Marginalia on sequent calculi
A.S.Troelstra

In this note we wish to draw attention to certain points of detail, concerning 
the subtle differences between several possible versions of Gentzen systems and
systems of natural deduction. In notation and terminology we conform to [TS96].
Gentzen([Gen35]) introduced the calculi LJ, LK with left­ and right introduction
rules, operating on sequents; systems of this kind are in the literature often 
called `sequent calculi'. Contrary to what many people think, natural deduction
did not originate with Gentzen (there is, for example, the earlier work by 
Jaskowski, [J'as34]) although Gentzen's work made it wellknown. Versions of 
natural deduction are sometimes also presented as calculi operating on sequents
(as in this note). For these reasons I have rejected the widely used designation
`sequent calculi' for systems of the LJ,LK­type and call them `Gentzen systems'
instead. (One might object that ``Gentzen systems'' might be interpreted as 
referring to all the formalisms discussed in Gentzen's paper, and that only 
`Gentzen sequent calculi' would be unambiguous. But in practice I find this too
long.)
On the side of the Gentzen systems, there are combinatorial differences between
systems with term labels and systems without term labels attached; the customary
presentation of Gentzen systems does not include term labels. Correspondingly, 
for natural deduction, there are the versions obeying the Complete Discharge 
Convention (the CDC): open assumptions are always discharged at the earliest 
opportunity, and versions with labelled assumption classes, isomorphic to 
(suitable extensions of) simple type theory. The correlation between natural 
deduction and Gentzen systems is usually described for Gentzen systems with term
labels on the one hand, and extensions of the simple type theory on the other 
hand. In this note we discuss the systems and their relationships primarily for
the unlabelled variants.
The deductions in the systems we want to consider are, with few exceptions,
represented as prooftrees with the nodes labeled by sequents $\Gamma \to A$, 
where $\Gamma$ is a multiset, but there are no term labels. To keep things 
simple, the whole discussion is carried out for intuitionistic implication 
logic, although there is no problem extending the discussion to full first­order
intuitionistic logic. Technically, the results are simple adaptations of work by
[Min96, Her95].
Possible reasons for being interested in these systems are (a) pedagogical: by
comparison we see what the advantages/disadvantages of term labels are, and (b)
for theorem­proving purposes, we are more interested in derivability than in 
derivations; natural deductions with the CDC permit representation by less data.
Maybe this is useful; in any case it might be worth looking into.
The version N_3 of natural deduction considered below may be seen as a presen­
tation of natural deduction under the CDC. Natural deduction under the CDC is
considered to have bad combinatorial properties (no strong normalization), when
compared with the standard version of natural deduction which is isomorphic to
simple type theory $\lambda_\rightarrow$. On the other hand, the results in 
this note show that there are natural counterparts to $N_3$ on the side of the 
Gentzen systems ($G_2$ ,$G_5$) such that known correlations of a more static 
nature for the term­labeled versions in an obvious way ``project down'' onto 
$N_3$, $G_2$, $G_5$. In particular we shall consider normal forms on the side 
of the Gentzen systems (in two versions) in 1­1­correspondence with normal 
deductions in $N_3$. These results may be regarded as a limited rehabilitation 
of deductions under the CDC. Also, the results seem to point to a natural 
correlation between standard Gentzen systems with contraction absorbed into the
rules (Kleene's G3) and natural deduction under the CDC.