Relevance Logic (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeRelevance LogicFirst published Wed Jun 17, 1998; substantive revision Mon Jan 2, 2006Relevance logics are non-classical logics. Called‘relevant logics” in Britain and Australasia, thesesystems developed as attempts to avoid the paradoxes of material andstrict implication. Among the paradoxes of material implicationare p → (q → p).¬p → (p → q).(p → q)  (q → r).Among the paradoxes of strict implication are the following: (p & ¬p) → q.p → (q → q). p → (q ¬q).Many philosophers, beginning with Hugh MacColl (1908), have claimedthat these theses are counterintuitive. They claim that these formulaefail to be valid if we interpret → as representing the concept ofimplication that we have before we learn classical logic. Relevancelogicians claim that what is unsettling about these so-calledparadoxes is that in each of them the antecedent seems irrelevant tothe consequent.In addition, relevance logicians have had qualms about certaininferences that classical logic makes valid. For example, consider theclassically valid inferenceThe moon is made of green cheese. Therefore, either it is raining inEcuador now or it is not.Again here there seems to be a failure of relevance. The conclusionseems to have nothing to do with the premise. Relevance logicians haveattempted to construct logics that reject theses and arguments thatcommit "fallacies of relevance".Relevant logicians point out that what is wrong with some of theparadoxes (and fallacies) is that is that the antecedents andconsequents (or premises and conclusions) are on completely differenttopics. The notion of a topic, however, would seem not to be somethingthat a logician should be interested in — it has to do with thecontent, not the form, of a sentence or inference. But there is aformal principle that relevant logicians apply to force theorems andinferences to “stay on topic”. This is the variablesharing principle. The variable sharing principle says that noformula of the form A → B can be proven in a relevance logic if Aand B do not have at least one propositional variable (sometimescalled a proposition letter) in common and that no inference can beshown valid if the premises and conclusion do not share at least onepropositional variable.At this point some confusion is natural about what relevant logicianshave attempted to do. The variable sharing principle is only anecessary condition that a logic must have to count as a relevancelogic. It is not sufficient. Moreover, this principle does not give usa criterion that eliminates all of the paradoxes and fallacies. Someremain paradoxical or fallacious even though they satisfy variablesharing. As we shall see, however, relevant logic does provide us witha relevant notion of proof in terms of the real use of premises (seethe section “Proof Theory” below), but it does not byitself tell us what counts as a true (and relevant) implication. It isonly when the formal theory is put together with a philosophicalinterpretation that it can do this (see the section“Semantics” below).In this article we will give a brief and relatively non-technicaloverview of the field of relevance logic.1. Semantics2. Proof Theory3. Systems of Relevance Logic4. Applications of Relevance LogicBibliography Books on Relevance Logic and Introductions to the Field: Other Works Cited: Other Internet ResourcesRelated Entries1. SemanticsOur exposition of relevant logic is backwards to most found in theliterature We will begin, rather than end, with the semantics, sincemost philosophers at present are semantically inclined.The semantics that I present here is the ternary relation semanticsdue to Richard Routley and Robert K. Meyer. This semantics is adevelopment of Alasdair Urquhart's “semilattice semantics”(Urquhart 1972). There is a similar semantics (which is also based onUrquhart's ideas), due to Kit Fine, that was developed at the sametime as the Routley-Meyer theory (Fine 1974). And there is analgebraic semantics due to J. Michael Dunn. Urquhart's, Fine's, andDunn's models are very interesting in their own right, but we do nothave 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 materialimplication. He added a new connective to classical logic, that ofstrict implication. In post-Kripkean semantic terms, A  B is true at a world w if and only if for