Editing Relevance logic (section)

==Axioms==
The early developments in relevance logic focused on the stronger systems. The development of the Routley–Meyer semantics brought out a range of weaker logics. The weakest of these logics is the relevance logic B. It is axiomatized with the following axioms and rules.
# <math>A\to A</math>
# <math>A\land B\to A</math>
# <math>A\land B\to B</math>
# <math>(A\to B)\land(A\to C)\to (A\to B\land C)</math>
# <math>A\to A\lor B</math>
# <math>B\to A\lor B</math>
# <math>(A\to C)\land(B\to C)\to (A\lor B\to C)</math>
# <math>A\land(B\lor C)\to (A\land B)\lor(A\land C)</math>
# <math>\lnot\lnot A\to A</math>
The rules are the following.
# <math>A, A\to B\vdash B</math>
# <math>A, B\vdash A\land B</math>
# <math>A\to B\vdash (C\to A)\to(C\to B)</math>
# <math>A\to B\vdash (B\to C)\to(A\to C)</math>
# <math>A\to \lnot B\vdash B\to\lnot A</math>
Stronger logics can be obtained by adding any of the following axioms.
# <math>(A\to B)\to (\lnot B\to\lnot A)</math>
# <math>(A\to B)\land(B\to C)\to (A\to C)</math>
# <math>(A\to B)\to((B\to C)\to(A\to C))</math>
# <math>(A\to B)\to((C\to A)\to(C\to B))</math>
# <math>(A\to(A\to B))\to(A\to B)</math>
# <math>(A\land (A\to B))\to B</math>
# <math>(A\to\lnot A)\to\lnot A</math>
# <math>(A\to (B\to C))\to(B\to(A\to C))</math>
# <math>A\to((A\to B)\to B)</math>
# <math>((A\to A)\to B)\to B</math>
# <math>A\lor\lnot A</math>
# <math>A\to(A\to A)</math>
There are some notable logics stronger than B that can be obtained by adding axioms to B as follows.
* For DW, add axiom 1.
* For DJ, add axioms 1, 2.
* For TW, add axioms 1, 2, 3, 4.
* For RW, add axioms 1, 2, 3, 4, 8, 9.
* For T, add axioms 1, 2, 3, 4, 5, 6, 7, 11.
* For R, add axioms 1-11.
* For E, add axioms 1-7, 10, 11, <math>((A\to A)\land(B\to B)\to C)\to C</math>, and <math>\Box A\land \Box B\to \Box (A\land B)</math>, where <math>\Box A</math> is defined as <math>(A\to A)\to A</math>.
* For RM, add all the additional axioms.