Relevance logic, also called relevant logic, is a kind of non-

classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...

requiring the antecedent and

consequent A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...

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

logician Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

s, 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 Clarence Irving Lewis (April 12, 1883 – February 3, 1964), usually cited as C. I. Lewis, was an American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted logic ...

was led to invent modal logic, and specifically

strict implication In logic, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessit ...

, on the grounds that classical logic grants

paradoxes of material implication The paradoxes of material implication are a group of tautology (logic), true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated as "if ... then ...". A material ...

such as the principle that a falsehood implies any proposition. 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. How does relevance logic formally capture a notion of relevance? In terms of a syntactical constraint for a

propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

, it is necessary, but not sufficient, that premises and conclusion share

atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...

e (formulae that do not contain any

logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...

s). In a

predicate calculus Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...

, 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 In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axiom ...

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 Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died o ...

-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

sequent In mathematical logic, a sequent is a very general kind of conditional assertion. : A_1,\,\dots,A_m \,\vdash\, B_1,\,\dots,B_n. A sequent may have any number ''m'' of condition formulas ''Ai'' (called " antecedents") and any number ''n'' of asse ...

s. A notable feature of relevance logics is that they are

paraconsistent logic A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" syste ...

s: the existence of a contradiction will not cause "

explosion An explosion is a rapid expansion in volume associated with an extreme outward release of energy, usually with the generation of high temperatures and release of high-pressure gases. Supersonic explosions created by high explosives are known ...

". 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).

History

Relevance logic was proposed in 1928 by Soviet philosopher

Ivan E. Orlov Ivan Efimovich Orlov (Russian: Иван Ефимович Орлов; Galich, Kostroma Oblast, – Moscow, after 1936 Ackermann,

Moh Moh (Punjabi: ਮੋਹ ''mōha''; Sanskrit: ''muh'': is a word in Punjabi and Sanskrit. Definition “to become stupefied, to be bewildered or perplexed, to err, to be mistaken”. It stands in ancient texts for perplexity or confusion and ...

, and

Church Church may refer to: Religion * Church (building), a building for Christian religious activities * Church (congregation), a local congregation of a Christian denomination * Church service, a formalized period of Christian communal worship * Chris ...

in ''Kontroliertes Denken: Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften'', Kommissions-Verlag Karl Alber, edited by A. Menne, A. Wilhelmy and H. Angsil, pp. 22–37. in the 1950s. Drawing on them,

Nuel Belnap Nuel Dinsmore Belnap Jr. (; born 1930) is an American logician and philosopher who has made contributions to the philosophy of logic, temporal logic, and structural proof theory. He taught at the University of Pittsburgh from 1963 until his ret ...

and

Alan Ross Anderson Alan Ross Anderson (1925–1973) was an American logician and professor of philosophy at Yale University and the University of Pittsburgh. A frequent collaborator with Nuel Belnap, Anderson was instrumental in the development of relevance lo ...

(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 Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...

and systems of relevance, where implications of the former kinds are supposed to be both relevant and necessary.

Axioms

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\to A

A\land B\to A

A\land B\to B

(A\to B)\land(A\to C)\to (A\to B\land C)

A\to A\lor B

B\to A\lor B

(A\to C)\land(B\to C)\to (A\lor B\to C)

A\land(B\lor C)\to (A\land B)\lor(A\land C)

\lnot\lnot A\to A

The rules are the following. #

A, A\to B\vdash B

A, B\vdash A\land B

A\to B\vdash (C\to A)\to(C\to B)

A\to B\vdash (B\to C)\to(A\to C)

A\to B\vdash \lnot B\to\lnot A

Stronger logics can be obtained by adding any of the following axioms. #

(A\to B)\to (\lnot B\to\lnot A)

(A\to B)\land(B\to C)\to (A\to C)

(A\to B)\to((B\to C)\to(A\to C))

(A\to B)\to((C\to A)\to(C\to B))

(A\to(A\to B))\to(A\to B)

(A\land (A\to B))\to B

(A\to\lnot A)\to\lnot A

(A\to (B\to C))\to(B\to(A\to C))

A\to((A\to B)\to B)

((A\to A)\to B)\to B

A\lor\lnot A

A\to(A\to A)

There are some notable logics stronger than B that can be obtained by adding axioms to B as follows. * For DW, add axiom 1. * For DJ, add axioms 1, 2. * For TW, add axioms 1, 2, 3, 4. * For RW, add axioms 1, 2, 3, 4, 8, 9. * For T, add axioms 1, 2, 3, 4, 5, 6, 7, 11. * For R, add axioms 1-11. * For E, add axioms 1-7, 10, 11,

((A\to A)\land(B\to B)\to C)\to C

, and

\Box A\land \Box B\to \Box (A\land B)

, where

\Box A

is defined as

(A\to A)\to A

. * For RM, add all the additional axioms.

