A logician who applies a supervaluational approach will continue to abide by LEM, but will abandon bivalence. Under this approach, the logician accepts that the problem resolves around the incompleteness of the words used, therefore the solution is to find some way to make the meaning more precise. This is done by saying that the borderline case lacks a truth value entirely, while still abiding by the LEM principle (that is, accepting that ‘p or not p’ is always true, provided that ‘p’ is interpreted in the exact same way throughout the argument).
When a word is vague, the logician creates three categories: (1) positive extension (the list of things that will make the predicate true), (2) negative extension (the list of things that will make the category false), and (3) penumbra (the grey area in the middle that is neither positive or negative). Then the logician will perform a ‘sharpening’, trying to reduce the list of possible meanings to a precise true or false (a sharpening that results in p being true will therefore also mean that not p is false, which retains LEM). Each possible interpretation will be narrowed down (sharpened) to either positive or negative.
If the proposition is true on every sharpening, it will be considered true. Similarly, if the proposition is false on every sharpening, it will be considered false. If however there is a mix of sharpenings, the proposition is neither true nor false. It lacks a truth-value entirely, so only by abandoning bivalence, have we reached some form of conclusion.
Returning to the sorites paradox, the logician will attempt to make a more precise meaning of the word ‘heap’. Perhaps in the circumstances she will decide that 200 grains is enough for a heap, but anything less than that isn’t a heap. So this means that the initial premise (‘if n grains of sand make a heap, then n – 1 grains also make a heap’) can be considered false on some sharpenings (when n is 200 or less).
Although this appears to resolve questions in logic, it’s unclear whether this really gets to the heart of vagueness. All we have done is tried to introduce a sharp dividing line into a concept where one previously did not exist. And given that whatever the sharp line happens to be (‘a heap is 200 grains’) will likely change in slightly different sets of circumstances, it could be argued that the definition is arbitrary at best, not really reflecting any idea of ‘truth’.
Another criticism of this approach is the splitting up of a proposition into three categories. Instead of just having one set of borderline case (between true and false), the multiple categories now introduce two sets of borderline cases, between positive and penumbra, and negative and penumbra. Again, it is unclear how this helps to make a word less vague.
This is similar to supervaluational logic, except in the case of subvaluational, the logician accepts that a proposition can be both true and false at the same time. This is known as a ‘glut’ logic.
The subvaluational logician will follow a similar approach in terms of ‘sharpening’, checking to see if the statement comes out true or false as the range of borderline cases are narrowed down. If the statement comes out true on at least one sharpening, that the proposition is considered to be true. Similarly, if the statement comes out false on at least one sharpening, then the proposition is considered false. So if you have a variety of sharpenings, and at least one true and one false result, then the proposition is both true and false.
A subvaluational logic can be criticised along similar lines to the supervaluationist logic. It does not really does with the question of vagueness, and merely tries to offer a ‘work-around’ solution, in order to dampen the effect of the vagueness.
A contextualist will argue that all premises need to be read in context, especially the fact that words change meaning depending on the current context. So in other words, when interpreting an iterative argument, a word may change meaning within the argument itself. We should not stick to just the one definition throughout. If for instance, we have an argument about the ‘horizon’, the meaning of ‘horizon’ might change as the argument progresses. This attempts to resolve the sorites paradox by recognising that the ‘heap’ changes meaning as the argument progresses. At some point, the induction step (take one grain away from the heap and you still have a heap) will turn out false.
Another good example of the contextualist approach is the word ‘tomorrow’. If you say ‘I will repay the debt tomorrow’, and then repeat the same declaration tomorrow, it cannot be interpreted in the same way. The ‘tomorrow’ has to be read in context.