## 1. Semantics for Relevant Implication



Our exposition of relevant logic is backwards to most found in the
literature We will begin, rather than end, with the semantics, since
most philosophers at present are semantically inclined.



The semantics that is presented here is the ternary relation semantics
due to Richard Routley and Robert K. Meyer. This semantics is a
development of Alasdair Urquhart's "semilattice
semantics" (Urquhart 1972). There is a similar semantics, due to
Kit Fine, that was developed at the same time as the Routley-Meyer
theory (Fine 1974). And there is an algebraic semantics due to
J. Michael Dunn. Urquhart's, Fine's, and Dunn's
models are very interesting in their own right, but we do not have
room to discuss them here.



The idea behind the ternary relation semantics is rather simple.
Consider C.I. Lewis' attempt to avoid the paradoxes of material
implication. He added a new connective to classical logic, that of
strict implication. In post-Kripkean semantic terms, \(A \prurel B\)
is true at a world \(w\) if and only if for all \(w'\) such that
\(w'\) is accessible to \(w\), either \(A\) fails in \(w'\) or \(B\)
obtains there. Now, in Kripke's semantics for modal logic, the
accessibility relation is a binary relation. It holds between pairs of
worlds. Unfortunately, from a relevant point of view, the theory of
strict implication is still irrelevant. That is, we still make valid
formulae like \(p \prurel (q \prurel q)\). We can see quite
easily that the Kripke truth condition forces this formula on us.



Like the semantics of modal logic, the semantics of relevance logic
relativises truth of formulae to worlds. But Routley and Meyer go
modal logic one better and use a three-place relation on worlds. This
allows there to be worlds at which \(q \rightarrow q\) fails
and that in turn allows worlds at which \(p \rightarrow(q \rightarrow q)\) fails. Their truth condition for \(\rightarrow\) on this
semantics is the following:



\(A \rightarrow B\) is true at a world \(a\) if and only
if for all worlds \(b\) and \(c\) such that \(Rabc\)
\((R\) is the accessibility relation) either \(A\) is false
at \(b\) or \(B\) is true at \(c\).




For people new to the field it takes some time to get used to this
truth condition. But with a little work it can be seen to be just a
generalisation of Kripke's truth condition for strict implication
(just set \(b = c\)).



The ternary relation semantics can be adapted to be a semantics for a
wide range of logics. Placing different constraints on the relation
makes valid different formulae and inferences. For example, if we
constrain the relation so that \(Raaa\) holds for all worlds
\(a\), then we make it true that if \((A \rightarrow B) \amp A\) is true at a world, then \(B\) is
also true there. Given other features of the Routley-Meyer semantics,
this makes the thesis \(((A \rightarrow B) \amp A) \rightarrow B\) valid. If we make the ternary relation
symmetrical in its first two places, that is, we constrain it so that,
for all worlds \(a, b\), and \(c\), if
\(Rabc\) then \(Rbac\), then we make valid the thesis
\(A \rightarrow ((A \rightarrow B) \rightarrow B)\).



The ternary accessibility relation needs a philosophical
interpretation in order to give relevant implication a real meaning on
this semantics. Recently there have been several interpretations
developed based on theories of information.



One interpretation is suggested in Jon Barwise (1993) and developed in
Restall (1996). On this view, worlds are taken to be
information-theoretic "sites" and "channels".
A site is a context in which information is received and a channel is
a conduit through which information is transferred. Thus, for example,
when the BBC news appears on the television in my living room, we can
consider the living room to be a site and the wires, satellites, and
so on, that connect my television to the studio in London to be a
channel. Using channel theory to interpret the Routley-Meyer
semantics, we take \(Rabc\) to mean that \(a\) is an
information-theoretic channel between sites \(b\) and \(c\).
Thus, we take \(A \rightarrow B\) to be true at \(a\) if
and only if, whenever \(a\) connects a site \(b\) at which
\(A\) obtains to a site \(c, B\) obtains at
\(c\).



Similarly, Mares (1997) uses a theory of information due to David
Israel and John Perry (1990). In addition to other information a world
contains informational links, such as laws of nature, conventions, and
so on. For example, a Newtonian world will contain the information
that all matter attracts all other matter. In information-theoretic
terms, this world contains the information that two things' being
material carries the information that they attract each other. On this
view, \(Rabc\) if and only if, according to the links in
\(a\), all the information carried by what obtains in \(b\)
is contained in \(c\). Thus, for example, if \(a\) is a
Newtonian world and the information that \(x\) and \(y\) are
material is contained in \(b\), then the information that
\(x\) and \(y\) attract each other is contained in
\(c\).



