On Intuitionistic Linear Logic - Microsoft Research

Chapter 2
Proof Theory
1 Sequent Calculus
As explained in Chapter 1, ILL arises from removing the structural rules of Weakening and
Contraction. This has the effect of distinguishing between different formulations of the familiar
connectives of IL. For example, in IL, we might formulate the ∧R rule in one of two ways.
Γ − A Γ − B ∧ ′ R
Γ − A ∧ B
Γ − A ∆ − B ∧ ′′ R
Γ, ∆ − A ∧ B
However, we can see that with rules of Weakening and Contraction with can simulate one with the
other.
Γ − A Γ − B ∧ ′′ R
Γ, Γ − A ∧ B Contraction ∗
Γ − A ∧ B
Γ − A Weakening ∗
Γ, ∆ − A
Γ, ∆ − A ∧ B
∆ − B Weakening ∗
Γ, ∆ − B ∧ ′ R
In ILL since we do not have the structural rules these two possible formulations become distinct
connectives. We shall use the terminology of Girard to describe these connectives: those where the
upper sequent contexts are disjoint (as in (∧ ′′ R )) are known as the multiplicatives and those where
the upper sequent contexts must be the same (as in (∧ ′ R )) are known as the additives. Thus for
ILL we shall consider the following connectives:
Connective Symbol
Multiplicative Implication −◦ “Linear Implication”
Conjunction ⊗ “Tensor”
Additive Conjunction & “With”
Disjunction ⊕ “Sum”
It can be seen that there are some obvious omissions from this table, namely multiplicative disjunction
and additive implication. Multiplicative disjunction ( . or “Par”) requires multiple conclusions
which is beyond the scope of this thesis. (It was thought that it only made sense as a classical
connective, but recent work by Hyland and de Paiva [43] shows how it can be considered as a
intuitionistic connective.) The additive implication, ⋄−, can be formulated as follows.
Γ − A B, Γ − C (⋄−L)
Γ, A⋄−B − C
Γ, A − B (⋄−R)
Γ − A⋄−B
However, this connective is generally ignored as its computational content seems minimal. 1 We shall
do likewise and not consider this connective further.
We have units for the two conjunctions and the (additive) disjunction. I is the unit for the
tensor, t is the unit for the With, and f is the unit for the Sum.
Of course the logic so far is extremely weak. Girard’s innovation was to introduce a new unary
connective, !, (the so-called ‘exponential’) to regain the logical power. The exponential allows a
1 Troelstra [75, Chapter 4] considers briefly additive implication in the context of a logic with just two implications.
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