Hello Kai,
Thanks for your timely response, of course I still disagree.
I am a contrarian 
> At least in the case of "premise" this definition does not match your
> intuition.
Yes, because the logician's "jargon" have taken over the commonly
construed sense of "premise" as a distal necessary condition and
replaced it by something along the line "a formula appearing on the
left of a turnstile", i.e. a proximal condition.
> You can find these definitions in almost all the papers on the CoS
> website (but my thesis is the only one which also has a large green
> friendly frog on it).
I fetchted your thesis so I won't be confused anymore by "fancy"
terminology but it appears that my initial misconstrual does not
detracts from the soundness of my critiscism, namely, that two
"obviously" equivalent logics (in the strong sense that all their true
formulas but a finite few are syntactically identical) will be seen as
formally non equivalent whenever their axioms sets are different but
entails each other (and are thus in some sense equivalent, am I wrong
on this?) because the leaves of the proofs (in your sense of proofs as
trees) are different.
It follows from this incommensurability that neither the proofs nor the
logics are even comparable whereas they are "trivially" equivalent to
common sense.
I think the catch lies in the selection of proof as "a concrete
syntactic object", this is probably not abstract enough and guarantees
that you (or anyone else) will be swamped forever in irrelevant tedious
"bureaucracy" with idiosyncratic details pertaining only to the
"current logic under consideration", no matter how you try to tackle
the problems of proofs equivalence/logics equivalence.
Not that this will not lead to a few more excellent thesis...
Best,
JLD
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