On Intuitionistic Linear Logic - Microsoft Research

§2. Overview of Logical Systems and The Curry-Howard Correspondence 3
[A x ]
·
M: B
(⊃I)x.
λx: A.M: A ⊃ B
It is common to annotate assumption packets with alphabetic identifiers rather than natural numbers.
Thus we have a system for term formation, where the terms have the property that they
uniquely encode the deduction.
Although natural deduction has many compelling qualities, it has several disadvantages of which
we shall mention just two (Girard [34, Page 74] gives a fuller criticism). Firstly it is distinctly
asymmetric; there is always a single deduction from a number of assumptions. 1 Secondly, some
connectives can only be formulated in an unsatisfactory way. Consider the rule for eliminating a
disjunction
·
A ∨ B
C
[A x ]
·
C
[B y ]
·
C (∨E)x,y.
The deduction C really has nothing to do with the connective being eliminated at all: it is often
dubbed parasitic.
The second system we consider is the sequent calculus, again introduced by Gentzen [73]. Deductions
consist of trees of sequents of the form Γ − ∆, where both Γ and ∆ represent collections
of propositions. Inference rules introduce connectives on the right and on the left of the ‘turnstile’
and rules have a more symmetric feel to them. For example the rules for introducing an implication
are
Γ − A, ∆ B, Γ ′ − ∆ ′
Γ, A ⊃ B, Γ ′ − ∆, ∆ ′
(⊃L), and
Γ, A − ∆, B (⊃R).
Γ − A ⊃ B, ∆
In this thesis we will be concerned only with intuitionistic logics. These can be obtained by restricting
the sequents to at most a single proposition on the right of the turnstile, Γ − A. There
are other, less restrictive ways of formulating intuitionistic logics but we shall not consider them
here. (For example, Hyland and de Paiva [43] have proposed a less restrictive (but more powerful)
formulation of ILL.)
Let us consider the form of Γ in a sequent Γ − A. We have a choice as to whether it represents
a set, multiset or sequence of propositions. Recall that in the natural deduction system, we had
multisets of assumptions, which could be empty. As the sequent calculus is an equivalent logical
system it must offer similar manipulations of its assumptions. These manipulations are provided by
so-called structural rules. 2 The structural rules needed depends on the chosen form of contexts and
on the way the inference rules are devised (as they can have the effect of structural rules ‘built-in’).
Generally we take the contexts to be multisets and then we have two structural rules 3
Γ − B Weakening, and
Γ, A − B
Γ, A, A − B Contraction.
Γ, A − B
1 A number of multiple conclusion formulations of natural deduction have been proposed, but they invariably
introduce more problems than they solve.
2 These rules really exist in the natural deduction formulation as well but they are often either embedded in informal
conditions concerning the assumption packets or built into the inference rules.
3 Were we interested in the order in which assumptions were used, then we would take contexts to be sequences of
assumptions and have an explicit Exchange rule:
Γ, A, B − C
Exchange
Γ, B, A − C