Editing Linear logic (section)

===Exponentials===

The exponentials are used to give controlled access to weakening and contraction. Specifically, we add structural rules of weakening and contraction for {{math|?}}'d propositions:{{sfn|Girard|1987|loc=p.25-26, Def.1.21}}

{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} Γ}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} Γ, ?<VAR>A</VAR>}}
|}
| width="50" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} Γ, ?<VAR>A</VAR>, ?<VAR>A</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} Γ, ?<VAR>A</VAR>}}
|}
|}

and use the following logical rules, in which {{math|?Γ}} stands for a list of propositions each prefixed with {{math|?}}:

{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} ?Γ, <VAR>A</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} ?Γ, !<VAR>A</VAR>}}
|}
| width="50" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} Γ, <VAR>A</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} Γ, ?<VAR>A</VAR>}}
|}
|}

One might observe that the rules for the exponentials follow a different pattern from the rules for the other connectives, resembling the inference rules governing modalities in sequent calculus formalisations of the [[normal modal logic]] [[S4 (modal logic)|S4]], and that there is no longer such a clear symmetry between the duals {{math|!}} and {{math|?}}.  
This situation is remedied in alternative presentations of CLL (e.g., the [[Logic of unity|LU]] presentation).