Classical Logic Solutions
In terms of classical logic, one way to resolve the problem is to abandon the concept of bivalence. No longer must any proposition have a strictly true or false conclusion. Rather, we can say that the proposition lacks a truth-value: we have no way of saying whether the proposition is true or false. However, by taking this approach, we may simply be shifting the question of vagueness from one area to another. It’s still not clear what we mean by ‘borderline case’. If we aren’t certain of the word to begin with, then we can’t be certain what constitutes a borderline case. A billion grains of sand would be considered a heap, but ten grains of sand would not, so what would be the ‘grey area’ somewhere in the middle?
A related option is to modify bivalence by introducing a third category, perhaps called ‘indeterminate’, which can be used for borderline cases. As we’ll see, a ‘multi-valued’ logic resolves the problem by assigning a probability-like value to each proposition to each proposition, ranging from ‘1’ (undoubtedly true) to ‘0’ (undoubtedly false). There are also logicians who assert that propositions can be both true and false at the same time. This is known as ‘glut’ logic, indicating the possibility of more than one logic value (in contrast to ‘gap’ logic, which introduces the possibility of there being no logic values at all). A glut solution does however have to consider how to avoid the possibility of ‘Explosion’ (if A and not A, then B, where B can be anything you wish).
A Many-Valued Logic (Degree Theory)
This resolves the sorites paradox by allowing for degrees of truth. You could introduce intermediate categories of truth (‘slightly true’), or more commonly, you could have an infinite range of categories by assigning a value from ‘1’ to ‘0’. A heap of 1000 grains for instance could receive a truth value of ‘1’, with a ‘heap’ of one grain receiving a truth value of ‘0’. Something intermediate like a heap of 500 grains might get a truth value of 0.5.
However, a many-valued logic does cause a number of logical anomalies, such as having to find a way to deal with the Law of Excluded Middle. For instance, if we say that a proposition has a truth value of 0.5, then its negation will have a truth value of 0.5 as well (1 minus 0.5). The conjunction of these propositions (p and not p) results in a truth value of 0.25. This contravenes the Law of Non-Contradiction as it suggests that there is a one-quarter chance that both results are true, which is clearly impossible. A conjunction of a proposition and its negative should always be negative (ie, have a truth-value of 0).
Another criticism of a multi-valued logic is that it doesn’t really resolve the problem of borderline cases. Rather than limiting your answers to ‘true’ or ‘false’, you now have to decide whether a truth is 0.3 or 0.4. Even with a more precise evaluation, the vagueness cannot be avoided. A counter-argument is that a value such as 0.3 might be more realistic than a simplistic ‘true’ or ‘false’. Roy Sorenson points out that a straight line would be a very poor way to model a circle, as would having two or three straight lines alone. But as you start to add more straight lines into your polygon, a circle-like shape starts to develop. The extra precision does in fact help to bring you closer to an idea of truth.
A multi-valued logic also requires you to include more complexity in your logical argument. A conjunction is no longer a simple case of evaluating the truth or A and B, but rather combining the two truth-values in a probability like calculation. Adding further variables would only complicate the argument further, and it is difficult to see how much value somebody could obtain from the argument.