Translations of classical logic

T-translation $A\imp B \mapsto \oc{\wn{A}}\limp\wn{B}$

Formulas are translated as:

$\begin{array}{rcl} X^T & = & X \\ (A\imp B)^T & = & \oc{\wn{A^T}}\limp\wn{B^T} \\ (A\wedge B)^T & = & \wn{A^T} \with \wn{B^T} \\ T^T & = & \top \\ (A\vee B)^T & = & \wn{A^T}\parr\wn{B^T} \\ F^T & = & \bot \\ (\neg A)^T & = & \wn{\oc{(A^T)\orth}} \\ (\forall\xi A)^T & = & \forall\xi \wn{A^T} \\ (\exists\xi A)^T & = & \exists\xi \oc{\wn{A^T}} \end{array}$

This is extended to sequents by $(\Gamma\vdash\Delta)^T = \oc{\wn{\Gamma^T}}\vdash\wn{\Delta^T}$.

This allows one to translate the rules of classical logic into linear logic:

$\LabelRule{\rulename{ax}} \NulRule{A\vdash A} \DisplayProof \qquad\mapsto\qquad \LabelRule{\rulename{ax}} \NulRule{\wn{A^T}\vdash\wn{A^T}} \LabelRule{\oc L} \UnaRule{\oc{\wn{A^T}}\vdash\wn{A^T}} \DisplayProof$

$\AxRule{\Gamma\vdash A,\Delta} \AxRule{\Gamma',A\vdash\Delta'} \LabelRule{\rulename{cut}} \BinRule{\Gamma,\Gamma'\vdash\Delta,\Delta'} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{\Delta^T}} \LabelRule{\oc R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash\oc{\wn{A^T}},\wn{\Delta^T}} \AxRule{\oc{\wn{\Gamma'^T}},\oc{\wn{A^T}}\vdash\wn{\Delta'^T}} \LabelRule{\rulename{cut}} \BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{\Gamma'^T}}\vdash\wn{\Delta^T},\wn{\Delta'^T}} \DisplayProof$

$\AxRule{\Gamma,A,A\vdash\Delta} \LabelRule{c L} \UnaRule{\Gamma,A\vdash\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}},\oc{\wn{A^T}}\vdash\wn{\Delta^T}} \LabelRule{\oc c L} \UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma\vdash A,A,\Delta} \LabelRule{c R} \UnaRule{\Gamma\vdash A,\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{A^T},\wn{\Delta^T}} \LabelRule{\wn c R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma\vdash\Delta} \LabelRule{w L} \UnaRule{\Gamma,A\vdash\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{\Delta^T}} \LabelRule{\oc w L} \UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma\vdash\Delta} \LabelRule{w R} \UnaRule{\Gamma\vdash A,\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{\Delta^T}} \LabelRule{\wn w R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma,A\vdash B,\Delta} \LabelRule{\imp R} \UnaRule{\Gamma\vdash A\imp B,\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash\wn{B^T},\wn{\Delta^T}} \LabelRule{\limp R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \oc{\wn{A^T}}\limp\wn{B^T},\wn{\Delta^T}} \LabelRule{\wn d R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{(\oc{\wn{A^T}}\limp\wn{B^T})},\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma\vdash A,\Delta} \AxRule{\Gamma',B\vdash\Delta'} \LabelRule{\imp L} \BinRule{\Gamma,\Gamma',A\imp B\vdash\Delta,\Delta'} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{\Delta^T}} \LabelRule{\oc R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash\oc{\wn{A^T}},\wn{\Delta^T}} \LabelRule{\rulename{ax}} \NulRule{\wn{B^T}\vdash\wn{B^T}} \LabelRule{\limp L} \BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\limp\wn{B^T}\vdash\wn{B^T},\wn{\Delta^T}} \LabelRule{\wn L} \UnaRule{\oc{\wn{\Gamma^T}},\wn{(\oc{\wn{A^T}}\limp\wn{B^T})}\vdash\wn{B^T},\wn{\Delta^T}} \LabelRule{\oc d L} \UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{(\oc{\wn{A^T}}\limp\wn{B^T})}}\vdash\wn{B^T},\wn{\Delta^T}} \LabelRule{\oc R} \UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{(\oc{\wn{A^T}}\limp\wn{B^T})}}\vdash\oc{\wn{B^T}},\wn{\Delta^T}} \AxRule{\oc{\wn{\Gamma'^T}},\oc{\wn{B^T}}\vdash\wn{\Delta'^T}} \LabelRule{\rulename{cut}} \BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{\Gamma'^T}},\oc{\wn{(\oc{\wn{A^T}}\limp\wn{B^T})}}\vdash\wn{\Delta^T},\wn{\Delta'^T}} \DisplayProof$

