I’ve encountered Moore’s Paradox in too many philosophical issues to count. The import of this paradox is that there are statements which could certainly be true, hence consistent in that sense, but cannot be believed, on pain of incoherence.
The form of the typical example is “A, and I do not believe that A” or “It is not the case that A, and I believe that A”. That is not precisely how Moore himself presented it, but rather Wittgenstein’s formulations in his Philosophical Investigations:
“Moore’s paradox can be put like this: the expression “I believe that this is the case” is used like the assertion “This is the case”; and yet the hypothesis that I believe that this is the case is not used like the hypothesis that this is the case.” ( p. 190 in the 2nd edition, Oxford 1998))
Many instances of this form are actually true. For example, I neither believe nor disbelieve that it is presently raining in Peking, but either it is or it is not. So one of the instances of Moore’s paradox is true: either “it is raining in Peking and I do not believe that”, or “it is not raining in Peking and I do not believe that” is true.
But I would reveal a serious incoherence in my state of opinion if I genuinely and correctly asserted either of these statements.
My take on this situation is that we are looking at two linguistic functions. The same words can be used either to state of fact (in this case, stating that I have a certain belief) or to express a belief.
The logic of statements is the very familiar one that we all learn in elementary logic courses. The logic of expression (or avowal) has drawn attention but has not been much explored.
I have thought of a parallel we can explore between these two logics, if we are careful about how we formulate the relevant consequence relations.
In the logic of statements A is a consequence of premise B, on pain of logical inconsistency, exactly if premise B cannot be true together with anything contrary to A. For example, that the cloth is red is a consequence of the premise that the cloth is scarlet because the cloth cannot be scarlet while having any color disjoint from red.
In the logic of belief expression I’ll say that A is a consequence of premise B, on pain of incoherence, exactly if premise B cannot be coherently believed together with anything contrary to A. You cannot coherently believe (the content of) “it is raining in Peking and I do not believe that it is raining in Peking”, nor “it is raining in Peking and I neither believe nor disbelieve that it is raining in Peking”. So, “I believe that it is raining in Peking” follows from “It is raining in Peking” on pain of incoherence, in the sense here defined.
What must the logic of expression be like then?
First, it must have a lot in common with the logic of statements. If A is tautology in the logic of statements I can certainly not coherently believe anything contrary to A. Therefore, if A is a tautology in the logic of statements it is also a tautology (defined analogously) in the logic of expression.
But the properties of the consequence relation are not the same all told. In the logic of statements there is the principle (sometimes called the Deduction Theorem) that if A is a consequence of premise B then the conditional “If B then A” (in the material sense, i.e. “Either not B, or A”) is a tautology. If that principle also held in the logic of expression then the following would be tautologies there:
“If it is raining in Peking then I believe that it is raining in Peking”
“If I believe that it is raining in Peking then it is raining in Peking”
which, if expressed by me as beliefs would show that I was convinced of my own powers of clairvoyance. Clearly they are not tautologies in the logic of expression.
This situation, that the analogue of the Deduction Theorem fails, is not unknown in philosophical logic. It appears quite typically in the logic of statements of a language in which there can be sentences which are neither true nor false (failure of the principle of Bivalence). So here, in the logic of expression, it clearly has to do with the fact that coherence does not require me to be highly opinionated: I can suspend both belief and disbelief, if I wish.
Bas van Fraassen's philosophy blog 