Another interpretation is developed in Mares (2004). This
interpretation takes the Routley-Meyer semantics to be a formalisation
of the notion of "situated implication". This
interpretation takes the "worlds" of the Routley-Meyer
semantics to be *situations*. A situation is a perhaps partial
representation of the universe. The information contained in two
situations, \(a\) and \(b\) might allow us to infer further
information about the universe that is contained in neither situation.
Thus, for example, suppose in our current situation that we have the
information contained in the laws of the theory of general relativity
(this is Einstein's theory of gravity). Then we hypothesise a
situation in which we can see a star moving in an ellipse. Then, on
the basis of the information that we have and the hypothesised
situation, we can infer that there is a situation in which there is a
very heavy body acting on this star.



We can model situated inference using a relation \(I\) (for
"implication"). Then we have \(IabP\), where
\(P\) is a proposition, if and only if the information in
\(a\) and \(b\) together license the inference to there
being a situation in which \(P\) holds. We can think of a
proposition itself as a set of situations. We set \(A \rightarrow B\) to hold at \(a\) if and only if, for all situations
\(b\) in which \(A\) holds, \(Iab|B|,\) where
\(|B\)| is the set of situations at which \(B\) is true. We
set \(Rabc\) to hold if and only if \(c\) belongs to every
proposition \(P\) such that \(IabP\). With the addition of
the postulate that, for any set of propositions \(P\) such that
\(IabP\), the intersection of that set \(X\) is such that
\(IabX\), we find that the implications that are made true on any
situation using the truth condition that appeals to \(I\) are the
same as those that are made true by the Routley-Meyer truth condition.
Thus, the notion of situated inference gives a way of understanding
the Routley-Meyer semantics. (This is a very brief version of the
discussion of situated inference that is in chapters 2 and 3 of Mares
(2004).)



Another informational interpretation is in Dunn (2015). This
interpretation neatly connects the concept of relevance in relevance
logic with the pragmatic notion of relevance in Grice's conversational
maxims and in Sperber and Wilson's pragmatic theory. The idea is that
the relation \(R\) holds between three "states",
\(a,b\), and \(c\) if in the context \(a,
b\) is relevant to \(c\). This means that using the
information both in \(a\) and \(b\) allows one to derive the
information in \(c\). The notion of derivation here is one that
Sperber and Wilson call "contextual implication". The result
is derivable from the information in \(a\) and \(b\), but
not from the information in either alone (Sperber and Wilson, 2002, p.
251). There are various ways to think of the combination of
information in \(a\) and \(b\). One that Dunn discusses is
to think of \(a\) and \(b\) as being similar to computer
programs and the combination of the two is their composition -- the
result of running \(b\) and taking the result of that as input
for \(a\), and then running \(a\). This combination is
relevant to \(c\) if the information produced by it is all
contained in \(c\).



A different sort of interpretation is given by Beall, et al. (2012).
In fact, Beall, et al (2012) presents three different interpretations
of the ternary relation. These interpretations link the ternary
relation with different notions of "conditionality". I
discuss only two of these interpretations here. The third is a bit too
involved for inclusion in an encyclopedia entry. On the first of these
interpretations, a conditional \(A\rightarrow B\) holds at a
world \(a\) if there is no counterexample to this conditional,
i.e. a place salient to \(a\) at which \(A\) holds and
\(B\) fails to hold. This "place" is not a single
world, but a pair of worlds, \(b, c\). A pair consists in
a counterexample to the conditional if \(A\) is true at the first
of the pair and \(B\) is not true at the second.



On the second interpretation, the points of Routley-Meyer models are
considered to be both operators (in the mathematical sense) and the
things that they operate on. On this understanding, \(Rabc\)
means that considered as an operator, \(a\) when applied to
\(b\) yields information all of which is contained in \(c\).
This interpretation makes the Routley-Meyer semantics very similar in
intent to Fine's operational semantics. And this interpretaion is very
closely related to Dunn's interpetation (in particular in his notion
of the combining of information in states). 



By itself, the use of the ternary relation is not sufficient to avoid
all the paradoxes of implication. Given what we have said so far, it
is not clear how the semantics can avoid paradoxes such as \((p \amp \neg p) \rightarrow q\) and \(p \rightarrow 
(q \vee \neg q)\). These paradoxes are avoided by the
inclusion of inconsistent and non-bivalent worlds in the semantics.
For, if there were no worlds at which \(p \amp \neg p\)
holds, then, according to our truth condition for the arrow,
\((p \amp \neg p) \rightarrow q\) would also hold
everywhere. Likewise, if \(q \vee \neg q\) held at every
world, then \(p \rightarrow(q \vee \neg q)\) would
be universally true.



