FOM: second-order logic is a myth

Stephen G Simpson simpson at math.psu.edu
Wed Mar 10 12:39:22 EST 1999

```Robert Black 9 Mar 1999 22:00:22
> is of course something with the power not of set theory but of
> second-order logic that, oddly, no-one (including me) has
> mentioned, namely unrestricted mereology, as used for example by
> Hartry Field in _Science Without Numbers_.

I for one am ignorant of mereology.  If you think mereology is
relevant to the discussion, why not describe it briefly?

> assuming that resistence to second-order logic comes from what
> Boolos calls its 'staggering' undecidability.

I don't think undecidability of second-order logic (staggering or
otherwise) is the worst thing about second-order logic.  What's really
bad about second-order logic is that it isn't a logic at all: it
doesn't provide a model of reasoning.  See also Harvey's posting of 24
Feb 1999 20:08:49.

Note: In the previous paragraph, I am referring to second-order logic
with the so-called `standard' set-theoretic semantics, i.e. with set
quantifiers interpreted as ranging over all subsets of the universe.
(Presumably this is what Boolos had in mind when he spoke of
staggering undecidability.)  This needs to be sharply distinguished
from certain first-order theories inspired by second-order logic.
These first-order theories are sometimes referred to as second-order
logic with Henkin semantics.  (These first-order theories do provide a
good model of reasoning.  Some mathematically important aspects of
this model are studied in my book `Subsystems of Second Order
Arithmetic' <http://www.math.psu.edu/simpson/sosoa/>.)

When discussing second-order logic, we need to constantly be aware of
the distinction between standard semantics and Henkin semantics.
Proponents of second-order logic sometimes blur this distinction.  For
example, the mathematical part of Shapiro's book spells out the
distinction in excruciating detail, but the philosophical part of the
book sometimes blurs it, with confusing results.

By the way, in your posting of 8 Mar 1999 19:14:37 you said that
Shapiro has published a reply to his critics in the current issue of
`Philosophia Mathematica'.  That journal is not easily available to me
and may not be available to all FOM subscribers.  Could you please
summarize Shapiro's reply here on FOM?

-- Steve

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