This post explores some variations of the sorites paradox. It takes the perspective that measurements may be multi-valued. As such it is very similar to the sub-valuational or dialethic responses. However I find this formulation conceptually clearer.

Let us warm up with some simpler examples.

The first example simply involves measuring a length.

Say we have a ruler with marks every centimetre. Let us measure the length a book that is somewhere between 27 and 28 centimetres long. If it is clearly closer to 28 then we say the length is 28cm. If it is clearly closer to 27 then we say the length is 27cm. Sometimes it is -- using only the human eye -- not possible to say which mark is closer. In this case we say that the length is both 27cm and 28cm. More precisely, we say that the result of the measurement is the two element set {27cm,28cm}.

Note that we are asking the question: what mark on the ruler is closer to the end of the object we are measuring? So it doesn't seem too strange to say that both of them are equally close. Now let us move on to the next example.

We analyse how the little-by-little paradox is overcome in two different situations.

Let us consider three rectangles all coloured with slightly different shades of red. We arrange them with the most yellow/green on the left and the most blue on the right. We choose the shades so that to the human eye:

- the middle one is indistinguishable from the other two;
- the colours of the left and right rectangles can be distinguished.

Now let L and R denote the colours of the left and right rectangles respectively. Now we ask -- for each rectangle in turn: which of the two colours L and R is closer to the colour of this rectangle? From left to right the answers are: {L} , {L,R} and {R}.

Note that we can make the following assertions:

- if a rectangle measures {L} then the rectangle to the right has a measurement containing L;
- if a rectangle measures {R} then the rectangle to the left has a measurement containing R.

However we cannot make the following assertions:

- if a rectangle has a measurement containing L then the rectangle to the right has a measurement containing L;
- if a rectangle has a measurement containing R then the rectangle to the right has a measurement containing R.

So we have structurally ruled out any little-by-little paradoxes.

Now consider a row of four rectangles. We insist that the colours of adjacent rectangles are indistinguishable. However the colour of any pair of rectangles that are not adjacent are distinguishable.

Again let L and R denote the colours of the far left and far right rectangles respectively. Again we ask -- for each rectangle in turn: which of the two colours L and R is closer to the colour of this rectangle? From left to right the answers are: {L} , {L}, {R} and {R}.

So now in this situation we cannot even assert:

- if a rectangle measures {L} then the rectangle to the right has a measurement containing L;
- if a rectangle measures {R} then the rectangle to the left has a measurement containing R.

In this case there is no multi-valued measurement and so the induction step fails.

In the full sorites paradox we also have some vagueness in the measurement taking.

Consider a tank of water. The vague question that we will ask is: is the tank of water closer to being nearly full or not nearly full? We have the possible answers: {Nearly Full}, {Nearly Full, Not Nearly Full} and {Not Nearly Full} where the second indicates that we are not able to distinguish.

It may be that different people give different answers to this question.

Let us also give ourselves a way of decreasing the water level in the tank. For instance by letting a drop of water escape from the tank. We insist on one important thing: the water level in the tank decreases by a non-zero but imperceptible amount every time we let a drop out.

It is most likely that we are in a situation similar to the three rectangle example above. (i.e. that there exist multi-valued measurements.) In this case we avoid the little-by-little paradox by asserting:

- if the measurement of the tank is {Nearly Full} then after letting a drop out of the tank the measurement of the tank contains Nearly Full.

We then deny that:

- if the measurement of the tank contains Nearly Full then after letting a drop out of the tank the measurement of the tank contains Nearly Full.

However, there is a chance that the situation is like the four rectangle example above. Where letting out a drop can take us directly from {Nearly Full} to {Not Nearly Full}. In this case we avoid the little-by-little paradox by denying the induction step.

Now note the following excluded middle: either there are multivalued measurements or there are not. This is true because of the way we have set up our measurement scheme. The confusing part of the sorites paradox is that we don't know which situation we are in. In short we don't know how letting a drop out lines up with our measurement scheme. (Even though we do know that in either case the little-by-little paradox is avoided.) In order to find out we would have to go through and take the measurements.

Indeed before we take the measurements we might say: the experiment itself is multi-valued...