Editing Relevance logic (section)

===Routley–Meyer models===
The standard model theory for relevance logics is the Routley-Meyer ternary-relational semantics developed by [[Richard Sylvan|Richard Routley]] and [[Bob Meyer (logician)|Robert Meyer]]. A Routley–Meyer frame F for a propositional language is a quadruple (W,R,*,0), where W is a non-empty set, R is a ternary relation on W, and * is a function from W to W, and <math>0\in W</math>. A Routley-Meyer model M is a Routley-Meyer frame F together with a valuation, <math>\Vdash</math>, that assigns a truth value to each atomic proposition relative to each point <math>a\in W</math>. There are some conditions placed on Routley-Meyer frames. Define <math>a\leq b</math> as <math>R0ab</math>.
* <math>a\leq a</math>.
* If <math>a\leq b</math>  and <math>b\leq c</math>, then <math>a\leq c</math>.
* If <math>d\leq a</math> and <math>Rabc</math>, then <math>Rdbc</math>.
* <math>a^{**}=a</math>.
* If <math>a\leq b</math>, then <math>b^*\leq a^*</math>.
Write <math>M,a\Vdash A</math> and <math>M,a\nVdash A</math> to indicate that the formula <math>A</math> is true, or not true, respectively, at point <math>a</math> in <math>M</math>. 
One final condition on Routley-Meyer models is the hereditariness condition.
* If <math>M,a\Vdash p</math> and <math>a\leq b</math>, then <math>M,b\Vdash p</math>, for all atomic propositions <math>p</math>.
By an inductive argument, hereditariness can be shown to extend to complex formulas, using the truth conditions below.
* If <math>M,a\Vdash A</math> and <math>a\leq b</math>, then <math>M,b\Vdash A</math>, for all formulas <math>A</math>.

The truth conditions for complex formulas are as follows.
* <math>M,a\Vdash A\land B \iff M, a\Vdash A</math> and <math>M,a\Vdash B</math>
* <math>M,a\Vdash A\lor B \iff M, a\Vdash A</math> or <math>M,a\Vdash B</math>
* <math>M,a\Vdash A\to B\iff \forall b,c((Rabc\land M,b\Vdash A)\Rightarrow M,c\Vdash B)</math>
* <math>M,a\Vdash\lnot A\iff M,a^*\nVdash A</math>

A formula <math>A</math> holds in a model <math>M</math> just in case <math>M,0\Vdash A</math>. A formula <math>A</math> holds on a frame <math>F</math> iff A holds in every model <math>(F,\Vdash)</math>. A formula <math>A</math> is valid in a class of frames iff A holds on every frame in that class. 
The class of all Routley–Meyer frames satisfying the above conditions validates that relevance logic B. One can obtain Routley-Meyer frames for other relevance logics by placing appropriate restrictions on R and on *. These conditions are easier to state using some standard definitions. Let <math>Rabcd</math> be defined as <math>\exists x(Rabx \land Rxcd)</math>, and let <math>Ra(bc)d</math> be defined as <math>\exists x(Rbcx \land Raxd)</math>. Some of the frame conditions and the axioms they validate are the following.
{| class="wikitable"
|-
|+
|-
! scope="col" | Name
! scope="col" | Frame condition
! scope="col" | Axiom
|-
! Pseudo-modus ponens 
| <math>Raaa</math>
| <math>(A\land (A\to B))\to B</math>
|-
! Prefixing 
| <math>Rabcd\Rightarrow Ra(bc)d</math>
| <math>(A\to B)\to((C\to A)\to(C\to B))</math>
|-
! Suffixing 
| <math>Rabcd\Rightarrow Rb(ac)d</math>
| <math>(A\to B)\to((B\to C)\to(A\to C))</math>
|-
! Contraction
| <math>Rabc\Rightarrow Rabbc</math>
| <math>(A\to(A\to B))\to(A\to B)</math>
|-
! Hypothetical syllogism 
| <math>Rabc\Rightarrow Ra(ab)c</math>
| <math>(A\to B)\land(B\to C)\to (A\to C)</math>
|-
! Assertion 
| <math>Rabc\Rightarrow Rbac</math>
| <math>A\to((A\to B)\to B)</math>
|-
! E axiom 
| <math>Ra0a</math>
| <math>((A\to A)\to B)\to B</math>
|-
! Mingle axiom
| <math>Rabc\Rightarrow a\leq c</math> or <math>b\leq c</math>
| <math>A\to(A\to A)</math>
|-
! Reductio
| <math>Raa^*a</math>
| <math>(A\to\lnot A)\to\lnot A</math>
|-
! Contraposition
| <math>Rabc\Rightarrow Rac^*b^*</math>
| <math>(A\to B)\to (\lnot B\to\lnot A)</math>
|-
! Excluded middle
| <math>0^*\leq 0</math>
| <math>A\lor\lnot A</math>
|-
! Strict implication weakening
| <math>0\leq a</math>
| <math>A\to(B\to B)</math>
|-
! Weakening
| <math>Rabc\Rightarrow b\leq c</math>
| <math>A\to(B\to A)</math>
|}
The last two conditions validate forms of weakening that relevance logics were originally developed to avoid. They are included to show the flexibility of the Routley–Meyer models.