An approach to relevance that does not require the ternary relation is
due to Routley and Loparic (1978) and Priest (1992) and (2008). This
semantics uses a set of worlds and a binary relation, \(S\).
Worlds are divided into two categories: normal worlds and non-normmal
worlds. An implication \(A \rightarrow B\) is true at a
normal world \(a\) if and only if for all worlds \(b\), if
\(A\) is true at \(b\) then \(B\) is also true true at
\(b\). At non-normal worlds, the truth values for implications
are random. Some may be true and others false. A formula is valid if
and only if it is true on every such model in its *normal*
worlds. This division of worlds into normal and non-normal and the use
of random truth values for implications at non-normal worlds enables
us to find countermodels for formulas such as \(p \rightarrow 
(q \rightarrow q)\).



Priest interprets non-normal worlds as the worlds that correspond to
"logic fictions". In a science fiction, the laws of nature
may be different than those in our universe. Similarly, in a logic
fiction the laws of logic may be different from our laws. For example,
\(A \rightarrow A\) may fail to be true in some logic
fiction. The worlds that such fictions describe are non-normal worlds.




One problem with the semantics without the ternary relation is that it
is difficult to use it to characterize as wide a range of logical
systems as can done with the ternary relation. In addition, the logics
determined by this semantics are quite weak. For example, they do not
have as a theorem the transitivity of implication -- \(((A
\rightarrow B) \amp(B \rightarrow C)) \rightarrow (A \rightarrow
C)\).



Like the ternary relation semantics, this semantics requires some
worlds to be inconsistent and some to be non-bivalent.


## 2. Semantics for Negation



The use of non-bivalent and inconsistent worlds requires a
non-classical truth condition for negation. In the early 1970s,
Richard and Val Routley invented their "star operator" to
treat negation (Routley & Routley 1972). The operator is an
operator on non-normal worlds. For each world \(a\), there is a world
\(a\)\*. And



\(\neg A\) is true at \(a\) if and only if \(A\) is
false at \(a\)\*.



Once again, we have the difficulty of interpreting a part of the
formal semantics. One interpretation of the Routley star is that of
Dunn (1993). Dunn uses a binary relation, \(C\), on worlds.
\(Cab\) means that \(b\) is compatible with \(a.
a\)\*, then, is the maximal world (the world containing the most
information) that is compatible with \(a\).



There are other semantics for negation. One, due to Dunn, is a
four-valued semantics. As is the case for truth tables for classical
logic, this semantics begins with the values T (true) and F (false). A
formula is given a set of these truth values. Thus, a formula \(A\)
can get the values \(\{\True\}\), \(\{\False\}\),
\(\{\True,\False\}\), or \(\varnothing\). If a formula gets the value
\(\{\True\}\), then it is just true; likewise, if it gets the value
\(\{\False\}\) it is just false; if it gets the value
\(\{\True,\False\}\) it is both true and false; if it gets the value
\(\varnothing\), it is neither true nor false.



Each formula is given a truth condition and a falsity condition. For
example, T is in the value of \(\neg A\) if and only if F is in
the value of \(A\) and F is in the value of \(\neg A\) if and
only if T is in the value of \(A\). With regard to conjunction, T
is in the value of \(A \amp B\) if and only if T is in the
value of \(A\) and T is in the value of \(B\) and F is in
the value of \(A \amp B\) if and only if F is the value
of \(A\) or F is in the value of \(B\). Disjunction has very
similar truth and falsity conditions. 



Dunn put forward his semantics to characterize the logic of First
Degree Entailment (FDE), which treats only entailments between
implication-free formulas. (For papers on the nature and development
of FDE, see Omori & Wansing 2017.) Richard Routley (1988) extended
the theory to treat some weak relevant logics, and Restall (1995)
extended the theory to treat logics just weaker than the promenant
relevant logics E and R (see sections 4 and 5 below). Mares (2004a)
used a neighbourhood semantics, together with four truth-values to
give a semantics for R. The four-valued semantics is treated in more
detail the entry on paraconsistent
logics. Other treatments of negation, some of which have been
used for relevance logics, can be found in Wansing (2001) and in the
entry on negation.


## 3. Semantics for Quantification



In (1980), Richard Routley conjectured that a constant domain
semantics, in the sense that is familiar from modal logic, will
characterize quantified relevant relevant logics. On this semantics, a
universally quantified formula \(\forall xA(x)\) is true
at a world if and only if \(A(x)\) is true on every
interpretation of \(x\).



