[FOM] Is first-order logic an adequate language for mathematics?

Randall Holmes m.randall.holmes at gmail.com
Thu Oct 26 13:38:31 EDT 2006

```[note to moderator:  how is this?]

This is a lovely question which we can go around and around on.

In response to Arnon Avron, I reply that his objections to first order
logic appear to suggest that the proper extension is to second-order
logic, which is, after all, simpler than the theories he favors.

The philosophical motivation for second-order logic is quite clear:
one wants to be able to talk not only about the domain elements
but also about properties of the domain elements (reified as classes
of domain elements).

Second order logic doesn't admit a complete axiomatization but
it does admit a natural incomplete axiomatization.  And it is
much simpler than the logic of the ancestral (while easily subsuming it).

The objection seems to be that second-order logic talks about
problematic entities (infinite sets).  For some of us, of course,
infinite sets simply aren't problematic.  But there is also the point
that what we are talking about is  universals (properties of the
objects of the domain that concerns us) and that viewing these
as sets may construed as metaphorical [even supposing that we
have a completed totality of these universals may be construed
as metaphorical]

Finally, we note that the language extension from first order logic
to second order logic uses no syntactical or indeed deductive
machinery which is not already found in first order logic...we do
first order logic on a larger domain.  So maybe first order logic
is the right language :-)

Personally, I favor higher-order logic of order omega (type theory).
This has the practical merit that the notion of equality (in any
type) becomes definable, and that properties of any object
under consideration belong to a type in the hierarchy and can
themselves be construed as objects.

The actual deductive machinery of higher order logic is
that of first order logic (applied in a specific context).  It is
incomplete but natural.

The intended semantics of higher order logic cannot be captured
by first order logic but can be expressed in second order logic.

Sincerely, Randall Holmes
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