Moore's paradox describes the so called Moorian sentences, like for example "It is raining but I don't believe it is raining". These are statements that can be true, and are logically consistent, yet seem (and are) absurd. But if we mathematically formalise what "believing" means, we can see the source of the paradox appear bright as day.
Lets start with some definitions:
$U$ is the set of all states of the universe. This is a set of custom objects, and does not have any other properties.
A statement is defined as the following.
$$U \rightarrow {T,F}$$
So, for example, 'it is raining', is a statement, so it is a function that takes the current state of the universe, and converts it to either true or false, dependant on the actual state (so if it is actually raining). Call the set of all statements $S$.
Statements have the following operations:
$$\forall u \in U, \forall s,t \in S$$
$$(s \land t)(u) := s(u) \land t(u)$$
$$(s \lor t)(u) := s(u) \lor t(u)$$
$$(\lnot s)(u) := \lnot s(u)$$
So the classical operations on booleans also apply on statements.
Now, consider two special statements, the Tautology and the Contradiction:
$$\forall u \in U: \bar{T}(u) := T$$
$$\forall u \in U: \bar{F}(u) := F$$
We call a statement undecided if it is not a Tautology nor a Contradiction.
A belief is a function who takes a statement, for example 'it is raining' and retuns a different statement, 'I believe it is raining'.
Formally:
$$M: S \rightarrow S$$
It has by axiom the following properties:
- Reasonability:
$$M(\bar{T}) = \bar{T}$$
One always believes a tautology to be true. - Consistency
$$M(s \land t) = M(s) \land M(t)$$
$$M(\lnot s) = \lnot M(s)$$
If I believe that "apples are red" and I believe that "bananas are yellow", I believe that "Both apples are red and bananas are yellow". In addition, if I believe apples are not red, I do not believe apples are red. - Self-conciousness
$$M(M(S)) = M(S)$$
If I believe that apples are red, I believe that I believe that apples are red.
Now, lets call a statement a triviality if the believe function of the statement is a tautology, and an absurdity if the belief of the statement is a contradiction.
Each contradiction is by the reasonability axiom an absurdity, but this property is not symmetric.
$$s \land \lnot M(s)$$
This statement can obviously be true. But if we apply the believe function to this statement it becomes:
$$M(s \land \lnot M(s) )= M(s) \land \lnot M(M(s)) = M(s) \land \lnot M(s) = \bar{F}$$
So, the statement, while it can be true, can never be believed, and is thus absurd. In short, the reason Moore's statement seems absurd is not because it cannot be true, but because the person stating it can never believe it.