FOM: Synonymous theories

[A.P. Hazen]

With reference to Alasdair Urquhart's post (and an anxious glance in the
direction of Harvey Friedman's):
  PA2 and FA are mutually interpretable.  Alasdair asks if they are also
"synonymous" in the sense of de Bouvere.  (Two theories are synonymous if
they have a common definitional extension.  A notion very similar to de
Bouvere's was defined, and a result very similar to his proven, by Stig
Kanger, "Equivalent theories," in "Theoria," v. 38 (1972), pp. 1-6.)  My
***guess*** is that they aren't.  Theoretical synonymy is a very strong
relation, implying at least mutual FAITHFUL interpretability, so I suspect
the existence of non-theorems of FA that map, under interpretation, to
theorems of PA2 is enough to rule out synonymy.  (This is not a proof.  For
one thing, I haven't shown that FA isn't faithfully interpreted in PA2 in
some devious manner: only that the obvious way of interpreting it isn't
faithful.)  Again, synonymous theories have isomorphic Lindenbaum algebras.
These two theories may, for all I know how to prove, have isomorphic
Lindenbaum algebras, but all the extra non-theorems on the one side make me
dubious.    ...  Harvey says he can show that (at least in application to
some interesting class of theories??) various kinds of mutual
interpretability come to the same thing, though, and suggests that maybe
this extends to synonymy.  (The end of Harvey's post asks if people would
be interested in details: I would.)
   >>>>Footnote: the literature on theoretical synonymy is frustrating.
Everybody refers to unpublished examples of Kaplan's that show that not all
pairs of mutually interpretable theories are synonymous, but I have never
seen the example discussed in detail.  Is there anyone in the FoM community
who can show me (or US, if you think it is of general enough FoM-relevant
interest!) how it goes?
 --
Allen Hazen
Philosophy Department
University of Melbourne