[FOM] Synonymity

[Alasdair Urquhart]

I recently wrote a paper with Jeff Pelletier
with the title "Synonymous Logics."  It is mostly 
in the context of propositional modal logics.
We give a few examples of non-trivial synonymities
between modal logics, as well as proving that 
various well known modal logics are NOT synonymous.
It is to appear in the Journal of Philosophical
Logic.

We also give examples of modal logics that are
mutually interpretable but not synonymous.
According to De Bouvere 1965, Montague's 1957
thesis contains the first example of this in
the context of first-order logic.  

An area of mathematics in which synonymous theories
are very important is the theory of matroids.
The book "Theory of Matroids" by Neil White
(CUP 1986) contains an appendix with many examples
of synonymous formulations of the theory.
Garrett Birkhoff's term "cryptomorphic" is used,
rather than "synonymous."  

I have a vague recollection that first order
Peano Arithmetic is synonymous with ZF + the
negation of the axiom of infinity.  Is that
right?

Alasdair Urquhart