allw′ such that w′ is accessible tow, either A fails in w′ or Bobtains there. Now, in Kripke's semantics for modal logic, theaccessibility relation is a binary relation. It holds between pairs ofworlds. Unfortunately, from a relevant point of view, the theory ofstrict implication is still irrelevant. That is, we still make validformulae like p (q q). We can see quite easily that the Kripke truth condition forces thisformula on us.Like the semantics of modal logic, the semantics of relevance logicrelativises truth of formulae to worlds. But Routley and Meyer gomodal logic one better and use a three-place relation on worlds. Thisallows there to be worlds at which q → q failsand that in turn allows worlds at which p → (q→ q) fails. Their truth condition for → on thissemantics is the following:A → B is true at a world a if and onlyif for all worlds b and c such that Rabc(R is the accessibility relation) either A is falseat b or B is true at c. For people new to the field it takes some time to get used to thistruth condition. But with a little work it can be seen to be just ageneralisation of Kripke's truth condition for strict implication(just set b = c).The ternary relation semantics can be adapted to be a semantics for awide range of logics. Placing different constraints on the relationmakes valid different formulae and inferences. For example, if weconstrain the relation so that Raaa holds for all worldsa, then we make it true that if (A →B) & A is true at a world, then B isalso true there. Given other features of the Routley-Meyer semantics,this makes the thesis ((A → B) &A) → B valid. If we make the ternary relationsymmetrical in its first two places, that is, we constrain it so that,for all worlds a, b, and c, ifRabc then Rbac, then we make valid the thesisA → ((A → B) →B).The ternary accessibility relation needs a philosophicalinterpretation in order to give relevant implication a real meaning onthis semantics. Recently there have been three interpretationsdeveloped based on theories about the nature of information. Oneinterpretation of the ternary relation, due to Dunn, develops the ideabehind Urquhart's semilattice semantics. On Urquhart'ssemantics, instead of treating indices as possible (or impossible)worlds, they are taken to be pieces of information. In the semilatticesemantics, an operator ° combines the information of two states— a°b is the combination of theinformation in a and b. The Routley-Meyer semanticsdoes not contain a combination or “fusion” operator onworlds, but we can get an approximation of it using the ternaryrelation. On Dunn's reading, ‘Rabc’ saysthat “the combination of the information states a andb is contained in the information state c”(Dunn 1986).Another interpretation is suggested in Jon Barwise (1993) anddeveloped in Restall (1996). On this view, worlds are taken to beinformation-theoretic "sites" and “channels”. A site is acontext in which information is received and a channel is a conduitthrough which information is transferred. Thus, for example, when theBBC news appears on the television in my living room, we can considerthe living room to be a site and the wires, satellites, and so on,that connect my television to the studio in London to be achannel. Using channel theory to interpret the Routley-Meyersemantics, we take Rabc to mean that a is aninformation-theoretic channel between sites b andc. Thus, we take A → B to be true ata if and only if, whenever a connects a siteb at which A obtains to a site c,B obtains at c.Similarly, Mares (1997) uses a theory of information due to DavidIsrael and John Perry (1990). In addition to other information a worldcontains informational links, such as laws of nature, conventions, andso on. For example, a Newtonian world will contain the informationthat all matter attracts all other matter. In information-theoreticterms, this world contains the information that two things' beingmaterial carries the information that they attract each other. On thisview, Rabc if and only if, according to the links ina, all the information carried by what obtains in bis contained in c. Thus, for example, if a is aNewtonian world and the information that x and y arematerial is contained in b, then the information thatx and y attract each other is contained inc.Another interpretation is developed in Mares (2004). Thisinterpretation takes