Professor Linda Burns, from Australia, had an article called Vagueness and Coherence, an article from 1986, where she addressed The Sorites Paradox.
See: https://philpapers.org/rec/BURVAC
I went to the Philosopher’s Index, which should contain all reliable articles in Philosophy according to plenty, and did not find this article of hers upon using the word sorites in 2000.
As a consequence, I did not mention her work in my work.
Notwithstanding, she thought in the same way I did in what regards the underlying idea of the solution to The Sorites: she believed each one of us has their own line.
The way she put it is ‘tolerance rules’.
“Thus Wittgenstein’s view of vagueness may be vindicated. Vagueness of the kind connected with tolerance may be argued to be an essential feature of natural languages, and those languages may also be seen to be in perfect order, provided that tolerance (and therefore vagueness) is seen to be a more complex matter than had been thought.”
The statement above probably equates Intrinsic Vagueness: Intrinsic Vagueness
We disagree on main points:
“However, any series which actually varies, however smoothly, in the way required for the Sorites reasoning to apply to it, will also exhibit discontinuities.”
Our solution does consider it is possible having no discontinuities, which we think is the original proposal of the problem.
We could have an ideal series, containing ideal grains of sand (manufactured via machine, and all else).
“Only strict tolerance rules lead to paradox• However, it should be clear by now that these cannot be true, even where the predicates are highly observational, and there seems to be only one dimension to appearances. For suppose someone judges that a is red and b matches it perfectly in colour. They are committed by the strict rule for “red”
to saying that b is red also, whatever else may be the case. But b may be indiscernible from c, and they may judge c to be nonred. On strict tolerance rules they would be committed to contradictory conclusions over the colour of b.”
Here she makes a distinction between strict and lose tolerance rules. Our concept of red is strict, for it is classified in the textile industry through a logical code, unique for each small variation of the hue. Yet, our personal concept is lose, and that is why the confusion: it is a fight between the machine, which would be able to read that colour as, say, BX2, and our eyes, as well as preference or choices, which would read that colour even as yellow, perhaps YE2 instead.
The simplest argument to debunk hers is that, once you stick to one of those choices, so personal concepts or universal, you still have the paradox. If we choose a colour that is already labelled, we can always come up with intermediaries that are not yet labelled, and those would then form the paradox, this because the machine still works with discretion (countable colours), since that is the way we can see, and count so far.
Nature is not like that; it is continuous.
The main thesis of hers is described in the conclusion:
“Only strict tolerance rules lead to paradox• However, it should be clear by now that these cannot be true, even where the predicates are highly observational, and there seems to be only one dimension to appearances. For suppose someone judges that a is red and b matches it perfectly in colour. They are committed by the strict rule for “red”
to saying that b is red also, whatever else may be the case. But b may be indiscernible from c, and they may judge c to be nonred. On strict tolerance rules they would be committed to contradictory conclusions over the colour of b.”
What really leads to the paradox is the incapacity of the human mind to correspond the continuity of the natural universe to human speech. Our minds may be able to cope with continuity, but not our biological capabilities, so our mouth occlusion, emission of voice, and so on. Our translation processes, mind x language, are also faulty. Once more, the mind is continuous, and sees everything in ‘continuous mode’, but the discourse is discrete, since we can only emit a word at a time. Only if language were continuous when it comes to speech, would we be able to deal with the continuity of nature through it.
Verbalising seems to be a necessity for us to be able to write. Both activities seem to be intrinsically connected, almost equivalent.
Even with lose rules, and those are the ones considered by everyone addressing The Sorites in Science so far, we still have the paradox, so that Linda’s account of The Sorites Paradox is not a solution: it is just an alternative way of explaining the problem.
In conclusion, our solution, and hers have little or nothing in common.
Ours:
“The Sorites, and the Paradox of the Bald can only be modeled in 3D, where they exist. In this case, we must have a three-axes system to describe human actions involving ‘heap’ or ‘bald’. Let (x,y,z) hold place for the position of ‘heap’ or ‘bald’ in our discourse. It is easy to see that all three dimensions will change each time we introduce a new object or each time we change an already placed evaluation, say we got non-heap on (1,2,3). The three dimensions change, and therefore we put one more grain, and (x,y,z) changed. Now we think one grain made a difference when we see the point moving. So far we have one dimension, x, in our puzzle, and one to use as increment. In Maths, even infinity can be accounted for, and therefore we will always be able to count the infinitesimals, for instance, if needed. Our grains have to all be guaranteed to have exactly the same shape, size, and even aspect, so that we can abstract from reality, and land in the World of Mathematics. Because we were able to place that point in our 3D-graph, we will be able to add one more grain, and tell the result in a precise way. Did one grain make a difference for the observer? Sure. As it was when we learned Calculus…
Adding one grain always made a difference now. If the answer is ‘yes’, there is no paradox, so that the difference was simply that so far we had wrong translation of the real-life problem into Mathematics.
Notice that one of the coordinates, say x, may not change, but we will always have change somewhere unless it is taking the added grain, and position it again. The process of the mathematical induction was then, so far, being applied to one coordinate only, making us think there was no change happening through the addition or withdrawal of one grain or one hair.
Whilst (1,2,3), and (1, 2.001, 3) are distinct points, the first, and the last coordinate are exactly the same, proving to us that it could not be in 2D or 1D unless we wanted mistakes.”
Articles containing her solution:
Dr. Marcia Pinheiro 