FOM: Boolos on the Analyticity of HP

[Roger Bishop Jones]

In his paper entitled "Is Hume's Principle Analytic?"
(reprinted in Logic, Logic, Logic)
George Boolos concludes:

    "It is thus difficult to see how on any sense of the word 'analytic',
     the key axiom of a theory that we don't know to be consistent
     and that contradicts our best established theory of number
     (on the natural readings of its primitives)
     can be thought of as analytic."

In this note I propose to show, contra-Boolos, how easy this can be.

The "key axiom" referred to is "HP" or "the cardinality principle".
It is indeed difficult to see how this can be analytic if it does
contradict the standard theory of cardinals, for it would then be in
all probability false.
I therefore propose to disregard Boolos's prejudicial referring
phrase, and to show that HP understood as involving some
concept of number distinct from the now standard notion of
cardinal number, can easily be "thought of as analytic".

Though Boolos talks about "any sense" of the word analytic,
it is clear from his paper that one sense he particularly has in
mind is that in which analytic truths are described as
"true in virtue of their meaning".  I therefore proceed first
by making this notion sufficiently precise that it is clearly well
defined for certain classes of language which include at least
one in which HP is analytic.

Clearly this sense of analyticity is only applicable to languages
in which all the relevant parts are given a meaning, and is
relative to a language and its semantics.
It is possible that a sentence expressing HP
(say "forall X,Y. #X=#Y <=> X =~ Y")
may belong to more than language, may have a meaning
which varies from one to another, and may be analytic
in some and false in others.

I therefore begin by defining the notion of "a semantics" for a
first order language.

A semantics for a first order language is a non-empty set of
interpretations of that language, which we will call "the intended"
interpretations, and which are exactly those interpretations of
the non-logical constants which are consistent with the
(informally intended) semantics.
A first order language together with a semantics for that language
may be called an interpreted first order language.

We may now define analyticity for an interpreted first order
language.
A sentence in an interpreted first order language is analytic
iff it is true in all the intended intepretations.

For the sake of discussing HP the above definition of analyticity
can be extended to higher order logics with their standard semantics
by replacing "first" by "higher-order" throughout.

I claim at this point that the concept of analyticity is as well defined
as any concept in logic, and a great deal better defined than most
in philosophy, and that this definition is, for the classes of languages
in question, a very good (and the obvious) precise rendering of
the informal notion "true in virtue of its meaning".

Though the concept of analyticity is now precisely and completely
defined, its application depends upon the semantics of relevant
languages being defined.
In this I have in effect given a criterion for when the semantics is
defined (i.e. when a certain non-empty set has been defined)
without saying anything about how this should be done.
It will be the case that the semantics of some languages (e.g. that
of first order set theory) is not so easy to precisely define
(or at least, to agree on how it should be defined).
The concept of analyticity relative to a particular language will
inherit any imprecision found in the semantics of the language.

A precise way of defining the semantics of a first or
higher-order language is that advocated by David Hilbert.
It is by giving a satisfiable set of ("non-logical") axioms
which characterise the non-logical constants.
The semantics is then the set of models of the axioms,
and it follows immediately that all the theorems derivable
in the logic from the axioms are analytic.
In first order languages theoremhood and analyticity will
then coincide, which tidy outcome is bought at a very heavy
price, for most interesting mathematical concepts cannot
be given quite the right semantics by a recursive axiomatisation
in a first order language.
(though an incomplete axiomatisation, i.e. one in which
the set of "intended interpretations" contains some which
are not really intended still yields a theory in which all
theorems are analytic in the semantics you really wanted.
The semantic gap corresponds to the proof theoretic one,
there are some theorems which are true but not provable
and it is these whose analyticity is misrepresented by an
incomplete axiomatic semantics).

Alternative ways of defining the semantics are informal,
or the use of an axiomatic semantics with informal codicils.

A classic example of a precise but informal semantics
is that of the first order language of "true arithmetic".
The semantics of this language may be specified by saying that
it has just one intended interpretation (up to isomorphism)
which is the natural numbers.
The sentences of true arithmetic, under this semantics and
the definition of analyticity given above, are analytic iff true,
and under this terminology the truth of the logicist doctrine
that "arithmetic is analytic" is established without any consideration
of axioms.

A more full blooded logicism, the doctrine that
"mathematics is analytic" can be approached through set theory.
The simplest but semantically unsatisfactory method is to
give a semantics to the language of first order set theory
using the axioms of ZFC.
Set theory and all the mathematics derivable in it then
becomes analytic.
Unfortunately this semantics is just plain wrong (in terms
of the meaning of the mathematical concepts defined using
it).
There are at least two way of fixing the semantics, each of
which involves the injection of a degree of informality into
the definition of the semantics.
The first is to do set theory in second order logic (the
"informality" here comes in the account of standard semantics)..
A second is to add an informal supplement to the first order
axioms.
For example, one might define the intended interpretations
of first order set theory as "the well-founded models of ZFC".
This is about the weakest/loosest semantics which is not
obviously wrong. Doubtless set theorists, if they were interested,
could come up with better definitions.

Returning to HP, it is now clear that under the proposed
definitions we can define a language (call it "Frege arithmetic")
which is a second order language with a "non-logical" function
symbol "#", whose intended interpretations are exactly those
in which HP is true under the standard second-order semantics.
In this language, HP and all the truths of arithmetic which
are derivable from it, are analytic (together with some which
are not derivable).
My own personal opinion is that this is a bad idea,
HP providing a good basis neither for the semantics nor
for the proof of arithmetic sentences.
But it can be done.

Two subsidiary issues arise from the manner in which
Boolos refers to HP in the quoted paragraph.
The first is the mismatch between Frege's # and the usual
notion of cardinality.
This has no bearing on the analyticity of HP (under the
above definitions) in any language in which it is true,
but it does suggest that the recent tendency to prefer
"the cardinality principle" as a name for HP may be
misguided.
The second is consistency.
Under the above definitions there will be an interpreted
second order language in which HP is analytic iff SOL+HP
is consistent.

There are of course many other considerations raised in
the paper which I have not here addressed, my aim here
has been exclusively to comment on its final paragraph.

I invite philosophers of mathematics who subscribe to
the alleged contemporary consensus against logicism to
nominate the argument which they consider most decisively
to refute that for logicism given in this note, and those philosophers
who consider that HP provides a case for logicism which
is otherwise absent to explain how and why the definition of
analyticity provided here should be modified (or replaced)
to make HP appear to have some relevance to the case for logicism.

Roger Jones