the Routley-Meyer semantics to be a formalisationof the notion of “situated implication”. Thisinterpretation takes the “worlds” of the Routley-Meyersemantics to be situations. A situation is a perhaps partialrepresentation of the universe. The information contained in twosituations, a and b might allow us to infer furtherinformation about the universe that is contained in neither situation.Thus, for example, suppose in our current situation that we have theinformation contained in the laws of the theory of general relativity(this is Einstein's theory of gravity). Then we hypothesise asituation in which we can see a star moving in an ellipse. Then, on thebasis of the information that we have and the hypothesised situation,we can infer that there is a situation in which there is a very heavybody acting on this star.We can model situated inference using a relation I (for“implication”). Then we have IabP, whereP is a proposition, if and only if the information ina and b together license the inference to therebeing a situation in which P holds. We can think of aproposition itself as a set of situations. We set A →B to hold at a if and only if, for all situationsb in which A holds, Iab|B|, where|B| is the set of situations at which B is true. Weset Rabc to hold if and only if c belongs to everyproposition P such that IabP. With the addition ofthe postulate that, for any set of propositions P such thatIabP, the intersection of that set X is such thatIabX, we find that the implications that are made true on anysituation using the truth condition that appeals to I are thesame as those that are made true by the Routley-Meyer truthcondition. Thus, the notion of situated inference gives a way ofunderstanding the Routley-Meyer semantics. (This is a very briefversion of the discussion of situated inference that is in chapters 2and 3 of Mares (2004).)By itself, the use of the ternary relation is not sufficient toavoid 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 & ¬p)→ q and p→ (q ¬q). These paradoxes are avoided by the inclusion of inconsistent andnon-bivalent worlds in the semantics. For, if there were no worlds atwhich p & ¬p holds, then, according to ourtruth condition for the arrow, (p & ¬p)→ q would also hold everywhere. Likewise, if q ¬q held at every world, then p → (q ¬q) would be universally true.This brings us to the semantics for negation. The use of non-bivalentand inconsistent worlds requires a non-classical truth condition fornegation. In the early 1970s, Richard and Val Routley invented their"star operator" to treat negation. The operator is an operator onworlds. For each world a, there is a world a*.And¬A is true at a if and only if A is falseat a*.Once again, we have the difficulty of interpreting a part of theformal semantics. One interpretation of the Routley star is that ofDunn (1993). Dunn uses a binary relation, C, onworlds. Cab means that b is compatible witha. a*, then, is the maximal world (the worldcontaining the most information) that is compatible witha.There are other semantics for negation. One, due to Dunn anddeveloped by Routley, is a four-valued semantics. This semantics istreated in the entry on paraconsistent logics. Other treatments of negation, some of which have been used forrelevant logics, can be found in Wansing (2001).2. Proof TheoryThere is now a large variety of approaches to proof theory forrelevant logics. There is a sequent calculus for the negation-freefragment of the logic R due to Gregory Mints (1972) and J.M. Dunn (1973) and an elegant and very general approachcalled "Display Logic" developed by Nuel Belnap (1982). For theformer, see the supplementary document: Logic R But here I will only deal with the natural deduction system for therelevant logic R due to Anderson and Belnap.Anderson and Belnap's natural deduction system is based on Fitch'snatural deduction systems for classical and intuitionistic logic. Theeasiest way to understand this technique is by looking at anexample.1. A{1}Hyp2. (A → B){2}Hyp3. B{1,2}1,2, → EThis is a simple case of modus ponens. The numbers in set bracketsindicate the hypotheses used to prove the formula. We will call them‘indices’. The indices in the conclusion indicate whichhypotheses are really used in the derivation of the conclusion. In thefollowing “proof” the second premise is not reallyused:1. A{1}Hyp2. B{2}Hyp3. (A → B){3}Hyp4. B{1,3}1,3, → EThis “proof” really just shows that the inference fromA and A → B to B isrelevantly valid. Because the number 2 does not appear in thesubscript on the conclusion, the second “premise” does notreally count as a premise.Similarly, when an implication is proven relevantly, the assumption ofthe antecedent must really be used to prove the conclusion. Here is anexample of the proof of an implication:1. A{1}Hyp2. (A → B){2}Hyp3. B{1,2}1,2, → E4. ((A → B) → B){1}2,3, → I5. A → ((A → B) → B)1,4, → IWhen 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 theformula that is to become the consequent of the implication.Now, it might seem that the system of indices allows irrelevantpremises to creep in. One way in which it might appear thatirrelevances can intrude is through the use of a rule of conjunctionintroduction. That is, it might seem that we can always add in anirrelevant premise by doing, say, the following:1. A{1}Hyp2. B{2}Hyp3. (A & B){1,2}1,2, &I4. B{1,2}3, &E5. (B → B){1}2,4, → I6. A → (B → B)1,5, → ITo a relevance logician, the first premise is completely out ofplace here. To block moves like this, Anderson and Belnap give thefollowing conjunction introduction rule:From Ai and Bi to infer(A & B)i.This rule says that two formulae to be conjoined must have the sameindex before the rule of conjunction introduction can be used.There is, of course, a lot more to the natural deduction system (seeAnderson and Belnap 1975 and Anderson, Belnap, and Dunn 1992), butthis will suffice for our purposes. The theory of relevance that iscaptured by at least some relevant logics can be understood in terms ofhow the corresponding natural deduction system records the real use ofpremises.3. Systems of Relevance LogicIn the work of Anderson and Belnap the central systems of relevancelogic were the logic E of relevant entailment and thesystem R of relevant implication. The relationshipbetween the two systems is that the entailment connective ofE was supposed to be a strict (i.e. necessitated)relevant implication. To compare the two, Meyer added a necessityoperator to R (to produce the logicNR). Larisa Maksimova, however, discovered thatNR and E are importantly different— that there are theorems of NR (on the naturaltranslation) that are not theorems of E. This hasleft some relevant logicians with a quandary. They have to decidewhether to take NR to be the system of strictrelevant implication, or to claim that NR was somehowdeficient and that E stands as the system of strictrelevant implication. (Of course, they can accept both systems andclaim that E and R have a differentrelationship to one another.)On the other hand, there are those relevance logicians who reject bothR and E. There are those, likeArnon Avron, who accept logics stronger than R (Avron1990). And there are those, like Ross Brady, John Slaney, SteveGiambrone, Richard Sylvan, Graham Priest, Greg Restall, and others,who have argued for the acceptance of systems weaker thanR or E. One extremely weak systemis the logic S of Robert Meyer and Errol Martin. AsMartin has proven, this logic contains no theorems of the formA → A. In other words, according toS, no proposition implies itself and no argument ofthe form ‘A, therefore A’ isvalid. Thus, this logic does not make valid any circulararguments.For more details on these logics see supplements on the logic E, logic R, logic NR, and logic S.Among the points in favour of weaker systems is that, unlikeR or E, many of them are decidable.Another feature of some of these weaker logics that makes themattractive is that they can be used to construct a naïve settheory. A naïve set theory is a theory of sets that includes as atheorem the naïve comprehension axiom, viz., for all formulaeA(y),∃x∀y(y ∈ x ↔ A(y)).In set theories based on strong relevant logics, likeE and R, as well as in classical settheory, if we add the naïve comprehension axiom, we are able toderive any formula at all. Thus, naïve set theories based onsystems such as E and R are said tobe “trivial”. Here is an intuitive sketch of the proof ofthe triviality of a naïve set theory using principles ofinference from the logic R. Let p be anarbitrary proposition:1. ∃x∀y(y ∈x ↔ (y ∈ y → p))Naïve Comprehension2. ∀y(y ∈ z ↔(y ∈ y → p))1, Existential Instantiation3. z ∈ z ↔ (z ∈z → p) 2, Universal Instantiation4. z ∈ z → (z ∈ z → p)3, definition of ↔ , &-Elimination5. (z ∈ z → (z ∈ z → p))→ (z ∈ z → p)Axiom of Contraction6. z ∈ z → p4,5, Modus Ponens7. (z ∈ z → p)) →z ∈ z 3, definition of ↔ , &-Elimination8. z ∈ z6,7, Modus Ponens9. p6,8, Modus PonensThus we show that any arbitrary proposition is derivable in thisnaïve set theory. This is the infamous Curry Paradox. Theexistence of this paradox has led Grishen, Brady, Restall, Priest, andothers to abandon the axiom of contraction ((A →(A → B)) → (A →B)). Brady has shown that by removing contraction, plus someother key theses, from R we obtain a logic that canaccept naïve comprehension without becoming trivial (Brady2005).In terms of the natural deduction system, the presence ofcontraction corresponds to allowing premises to be used more than once.Consider the following proof:1. A → (A → B){1}Hyp2. A{2}Hyp3. A → B{1,2}1,2, → E4. B{1,2}2,3, → E5. A → B{1}2-4, → I6. (A → (A → B))→ (A → B)1-5, → IWhat enables the derivation of contraction is the fact that oursubscripts are sets. We do not keep track of how many times (more thanonce) that a hypothesis is used in its derivation. In order to rejectcontraction, we need a way of counting the number of uses ofhypotheses. Thus natural deduction systems for contraction-freesystems use “multisets” of relevance numerals instead ofsets — these are structures in which the number of occurrencesof a particular numeral counts, but the order in which they occursdoes not. Even weaker systems can be constructed, which keep trackalso of the order in which hypotheses are used (see Read 1986 andRestall 2000).4. Applications of Relevance LogicApart from the motivating applications of providing betterformalisms of our pre-formal notions of implication and entailment andproviding a basis for naïve set theory, relevance logic has beenput to various uses in philosophy and computer science. Here I willlist just a few.Dunn has developed a theory of intrinsic and essential propertiesbased on relevant logic. This is his theory of relevantpredication. Briefly put, a thing i has a propertyF relevantly iff ∀x(x=i →F(x)). Informally, an object has a propertyrelevantly if being that thing relevantly implies having thatproperty. Since the truth of the consequent of a relevant implicationis by itself insufficient for the truth of that implication, thingscan have properties irrelevantly as well as relevantly. Dunn'sformulation would seem to capture at least one sense in which we usethe notion of an intrinsic property. Adding modality to the languageallows for a formalisation of the notion of an essential property as aproperty that is had both necessarily and intrinsically (see Anderson,Belnap, and Dunn 1992, §74).Relevant logic has been used as the basis for mathematical theoriesother than set theory. Meyer has produced a variation of Peanoarithmetic based on the logic R. Meyer gave afinitary proof that his relevant arithmetic does not have 0 = 1 as atheorem. Thus Meyer solved one of Hilbert's central problems in thecontext of relevant arithmetic; he showed using finitary means thatrelevant arithmetic is absolutely consistent. This makes relevantPeano arithmetic an extremely interesting theory. Unfortunately, asMeyer and Friedman have shown, relevant arithmetic does not containall of the theorems of classical Peano arithmetic. Hence we cannotinfer from this that classical Peano arithmetic is absolutelyconsistent (see Meyer and Friedman 1992).Anderson (1967) formulated a system of deontic logic based onR and, more recently, relevance logic has been usedas a basis for deontic logic by Mares (1992) and Lou Goble(1999). These systems avoid some of the standard problems with moretraditional deontic logics. One problem that standard deontic logicsface is that they make valid the inference from A'sbeing a theorem to OA's being a theorem, where‘OA’ means ‘it ought to be thatA’. The reason that this problem arises is that it isnow standard to treat deontic logic as a normal modal logic. On thestandard semantics for modal logic, if A is valid, then it istrue at all possible worlds. Moreover, OA is true at a worlda if and only if A is true at every world accessibleto a. Thus, if A is a valid formula, then so isOA. But it seems silly to say that every valid formula oughtto be the case. Why should it be the case that either it is nowraining