Stewart Shapiro wrote:
"Typical vague predicates give rise to the ancient *sorites paradox*,
sometimes called the *paradox of the heap*. Here is an example:
Premise: A man with no hair at all is bald.
Premise: If a man has only n hairs on his head is bald, then so is a man
with n+1 hairs.
Conclusion: A man with 50,000 hairs on his head is bald.
[Things skipped]...
If a man with 0 hairs is bald, then so is a man with 1;
if a man with 1 hair is bald, then so is a man with 2; ...
This argument has 50,002 premises."
****
In "Zooming Down The Slippery Slope," George
Boolos shows how in a standard natural deduction
system it is possible to infer essentially the same
conclusion for 1,000,000 (one billion) hairs
using fewer than 70 premises. He then extends
this reasoning to various other numbers of hairs--
for instance 2^30--while indicating how derivations
can be compressed. He says: "2^40 is greater than
one trillion, and f(40)=7072; thus it would be perfectly
feasible, if rather boring, for someone to write down
a derivation...[for] any number less than a trillion."
He also compares methods of compressing the size
of derivations using standard formalizations of natural
deduction with a compression method that can
be attached to a standard tree-style system, and then
draws some conclusions about reasoning in natural
deduction vs. reasoning in tree-style systems.
[_Logic, Logic, and Logic_, pp. 354-369.]
Charlie Silver
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