FOM: Universal Generalization

[A.P. Hazen]

((I've been away, I haven't read all the "arbitrary objects" posts:
apologies if I am rehashing stuff that others have said!))
-
What differentiates rules like Universal Generalization (called Universal
Quantifier Introduction in the Natural Deduction literature) from the sorts
of rules used in propositional logic is that its applicability depends not
just on the form of premiss (the "Phi(x)" from which one "infers" the
conclusion "For all x, Phi(x)" by the rule), but also on the
premisses/assumptions used in deriving that premiss.  Because of this
difference-- which I think is of some  philosophical importance-- I would
prefer to say that this is  not a rule of INFERENCE at all.
   I find the notation for Natural Deduction developed by F.B. Fitch (used
in his "Symbolic Logic: an introduction" and many later American logic
textbooks, such as R.H. Thomason's "Symbolic Logic: an introduction")
helpfully perspicuous in this area.  Fitch writes the formulas of a
derivation (either a "categorical" derivation or a derivation from a
hypothesis) in a column with a vertical line at the left-- the hypothesis,
if any, is marked, and there is a special notation marking the re-use in a
new column of a formula derived earlier ("reiteration").  The column of
formulas beside an unbroken vertical line (representing the part of a long
proof dependent on a particular hypothesis) is called a "subproof."
   Fitch's version of the rule of Universal Generalization is that "For all
x, Phi(x)" can be written in a column after a certain kind of subproof: a
subproof containing "Phi(a)," having no hypothesis, and into which no
formula containing "a" has been reiterated.  His convention for annotating
derivations is to mark "For all x  Phi(x)" as a consequence (NOT of the
formula "Phi(a)," but) of the subproof as a whole.
   The story I tell about this rule (I think it is PHILOSOPHICALLY
plausible, and also useful in explaining the formal rule to beginning logic
students) is this:
     STORY: The constant "a" in the subproof is treated syntactically like
a name, but we haven't specified that it is the name OF any particular
object.  The formulas containing it, therefore, LOOK like sentences, but
aren't fully interpreted.  The whole subproof, therefore, LOOKS like a
deduction of the conclusion "Phi(a)," but it is as underinterpreted as its
conclusion!  Still, it has the FORM of a deduction.  We could turn it into
one by stipulating which object in the domain "a" names, and if we did that
it would turn into a proof that that object was Phi.  The restrictions on
the subproof (no hypothesis, no "a"-containing imported premisses) ensure
that we can do this for any object in the domain.  We can imagine ourselves
(if the domain is very small) or a very fast angel (otherwise) going and
examining each object in the domain in turn, re-interpreting "a" as a
temporary name for that object, and using the temporarily interpreted
subproof to deduce that that object is Phi: the subproof is a "re-usable
argument template," and our conlusion "For all x, Phi(x)" follows from the
EXISTENCE of such a subproof.  When we conclude that eveything in the
domain is Phi, we are inferring this, not from a line in the derivation,
but from the observed existence of the subproof." END STORY.
   Comments: (i) This provides an interpretation-- a "semantics" -- for the
rule of Universal Generalization that does not invoke "arbitrary" objects.
   (ii) Whatever one thinks of my semantics, Kit Fine's interpretation is
for some technical purposes a very fruitful one.  See, e.g., his proof, in
"Journal of Philosophical Logic" 12 (1983) of the independence of the
quantifier-permutation axiom in Quine-style axiomatizations of logic, which
is obtained by fiddling with his definition of arbitrary objects.
   (iii)  Anyone interested in my philosophical reflections on natural
deduction rules can find more in my paper "Logic and Analyticity" in
"European Review of Philosophy, v. 4: The Nature of Logic," edited by
Achille Varzi (Stanford: CSLI Publications, 1999), where I discuss this
interpretation at greater length.
--
Allen Hazen
Philosophy Department
University of Melbourne