Reduction and Elimination in Philosophy and the Sciences

S4) the necessity of arithmetic is secured. It is also well
placed to explain the apriority and applicability of arithmetic
sums.
It might be objected that it is still problematic to
mesh this reduction with the beliefs of ordinary speakers.
For suppose there could not be more than m objects. Then
since counterfactuals with impossible antecedents are
vacuously true, the paraphrase of m + n = p will be true
even in a case where the paraphrased utterance is
standardly taken to be false. The willingness of speakers
to assert or deny arithmetic sums in line with the standard
distribution of truth values should therefore not be
independent of their belief in the potential infinitude of the
world.
There are at least two possible lines of response to
this objection. It could be argued that a willingness to
make standard arithmetical assertions is indeed tied to a
belief in the potential infinitude of the world. Alternatively, it
could be argued that counterfactuals with impossible
antecedents are not uniformly (vacuously) true. A common
proposal is that the counterfactual conditional if it were the
case that A, then it would be the case that B may be
analysed as all nearby A-worlds are B-worlds. One might
then follow, for example, (Nolan 1997) in accepting
impossible worlds at which not everything is the case. On
that view, some impossible worlds are nearer to the actual
world than others, and so it is not the case that all
counterfactual conditionals with impossible antecedents
are true. Someone disposed to accept that view will
arguably agree that on the assumption that there could not
be 100 objects, the counterfactual ‘if there were exactly
100 objects that fell under F, exactly 0 objects that fell
under G, and no objects fell under both F and G, then
there would be exactly 100 objects which fell either under
F or under G’ is true and that the counterfactual ‘if there
were exactly 100 objects that fell under F, exactly 0 objects
that fell under G, and no objects fell under both F and G,
then there would be exactly 999 objects which fell either
under F or under G’ is not. The willingness of speakers to
assert or deny arithmetic sums in line with the standard
distribution of truth values can therefore be consistently
held to be independent of their beliefs regarding the
potential infinitude of the world.
3.
Can this approach be extended to capture more of arithmetic?
One might consider directly paraphrasing quantified
arithmetic sentences by introducing third-order quantifiers
into the paraphrases. Alternatively, one might claim that
arithmetic quantification is substitutional.
The latter option may have more philosophical
appeal. It is difficult to see how to make a case for ordinary
speakers’ belief in something like a third-order
counterfactual paraphrase of even the simplest quantified
sentence. However, the willingness of ordinary speakers to
assert quantified sentences is arguably not independent of
their beliefs concerning the truth values of substitution
instances of those quantified sentences. One might see in
this the beginnings of a case for construing arithmetic
quantifiers substitutionally.
There is not enough space here to develop such a
case. However, it may be objected that if arithmetic
quantifiers are substitutional then arithmetic must after all
involve epistemologically troublesome ontological
commitment. For an explanation of the truth of sentences
Counterfactuals, Ontological Commitment and Arithmetic — Paul McCallion
involving substitutional quantifiers is commonly given in
terms of the existence of expression types (or the possible
existence of expression tokens) and the truth of
substitution instances. For one thing, it is worth noting that
the problem of the epistemology of expression types (or
the possibility of expression tokens) is of a different sort
from the problem of the epistemology of numbers, as on a
standard construal numbers are not types. But more
importantly, such commitments would belong to the metatheory
of arithmetic sentences, not to arithmetic sentences
themselves. Substitutional quantifiers may be taken to be
primitive devices of infinite conjunction/disjunction (see
(Field 1984)), or to have their meaning given by their
inferential role (see (Rossberg 2006) and (Wright 2007) for
an inferentialist treatment of higher-order quantifiers).
4.
On the understanding of arithmetic sentences just
sketched it is natural to think of syntactically singular occurrences
of numerals and number-words as occurrences
of mere pseudo-singular terms. Such terms do not behave
semantically like genuine singular terms; they do not purport
to refer to objects. It is likewise natural to think of numerical
predicates as pseudo-predicates. Such predicates
do not behave semantically like genuine predicates; they
do not purport to ascribe properties to objects.
A consequence of this would be that second-order
arithmetic quantifiers (such as those that occur in the
second-order version of the induction axiom) should also
be construed substitutionally. This would in turn entail that
arithmetic predicates may not be impredicatively defined,
on pain of circularity. The moral is that commitment to
numbers as objects is therefore unexpectedly revealed not
by the assertion of sentences involving numerical singular
terms or quantifiers but rather by the acceptance of
impredicative definitions of numerical predicates.
Literature
Bostock, David 1974 Logic and Arithmetic. Volume 1. Natural
Numbers, The Clarendon Press, Oxford University Press, Oxford.
Gottlieb, Dale 1980 Ontological Economy: Substitutional Quantification
and Mathematics, New York: Oxford University Press.
Field, Hartry 1984 Review of Dale Gottlieb, Ontological Economy:
Substitutional Quantification and Mathematics, Nous 18.
Hodes, Harold 1984 “Logicism and the Ontological Commitments
of Arithmetic”, Journal of Philosophy LXXXI, 123-49.
Nolan, Daniel 1997 “Impossible Worlds: a modest approach”, Notre
Dame Journal of Formal Logic, 38(4).
Rayo, Agustin 2002 “Frege’s Unofficial Arithmetic”, The Journal of
Symbolic Logic 67, 1623–1638.
Rayo, Agustin forthcoming “On Specifying Truth Conditions”, The
Philosophical Review.
Rossberg, Marcus 2006 Second-Order Logic: Ontological and
Epistemological Problems, (Ph.D dissertation, University of St.
Andrews).
Wright, Crispin 2007 ‘On Quantifying into Predicate Position: Steps
towards a New(tralist) Perspective’, in: Mary Leng, Alexander
Paseau, and Michael Potter, (eds.) Mathematical Knowledge, Oxford
University Press.
Yablo, Stephen 2001 “Go figure: a path through fictionalism”, Midwest
Studies in Philosophy 25, 72-102.
Yablo, Stephen 2005 “The myth of the seven”, in: Kalderon, Mark
(ed.) 2005 Fictionalism in Metaphysics, Oxford: Clarendon Press,
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