Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but not universally, called relevant logic by British and, especially, Australian logicians, and relevance logic by American logicians.
Relevance logic aims to capture aspects of implication that are ignored by the "material implication" operator in classical truth-functional logic, namely the notion of relevance between antecedent and conditional of a true implication. This idea is not new: C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication such as the principle that a falsehood implies any proposition.[1] [2] Hence "if I'm a donkey, then two and two is four" is true when translated as a material implication, yet it seems intuitively false since a true implication must tie the antecedent and consequent together by some notion of relevance. And whether or not the speaker is a donkey seems in no way relevant to whether two and two is four.
In terms of a syntactical constraint for a propositional calculus, it is necessary, but not sufficient, that premises and conclusion share atomic formulae (formulae that do not contain any logical connectives). In a predicate calculus, relevance requires sharing of variables and constants between premises and conclusion. This can be ensured (along with stronger conditions) by, e.g., placing certain restrictions on the rules of a natural deduction system. In particular, a Fitch-style natural deduction can be adapted to accommodate relevance by introducing tags at the end of each line of an application of an inference indicating the premises relevant to the conclusion of the inference. Gentzen-style sequent calculi can be modified by removing the weakening rules that allow for the introduction of arbitrary formulae on the right or left side of the sequents.
A notable feature of relevance logics is that they are paraconsistent logics: the existence of a contradiction will not necessarily cause an "explosion." This follows from the fact that a conditional with a contradictory antecedent that does not share any propositional or predicate letters with the consequent cannot be true (or derivable).
Relevance logic was proposed in 1928 by Soviet philosopher Ivan E. Orlov (1886 – circa 1936) in his strictly mathematical paper "The Logic of Compatibility of Propositions" published in Matematicheskii Sbornik. The basic idea of relevant implication appears in medieval logic, and some pioneering work was done by Ackermann, Moh,[3] and Church[4] in the 1950s. Drawing on them, Nuel Belnap and Alan Ross Anderson (with others) wrote the magnum opus of the subject, Entailment: The Logic of Relevance and Necessity in the 1970s (the second volume being published in the nineties). They focused on both systems of entailment and systems of relevance, where implications of the former kinds are supposed to be both relevant and necessary.
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.
A\toA
A\landB\toA
A\landB\toB
(A\toB)\land(A\toC)\to(A\toB\landC)
A\toA\lorB
B\toA\lorB
(A\toC)\land(B\toC)\to(A\lorB\toC)
A\land(B\lorC)\to(A\landB)\lor(A\landC)
lnotlnotA\toA
A,A\toB\vdashB
A,B\vdashA\landB
A\toB\vdash(C\toA)\to(C\toB)
A\toB\vdash(B\toC)\to(A\toC)
A\toB\vdashlnotB\tolnotA
(A\toB)\to(lnotB\tolnotA)
(A\toB)\land(B\toC)\to(A\toC)
(A\toB)\to((B\toC)\to(A\toC))
(A\toB)\to((C\toA)\to(C\toB))
(A\to(A\toB))\to(A\toB)
(A\land(A\toB))\toB
(A\tolnotA)\tolnotA
(A\to(B\toC))\to(B\to(A\toC))
A\to((A\toB)\toB)
((A\toA)\toB)\toB
A\lorlnotA
A\to(A\toA)
((A\toA)\land(B\toB)\toC)\toC
\BoxA\land\BoxB\to\Box(A\landB)
\BoxA
(A\toA)\toA
The standard model theory for relevance logics is the Routley-Meyer ternary-relational semantics developed by Richard Routley and 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
0\inW
\Vdash
a\inW
a\leqb
R0ab
a\leqa
a\leqb
b\leqc
a\leqc
d\leqa
Rabc
Rdbc
a**=a
a\leqb
b*\leqa*
M,a\VdashA
M,a\nVdashA
A
a
M
M,a\Vdashp
a\leqb
M,b\Vdashp
p
M,a\VdashA
a\leqb
M,b\VdashA
A
The truth conditions for complex formulas are as follows.
