Logic and Philosophy of Logic > Logics > Nonclassical Logics > Relevance Logic

Relevance Logic

Edited by Mark Jago (Nottingham University)

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Summary

Relevant logics are a group of logics which attempt to block irrelevant conclusions being drawn from a set of premises. The following inferences are all valid in classical logic, where A and B are any sentences whatsoever: from A to B → A, B → B and B ∨ ¬B; from ¬A to A→B; and from A ∧ ¬A to B. But if A and B are utterly irrelevant to one another, many feel reluctant to call these inferences acceptable. Similarly for the validity of the corresponding material implications, often called ‘paradoxes’ of material implication. Relevant logic can be seen as the attempt to avoid these ‘paradoxes’.

Key works

Many trace the beginnings of relevant logic to Anderson & Belnap 1962. Anderson & Belnap 1975 is a key early book-length exposition of relevant logics. Routley & Meyer 1972 and Routley & Meyer 1972 develop the relational ‘Routley-Meyer’ semantics for relevant implication, which has proved vital to the success of relevant logics. Read 1988 and Mares 2004 set out the philosophy of relevant logics. Brady 2006 contains much of Brady's work on relevant logics (which has been important throughout their development). Restall 1995 explores using 4-valued semantics for relevant logics.

Introductions	Mares 2012 is a recent introduction to the area. Jago 2013 surveys some of the most important recent work (2003–13) in relevant logic. The chapter on relevant logic in Priest 2001 introduces the logical details in a concise way.

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A Note on "Recent Work in Relevant Logic".José M. Méndez - manuscriptdetails
In his paper “Recent work in relevant logic”, Jago includes a section on Disjunctive Syllogism . The content of the section essentially consists of (a) a valuation of some work by Robles and Méndez on the topic as “not particularly interesting in itself”; (b) a statement establishing that “What would be interesting is to discover just how weak a relevant logic needs to be before disjunctive syllogism becomes inadmissible”. The main problem with this section of Jago’s paper on DS is (...)
Relevance Logic in Logic and Philosophy of Logic

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Meaning, Presuppositions, Truth-Relevance, Gödel's Sentence and the Liar Paradox.X. Y. Newberry - manuscriptdetails
Section 1 reviews Strawson’s logic of presuppositions. Strawson’s justification is critiqued and a new justification proposed. Section 2 extends the logic of presuppositions to cases when the subject class is necessarily empty, such as (x)((Px & ~Px) → Qx) . The strong similarity of the resulting logic with Richard Diaz’s truth-relevant logic is pointed out. Section 3 further extends the logic of presuppositions to sentences with many variables, and a certain valuation is proposed. It is noted that, given this valuation, (...)
Analyticity in Mathematics in Philosophy of Mathematics

Liar Paradox in Logic and Philosophy of Logic

Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Logicism in Mathematics in Philosophy of Mathematics

Relevance Logic in Logic and Philosophy of Logic

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An Epimorphism Between Fine and Ferguson's Matrices for Angell's AC.Richard Zach - manuscriptdetails
Angell's logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. We show that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. The epimorphism was found with the help of MUltlog, which also provides a tableau calculus for NC extended by quantifiers that generalize conjunction and disjunction.
Many-Valued Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Quantified Modal Relevant Logics.Nicholas Ferenz - forthcoming - Review of Symbolic Logic:1-32.details
Modal and Intensional Logic in Logic and Philosophy of Logic

Quantifiers in Philosophy of Language

Relevance Logic in Logic and Philosophy of Logic

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Disjunctive Parts.Mark Jago - forthcoming - In Federico L. G. Faroldi & Frederik Van De Putte (eds.), Outstanding Contributions to Logic: Kit Fine. Springer.details
Fine (2017a) sets out a theory of content based on truthmaker semantics which distinguishes two kinds of consequence between contents. There is entailment, corresponding to the relationship between disjunct and disjunction, and there is containment, corresponding to the relationship between conjunctions and their conjuncts. Fine associates these with two notions of parthood: disjunctive and conjunctive. Conjunctive parthood is a very useful notion, allowing us to analyse partial content and partial truth. In this chapter, I extend the notion of disjunctive parthood (...)
Determinates and Determinables in Metaphysics

