The Sorites Paradox

The coloured strip to the left is actually made up of a series of rectangles each about 40 times wider than it is high and each a different colour. The top one is clearly red (its rgb specification is "255, 0, 0"). The bottom one is absolute yellow (255, 255, 0), and therefore clearly not red. Each ith one in between is rgb(255, i, 0). Now each of these little rectangles is indistinguishable to the naked eye from its neighbours, so if any one of them looks red then so does the one immediately below it:
∀x (LR(x) ⇒ LR(x+1))
But from this, and the observation LR(0) there follows by 255 applications of instantiation and modus ponens, LR(255) ⎯ which is not true, because the colour at the bottom of the strip matches lemons, not strawberries.

This is a famous old paradox, known as the sorites paradox or paradox of the heap: the applicability of a vague predicate such as "looks red" is insensitive to small changes in some respect, but sensitive to large ones, and yet the large ones are made up of small ones. There are only three ways out of it. One can:

  1. Deny that it is legitimately set up. That is, deny that logic applies to vague statements.
  2. Deny the major premise. That is, hold that there is a precise boundary to redness, so there is a last red rectangle and a first non-red one.
  3. Deny the validity of the inference. That is, take it to refute orthodox logic, even in such basic parts as the rule of detachment.
None of these is especially attractive or easy to maintain. Option 1 was suggested by Frege back in the 19th century, and reiterated by the later Wittgenstein, with a superficial difference of attitude: whereas Frege says "So much the worse for ordinary language", Wittgenstein and his followers say "So much the worse for formal logic." Either way, they cut logic off from most of its subject matter, holding that even our most careful and accurate descriptions of the world obey no statable rules and have no specifiable semantics or structure. This is far too much of a retreat. Opion 2 is popular with formally-minded philosophers today, usually as part of the view that vaguness is only expressible in the meta-metalanguage. It is held that there is a last red rectangle, but no precise answer to the question of which one this is. The first half of this, unfortunately, just doesn't look true: why say there is a last red rectangle when everyone can see there isn't? Perhaps because behind this view is again the doctrine that "red" can't have any extension unless it is precise, so a vague extension has to be cashed as a set of precise ones.

Option 3 deserves to be taken more seriously than it usually has been: if the phenomena provide counterexamples to logical theory, perhaps it is the theory rather than the world that needs adjustment. Merely introducing "degrees of truth" into the semantics of logic, however, is not enough. I started working on a substructural logic suitable for vague reasoning in the early 1980s and have returned to it sporadically since. While I am almost the only philosopher who doesn't claim to have all the answers about this thoroughly intractable paradox, I continue to see nonclassical logic as holding the best hope for a satisfactory account.

Links

Dominic Hyde, 'Sorites Paradox', Stanford Encyclopedia of Philosophy.

John Slaney, 'A Logic for Vagueness', Australasian Journal of Logic. 

Dr J K Slaney                      Phone (Aus.):  (026) 125 8607
Automated Reasoning Group, CSL     Phone (Int.): +61 26 125 8607
Australian National University     Fax (Aus.):    (026) 125 8651
Canberra, ACT, 0200, AUSTRALIA     Fax (Int.):   +61 26 125 8651
John.Slaney@anu.edu.au