the contradiction - while 4 (which is 2) does not appear the contradiction, symmetrically reversing what was stated about 1 and 2 immediately above.
Suppose that we can decide which sentences produce contradictions - only when their logical structure is identical. After converting the assertions into "relations of belief", they yield:
5. Br ^ -Br
6. Br ^ B-r
A fortiori and assuming that a belief relation adequately captures what an assertion is about, 5 (which is 1 and 3) is definitely a contradiction. And, 6 (which is 2 and 4) is not a contradiction.
Below - when 5 (a contradiction) is negated, it yields a tautology (A v -A) true in all cases - which it must be for it to be a contradiction - once negated:
Further, negating 6 (a non-contradiction) does not produces a tautology, verifying that it is a non-contradiction:
10. -(Br ^ B-r)
11. -Br v -B-r
From the analysis above - contrary to what has been written about Moore's Paradox for decades, the omissive version shows-up in more instances to be contradictory - than the commissive one.
5 above is not a contradiction - unless it's assumed that assertions can be translated simpliciter into belief-relations, which is problematic. Why should assertions be interpreted as (or reduced to) beliefs? If a person believes x, he may never assert x, and if he asserts x, does it follow that he always believes x? Can't beliefs just be intensional, containing an inherently undecidable nature (a "referential opacity" as Quine put it) built-into them - which somehow explains (or excuses) their inconsistent results, when embedded into natural (and formal) language sentences? 
The prevailing view among philosophers (against the structural above showing the omissive version to be a contradiction) holds that the commissive version: "It's raining but I believe that it is not raining." is a dyed-in-the-wool contradiction.
However, it's logically possible (though empty) for a person to assert that it's raining, but believe that it is not raining. Suppose that Moore succumbs to a brain injury. Moore could authentically believe that it's not raining - after asserting that it is raining should he fail to consistently understand what the phenomenon of rain is.
(1) "In logic, the formal properties of verbs like assert, believe, command, consider, deny, doubt, imagine, judge, know, want, wish, and a host of others that involve attitudes or intentions toward propositions are notorious for their recalcitrance to analysis." - Quine (1956)