QUALITATIVE MEASURE IS NOT OBJECTIVE

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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