in Ecuador or it is not? In the semantics for relevantlogics, not every world makes true every valid formula. Only a specialclass of worlds (sometimes called “base worlds” andsometimes called “normal worlds”) make true the validformulae. Any valid formula can fail at a world. By allowing these“non-normal worlds” in our models, we invalidate thisproblematic rule.Other sorts of modal operators have been added to relevant logic aswell. See, Fuhrmann (1990) for a general treatment of relevant modallogic and Wansing (2002) for a development and application of relevantepistemic logic.Routley and Val Plumwood (1989) and Mares and André Fuhrmann(1995) present theories of counterfactual conditionals based onrelevant logic. Their semantics adds to the standard Routley-Meyersemantics an accessibility relation that holds between a formula andtwo worlds. On Routley and Plumwood's semantics,A>B holds at a world a if and only iffor all worlds b such that SAab, B holds atb. Mares and Fuhrmann's semantics is slightly morecomplex: A>B holds at a world a if andonly if for all worlds b such that SAab, A→ B holds at b (also see Brady (ed.) 2002,§10 for details of both semantics). Mares (2004) presents a morecomplex theory of relevant conditionals that includes counterfactualconditionals. All of these theories avoid the analogues of theparadoxes of implication that appear in standard logics ofcounterfactuals.Relevant logics have been used in computer science as well as inphilosophy. Linear logics — a branch of logic initiated byJean-Yves Girard — is a logic of computational resources. Linearlogicians read an implication A → B as sayingthat having a resource of type A allows us to obtainsomething of type B. If we have A → (A→ B), then, we know that we can obtain a B fromtwo resources of type A. But this does not mean that we canget a B from a single resource of type A, i.e. wedon't know whether we can obtain A → B. Hence,contraction fails in linear logic. Linear logics are, in fact,relevant logics that lack contraction and the distribution ofconjunction over disjunction ((A & (B C)) → ((A & B) (A & C))). They also include two operators (! and ?) that are known as“exponentials”. Putting an exponential in front of aformula gives that formula the ability to act classically, so tospeak. For example, just as in standard relevance logic, we cannotusually merely add an extra premise to a valid inference and have itremain valid. But we can always add a premise of the form !Ato a valid inference and have it remain valid. Linear logic also hascontraction for formulae of the form !A, i.e., it is atheorem of these logics that (!A → (!A →B)) → (!A → B) (see Troelstra1992). The use of ! allows for the treatment of resources “thatcan be duplicated or ignored at will” (Restall 2000, p 56). Formore about linear logic, see the entry on substructural logic.BibliographyAn extremely good, although slightly out of date, bibliography onrelevance logic was put together by Robert Wolff and is in Anderson,Belnap, and Dunn (1992). What follows is a brief list of introductionsto and books about relevant logic and works that are referred toabove.Books on Relevance Logic and Introductions to the Field:Anderson, A.R. and N.D. Belnap, Jr. (1975) Entailment: TheLogic of Relevance and Necessity, Princeton, Princeton UniversityPress, Volume I. Anderson, A.R. N.D. Belnap, Jr. and J.M. Dunn (1992)Entailment, Volume II. [These are both collections of slightlymodified articles on relevance logic together with a lot of materialunique to these volumes. Excellent work and still the standard books onthe subject. But they are very technical and quite difficult.]Brady, R.T. (2005), Universal Logic, Stanford: CSLI, 2005.[A difficult, but extremely important book, which gives details ofBrady's semantics and his proofs that naïve set theory andhigher order logic based on his weak relevant logic areconsistent.]Dunn, J.M. (1986) "Relevance Logic and Entailment" in F. Guenthnerand D. Gabbay (eds.), Handbook of Philosophical Logic, Volume3, Dordrecht: Reidel pp 117-24.