[FOM] Formation Rules

Richard Heck rgheck at brown.edu
Mon Oct 16 10:29:13 EDT 2006

Edwin Mares wrote:
> Who was the first logician to present rigorous formation rules for a formal language? And where (and when) did they do it?
The very first, I am not absolutely sure. But Frege does so in 
/Grundgesetze/, published in 1893: See, in particular, sections 28 and 
30. Frege is there characterizing "correctly formed names", and the 
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. Frege's 
specification is, to be sure, not as rigorous as what one finds in 
Goedel 1930, but one shouldn't expect it to be. What makes it 
particularly messy is that the specification is spread out over two 
sections, and perhaps the most interesting part of it---Frege's 
explanation of how "complex predicates" may be formed---is separated 
from the main exposition. There's also a slight inaccuracy there, since 
Frege explicitly mentions this method of formation only in connection 
with first-level predicates when it is clearly needed at all levels. But 
what he says generalizes smoothly up through the hierarchy, so we can 
let that pass.

It's possible that one could find something earlier in the work of the 
Booleans, but I don't know of anything---Boole seems simply to assume, 
not unreasonably, that one understands how to form simple algebraic 
expressions, and he's not really interested in the language itself---and 
then there's the Indian tradition, of which I know essentially nothing. 
And of course the languages in which these groups were interested lacked 
the expressive power of Frege's, and so some of the problems that arise 
for Frege---in particular, nesting of bound variables---simply don't 
arise in that context.

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.

Richard Heck

Richard G Heck Jr
rgheck at brown.edu

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