Unfortunately, Kit Fine (1988a) proved that the logic RQ (the logic R
of relevant implication together with some standard quantificational
axioms) is incomplete over the constant domain semantics. 



Fine (1988b) also developed a semantics over which RQ is complete and
which can be modified to accommodate any of the mainstream relevance
logics. Fine's semantics is "stratified". This means that a
model is made up of a collection of miniature models, each with its
own domain of individuals. Each of these mini-models is related to
models with larger domains, and each world in a model is similarly
related to worlds in these models with larger domains. A very clear
explanation of how Fine's semantics works is given in Shay Logan
(2019). 



Another semantics for quantified relevance logics is given by Mares
and Robert Goldblatt (2006) and further developed in Goldblatt (2011).
In addition to worlds and domains, a model on this theory contains a
set of propositions, which are designated sets of words. A formula
\(\forall xA(x)\) is true at a world \(w\) if and
only if there is a proposition \(\pi\) true at \(w\) such that \(\pi\)
entails every instance of \(A(x)\). This means that at
every world in the set \(\pi\), every instance of \(A(x)\)
is true. 



The Mares-Goldblatt semantics is called an "admissible set"
semantics. The propositions are the admissible sets. In some models at
least, not every set of worlds counts as a proposition. One rationale
for that comes from reflection on how humans relate circumstances
together as similar. Not every set of situations are such that we
would see a similarity that the members of the set have to one another
and to no other situations outside the set. It seems reasonable to
think of a proposition as a set of worlds that could act as a content
for some person. (Perhaps if we were to construct a language to talk
about what people could think in other words, we could index sets of
propositions to worlds. But that is a topic to be left for some other
time.) 



One useful addition to the relevant theory of quantification is a
conditional that is used to represent restricted quantification. This
is developed by Jc Beall, et al. (2006). Consider the categorial
scheme, "All \(A\)s are \(B\)s". This scheme is
translated into the language of classical logic as
\(\forall x(A(x)\supset B(x))\).
The material conditional, \(\supset\), is too weak to do this job in
relevant logic (where \(A\supset B\) is understood as
\(\neg A\vee B)\). If we were to use material implication
is this manner, in a model for relevant logic, we could have a world
in which all \(A\)s are \(B\)s, and some \(i\) is
\(A\), but where \(i\) is not \(B\). The material
conditional is too weak, but relevant implication is too strong. When
one says, for example, "Everyone in this room owns a dog",
she does not mean that it follows from being in this room that people
own a dog. Rather, it just happens that every person in this room owns
a dog. It is this connection, that lies somewhere between material and
relevant implication that the restricted quantificational conditional
is supposed to capture. 



It is unclear, however, that the conditional of Beall, et al. (2006)
is the right connective to use, at least with regard to some weaker
relevance logics. One virtue of some weaker systems is that they can
be used to formalise naive set theory (see
 section 6).
 Zach Weber (2010) has formalised naive set theory, using this
conditional to define the subset relation. The resulting system,
unfortunately, is trivial in that every formula is provable in it.


## 4. Proof Theory



The logic that is often taken to be the paradigm relevance logic is
the logic \(\mathbf{R}\). For an axiomatisation of
\(\mathbf{R}\), see
 Logic \(\mathbf{R}\).



There are now several approaches to the proof theory for
\(\mathbf{R}\). There is a sequent calculus for the negation-free
fragment of the logic \(\mathbf{R}\) due to Gregory Mints (1972)
and J.M. Dunn (1973) and an elegant and very general approach called
"Display Logic" developed by Nuel Belnap (1982). But here
I will only deal with the natural deduction system for the relevant
logic \(\mathbf{R}\) due to Anderson and Belnap.



Anderson and Belnap's natural deduction system is based on Fitch's
natural deduction systems for classical and intuitionistic logic. The
easiest way to understand this technique is by looking at an
example.




|  |  |  |
| --- | --- | --- |
| 1. | \(A\_{\{1\}}\) | Hyp |
| 2. | \((A \rightarrow B)\_{\{2\}}\) | Hyp |
| 3. | \(B\_{\{1,2\}}\) | \(1,2, \rightarrow\) E |



This is a simple case of modus ponens. The numbers in set brackets
indicate the hypotheses used to prove the formula. We will call them
'indices'. The indices in the conclusion indicate which
hypotheses are really used in the derivation of the conclusion. In the
following "proof" the second premise is not really
used:




|  |  |  |
| --- | --- | --- |
| 1. | \(A\_{\{1\}}\) | Hyp |
| 2. | \(B\_{\{2\}}\) | Hyp |
| 3. | \((A \rightarrow B)\_{\{3\}}\) | Hyp |
| 4. | \(B\_{\{1,3\}}\) | \(1,3, \rightarrow\) E |



This "proof" really just shows that the inference from
\(A\) and \(A \rightarrow B\) to \(B\) is
relevantly valid. Because the number 2 does not appear in the
subscript on the conclusion, the second "premise" does not
really count as a premise.