Models

Routley–Meyer models

The standard model theory for relevance logics is the Routley-Meyer ternary-relational semantics developed by

Richard Routley Richard Sylvan (13 December 1935 – 16 June 1996) was a New Zealand–born philosopher, logician, and environmentalist. Biography Sylvan was born Francis Richard Routley in Levin, New Zealand, and his early work is cited with this surname. ...

and

Robert Meyer Robert Meyer (born October 2, 1945 in Oslo, Norway) is a Norwegian art photographer, professor, photo historian, collector, writer and publicist. He is the son of journalist Robert Castberg Meyer and homemaker Edel Nielsen; and brother of the in ...

. 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\in W

. A Routley-Meyer model M is a Routley-Meyer frame F together with a valuation,

\Vdash

, that assigns a truth value to each atomic proposition relative to each point

a\in W

. There are some conditions placed on Routley-Meyer frames. Define

a\leq b

R0ab

. *

a\leq a

. * If

a\leq b

and

b\leq c

, then

a\leq c

. * If

d\leq a

and

Rabc

, then

Rdbc

. *

a^=a

. * If

a\leq b

, then

b^*\leq a^*

. Write

M,a\Vdash A

and

M,a\nVdash A

to indicate that the formula

A

is true, or not true, respectively, at point

a

M

. One final condition on Routley-Meyer models is the hereditariness condition. * If

M,a\Vdash p

and

a\leq b

, then

M,b\Vdash p

, for all atomic propositions

p

. By an inductive argument, hereditariness can be shown to extend to complex formulas, using the truth conditions below. * If

M,a\Vdash A

and

a\leq b

, then

M,b\Vdash A

, for all formulas

A

. The truth conditions for complex formulas are as follows. *

M,a\Vdash A\land B \iff M, a\Vdash A

and

M,a\Vdash B

M,a\Vdash A\lor B \iff M, a\Vdash A

M,a\Vdash B

M,a\Vdash A\to B\iff \forall b,c((Rabc\land M,b\Vdash A)\Rightarrow M,c\Vdash B)

M,a\Vdash\lnot A\iff M,a^*\nVdash A

A formula

A

holds in a model

M

just in case

M,0\Vdash A

. A formula

A

holds on a frame

F

iff A holds in every model

(F,\Vdash)

. A formula

A

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

Rabcd

be defined as

\exists x(Rabx \land Rxcd)

, and let

Ra(bc)d

be defined as

\exists x(Rbcx \land Raxd)

. Some of the frame conditions and the axioms they validate are the following. 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 Alasdair Ian Fenton Urquhart (; born 20 December 1945) is a Scottish–Canadian philosopher and emeritus professor of philosophy at the University of Toronto. He has made contributions to the field of logic, especially non-classical logic. On ...

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

is a triple

(K,\cdot,0)

, where

K

is a non-empty set,

0\in K

, and

\cdot

is a binary operation on

K

. 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. *

x\cdot x=x

(x\cdot y)\cdot z=x\cdot(y\cdot z)

x\cdot y=y\cdot x

0\cdot x=x

Under these conditions, the operational frame is a

join-semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a mee ...

. An operational model

M

is a frame

F

with a valuation

V

that maps pairs of points and atomic propositions to truth values, T or F.

V

can be extended to a valuation

\Vdash

on complex formulas as follows. *

M,a\Vdash p \iff V(a,p)=T

, for atomic propositions *

M,a\Vdash A\land B \iff M, a\Vdash A

and

M,a\Vdash B

M,a\Vdash A\lor B \iff M, a\Vdash A

M,a\Vdash B

M,a\Vdash A\to B\iff \forall b(M,b\Vdash A\Rightarrow M,a\cdot b\Vdash B)

A formula

A

holds in a model

M

iff

M,0\Vdash A

. A formula

A

is valid in a class of models

C

iff it holds in each model

M\in C

. 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\lor C))\land(B\to C)\to (A\to C)

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 Kit Fine (born 26 March 1946) is a British philosopher, currently university professor and Silver Professor of Philosophy and Mathematics at New York University. Prior to joining the philosophy department of NYU in 1997, he taught at the Uni ...

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 Dag Prawitz (born 1936, Stockholm) is a Swedish philosopher and logician. He is best known for his work on proof theory and the foundations of natural deduction. Prawitz is a member of the Norwegian Academy of Science and Letters, of the Royal Sw ...

. The operational semantics can be adapted to model the conditional of E by adding a non-empty set of worlds

W

and an accessibility relation

\leq

W\times W

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. *

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)

The operational semantics can be adapted to model the conditional of T by adding a relation

\leq

K\times K

. The relation is required to obey the following conditions. *

0\leq x

* If

x\leq y

and

y\leq z

, then

x\leq z

* If

x\leq y

, then

x\cdot z\leq y\cdot z

The truth condition for the conditional is changed to the following. *

M,a\Vdash A\to B\iff \forall b((a\leq b\land M,b\Vdash A)\Rightarrow M,a\cdot b\Vdash B)

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\cdot x=x

. The second way is to keep the semilattice conditions on frames and add a binary relation,

J

, 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. *

M,a\Vdash A\to B\iff \forall b((Jab \land M,b\Vdash A)\Rightarrow M,a\cdot b\Vdash B)

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

F

is a quadruple

(K,\cdot,+,0)

, where

K

is a non-empty set,

0\in K

, and are binary operations on

K

. Let

a\leq b

be defined as

\exists x(a+x=b)

. The frame conditions are the following. An operational model

M

is a frame

F

with a valuation

V

that maps pairs of points and atomic propositions to truth values, T or F.