$\AxRule{\Gamma\vdash A,\Delta} \AxRule{\Gamma\vdash B,\Delta} \LabelRule{\wedge R} \BinRule{\Gamma\vdash A\wedge B,\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T},\wn{\Delta^T}} \AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{B^T},\wn{\Delta^T}} \LabelRule{\with R} \BinRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T}\with \wn{B^T},\wn{\Delta^T}} \LabelRule{\wn d R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{(\wn{A^T}\with \wn{B^T})},\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma,A\vdash \Delta} \LabelRule{\wedge_1 L} \UnaRule{\Gamma,A\wedge B\vdash \Delta} \DisplayProof \qquad\mapsto\qquad \LabelRule{\rulename{ax}} \NulRule{\wn{A^T}\vdash \wn{A^T}} \LabelRule{\with_1 L} \UnaRule{\wn{A^T}\with \wn{B^T}\vdash \wn{A^T}} \LabelRule{\wn L} \UnaRule{\wn{(\wn{A^T}\with \wn{B^T})}\vdash \wn{A^T}} \LabelRule{\oc d L} \UnaRule{\oc{\wn{(\wn{A^T}\with \wn{B^T})}}\vdash \wn{A^T}} \LabelRule{\oc R} \UnaRule{\oc{\wn{(\wn{A^T}\with \wn{B^T})}}\vdash \oc{\wn{A^T}}} \AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash \wn{\Delta^T}} \LabelRule{\rulename{cut}} \BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{(\wn{A^T}\with \wn{B^T})}}\vdash \wn{\Delta^T}} \DisplayProof$

$\LabelRule{T R} \NulRule{\Gamma\vdash T,\Delta} \DisplayProof \qquad\mapsto\qquad \LabelRule{\top R} \NulRule{\oc{\wn{\Gamma^T}}\vdash \top,\wn{\Delta^T}} \LabelRule{\wn d R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\top},\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma\vdash A,B,\Delta} \LabelRule{\vee R} \UnaRule{\Gamma\vdash A\vee B,\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T},\wn{B^T},\wn{\Delta^T}} \LabelRule{\parr R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T}\parr\wn{B^T},\wn{\Delta^T}} \LabelRule{\wn d R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{(\wn{A^T}\parr\wn{B^T})},\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma,A\vdash \Delta} \AxRule{\Gamma',B\vdash \Delta'} \LabelRule{\vee L} \BinRule{\Gamma,\Gamma',A\vee B\vdash \Delta,\Delta'} \DisplayProof \qquad\mapsto\qquad \LabelRule{\rulename{ax}} \NulRule{\wn{A^T}\vdash \wn{A^T}} \LabelRule{\rulename{ax}} \NulRule{\wn{B^T}\vdash \wn{B^T}} \LabelRule{\parr L} \BinRule{\wn{A^T}\parr \wn{B^T}\vdash \wn{A^T},\wn{B^T}} \LabelRule{\wn L} \UnaRule{\wn{(\wn{A^T}\parr \wn{B^T})}\vdash \wn{A^T},\wn{B^T}} \LabelRule{\oc d L} \UnaRule{\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \wn{A^T},\wn{B^T}} \LabelRule{\oc R} \UnaRule{\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \wn{A^T},\oc{\wn{B^T}}} \AxRule{\oc{\wn{\Gamma'^T}},\oc{\wn{B^T}}\vdash \wn{\Delta'^T}} \LabelRule{\rulename{cut}} \BinRule{\oc{\wn{\Gamma'^T}},\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \wn{A^T},\wn{\Delta'^T}} \LabelRule{\oc R} \UnaRule{\oc{\wn{\Gamma'^T}},\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \oc{\wn{A^T}},\wn{\Delta'^T}} \AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash \wn{\Delta^T}} \LabelRule{\rulename{cut}} \BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{\Gamma'^T}},\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \wn{\Delta^T},\wn{\Delta'^T}} \DisplayProof$

$\AxRule{\Gamma\vdash \Delta} \LabelRule{F R} \UnaRule{\Gamma\vdash F,\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{\Delta^T}} \LabelRule{\bot R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \bot,\wn{\Delta^T}} \LabelRule{\wn R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\bot},\wn{\Delta^T}} \DisplayProof$

$\LabelRule{F L} \NulRule{F\vdash {}} \DisplayProof \qquad\mapsto\qquad \LabelRule{\bot L} \NulRule{\bot\vdash {}} \LabelRule{\wn L} \UnaRule{\wn{\bot}\vdash {}} \LabelRule{\oc d L} \UnaRule{\oc{\wn{\bot}}\vdash {}} \DisplayProof$