Similarly, when an implication is proven relevantly, the assumption of
the antecedent must really be used to prove the conclusion. Here is an
example of the proof of an implication:




|  |  |  |
| --- | --- | --- |
| 1. | \(A\_{\{1\}}\) | Hyp |
| 2. | \((A \rightarrow B)\_{\{2\}}\) | Hyp |
| 3. | \(B\_{\{1,2\}}\) | \(1,2, \rightarrow\) E |
| 4. | \(((A \rightarrow B) \rightarrow B)\_{\{1\}}\) | \(2,3, \rightarrow\) I |
| 5. | \(A \rightarrow((A \rightarrow B) \rightarrow B)\) | \(1,4, \rightarrow\) I |



When we discharge a hypothesis, as in lines 4 and 5 of this proof, the
number of the hypothesis must really occur in the subscript of the
formula that is to become the consequent of the implication.



Now, it might seem that the system of indices allows irrelevant
premises to creep in. One way in which it might appear that
irrelevances can intrude is through the use of a rule of conjunction
introduction. That is, it might seem that we can always add in an
irrelevant premise by doing, say, the following:




|  |  |  |
| --- | --- | --- |
| 1. | \(A\_{\{1\}}\) | Hyp |
| 2. | \(B\_{\{2\}}\) | Hyp |
| 3. | \((A \amp B)\_{\{1,2\}}\) | 1,2, &I |
| 4. | \(B\_{\{1,2\}}\) | 3, &E |
| 5. | \((B \rightarrow B)\_{\{1\}}\) | \(2,4, \rightarrow\) I |
| 6. | \(A \rightarrow(B \rightarrow B)\) | \(1,5, \rightarrow\) I |



To a relevance logician, the first premise is completely out of place
here. To block moves like this, Anderson and Belnap give the following
conjunction introduction rule:



From \(A\_i\) and \(B\_i\) to infer \((A \amp B)\_i\).




This rule says that two formulae to be conjoined must have the same
index before the rule of conjunction introduction can be used.



There is, of course, a lot more to the natural deduction system (see
Anderson and Belnap 1975 and Anderson, Belnap, and Dunn 1992), but
this will suffice for our purposes. The theory of relevance that is
captured by at least some relevant logics can be understood in terms
of how the corresponding natural deduction system records the real use
of premises.


## 5. Some Systems of Relevance Logic



In the work of Anderson and Belnap the central systems of relevance
logic were the logic \(\mathbf{E}\) of relevant entailment and the
system \(\mathbf{R}\) of relevant implication. The relationship
between the two systems is that the entailment connective of
\(\mathbf{E}\) was supposed to be a strict (i.e. necessitated)
relevant implication. To compare the two, Meyer added a necessity
operator to \(\mathbf{R}\) (to produce the logic
\(\mathbf{NR})\). Larisa Maksimova, however, discovered that
\(\mathbf{NR}\) and \(\mathbf{E}\) are importantly different
-- that there are theorems of \(\mathbf{NR}\) (on the natural
translation) that are not theorems of \(\mathbf{E}\). This has
left some relevant logicians with a quandary. They have had to decide
whether to take \(\mathbf{NR}\) or \(\mathbf{E}\) to be the
system of relevant entailment. If \(\mathbf{E}\) is chosen, then
perhaps it is not reasonable to say that entailment is just relevant
implication together with logical necessity. It may be that entailment
and implication are related in some other way.



On the other hand, there are those relevance logicians who reject both
\(\mathbf{R}\) and \(\mathbf{E}\). There are those, like Arnon
Avron, who accept logics stronger than \(\mathbf{R}\) (Avron
1990). And there are those, like Ross Brady, John Slaney, Steve
Giambrone, Richard Sylvan, Graham Priest, Greg Restall, and others,
who have argued for the acceptance of systems weaker than
\(\mathbf{R}\) or \(\mathbf{E}\). One extremely weak system is
the logic \(\mathbf{S}\) of Robert Meyer and Errol Martin. As
Martin has proven, this logic contains no theorems of the form
\(A \rightarrow A\). In other words, according to
\(\mathbf{S}\), no proposition implies itself and no argument of
the form '\(A\), therefore \(A\)' is valid.
Thus, this logic does not make valid any circular arguments.



