Editing Three-valued logic (section)

=== RM3 logic ===

The truth table for the material implication of R-mingle 3 (RM3) is
{| style="border-spacing: 10px 0;" align="center"
|- valign="bottom"
|
{| class="wikitable" style="text-align:center;"
|+ IMP{{sub|RM3}}(A, B)
! rowspan="2" colspan="2" | A → B
! colspan="3" |B
|-
! width="25" {{no|F}}
! width="25" | U
! width="25" {{yes|T}}
|-
! rowspan="3" |A
! scope="row" {{no|F}}
| {{yes|T}}
| {{yes|T}}
| {{yes|T}}
|-
! scope="row" | U
| {{no|F}}
| U
| {{yes|T}}
|-
! scope="row" width="25" {{yes|T}}
| {{no|F}}
| {{no|F}}
| {{yes|T}}
|}
|}

A defining characteristic of RM3 is the lack of the axiom of Weakening:
* {{math|(''A'' → (''B'' → ''A''))}}
which, by adjointness, is equivalent to the projection from the product:
* {{math|(''A'' ⊗ ''B'') → ''A''}}
RM3 is a non-cartesian symmetric monoidal closed category; the product, which is left-adjoint to the implication, lacks valid projections, and has {{math|''U''}} as the monoid identity.
This logic is equivalent to an [[Paraconsistent logic#An ideal three-valued paraconsistent logic|"ideal" paraconsistent logic]] which also obeys the contrapositive.