[FOM] primer on vagueness

[Charles Silver]

> 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.]
> >
Michael Sheard: 
> 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.

    About how many lines would be required by your analysis for a standard 
natural deduction system for Sazonov's number, 2^1000?   Would the 
number of lines (and characters on each line) be "perfectly feasible, if rather 
boring," like Boolos's derivations?   

    There must be some well-known results giving lower limits to how 
much compression is theoretically possible, given certain types of 
deductive systems.  Can anyone cite some of these results?

Charlie Silver