FOM: Re: logical vs mathematical
Roger Bishop Jones
rbjones at rbjones.com
Mon Sep 11 11:21:52 EDT 2000
The status of ontological propositions has long been considered a problem in
the relationship between logic and mathematics.
Not unreasonably, absolute claims about abstract ontology are thought not to
be logically necessary, and hence not to be part of logic.
Using this to mark the division between logic and mathematics is however
unsatisfactory, for they have even less claim to be a part of mathematics
than of logic.
There is no need to consider the sentences of arithmetic to be asserting the
existence of the natural numbers, and if that interpretation were placed
upon them neither mathematicians nor anyone else would have any way of
establishing their truth.
Ontology is more plausibly a part of philosophy, but this kind of
ontological claim (e.g. "do the natural numbers exist?" without any
ontological context) is doubtful even as a part of philosophy.
Absolute claims about abstract ontology are strong contenders for being
considered "metaphysical" in the most pejorative sense of the logical
positivists, saved only by the possibility that they can be falsified by
being shown (logically) inconsistent.
The problem is then, not just that there may be doubt about their necessity,
but that there is considerable doubt about whether they have any meaning.
If we make the mistake of concluding from difficulties with absolute
abstract ontology, that any proposition which appears to involve ontology
cannot be logically necessary, then we discard most of what has been called
logic for the entire history of humanity, including of course, many theorems
of first order logic.
There is an alternative, which involves careful consideration of the meaning
of the propositions of logic and mathematics.
This alternative is not in the least bit novel or revolutionary, but is
reflected in common practices of logicians when they devise and study
Except in the case of exteme formalists who study formal systems while
denying that the formulae of these systems have any meaning, it is usual to
seek two kinds of assurance about a logical system at the earliest possible
The first is that the logic is consistent, and the second, desirable but not
essential, that it is complete.
In demonstrating consistency often the most straightforward method is to
give a semantics to the notation, utlimately defining what it is for a
proposition to be true (often the term "valid" is used), and then to
demonstrate inductively that only true propositions are provable by showing
that all the axioms are true and that the inference rules preserve truth.
Even where consistency is demonstrated by proof theoretic rather than
semantic methods, a semantics can subsequently be reverse engineered under
which semantics the theorems are then known to be true.
This practice is a wholly satisfactory method of demonstrating that the
theorems of the logic are all necessarily true under the defined semantics
for the language, and hence that they are logically true.
(All statements which are exclusively about abstract entities must if true
be necessarily true, since they make no reference to any entity which can be
expected to change from one possible world to the next or which we could
detect a change in if it did).
In formulating the semantics, some "presuppositions" about abstract ontology
must be made.
However, we need not hold that the content of these presuppositions is part
of the content of the propositions of the language defined.
Our consistency demonstration is of course relative, if the ontological
presuppositions used in formulating the semantics are incoherent then the
object logic may well be inconsistent.
The point of view here expressed has the following convenient consequences:
1. That almost all the theorems of those formal systems which are commonly
called logics are indeed logical truths (though some semantic reverse
engineering may be necessary to discover which necessary propositions they
2. That set theory consists also of logical truths.
There is some vagueness about which logical truths are expressed by the
sentences of (say) ZFC since these sentences can be construed either as
claims about all the models of ZFC or about certain special models
("standard" we might call them), e.g. the well-founded models, or those
which are V(alpha) for some alpha.
However, under any of these interpretations they all turn out true, and
since not contingent (making no reference to the material world),
3. That mathematics consists of logical truths.
The doctrine that "mathematics is analytic" can be reasonably so construed
(as it was by some of the logical positivists) that it states no more than
that the theorems of mathematics are derived in sound (not necessarily
complete) logical systems.
Or better, simply that they are propositions about abstract entities in well
defined languages, which happen to be true of their intended subject matter.
I do not for one second imagine that this (rather old) point of view will
receive mass assent after half a century of derision.
However, I believe it to be sensible, pragmatic, and much less problematic
than any other account known to me of the scope, nature and status of the
propositions of logic and mathematics.
(and am happy to defend it against all comers)
The only way we have to settle such absolute claims (by contrast with
relative ontological claims which are controversial neither as logic nor as
mathematics), can only be settled if they are logically inconsistent.
Note however, that a fact about logical consistency is a truth of
It is in fact a claim about the existence of a number, in my parlance a
relative ontological claim, i.e. a claim about the natural numbers which
does not assert
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