Editing Relevance logic (section)

==Models==
===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.

===Operational models===
====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>

====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
|}

===Algebraic models===
Some relevance logics can be given algebraic models, such as the logic R. The algebraic structures for R are de Morgan monoids, which are sextuples <math>(D,\land,\lor,\lnot,\circ,e)</math> where
* <math>(D,\land,\lor,\lnot)</math> is a distributive [[Lattice (order)|lattice]] with a unary operation, <math>\lnot</math> obeying the laws <math>\lnot\lnot x=x</math> and if <math>x\leq y</math> then <math>\lnot y\leq \lnot x</math>;
* <math>e\in D</math>, the binary operation <math>\circ</math> is [[Commutative property|commutative]] (<math>x\circ y=y\circ x</math>) and [[Associative property|associative]] (<math>(x\circ y)\circ z=x\circ (y\circ z)</math>), and <math>e\circ x=x</math>, i.e. <math>(D,\circ,e)</math> is an [[Monoid#Commutative monoid|Abelian monoid]] with [[Identity element|identity]] <math>e</math>;
* the monoid is lattice-ordered and satisfies <math>x\circ(y\lor z)=(x\circ y)\lor(x\circ z)</math>;
* <math>x\leq x\circ x</math>; and 
* if <math>x\circ y\leq z</math>, then <math>x\circ\lnot z\leq \lnot y</math>.
The operation <math>x\to y</math> interpreting the conditional of R is defined as <math>\lnot(x\circ\lnot y)</math>. 
A de Morgan monoid is a [[residuated lattice]], obeying the following residuation condition. 
: <math>x \circ y\leq z \iff x\leq y\to z</math>

An interpretation <math>v</math> is a [[homomorphism]] from the propositional language to a de Morgan monoid <math>M</math> such that 
* <math>v(p)\in D</math> for all atomic propositions, 
* <math>v(\lnot A)=\lnot v(A)</math>
* <math>v(A\lor B)=v(A)\lor v(B)</math>
* <math>v(A\land B)=v(A)\land v(B)</math>
* <math>v(A\to B)=v(A)\to v(B)</math>

Given a de Morgan monoid <math>M</math> and an interpretation <math>v</math>, one can say that formula <math>A</math> holds on <math>v</math> just in case <math>e\leq v(A)</math>. A formula <math>A</math> is valid just in case it holds on all interpretations on all de Morgan monoids. The logic R is sound and complete for de Morgan monoids.