Grounding in Metaphysics

Relevance Logic in Logic and Philosophy of Logic

Truthmaker Semantics in Philosophy of Language

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La Physique Quantique,(Michel MEYER).E. Klein - forthcoming - Revue Internationale de Philosophie.details
Relevance Logic in Logic and Philosophy of Logic

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Depth Relevance and Hyperformalism.Shay Logan - forthcoming - Journal of Philosophical Logic.details
Formal symptoms of relevance usually concern the propositional variables shared between the antecedent and the consequent of provable conditionals. Among the most famous results about such symptoms are Belnap's early results showing that for sublogics of the strong relevant logic R, provable conditionals share a signed variable between antecedent and consequent. -/- For logics weaker than R stronger variable sharing results are available. In 1984, Ross Brady gave one well-known example of such a result. As a corollary to the main (...)
Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Deep Fried Logic.Shay Allen Logan - forthcoming - Erkenntnis:1-30.details
There is a natural story about what logic is that sees it as tied up with two operations: a ‘throw things into a bag’ operation and a ‘closure’ operation. In a pair of recent papers, Jc Beall has fleshed out the account of logic this leaves us with in more detail. Using Beall’s exposition as a guide, this paper points out some problems with taking the second operation to be closure in the usual sense. After pointing out these problems, I (...)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Pure Variable Inclusion Logics.Francesco Paoli, Michele Pra Baldi & Damian Szmuc - forthcoming - Logic and Logical Philosophy:1-22.details
The aim of this article is to discuss pure variable inclusion logics, that is, logical systems where valid entailments require that the propositional variables occurring in the conclusion are included among those appearing in the premises, or vice versa. We study the subsystems of Classical Logic satisfying these requirements and assess the extent to which it is possible to characterise them by means of a single logical matrix. In addition, we semantically describe both of these companions to Classical Logic in (...)
History of Logic in Logic and Philosophy of Logic

Logical Consequence and Entailment in Logic and Philosophy of Logic

Many-Valued Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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The Universal Theory Tool Building Toolkit Is Substructural.Logan Shay - forthcoming - In Ivo Duntsch & Edwin D. Mares (eds.), Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs. Springer, Cham.details
Consider the set of inferences that are acceptable to use in all our theory building endeavors. Call this set of inferences the universal theory building toolkit, or just ’the toolkit’ for short. It is clear that the toolkit is tightly connected to logic in a variety of ways. Beall, for example, has argued that logic just is the toolkit. This paper avoids making a stand on that issue and instead investigates reasons for thinking that, logic or not, the toolkit is (...)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Neighbourhood Semantics for Quantified Relevant Logics.Andrew Tedder & Nicholas Ferenz - forthcoming - Journal of Philosophical Logic:1-28.details
The Mares-Goldblatt semantics for quantified relevant logics have been developed for first-order extensions of R, and a range of other relevant logics and modal extensions thereof. All such work has taken place in the the ternary relation semantic framework, most famously developed by Sylvan and Meyer. In this paper, the Mares-Goldblatt technique for the interpretation of quantifiers is adapted to the more general neighbourhood semantic framework, developed by Sylvan, Meyer, and, more recently, Goble. This more algebraic semantics allows one to (...)
Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Revisiting Constructive Mingle: Algebraic and Operational Semantics.Yale Weiss - forthcoming - In Katalin Bimbo (ed.), Essays in Honor of J. Michael Dunn. London: College Publications.details
Among Dunn’s many important contributions to relevance logic was his work on the system RM (R-mingle). Although RM is an interesting system in its own right, it is widely considered to be too strong. In this chapter, I revisit a closely related system, RM0 (sometimes known as ‘constructive mingle’), which includes the mingle axiom while not degenerating in the way that RM itself does. My main interest will be in examining this logic from two related semantical perspectives. First, I give (...)
Intuitionistic Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Confused Entailment.Tore Fjetland Øgaard - forthcoming - Topoi:1-13.details
Priest argued in Fusion and Confusion (Priest in Topoi 34(1):55–61, 2015a) for a new concept of logical consequence over the relevant logic B, one where premises my be “confused” together. This paper develops Priest’s idea. Whereas Priest uses a substructural proof calculus, this paper provides a Hilbert proof calculus for it. Using this it is shown that Priest’s consequence relation is weaker than the standard Hilbert consequence relation for B, but strictly stronger than Anderson and Belnap’s original relevant notion of (...)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

