[FOM] Formation Rules

Richard Zach rzach at ucalgary.ca
Mon Oct 16 16:19:58 EDT 2006

> 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.  But if what you mean is something that
explicitly involves "If A and B are formulas, then A & B is a formula" a
probable answer is: Hilbert in the lecture course "Prinzipien der
Mathematik" (Göttingen, Winter Semester 1917/18) (pp. 130-131 of the
lecture notes). Post's 1921 "Introduction to a general theory of
elementary propositions" gives such a definition for a propositional
language (van Heijenoort, p. 266).  Von Neumann's 1927 (submitted in
1925) "Zur Hilbertschen Beweistheorie" (Math. Z. 26) has a fully formal
set of formation rules for a first-order language and states that one
can prove (without, however, giving the proof) that on the basis of the
rules it is possible to decide whether a given string of symbols is
well-formed, how it can be constructed using the formation rules if it
is, and that the construction is unique (ie, unique readability) on p. 7
of the paper. 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".

Richard Heck mentioned Gödel 1930--neither the dissertation nor the
published version contain formation rules for formula, as far as I can
tell.  In the dissertation, Gödel just points to Hilbert & Ackermann
(III.4, i.e., p. 52 of the first ed.).

Richard Zach ...... http://www.ucalgary.ca/~rzach/
Associate  Professor,   Department  of  Philosophy
University of Calgary, Calgary, AB T2N 1N4, Canada

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