Editing Relevance logic (section)

====Humberstone models====
Urquhart showed that the semilattice logic for R is properly stronger than the positive fragment of R. Lloyd Humberstone provided an enrichment of the operational models that permitted a different truth condition for disjunction. The resulting class of models generates exactly the positive fragment of R.

An operational frame <math>F</math> is a quadruple <math>(K,\cdot,+,0)</math>, where <math>K</math> is a non-empty set, <math>0\in K</math>, and {<math>\cdot</math>, <math>+</math>} are binary operations on <math>K</math>. Let <math>a\leq b</math> be defined as <math>\exists x(a+x=b)</math>. The frame conditions are the following.
{{ordered list|start=1
| <math>0\cdot x=x</math>
| <math>x\cdot y=y\cdot x</math>
| <math>(x\cdot y)\cdot z=x\cdot(y\cdot z)</math>
| <math>x\leq x\cdot x</math>
| <math>x+y=y+x</math>
| <math>(x+y)+z=x+(y+z)</math>
| <math>x+x=x</math>
| <math>x\cdot(y+z)=x\cdot y+x\cdot z</math>
| <math>x\leq y+z\Rightarrow \exists y',z'\in K(y'\leq y</math>, <math>z'\leq z</math> and <math>x=y'+z')</math>
}}

An operational model <math>M</math> is a frame <math>F</math> with a valuation <math>V</math> that maps pairs of points and atomic propositions to truth values, T or F. <math>V</math> can be extended to a valuation <math>\Vdash</math> on complex formulas as follows.
* <math>M,a\Vdash p \iff V(a,p)=T</math>, for atomic propositions
* <math>M,a+b\Vdash p \iff M,a\Vdash p</math> and <math>M,b\Vdash p</math>
* <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> or <math>\exists b,c(a=b+c</math>; <math>M,b\Vdash A</math> and <math>M,c\Vdash B)</math>
* <math>M,a\Vdash A\to B\iff \forall b(M,b\Vdash A\Rightarrow M,a\cdot b\Vdash B)</math>

A formula <math>A</math> holds in a model <math>M</math> iff <math>M,0\Vdash A</math>. A formula <math>A</math> is valid in a class of models <math>C</math> iff it holds in each model <math>M\in C</math>.

The positive fragment of R is sound and complete with respect to the class of these models. Humberstone's semantics can be adapted to model different logics by dropping or adding frame conditions as follows.
{| class="wikitable"
|-
|+
|-
! scope="col" | System
! scope="col" colspan="2" | Frame conditions
|-
! scope="row" | B
| 1, 5-9, 14
| rowspan="8" | {{ordered list|start=10
| <math>x\leq x\cdot 0</math>
| <math>(x\cdot y)\cdot z\leq y\cdot(x\cdot z)</math>
| <math>(x\cdot y)\cdot z\leq x\cdot(y\cdot z)</math>
| <math>x\cdot y\leq(x\cdot y)\cdot y</math>
| <math>(y+z)\cdot x=y\cdot x+z\cdot x</math>
| <math>x\cdot x=x</math>
}}
|-
! scope="row" | TW
| 1, 11, 12, 5-9, 14
|-
! scope="row" | EW
| 1, 10, 11, 5-9, 14
|-
! scope="row" | RW
| 1-3, 5-9
|-
! scope="row" | T
| 1, 11, 12, 13, 5-9, 14
|-
! scope="row" | E
| 1, 10, 11, 13, 5-9, 14
|-
! scope="row" | R
| 1-9
|-
! scope="row" | RM
| 1-3, 5-9, 15
|}