[Dunn has rewritten this piece togetherwith Greg Restall and the new version has appeared in volume 6 of thenew edition of the Handbook of Philosophical Logic(Dordrecht: Kluwer, 2002, pp 1-128)]Mares, E.D. (2004) Relevant Logic: A PhilosophicalInterpretation, Cambridge: Cambridge University PressMares, E.D. and R.K. Meyer (2001) “Relevant Logics” inL. Goble (ed.), The Blackwell Guide to Philosophical Logic,Oxford BlackwellPaoli, F (2002), Substructural Logics: A Primer,Dordrecht: Kluwer. [Excellent and clear introduction to a field oflogic that includes relevance logic.]Read, S. (1988), Relevant Logic, Oxford: Blackwell. [Avery interesting and fun book. Idiosyncratic, but philosophically adeptand excellent on the pre-history and early history of relevancelogic.]Restall, G. (2000), An Introduction to SubstructuralLogics, London: Routledge. [Excellent and clear introduction to afield of logic that includes relevance logic.]Rivenc, François (2005), Introduction à lalogique pertinente, Paris: Presses Universitaires de France. [InFrench. Gives a "structural" interpretation of relevant logic, whichis largely proof theoretic. The structures involved are structures ofpremises in a sequent calculus.]Routley, R., R.K. Meyer, V. Plumwood and R. Brady (1983),Relevant Logics and its Rivals, Volume I, Atascardero, CA:Ridgeview. [A very useful book for formal results especially about thesemantics of relevance logics. The introduction and philosophicalremarks are full of "Richard Routleyisms". They tend to be Routley'sviews rather than the views of the other authors and are fairly radicaleven for relevant logicians. Volume II is now out: R.Brady (ed.),Relevant Logics and their Rivals II, Aldershot: Ashgate, 2003.]Other Works Cited:Anderson, A.R. (1967) "Some Nasty Problems in the Formal Logic ofEthics" Nous 1 pp 354-360Avron, Arnon (1990) “Relevance and Paraconsistency — ANew Approach” The Journal of Symbolic Logic 55 pp707-732Barwise, J. (1993) "Constraints, Channels and the Flow ofInformation" in P.Aczel, et al. (eds), Situation Theory and ItsApplications, Volume 3, Stanford: CSLI pp 3-27Belnap, N.D. (1982) “Display Logic” Journal ofPhilosophical Logic 11 pp 375-417Brady, R.T. (1989) "The Non-Triviality of Dialectical Set Theory"in G. Priest, R. Routley and J. Norman (eds.), ParaconsistentLogic, Munich: Philosophia Verlag pp 437-470Dunn, J.M. (1973) (Abstract) “A ‘Gentzen System’for Positive Relevant Implication” The Journal of SymbolicLogic 38 pp 356-357Dunn, J.M. (1993) "Star and Perp" PhilosophicalPerspectives 7 pp 331-357Fine, K. (1974) “Models for Entailment” Journal ofPhilosophical Logic 3 pp 347-372Fuhrmann, A. (1990) “Models for Relevant Modal Logics”Studia Logica 49 pp 501-514Goble, L. (1999) “Deontic Logic with Relevance” in P.McNamara and H. Prakken (eds.), Norms, Logis and InformationSystems, Amsterdam: ISO Press, pp 331-346Grishen, V.N. (1974) “A Non-Standard Logic and itsApplication to Set Theory” Studies in Formalized Languagesand Non-Classical Logics (Russian), Moscow: NaukaIsrael, D. and J. Perry (1990) "What is Information?" in P.P.Hanson (ed.), Information, Language, and Cognition, Vancouver:University of British Columbia Press pp 1-19MacColl, H. (1908) “‘If’ and‘imply’” Mind 17 pp 151-152, 453-455Mares, E.D. (1992) “Andersonian Deontic Logic”Theoria 58 pp 3-20Mares, E.D. (1997) "Relevant Logic and the Theory of Information"Synthese 109 pp 345-360Mares, E.D. and A. Fuhrmann (1995) "A Relevant Theory ofConditionals" Journal of Philosophical Logic 24 pp645-665Meyer, R.K. and H. Friedman (1992) "Whither Relevant Arithmetic?"The Journal of Symbolic Logic 57 pp 824-831Restall, G. (1996) "Information Flow and Relevant Logics" in J.Seligman and D. Westerstahl (eds), Logic, Language andComputation, Volume 1, Stanford: CSLI pp 463-478Troelstra, A.S. (1992), Lectures on Linear Logic,Stanford: CSLIUrquhart, A. (1972) “Semantics for Relevant Logics”The Journal of Symbolic Logic 37 pp 159-169Wansing, H. (2001) “Negation” in L. Goble (ed.),The Blackwell Guide to Philosophical Logic, Oxford: Blackwell,pp 415-436Wansing, H. (2002) “Diamonds are a Philosopher's BestFriends” Journal of Philosophical Logic 31 pp591-612Other Internet ResourcesAn Alternative Semantics for Quantified Relevant Logic[PDF], by Edwin D. Mares and Robert Goldblatt, Victoria University ofWellington, provides a new semantics for quantified relevantlogic.[Please contact the author with other suggestions.]Related Entries logic: modal | logic: paraconsistent | logic: substructural | mathematics: inconsistent Copyright © 2006 byEdwin Mares <Edwin.Mares@vuw.ac.nz> |
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