V

can be extended to a valuation

\Vdash

on complex formulas as follows. *

M,a\Vdash p \iff V(a,p)=T

, for atomic propositions *

M,a+b\Vdash p \iff M,a\Vdash p

and

M,b\Vdash p

M,a\Vdash A\land B \iff M,a\Vdash A

and

M,a\Vdash B

M,a\Vdash A\lor B \iff M, a\Vdash A

M,a\Vdash B

\exists b,c(a=b+c

;

M,b\Vdash A

and

M,c\Vdash B)

M,a\Vdash A\to B\iff \forall b(M,b\Vdash A\Rightarrow M,a\cdot b\Vdash B)

A formula

A

holds in a model

M

iff

M,0\Vdash A

. A formula

A

is valid in a class of models

C

iff it holds in each model

M\in C

. 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.

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

(D,\land,\lor,\lnot,\circ,e)

where *

(D,\land,\lor,\lnot)

is a distributive

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

with a unary operation,

\lnot

obeying the laws

\lnot\lnot x=x

and if

x\leq y

then

\lnot y\leq \lnot x

; *

e\in D

, the binary operation

\circ

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

(

x\circ y=y\circ x

) and

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...

(

(x\circ y)\circ z=x\circ (y\circ z)

), and

e\circ x=x

, i.e.

(D,\circ,e)

is an

Abelian monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...

with

identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...

e

; * the monoid is lattice-ordered and satisfies

x\circ(y\lor z)=(x\circ y)\lor(x\circ z)

; *

x\leq x\circ x

; and * if

x\circ y\leq z

, then

x\circ\lnot z\leq \lnot y

. The operation

x\to y

interpreting the conditional of R is defined as

\lnot(x\circ\lnot y)

. A de Morgan monoid is a

residuated lattice In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice ''x'' ≤ ''y'' and a monoid ''x''•''y'' which admits operations ''x''\''z'' and ''z''/''y'', loosely analogous to division or implication, when ' ...

, obeying the following residuation condition. :

x \circ y\leq z \iff x\leq y\to z

An interpretation

v

is a

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

from the propositional language to a de Morgan monoid

M

such that *

v(p)\in D

for all atomic propositions, *

v(\lnot A)=\lnot v(A)

v(A\lor B)=v(A)\lor v(B)

v(A\land B)=v(A)\land v(B)

v(A\to B)=v(A)\to v(B)

Given a de Morgan monoid

M

and an interpretation

v

, one can say that formula

A

holds on

v

just in case

e\leq v(A)

. A formula

A

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.

References

Bibliography

and

, 1975. ''Entailment: the logic of relevance and necessity, vol. I''. Princeton University Press. * ------- and J. M. Dunn, 1992. ''Entailment: the logic of relevance and necessity, vol. II'', Princeton University Press. * Mares, Edwin, and Meyer, R. K., 2001, "Relevant Logics", in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell. * Richard Routley, Val Plumwood, Robert K. Meyer, and Ross T. Brady. ''Relevant Logics and their Rivals''. Ridgeview, 1982. * R. Brady (ed.), ''Relevant Logics and their Rivals (Volume II)'', Aldershot: Ashgate, 2003. * * Alasdair Urquhart.
The Semantics of Entailment
'. PhD thesis, University of Pittsburgh, 1972. * Katalin Bimbó, Relevance logics, in ''Philosophy of Logic'', D. Jacquette (ed.), (volume 5 of ''Handbook of the Philosophy of Science'', D. Gabbay, P. Thagard, J. Woods (eds.)), Elsevier (North-Holland), 2006, pp. 723–789. * J. Michael Dunn and Greg Restall. Relevance logic. In ''Handbook of Philosophical Logic'', Volume 6, F. Guenthner and D. Gabbay (eds.), Dordrecht: Kluwer, 2002, pp. 1–136. * Stephen Read, ''Relevant Logic'', Oxford: Blackwell, 1988. *

External links

Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...

:
Relevance logic
– by Edwin Mares. *
Relevance logic
' – by J. Michael Dunn and Greg Restall *
Relevant Logic
' – by Stephen Read {{DEFAULTSORT:Relevance Logic Substructural logic Non-classical logic Paraconsistent logic

History

Axioms

Models

Routley–Meyer models

Operational models

Urquhart models

Humberstone models

Algebraic models

See also

References

Bibliography

External links