[FOM] Formation Rules

[Richard Heck]

Max Weiss wrote:
> 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  
> ancestral.
>   
As has been pointed out, there are different standards of `rigor'. The
one in play here seems to have to do with formalization, as the
definition of the ancestral is a /formal/ definition. But I didn't
understand the question that way. A definition can be rigorous without
being formal. I take it, for example, that Frege gives us a reasonably
rigorous definition of /proof in begriffsschrift/, though he doesn't
formalize it and he doesn't explicitly invoke the ancestral in
connection with it. But, given the structure of that definition and
Frege's interest in /Grundgesetze/ in inductive definitions, it's hard
to imagine he wouldn't have realized that he could have done so had he
wanted to do so. Indeed, that Frege was aware that some such definition
could be given emerges pretty clearly in his discussion of formalism in
Part III of /Grundgesetze/.
> Heck says Frege's characterization
>> as 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  
> concatenation.
>   
I don't understand the relevance of this remark. Frege's conception of
syntax is idiosyncratic, to be sure, but what does that have to do with
whether he gave `rigorous formation rules for a formal language'? My
point was that without some reasonably precise account of what the
well-formed expressions of the language are, you can't argue by
induction on the complexity of the expressions of that language, as
Frege does. That, indeed, is why we find the discussion of what (in
Frege's broad sense of the term) a `correctly formed name' is, where we
find it. What Frege says is that a correctly formed name is one that is
generated from the primitives by means of certain syntactic operations,
and he is very clear about what those operations are and is extremely
careful with the details. (See, for example, his various discussions of
what an argument-place is and, to mention it yet again, the discussion
of (what we would call) complex predicates in section 30.) Now, it's
true, as I've said before, that Frege doesn't explicitly mention a
closure clause. But what we have is an inductive specification, and
converting inductive specifications into explicit definitions is the
kind of thing the ancestral is good for. I suppose Frege might have been
blind to the possibility of using the ancestral here, but it's hard to
imagine.

Perhaps the worry is that Frege isn't as clear as one might wish about
what functional expressions are. But my own view is that there isn't
anything terribly mysterious about this, and a formal treatment isn't
all that hard, anyway: We can make use of placeholders, which is
precisely what Frege does in his informal but nonetheless mathematical
discussion. And it is a confusion---one of which Frege may himself be
guilty---to think that, in using these placeholders, we are
"approximating" anything. In a theory of the syntax of Frege's formal
language, the placeholders will appear in NAMES of functional
expressions. We need not suppose that placeholders occur in functional
expressions themselves, even if our names of functional expressions have
quotation marks at both ends. To think otherwise would be to have far
too naive a conception of how quotation marks work. Nor need we need
have any metaphysical conception of what `gaps' are nor, so far as I can
see, any substantial conception of what placeholders represent, any more
than we need to have such a metaphysical conception of what
concatenation is or what exactly sentences and their parts are, anyway:
The placeholders' meaning is simply fixed by their role within the
syntactic and semantic theory formulated using them. It is just a
mistake---and again, one Frege may have committed---to think syntax and
semantics need answers to these sorts of metaphysical questions. That is
not to say there are not substantial issues here about which conception
of syntax---Frege's or the more familiar Tarskian one---is correct and
for what sort of purpose. It is rather to say that these are issues
within the theory of language itself. (See the discussion in Heck and
May, "Frege's Contribution to Philosophy of Language", for remarks along
roughly the same lines.)

Richard Heck

-- 
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Richard G Heck, Jr
Professor of Philosophy
Brown University
http://bobjweil.com/heck/
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