Editing Relevance logic (section)

====Urquhart models====
Operational models for negation-free fragments of relevance logics were developed by [[Alasdair Urquhart]] in his PhD thesis and in subsequent work. The intuitive idea behind the operational models is that points in a model are pieces of information, and combining information supporting a conditional with the information supporting its antecedent yields some information that supports the consequent. Since the operational models do not generally interpret negation, this section will consider only languages with a conditional, conjunction, and disjunction.

An operational frame <math>F</math> is a triple <math>(K,\cdot,0)</math>, where  <math>K</math> is a non-empty set, <math>0\in K</math>, and <math>\cdot</math> is a binary operation on <math>K</math>. Frames have conditions, some of which may be dropped to model different logics. The conditions Urquhart proposed to model the conditional of the relevance logic R are the following.
* <math>x\cdot x=x</math>
* <math>(x\cdot y)\cdot z=x\cdot(y\cdot z)</math>
* <math>x\cdot y=y\cdot x</math>
* <math>0\cdot x=x</math>
Under these conditions, the operational frame is a [[join-semilattice]].

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\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(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 conditional fragment of R is sound and complete with respect to the class of semilattice models. The logic with conjunction and disjunction is properly stronger than the conditional, conjunction, disjunction fragment of R. In particular, the formula <math>(A\to(B\lor C))\land(B\to C)\to (A\to C)</math> is valid for the operational models but it is invalid in R. The logic generated by the operational models for R has a complete axiomatic proof system, due [[Kit Fine]] and to Gerald Charlwood. Charlwood also provided a natural deduction system for the logic, which he proved equivalent to the axiomatic system. Charlwood showed that his natural deduction system is equivalent to a system provided by [[Dag Prawitz]].

The operational semantics can be adapted to model the conditional of E by adding a non-empty set of worlds <math>W</math> and an accessibility relation <math>\leq</math> on <math>W\times W</math> to the frames. The accessibility relation is required to be reflexive and transitive, to capture the idea that E's conditional has an S4 necessity. The valuations then map triples of atomic propositions, points, and worlds to truth values. The truth condition for the conditional is changed to the following. 
* <math>M,a, w\Vdash A\to B\iff \forall b, \forall w'\geq w(M,b, w'\Vdash A\Rightarrow M,a\cdot b,w'\Vdash B)</math>

The operational semantics can be adapted to model the conditional of T by adding a relation <math>\leq</math> on <math>K\times K</math>. The relation is required to obey the following conditions.
* <math>0\leq x</math>
* If <math>x\leq y</math> and <math>y\leq z</math>, then <math>x\leq z</math>
* If <math>x\leq y</math>, then <math>x\cdot z\leq y\cdot z</math>
The truth condition for the conditional is changed to the following.
* <math>M,a\Vdash A\to B\iff \forall b((a\leq b\land M,b\Vdash A)\Rightarrow M,a\cdot b\Vdash B)</math>

There are two ways to model the contraction-less relevance logics TW and RW  with the operational models. The first way is to drop the condition that <math>x\cdot x=x</math>. The second way is to keep the semilattice conditions on frames and add a binary relation, <math>J</math>, of disjointness to the frame. For these models, the truth conditions for the conditional is changed to the following, with the addition of the ordering in the case of TW.
* <math>M,a\Vdash A\to B\iff \forall b((Jab \land M,b\Vdash A)\Rightarrow M,a\cdot b\Vdash B)</math>