M,a\VdashA\landB\iffM,a\VdashA
M,a\VdashB
M,a\VdashA\lorB\iffM,a\VdashA
M,a\VdashB
M,a\VdashA\toB\iff\forallb,c((Rabc\landM,b\VdashA) ⇒ M,c\VdashB)
M,a\VdashlnotA\iffM,a*\nVdashA
A formula
A
M
M,0\VdashA
A
F
(F,\Vdash)
A
Rabcd
\existsx(Rabx\landRxcd)
Ra(bc)d
\existsx(Rbcx\landRaxd)
+ | |||
Name | Frame condition | Axiom | |
---|---|---|---|
Pseudo-modus ponens | Raaa | (A\land(A\toB))\toB | |
Prefixing | Rabcd ⇒ Ra(bc)d | (A\toB)\to((C\toA)\to(C\toB)) | |
Suffixing | Rabcd ⇒ Rb(ac)d | (A\toB)\to((B\toC)\to(A\toC)) | |
Contraction | Rabc ⇒ Rabbc | (A\to(A\toB))\to(A\toB) | |
Hypothetical syllogism | Rabc ⇒ Ra(ab)c | (A\toB)\land(B\toC)\to(A\toC) | |
Assertion | Rabc ⇒ Rbac | A\to((A\toB)\toB) | |
E axiom | Ra0a | ((A\toA)\toB)\toB | |
Mingle axiom | Rabc ⇒ a\leqc b\leqc | A\to(A\toA) | |
Reductio | Raa*a | (A\tolnotA)\tolnotA | |
Contraposition | Rabc ⇒ Rac*b* | (A\toB)\to(lnotB\tolnotA) | |
Excluded middle | 0*\leq0 | A\lorlnotA | |
Strict implication weakening | 0\leqa | A\to(B\toB) | |
Weakening | Rabc ⇒ b\leqc | A\to(B\toA) |
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
F
(K, ⋅ ,0)
K
0\inK
⋅
K
x ⋅ x=x
(x ⋅ y) ⋅ z=x ⋅ (y ⋅ z)
x ⋅ y=y ⋅ x
0 ⋅ x=x
An operational model
M
F
V
V
\Vdash
M,a\Vdashp\iffV(a,p)=T
M,a\VdashA\landB\iffM,a\VdashA
M,a\VdashB
M,a\VdashA\lorB\iffM,a\VdashA
M,a\VdashB
M,a\VdashA\toB\iff\forallb(M,b\VdashA ⇒ M,a ⋅ b\VdashB)
A formula
A
M
M,0\VdashA
A
C
M\inC
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
(A\to(B\lorC))\land(B\toC)\to(A\toC)
The operational semantics can be adapted to model the conditional of E by adding a non-empty set of worlds
W
\leq
W x W
M,a,w\VdashA\toB\iff\forallb,\forallw'\geqw(M,b,w'\VdashA ⇒ M,a ⋅ b,w'\VdashB)
The operational semantics can be adapted to model the conditional of T by adding a relation
\leq
K x K
0\leqx
x\leqy
y\leqz
x\leqz
x\leqy
x ⋅ z\leqy ⋅ z
M,a\VdashA\toB\iff\forallb((a\leqb\landM,b\VdashA) ⇒ M,a ⋅ b\VdashB)
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
x ⋅ x=x
J
M,a\VdashA\toB\iff\forallb((Jab\landM,b\VdashA) ⇒ M,a ⋅ b\VdashB)
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
F
(K, ⋅ ,+,0)
K
0\inK
K
a\leqb
\existsx(a+x=b)
An operational model
M
F
V
V
\Vdash
M,a\Vdashp\iffV(a,p)=T
M,a+b\Vdashp\iffM,a\Vdashp
M,b\Vdashp
M,a\VdashA\landB\iffM,a\VdashA
M,a\VdashB
M,a\VdashA\lorB\iffM,a\VdashA
M,a\VdashB
\existsb,c(a=b+c
M,b\VdashA
M,c\VdashB)
M,a\VdashA\toB\iff\forallb(M,b\VdashA ⇒ M,a ⋅ b\VdashB)
A formula
A
M
M,0\VdashA
A
C
M\inC
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.
+ | ||
System | Frame conditions | |
---|---|---|
B | 1, 5-9, 14 | |
TW | 1, 11, 12, 5-9, 14 | |
EW | 1, 10, 11, 5-9, 14 | |
RW | 1-3, 5-9 | |
T | 1, 11, 12, 13, 5-9, 14 | |
E | 1, 10, 11, 13, 5-9, 14 | |
R | 1-9 | |
RM | 1-3, 5-9, 15 |
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
(D,\land,\lor,lnot,\circ,e)
(D,\land,\lor,lnot)
lnot
lnotlnotx=x
x\leqy
lnoty\leqlnotx
e\inD
\circ
x\circy=y\circx
(x\circy)\circz=x\circ(y\circz)
e\circx=x
(D,\circ,e)
e
x\circ(y\lorz)=(x\circy)\lor(x\circz)
x\leqx\circx
x\circy\leqz
x\circlnotz\leqlnoty
x\toy
lnot(x\circlnoty)
x\circy\leqz\iffx\leqy\toz
An interpretation
v
M
v(p)\inD
v(lnotA)=lnotv(A)
v(A\lorB)=v(A)\lorv(B)
v(A\landB)=v(A)\landv(B)
v(A\toB)=v(A)\tov(B)
Given a de Morgan monoid
M
v
A
v
e\leqv(A)
A
"Relevance logic" – by Edwin Mares.