Talk:Intuitionistic linear logic

This is what used to be there.

Natural deduction formulation
We shall follow Martin L&ouml;f's style of hypothetical judgments. A linear hypothetical judgment $$A_1, A_2, \ldots, A_n \vdash A$$, abbreviated as $$\Delta \vdash A$$, defines the logical fact that the goal $$A$$ is obtained from the resources $$A_i$$ by consuming each resource exactly once. The simplest judgment is that of linear consumption of hypothesis, which is written as an axiomatic rule: $$\frac{}{A \vdash A}$$

(Note that this is different from the usual hypothesis rule of ordinary logic, $$\frac{}{\Gamma, A \vdash A}$$, because the sole allowed resource is the goal $$A$$, which must be present; thus consumed exactly once.)

Dual to the hypothesis rule is a substitution principle that describes how to chain a judgment concluding $$A$$ with a judgment having a linear resource $$A$$.

substitution principle
If $$\Delta \vdash A$$ and $$\Delta', A \vdash C$$, then $$\Delta, \Delta' \vdash C$$.

The notation $$\Delta,\Delta'$$ denotes multiset union.

Logical connectives
The connectives of ILL are of two major kinds: multiplicative and additive. Let us visit them in sequence.

Multiplicative conjunction
$$ \frac{\Delta \vdash A \quad \Delta' \vdash B}    {\Delta, \Delta' \vdash A \otimes B}\otimes_I \qquad \frac{\Delta \vdash A \otimes B \quad \Delta', A, B \vdash C}{\Delta, \Delta' \vdash C}\otimes_E $$

Multiplicative implication
I'll use $$\rightarrow$$ to denote this connective, though the traditional glyph is.

$$ \frac{\Delta, A \vdash B}{\Delta \vdash A \rightarrow B}\rightarrow_I \qquad \frac{\Delta \vdash A \rightarrow B \quad \Delta' \vdash A}    {\Delta, \Delta' \vdash B}\rightarrow_E $$

Multiplicative disjunction
Multiplicative disjuntion or par is impossible in ILL.

Next the additive connectives.

Additive conjunction
I had to stop here because I can't for the life of me get the ampersand to work inside. Time to upload images.

--Kaustuv Chaudhuri 07:22, 31 May 2004 (UTC)