A) The next statement is true.
B) The preceding statement is false.
Taken together, A and B form an irresolvable logical paradox. For if A is true then B is true, making A false; and if A is false then B is false, making A true. We must conclude that A is neither true nor false, but simply meaningless. It has the form of a sensible statement but not the content. Essentially both variants of the paradox say, "I deny what I assert" but do not actually assert anything.
My solution is that the liar paradox is empty of content - there is nothing in it that could be either true or false. However, not all self-reference is nonsensical or empty of content, eg "This sentence contains 7 words." This is self-referential but causes no problems - it is simply false. What's needed is a criterion to decide when self-reference is invalid, ie empty of content. This example is OK because it refers to a property that can be looked at, rather than a conclusion that is based on nothing except self-reference. If a statement or group of statements only talks about their own truth value then it is empty of content.
The moral of the story is that self-reference in logic is almost as dangerous as division by zero in arithmetic. In many cases it simply does not make sense.
Another kind of meaningless statement that is neither true nor false is the statement with no referent, eg "All the horses inside this matchbox are green". One cannot say of zero horses that they are green, say. On the face of it this assertion is not false, since it can only be false if there is a horse in the matchbox which is not green. Since the statement is not false we are led to believe it must be true. In fact, the statement that every horse in the matchbox is green is neither true nor false, but simply meaningless. A statement without a referent is without meaning.
Otherwise, we could subtract one horse (Tracey) from the set having just one horse (Tracey) and observe that the remaining horses (ie none) have the colour of Tracey. We can also state that all the horses in the resulting (empty) set had any colour we cared to name, for instance the colour of whatever other horse we cared to think of, eg Jules. So the horse, Jules, would have the same colour as the non-existent horses in the empty set, which also have the same colour as Tracey. Ergo Tracey and Jules have the same colour. We can only avoid this paradox by not allowing the statement that a non-horse has a colour. In effect, non-existent things do not have properties and hence are not fit for discussion.
My conclusion is that to avoid logical contradictions we need to avoid discussing the truth value of statements that are empty or meaningless.
Postscript December 2015
Someone pointed out that the horses question given above can be seen in two different ways: on the level of formal logic and on that of ordinary discourse. By "ordinary discourse" I mean the way rational people argue in normal life. This is not the same as formal logic, where we have the so-called
paradoxes of material implication, ie statements that are true but which appear to be nonsensical.
In formal logic statements about non-existent things are not meaningless. It is true that “all the horses in this matchbox are green”. We can say that they all have the same colour as Tracey and that they all have the same colour as Jules, but that does not mean that Tracey and Jules are the same colour, because to make that deduction you would have to know that there exists a horse in the matchbox.