[FOM] Formation Rules

Neil Tennant neilt at mercutio.cohums.ohio-state.edu
Tue Oct 17 10:37:34 EDT 2006

On Mon, 16 Oct 2006, Richard Zach wrote:

> > Who was the first logician to present rigorous formation rules for a
> > formal language? And where (and when) did they do it?
> It depends on your standards for "rigorous".  Neil Tennant's standards
> are apparently higher than Richard Heck's, so you may or may not be
> willing to count Frege.  

The study from which I sent a snippet is devoted to the problem of how
rigorous one has to be in order to avoid non-standard interpretations of
one's theory of syntax, even at the propositional level. (Quick answer:
very!) These concerns were not uppermost in the minds of earlier logicians
trying to work out basic definitions so that they could secure uptake and
get on with their work. Any comparison of standards of "rigor" needs to
bear that in mind. 

> Hilbert & Ackermann 1928, "Grundzüge der theoretischen
> Logik" also give rigorous formation rules for formulas of first-order
> logic (you may quibble about the treatment of parentheses there,
> though).  I don't know why Neil says they don't, but it's on p. 52 of
> the first edition. Maybe he's making a distinction between "recursive
> formation rules" and "inductive definition".

Thank you for pointing this out. I had confined my attention to their
discussion of the sentential calculus in Ch. I, sec. 1-13, which covers
the whole subject without entering a formal inductive definition of
"sentence of propositional logic". But (quibbles about parentheses aside)
their later definition of "formula" in Ch. III, sec. 4 ("The Restricted
[i.e., first-order] Predicate Calculus") makes up for this oversight.
Thank you for pointing this out.

BTW, their clause (3) in the English edition (p.66) is missing the
negation sign:

	If any combination [gothic] A of symbols is a formula,
	then [gothic] A is also a formula

It's also strange that they proceed to remark immediately on conventions
for suppressing parentheses, when their own definition provided for no
insertion of parentheses at all, except around quantifier prefixes. One
cannot suppress what is not there. Might they have been influenced by
Polish ways of thinking, not realizing that with infix notation one needs
parentheses when applying two-place connectives? Or did Polish notation
itself only come later, by way of departure from Russellian notation?

Neil Tennant

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