For more details on these logics see supplements on the
 logic \(\mathbf{E}\),
 logic \(\mathbf{R}\),
 logic \(\mathbf{NR}\),
 and
 logic \(\mathbf{S}\).
 



Among the points in favour of weaker systems is that, unlike
\(\mathbf{R}\) or \(\mathbf{E}\), many of them are decidable.
Another feature of some of these weaker logics that makes them
attractive is that they can be used to construct a naive set
theory. A naive set theory is a theory of sets that includes as a
theorem the naive comprehension axiom, viz., for all formulae
\(A(y)\),



\(\exists x\forall y(y \in x \leftrightarrow A(y))\).




In set theories based on strong relevant logics, like
\(\mathbf{E}\) and \(\mathbf{R}\), as well as in classical set
theory, if we add the naive comprehension axiom, we are able to
derive any formula at all. Thus, naive set theories based on
systems such as \(\mathbf{E}\) and \(\mathbf{R}\) are said to
be "trivial". Here is an intuitive sketch of the proof of
the triviality of a naive set theory using principles of
inference from the logic **R.** Let \(p\) be an
arbitrary proposition:




|  |  |  |
| --- | --- | --- |
| 1. | \(\exists x\forall y(y \in x \leftrightarrow(y \in y \rightarrow p))\) | Naive Comprehension |
| 2. | \(\forall y(y \in z \leftrightarrow (y \in y \rightarrow p))\) | 1, Existential Instantiation |
| 3. | \(z \in z \leftrightarrow(z \in z \rightarrow p)\)  | 2, Universal Instantiation |
| 4. | \(z \in z \rightarrow(z \in z \rightarrow p)\)  | 3, df of \(\leftrightarrow\) , &-Elimination |
| 5. | \((z \in z \rightarrow (z \in z \rightarrow p)) \rightarrow(z \in z \rightarrow p)\)  | Axiom of Contraction |
| 6. | \(z \in z \rightarrow p\)  | 4,5, Modus Ponens |
| 7. | \((z \in z \rightarrow p)) \rightarrow z \in z\) | 3, df of \(\leftrightarrow\) , &-Elimination |
| 8. | \(z \in z\) | 6,7, Modus Ponens |
| 9. | \(p\) | 6,8, Modus Ponens |



Thus we show that any arbitrary proposition is derivable in this
naive set theory. This is the infamous Curry Paradox. The
existence of this paradox has led Grishen, Brady, Restall, Priest, and
others to abandon the axiom of contraction \(((A \rightarrow 
(A \rightarrow B)) \rightarrow(A \rightarrow B))\). Brady has shown that by removing contraction, plus some
other key theses, from \(\mathbf{R}\) we obtain a logic that can
accept naive comprehension without becoming trivial (Brady
2005).



In terms of the natural deduction system, the presence of contraction
corresponds to allowing premises to be used more than once. Consider
the following proof:




|  |  |  |
| --- | --- | --- |
| 1. | \(A \rightarrow(A \rightarrow B)\_{\{1\}}\) | Hyp |
| 2. | \(A\_{\{2\}}\) | Hyp |
| 3. | \(A \rightarrow B\_{\{1,2\}}\) | \(1,2, \rightarrow\) E |
| 4. | \(B\_{\{1,2\}}\) | \(2,3, \rightarrow\) E |
| 5. | \(A \rightarrow B\_{\{1\}}\) | 2-\(4, \rightarrow\) I |
| 6. | \((A \rightarrow(A \rightarrow B)) \rightarrow (A \rightarrow B)\) | 1-\(5, \rightarrow\) I |



What enables the derivation of contraction is the fact that our
subscripts are sets. We do not keep track of how many times (more than
once) that a hypothesis is used in its derivation. In order to reject
contraction, we need a way of counting the number of uses of
hypotheses. Thus natural deduction systems for contraction-free
systems use "multisets" of relevance numerals instead of
sets -- these are structures in which the number of occurrences
of a particular numeral counts, but the order in which they occurs
does not. Even weaker systems can be constructed, which keep track
also of the order in which hypotheses are used (see Read 1986 and
Restall 2000).



For three of the better known and more widely used weak relevant
logics, \(\mathbf{B}, \mathbf{DK}\), and
\(\mathbf{DJ}\), see the supplement on them: 



The logics \(\mathbf{B}, \mathbf{DJ}\), and \(\mathbf{DK}\).
 


## 6. Systems Closely Related to Mainstream Relevance Logic



There are some systems that deserve to be called *relevant*
that are not mainstream relevant logic. One such system is Graham
Priest's logic N\(\_4\). The easiest way to present this logic is
to explain its semantics.



