FOM: Re: The Logical, the set-theoretical, the mathematical

[Allen Hazen]

    Jeffrey Ketland and Edwin Mares have raised the issue of whether LOGIC
should be "ontologically committed": whether any existentially quantified
statements should be thought of as LOGICALLY true.
   (i) Part of me says NO, that (though for sufficient practical reasons as
well as tradition standard formulations of First Order Logic ARE committed)
Universally Free Logic (quantificational logic allowing both non-denoting
terms and the empty domain) is what really deserves the honorific name
LOGIC.  After all, serious mathematical reasoning has to tolerate partial
functions, so a logic allowing terms that don't stand for objects in the
domain of the quantified variables is practically important.  ...  It is
interesting to compare the conceptual or philosophical sophistication (as
well as the technical competence) of the early papers by Hailperin and
Quine (JSL 1954; last chapter of "From a Logical Point of View") with the
appalling nonsense published on the topic in "Analysis" (the English
philosophical journal, not the math journal!) vol. 14 (1953), pp. 1-5.
    (ii) Part of me wants to say YES.  Russell's Ramified Type Theory (cf.
Church in JSL v. 41 (1976), pp. 747-760 for modern formulation)
incorporates comprehension principles, so can be seen as ontologically
committed to an infinite universe of properties.  Still, in lots of ways it
seems to be "LOGICAL" in character.  Formulated with a substitution
operation instead of comprehension axioms, its "machinery" reduces to
quantificational rules exactly parallel to those of First Order Logic
except for the category of expression substitutable for variables.
Gentzen's Hauptsatz extends from First Order Logic to it with virtually the
same proof (redefine "grade" to allow grades of ordinal less than omega^2:
induction on these ordinals is finitistic (=primitive recursive)), so
conservatism of a Ramified theory over the First Order theory with the same
logical axioms is immediate (a point made by Ian Hacking, "Journal of
Philosophy," vol. 76 (1979).)  It has a transparent semantics in terms of
"substitutional quantification" (cf. Charles Parsons, "Journal of
Philosophy" v. 68 (1971), or Quine's "Roots of Reference.")  I once argued
("Analysis" vol. 45 (1985), pp. 65-68) that, though it provided a theory of
properties of some use in semantics or the philosophy of science, its use
was compatible with views which-- as such things are ranked in the
philosophy of mathematics-- ought to count as fairly strict nominalism:
indeed Wilfrid Sellars ("Review of Metaphysics" v. 17 (1963), pp. 67-90)
presented it as the result of a reductive, nominalistic, analysis of set
theory!