Two considerations might let allow us to understand Wittgenstein's analysis of the Sorites Paradox. The first has to do with Wittgenstein's notes on vagueness in natural languages. The second arises from his remarks on purity.
Contra Frege, Wittgenstein considers vagueness to be pervasive in natural language, unproblematic (at points, useful), and unshakable. Its pervasiveness is handled directly in §§104-108 of the Investigations. Its usefulness, and that vague sentences do not lack sense, is defended par example in §75 and §76. And Wittgenstein speaks about the unavoidability of vagueness most centrally at §§72-73, §§77-81, §§104-108.
Wittgenstein asks us to consider (§72) the narrow definition of an item by ostension (I point to a color patch and say «this color patch is called `yellow ochre'.») This is akin to giving someone a color patch with a word written under it. As of yet the person has no fixed rule at his disposal which tells him whether or not some new, similar color, may be called `yellow ochre' -- he has only the rule: this color is called `yellow ochre'.. The respondent complains: «here is this new color, and I have no way of saying whether it is called `yellow ochre'.» And so I give him a range of color patches lined up on a chart, each slightly different from the one next to it, and say: «only color patches falling within this range may be called `yellow ochre'.» Now I have given a precise definition of the term, by giving examples, by marking off the territory in which the term is applicable (the instances under which the predication is appropriate).
Here we may be tempted to say that we have `discovered yellow ochre', Wittgenstein tells us, or that we have resolved the fuzziness of the term's application tout court. The first temptation arises when we assume that we have seen our language's horizon, and therefore caught a glimpse of what lies beyond it -- that we have glimpsed some fact that is pure and exhaustive. The second, connected, temptation stems from our mistaken assumption that a localized use of a term (hence, the local language game itself), exists in isolation from all others and is thereby not threatened by the possibility they will infect the game under consideration.
In the first case we have not made a discovery but a decision -- «To repeat, we can draw a boundary» (§69). We have decided that the boundary goes like this, or the constraints relative to a «special purpose» (painting the house, say) have made the decision ahead of time. In seeking precision, or in attaining it in the case of a narrow domain such as logic, we want to say that we have seen past the bounds of our language game and stared in the face of the ideal or true predication. But when we sense that we have made a choice to bound the extension of the term in this way rather than that, we become frustrated -- «This means: it has impurities, and what I am interested in at present is the pure article» (§100).
In the second case, we are all the more apt, especially in making a choice as to the use of a word or sentence, to think that this choice has resolved all (or orthogonal) ambiguities, that this resolve gets us closer to the meaning of the term. We may say of «Moses» (§79) that he was so and so, and decide that such a set of criteria for the ascription of «Moses» are necessary and sufficient. But this is no stopgap to the possibility of appropriating the term for different uses, to fill the spaces for different language games. Hence, the business of identifying necessary and sufficient conditions for a term's application is as slippery as tracking its use, its various functions. We should not mistake the resolution of an ambiguity within a local neighborhood of our language as a definitive victory over vagueness, so much as a relocation of it. But again, that we cannot isolate necessary and sufficient conditions for the application of a term or concept, save by making a choice (or by having a choice foisted upon us by customs and practices surrounding our particular endeavor), does not render the term or concept useless, lacking in sense.
If we approach the Sorites Paradox with these considerations at hand, we come to see that the «paradox» is rather a «grammatical joke» (cf. Culture and Value). We start by using our language in the ordinary way, saying: «well, I know that by `child' I am talking about a human that is closer to birth than to death, given the average life span these days.» And then I say: «surely a one day-old human is a child, by my lights.» And then I ask: «If a human were a child and n days-old, then surely he would be a child at n+1 days-old?» But by inductive inference I have apparently committed myself to saying that a 100,001-day-old human being is a child, which I clearly would not like to do. My first reaction might be to reject the second premise. But by doing this I apparently commit myself to an incredibly stringent definition of «is a child,» which is again an unhappy consequence.
Wittgenstein's response, however indirectly, is to suggest that the problem is fiat.. «Instead of philosophizing,» we might say, «look and see how the language functions.» Let us trace the use of the terms being employed, trace it back to its familiar application. We might find that the predicate «is a child» is usefully vague. It might turn out that «child» is sometimes used in this context, and now that, and all the while there is no necessary and sufficient set of criteria for application. It might turn out that all we have, as it were, are instances of use and accepted application. The problem looks like a philosophical one because we mistake the broader problem of predication as one relating propositions to states of affairs, wherein we actually are in a continual process of relating propositions to propositions. It might turn out, that is, that «is a child» simply does not admit of the kind of highest generality and greatest precision that we want. And so our only recourse is to make the term do what we want, give it a strict definition (or such a general one as to come out empty)--but this is to simply change the rules arbitrarily, and of course we can do that. The use of the predicate may have definite applications on either end of a spectrum, but in the middle it is unclear: «is this a child? I don't know, the question hasn't come up -- I've never used the term in this way.»
Bringing ordinary language into philosophical waters in this regard is like training a microscope on the boundary between black and white squares of a chess board--we are now lost because the terrain is somehow extraordinary. It is precisely the open texture of certain predicates, and the demand that they function with sharper boundaries, that allows the sorites paradox to get off the ground. Hence, Wittgenstein suggests of the Sorites problem that «there is no problem,» and would say of vagueness generally that «the problem, insofar as there is one, should only be of practical interest--otherwise we are all on holiday.» Insofar as this is a «solution,» it is the kind that only claims to trace the steps of how we arrived at the supposed problem in the first place--to show the fly the way out of the bottle.