Talk:Intuitionistic linear logic

message: We shall follow Martin Löf'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 \multimap.
\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 <math>. Time to upload images.
--Kaustuv Chaudhuri 07:22, 31 May 2004 (UTC)
thread_title: [[Natural deduction]] formulation

message: 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 \multimap.
\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 <math>. Time to upload images.
--Kaustuv Chaudhuri 07:22, 31 May 2004 (UTC)
thread_title:  Logical connectives 

message: I had to stop here because I can't for the life of me get the ampersand to work inside <math>. Time to upload images.
--Kaustuv Chaudhuri 07:22, 31 May 2004 (UTC)
thread_title:  Additive conjunction