Value Theory

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Relevance and Verification.Ben Blumson - 2021 - Philosophical Quarterly 71 (3):457-480.details
A. J. Ayer’s empiricist criterion of meaning was supposed to have sorted all statements into nonsense on the one hand, and tautologies or genuinely factual statements on the other. Unfortunately for Ayer, it follows from classical logic that his criterion is trivial—it classifies all statements as either tautologies or genuinely factual, but none as nonsense. However, in this paper, I argue that Ayer’s criterion of meaning can be defended from classical proofs of its triviality by the adoption of a relevant (...)
A. J. Ayer in 20th Century Philosophy

Relevance Logic in Logic and Philosophy of Logic

Verificationist Theories of Meaning in Philosophy of Language

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Variable Sharing in Connexive Logic.Luis Estrada-González & Claudia Lucía Tanús-Pimentel - 2021 - Journal of Philosophical Logic 50 (6):1377-1388.details
However broad or vague the notion of connexivity may be, it seems to be similar to the notion of relevance even when relevance and connexive logics have been shown to be incompatible to one another. Relevance logics can be examined by suggesting syntactic relevance principles and inspecting if the theorems of a logic abide to them. In this paper we want to suggest that a similar strategy can be employed with connexive logics. To do so, we will suggest some properties (...)
Logical Connectives in Logic and Philosophy of Logic

Logics, Misc in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Two Pretabular Linear Extensions of Relevance Logic R.Asadollah Fallahi - 2021 - Journal of Applied Non-Classical Logics 31 (2):154-179.details
Pretabularity is the attribute of logics that are not characterised by finite matrices, but all of whose proper extensions are. Two of the first-known pretabular logics were Dummett’s famous super-...
Logics, Misc in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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A Cut-Elimination Proof in Positive Relevant Logic with Necessity.Mirjana Ilić - 2021 - Studia Logica 109 (3):607-638.details
This paper presents a sequent calculus for the positive relevant logic with necessity and a proof that it admits the elimination of cut.
Modal and Intensional Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Strong Depth Relevance.Shay Logan - 2021 - Australasian Journal of Logic 18 (6):645-656.details
Relevant logics infamously have the property that they only validate a conditional when some propositional variable is shared between its antecedent and consequent. This property has been strengthened in a variety of ways over the last half-century. Two of the more famous of these strengthenings are the strong variable sharing property and the depth relevance property. In this paper I demonstrate that an appropriate class of relevant logics has a property that might naturally be characterized as the supremum of these (...)
Intuitionistic Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Hyperdoctrines and the Ontology of Stratified Semantics.Shay Logan - 2021 - In Davide Fazio, Antonio Ledda & Francesco Paoli (eds.), Algebraic Perspectives on Substructural Logics. Springer International Publishing. pp. 169-193.details
I present a version of Kit Fine's stratified semantics for the logic RWQ and define a natural family of related structures called RW hyperdoctrines. After proving that RWQ is sound with respect to RW hyperdoctrines, we show how to construct, for each stratified model, a hyperdoctrine that verifies precisely the same sentences. Completeness of RWQ for hyperdoctrinal semantics then follows from completeness for stratified semantics, which is proved in an appendix. By examining the base category of RW hyperdoctrines, we find (...)
Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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On Not Saying What We Shouldn't Have to Say.Shay Logan & Leach-Krouse Graham - 2021 - Australasian Journal of Logic 18 (5):524-568.details
In this paper we introduce a novel way of building arithmetics whose background logic is R. The purpose of doing this is to point in the direction of a novel family of systems that could be candidates for being the infamous R#1/2 that Meyer suggested we look for.
Number Theory in Philosophy of Mathematics

