Editing Linear logic (section)

===Additives===

The rules for additive conjunction (&) and disjunction (⊕) :

{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} Γ, <VAR>A</VAR>}} || &nbsp;&nbsp; || &nbsp;&nbsp; || {{math|{{tee}} Γ, <VAR>B</VAR>}}
|-
| colspan=4 style="border-top:2px solid black;" |
|-
| colspan=4 align="center" | {{math|{{tee}} Γ, <VAR>A</VAR> & <VAR>B</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> ⊕ <VAR>B</VAR>}}
|}
| width="25" |
| style="text-align: center;" |
{| border="0"
|-
| {{math|{{tee}} Γ, <VAR>B</VAR>}}
|-
| style="border-top:2px solid black;" |
|-
| {{math|{{tee}} Γ, <VAR>A</VAR> ⊕ <VAR>B</VAR>}}
|}
|}

and for their units:

{| style="margin:auto"
|-
| style="text-align: center;" |
{| border="0"
|-
| &nbsp;
|-
| colspan=4 style="border-top:2px solid black;" |
|-
| colspan=4 align="center" | {{math|{{tee}} Γ, ⊤}}
|}
| width="50" |
| style="text-align: center;" | (no rule for {{math|0}})
|}

Observe that the rules for additive conjunction and disjunction are again admissible under a classical interpretation. 
But now we can explain the basis for the multiplicative/additive distinction in the rules for the two different versions of conjunction: for the multiplicative connective (⊗), the context of the conclusion ({{math|Γ, Δ}}) is split up between the premises, whereas for the additive case connective (&) the context of the conclusion ({{math|Γ}}) is carried whole into both premises.