A model for N\(\_4\) consists in a set of world that is
partitioned into normal and non-normal worlds. At every world,
formulas are given one of four truth values, in accordance with Dunn's
semantics explained in section 2 above, with regard to conjunction,
disjunction, and negation. But the treatment of implication is rather
interesting. At normal worlds, an implication
\(A\rightarrow B\), is true if and only if in every world w,
if \(A\) is true in w, then \(B\) is also true in w. The
implication is false if there is at least one world in which
\(A\) is true and \(B\) is false. At non-normal worlds,
implications are made true and false randomly. 



N\(\_4\) is a relevant logic. It has the variable-sharing
property. And it has a very simple and intuitive semantics. It is,
however, a very weak logic. It does not contain any transitivity
axioms for implication. It has a transitivity rule. It does not
contain either the contraposition axiom nor the rule form of
contraposition.



Another very interesting logic is Neil Tennant's Core Logic. One of
the "fallacies" that relevance logic was created to avoid
is ex falso quodlibet, or explosion - the inference from a
contradiction to any proposition whatsoever. C.I. Lewis justified
explosion by means of a little argument. He started with the
premise \(p \amp \neg p\). By conjunction elimination he
derived, \(p\), and by disjunction
introduction, \(p\vee q\). From the premise, he also
derived \(\neg p\), by conjunction elimination. Thus, he
had \(p\vee q\) and \(\neg p\). From these, by
disjunctive syllogism, Lewis derived \(q\). Mainstream relevance
logicians block this argument by rejecting disjunctive syllogism. The
rejection of disjunctive syllogism, however, has become one of the
most controversial aspects of relevance logic.




Tennant's core logic, however, accepts disjunctive syllogism. It also
accepts conjunction elimination and disjunction introduction. In fact,
Core Logic supports all the standard primitive rules that we find in
the proof theory of 
 intuitionist logic. Thus, one could say
that the meanings of the connectives in Core logic are just their
meanings in intuitionist logic. What is different is its treatment of
one of the structural rules of proof -- it rejects the transitivity of
logical consequence in its most general form. 



Yet another logical system that is closely related to relevance logic
is William Parry's logic of Analytic Implication. Analytic Implication
is motivated by the desire to satisfy a very strong form of
variable-sharing. No implication \(A\rightarrow B\) is
provable in this logic unless \(\mathbf{all}\) the variables in \(B\)
are contained in \(A\). In order to satisfy this strong
variable-sharing principle, the principle of disjunction introduction
needs to be restricted. So, instead of having
\(A\rightarrow(A\vee B)\) as a theorem for all
formulas \(A\) and \(B\), this schema is valid only when all
the propositional variables in \(B\) are also in \(A\). The
principle of contraposition and some transitivity principles for
implication also have to be restricted. 



Analytic Implication has been given an elegant possible worlds
semantics by Kit Fine. Fine adds to a possible worlds model a domain
of subject matters. An implication holds at a world if and only if it
both preserves truth at all accessible worlds and also every subject
matter of the consequent is also a subject matter of the antecedent
(Fine 1986).



For a comparison of Analytic Implication and relevance logic, see
Routley, et al., 1982, pages 96-101. For a detailed examination and
defense of Analytic Implication, see Ferguson (2017).


## 7. Applications and Extensions of Relevance Logic



Apart from the motivating applications of providing better formalisms
of our pre-formal notions of implication and entailment and providing
a basis for naive set theory, relevance logic has been put to
various uses in philosophy and computer science. Here I will list just
a few.



Dunn has developed a theory of intrinsic and essential properties
based on relevant logic. This is his theory of *relevant
predication*. Briefly put, a thing \(i\) has a property
\(F\) relevantly iff \(\forall x(x{=}i \rightarrow F(x))\). Informally, an object has a property
relevantly if being that thing relevantly implies having that
property. Since the truth of the consequent of a relevant implication
is by itself insufficient for the truth of that implication, things
can have properties irrelevantly as well as relevantly. Dunn's
formulation would seem to capture at least one sense in which we use
the notion of an intrinsic property. Adding modality to the language
allows for a formalisation of the notion of an essential property as a
property that is had both necessarily and intrinsically (see Anderson,
Belnap, and Dunn 1992, SS74).



Relevant logic has been used as the basis for mathematical theories
other than set theory. Meyer has produced a variation of Peano
arithmetic based on the logic \(\mathbf{R}\). Meyer gave a
finitary proof that his relevant arithmetic does not have \(0 = 1\) as a
theorem. Thus Meyer solved one of Hilbert's central problems in the
context of relevant arithmetic; he showed using finitary means that
relevant arithmetic is absolutely consistent. This makes relevant
Peano arithmetic an extremely interesting theory. Unfortunately, as
Meyer and Friedman have shown, relevant arithmetic does not contain
all of the theorems of classical Peano arithmetic. Hence we cannot
infer from this that classical Peano arithmetic is absolutely
consistent (see Meyer and Friedman 1992).



