[FOM] Formation Rules

Max Weiss mmweiss at sfu.ca
Tue Oct 17 15:21:33 EDT 2006

Ed Mares asks

>> Who was the first logician to present rigorous formation rules for a
>> formal language? And where (and when) did they do it?

Richard Heck responds:

> So if one were to rephrase the question as "Who was the first  
> logician to give rigorous formation rules for a language of  
> reasonable expressive power?" then the answer is definitely: Frege.

As Richard Zach says, Frege did not rigorously define the notion:  
formula of Begriffsschrift.  That is, he did not do so up to the  
standards of rigor of e.g. Tarski---nor, for that matter, up to the  
standards of rigor exhibited in e.g. Frege's 1879 definition of the  

One can give a precise definition to the notion: formula of  
Begriffsschrift, in contrast e.g. to the notion: formula of  
Principia.  Perhaps Frege was the first person to use a system whose  
formulas can be specified by the sort of definitions Mares inquires  
about.  But this is not to have given such a definition.

Heck says Frege's characterization

>  has to be rigorous enough to underwrite the induction on the  
> complexity of expressions used in the the argument, given in  
> sections 30-31, that all correctly formed names denote.

The argument of sections 30-31 is an attempted induction on  
complexity of "names" in a peculiar sense of "name".  Not all such  
"names" are expressions of Begriffsschrift, for some of them are  
"names of functions".  Frege (Gg sections 1,2) says that such "names"  
of functions are, like functions themselves, "incomplete" or  
"unsaturated".  He conceives names of functions to have "gaps" that  
need saturation by other names.  Names of functions do not contain  
free variables but may be approximated using placeholders.  In  
contrast, "formulas" in a more familiar sense are built up by  

Max Weiss
Dept of Philosophy
U. British Columbia

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