QUALITATIVE MEASURE IS NOT OBJECTIVE

This topic concerns an invalid form of argument that commonly enters thought – perhaps in a careless way. Here is an example of the argument:

In what follows is assumed that a pinch is very small in comparison to a heap

If a pinch of salt is added to a small heap of salt the result remains a small heap                                               [A]

But [A] implies that any number of pinches added to a small heap results in a small heap                                   [B]

But this contradicts common sense according to which:

A sufficient number of pinches added to a small heap results in a heap that is not small.                                    [C]

# Resolution

The paradox is well known. It could be approached by “fuzzy” and many-valued logic. But the following is the essence. Consider:

Every heap is either small or not small                                                                                                           [D]

[D] and [A] are contradictory                                                                                                                         [E]

But the truth of [D] follows by tautology – since ~ [D] is a contradiction. Therefore [A] is not true – given a fixed size of pinch, there is some small heap which is not small when a pinch is added to it