Anderson (1967) formulated a system of deontic logic based on
\(\mathbf{R}\) and, more recently, relevance logic has been used
as a basis for deontic logic by Mares (1992) and Lou Goble (1999).
These systems avoid some of the standard problems with more
traditional deontic logics. One problem that standard deontic logics
face is that they make valid the inference from \(A\)'s being a
theorem to \(OA\)'s being a theorem, where
'\(OA\)' means 'it ought to be that
\(A\)'. The reason that this problem arises is that it is
now standard to treat deontic logic as a normal modal logic. On the
standard semantics for modal logic, if \(A\) is valid, then it is
true at all possible worlds. Moreover, \(OA\) is true at a world
\(a\) if and only if \(A\) is true at every world accessible
to \(a\). Thus, if \(A\) is a valid formula, then so is
\(OA\). But it seems silly to say that every valid formula ought
to be the case. Why should it be the case that either it is now
raining in Ecuador or it is not? In the semantics for relevant logics,
not every world makes true every valid formula. Only a special class
of worlds (sometimes called "base worlds" and sometimes
called "normal worlds") make true the valid formulae. Any
valid formula can fail at a world. By allowing these "non-normal
worlds" in our models, we invalidate this problematic rule.



Other sorts of modal operators have been added to relevance logic as
well. Fuhrmann (1990) adapts the usual axioms for the familiar
classical modal logics to the relevant context to produce a collection
of relevant modal logics and proves completeness results for them.
Epistemic modal operators have been added to relevance logics by
Heinrich Wansing (2002), Marta Bilkova, Ondrej Majer, Michal
Pelis, and Greg Restall (2010), among others. Shawn Standefer
(2019) has produced a relevant version of justification logic and has
very recently added an actuality opertor to relevance logic. There is
even a relevant logic of questions and answers (see
Puncochar 2020).



Routley and Val Plumwood (1989) and Mares and Andre Fuhrmann
(1995) present theories of counterfactual conditionals based on
relevant logic. Their semantics adds to the standard Routley-Meyer
semantics an accessibility relation that holds between a formula and
two worlds. On Routley and Plumwood's semantics, \(A\gt B\)
holds at a world \(a\) if and only if for all worlds \(b\) such that
\(SAab\), \(B\) holds at \(b\). Mares and Fuhrmann's semantics
is slightly more complex: \(A\gt B\) holds at a world \(a\) if and
only if for all worlds \(b\) such that \(SAab\), \(A \rightarrow B\)
holds at \(b\) (also see Brady (ed.) 2002, SS10 for details of
both semantics). Mares (2004) presents a more complex theory of
relevant conditionals that includes counterfactual conditionals. All
of these theories avoid the analogues of the paradoxes of implication
that appear in standard logics of counterfactuals.



Relevant logics have been used in computer science as well as in
philosophy. Linear logics -- a branch of logic initiated by
Jean-Yves Girard -- is a logic of computational resources. Linear
logicians read an implication \(A \rightarrow B\) as saying that
having a resource of type \(A\) allows us to obtain something of type
\(B\). If we have \(A \rightarrow(A \rightarrow B)\), then, we know
that we can obtain a \(B\) from two resources of type \(A\). But this
does not mean that we can get a \(B\) from a single resource of type
\(A\), i.e. we don't know whether we can obtain \(A \rightarrow
B\). Hence, contraction fails in linear logic. Linear logics are, in
fact, relevant logics that lack contraction and the distribution of
conjunction over disjunction \(((A \amp(B \vee C)) \rightarrow((A \amp
B) \vee(A \amp C)))\). They also include two operators (! and ?) that
are known as "exponentials". Putting an exponential in
front of a formula gives that formula the ability to act classically,
so to speak. For example, just as in standard relevance logic, we
cannot usually merely add an extra premise to a valid inference and
have it remain valid. But we can always add a premise of the form
\(!A\) to a valid inference and have it remain valid. Linear logic
also has contraction for formulae of the form \(!A\), i.e., it is a
theorem of these logics that \((!A \rightarrow(!A \rightarrow B))
\rightarrow(!A \rightarrow B)\) (see Troelstra 1992). The use of !
allows for the treatment of resources "that can be duplicated or
ignored at will" (Restall 2000, p 56). For more about linear
logic, see the entry on
 substructural logic.