$\AxRule{\Gamma,A\vdash \Delta} \LabelRule{\neg R} \UnaRule{\Gamma\vdash \neg A,\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash \wn{\Delta^T}} \LabelRule{(.)\orth R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\oc{(A^T)\orth}},\wn{\Delta^T}} \LabelRule{\wn d R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\wn{\oc{(A^T)\orth}}},\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma\vdash A,\Delta} \LabelRule{\neg L} \UnaRule{\Gamma,\neg A\vdash \Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T},\wn{\Delta^T}} \LabelRule{(.)\orth L} \UnaRule{\oc{\wn{\Gamma^T}},\oc{(A^T)\orth}\vdash \wn{\Delta^T}} \LabelRule{\wn L} \UnaRule{\oc{\wn{\Gamma^T}},\wn{\oc{(A^T)\orth}}\vdash \wn{\Delta^T}} \LabelRule{\wn L} \UnaRule{\oc{\wn{\Gamma^T}},\wn{\wn{\oc{(A^T)\orth}}}\vdash \wn{\Delta^T}} \LabelRule{\oc d L} \UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{\wn{\oc{(A^T)\orth}}}}\vdash \wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma\vdash A,\Delta} \LabelRule{\forall R} \UnaRule{\Gamma\vdash \forall\xi A,\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T},\wn{\Delta^T}} \LabelRule{\forall R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \forall\xi \wn{A^T},\wn{\Delta^T}} \LabelRule{\wn d R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\forall\xi \wn{A^T}},\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma,A[\tau/\xi]\vdash \Delta} \LabelRule{\forall L} \UnaRule{\Gamma,\forall\xi A\vdash \Delta} \DisplayProof \qquad\mapsto\qquad \LabelRule{\rulename{ax}} \NulRule{\wn{A^T}[\tau^T/\xi]\vdash \wn{A^T}[\tau^T/\xi]} \LabelRule{\forall L} \UnaRule{\forall\xi \wn{A^T}\vdash \wn{A^T}[\tau^T/\xi]} \LabelRule{\wn L} \UnaRule{\wn{\forall\xi \wn{A^T}}\vdash \wn{A^T}[\tau^T/\xi]} \LabelRule{\oc d L} \UnaRule{\oc{\wn{\forall\xi \wn{A^T}}}\vdash \wn{A^T}[\tau^T/\xi]} \LabelRule{\oc R} \UnaRule{\oc{\wn{\forall\xi \wn{A^T}}}\vdash \oc{\wn{A^T}}[\tau^T/\xi]} \AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{(A^T[\tau^T/\xi])}}\vdash \wn{\Delta^T}} \LabelRule{\rulename{cut}} \BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{\forall\xi \wn{A^T}}}\vdash \wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma\vdash A[\tau/\xi],\Delta} \LabelRule{\exists R} \UnaRule{\Gamma\vdash \exists\xi A,\Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T[\tau^T/\xi]},\wn{\Delta^T}} \LabelRule{\oc R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \oc{\wn{A^T[\tau^T/\xi]}},\wn{\Delta^T}} \LabelRule{\exists R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \exists\xi \oc{\wn{A^T}},\wn{\Delta^T}} \LabelRule{\wn d R} \UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\exists\xi \oc{\wn{A^T}}},\wn{\Delta^T}} \DisplayProof$

$\AxRule{\Gamma,A\vdash \Delta} \LabelRule{\exists L} \UnaRule{\Gamma,\exists\xi A\vdash \Delta} \DisplayProof \qquad\mapsto\qquad \AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash \wn{\Delta^T}} \LabelRule{\exists L} \UnaRule{\oc{\wn{\Gamma^T}},\exists\xi\oc{\wn{A^T}}\vdash \wn{\Delta^T}} \LabelRule{\wn L} \UnaRule{\oc{\wn{\Gamma^T}},\wn{\exists\xi\oc{\wn{A^T}}}\vdash \wn{\Delta^T}} \LabelRule{\oc d L} \UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{\exists\xi\oc{\wn{A^T}}}}\vdash \wn{\Delta^T}} \DisplayProof$

Alternative presentation

It is also possible to define $A^{\overline{T}}$ by:

$\begin{array}{rcl} X^{\overline{T}} & = & \wn{X} \\ (A\imp B)^{\overline{T}} & = & \wn{(\oc{A^{\overline{T}}}\limp B^{\overline{T}})} \\ (A\wedge B)^{\overline{T}} & = & \wn{(A^{\overline{T}} \with B^{\overline{T}})} \\ T^{\overline{T}} & = & \wn{\top} \\ (A\vee B)^{\overline{T}} & = & \wn{(A^{\overline{T}}\parr B^{\overline{T}})} \\ F^{\overline{T}} & = & \wn{\bot} \\ (\neg A)^{\overline{T}} & = & \wn{\wn{(A^{\overline{T}})\orth}} \\ (\forall\xi A)^{\overline{T}} & = & \wn{\forall\xi A^{\overline{T}}} \\ (\exists\xi A)^{\overline{T}} & = & \wn{\exists\xi \oc{A^{\overline{T}}}} \end{array}$

If we define $(\Gamma\vdash\Delta)^{\overline{T}} = \oc{\Gamma^{\overline{T}}}\vdash \Delta^{\overline{T}}$, we have $(\Gamma\vdash\Delta)^{\overline{T}} = (\Gamma\vdash\Delta)^T$ and thus we obtain the same translation of proofs.