[FOM] primer on vagueness

[Michael Sheard]

Charles Silver wrote:

>Stewart Shapiro wrote:
>" [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."
>...
>[_Logic, Logic, and Logic_,  pp. 354-369.]
>

There's a similar analysis in my expository paper "Induction the Hard 
Way"  (American Mathematical Monthly, Vol. 105, April 1998, pp. 
348-353), where an even more dramatic compression is discussed.  Under 
that approach, a complete derivation without induction of the conclusion 
for 2^30 or 2^40 -- or even 2^250 -- takes about 30 lines in informal 
logic, and some small multiple of that in a formal deductive system.

(By the way, the paper does not claim any originality of the technique 
other than its presentation.  The range of e-mails I received after its 
publication revealed that the approach has been re-discovered 
independently on several occasions.)

Michael Sheard