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Basic Quasi-Boolean Expansions of Relevance Logics.Gemma Robles & José M. Méndez - 2021 - Journal of Philosophical Logic 50 (4):727-754.details
The basic quasi-Boolean negation expansions of relevance logics included in Anderson and Belnap’s relevance logic R are defined. We consider two types of QB-negation: H-negation and D-negation. The former one is of paraintuitionistic or superintuitionistic character, the latter one, of dual intuitionistic nature in some sense. Logics endowed with H-negation are paracomplete; logics with D-negation are paraconsistent. All logics defined in the paper are given a Routley-Meyer ternary relational semantics.
Intuitionistic Logic in Logic and Philosophy of Logic

Logical Pluralism in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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An Incompleteness Theorem for Modal Relevant Logics.Shawn Standefer - 2021 - Notre Dame Journal of Formal Logic 62 (4):669 - 681.details
In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.
Intensional Modal Logic in Logic and Philosophy of Logic

Modal Logic in Logic and Philosophy of Logic

Nonclassical Logic, Misc in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Semantics for Modal Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Identity in Mares-Goldblatt Models for Quantified Relevant Logic.Shawn Standefer - 2021 - Journal of Philosophical Logic 50 (6):1389-1415.details
Mares and Goldblatt, 163–187, 2006) provided an alternative frame semantics for two quantified extensions of the relevant logic R. In this paper, I show how to extend the Mares-Goldblatt frames to accommodate identity. Simpler frames are provided for two zero-order logics en route to the full logic in order to clarify what is needed for identity and substitution, as opposed to quantification. I close with a comparison of this work with the Fine-Mares models for relevant logics with identity and a (...)
Logical Connectives in Logic and Philosophy of Logic

Quantifiers in Philosophy of Language

Relevance Logic in Logic and Philosophy of Logic

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A Simple Logical Matrix and Sequent Calculus for Parry’s Logic of Analytic Implication.Damian E. Szmuc - 2021 - Studia Logica 109 (4):791-828.details
We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints.
History of Logic in Logic and Philosophy of Logic

Logical Consequence and Entailment in Logic and Philosophy of Logic

Many-Valued Logic in Logic and Philosophy of Logic

Proof Theory in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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A Characteristic Frame for Positive Intuitionistic and Relevance Logic.Yale Weiss - 2021 - Studia Logica 109 (4):687-699.details
I show that the lattice of the positive integers ordered by division is characteristic for Urquhart’s positive semilattice relevance logic; that is, a formula is valid in positive semilattice relevance logic if and only if it is valid in all models over the positive integers ordered by division. I show that the same frame is characteristic for positive intuitionistic logic, where the class of models over it is restricted to those satisfying a heredity condition. The results of this article highlight (...)
Intuitionistic Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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A Reinterpretation of the Semilattice Semantics with Applications.Yale Weiss - 2021 - Logica Universalis 15 (2):171-191.details
In the early 1970s, Alasdair Urquhart proposed a semilattice semantics for relevance logic which he provided with an influential informational interpretation. In this article, I propose a BHK-inspired reinterpretation of the semantics which is related to Kit Fine’s truthmaker semantics. I discuss and compare Urquhart’s and Fine’s semantics and show how simple modifications of Urquhart’s semantics can be used to characterize both full propositional intuitionistic logic and Jankov’s logic. I then present (quasi-)relevant companions for both of these systems. Finally, I (...)
Intuitionistic Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Truthmaker Semantics in Philosophy of Language

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Non-Boolean Classical Relevant Logics II: Classicality Through Truth-Constants.Tore Fjetland Øgaard - 2021 - Synthese (3-4):1-33.details
This paper gives an account of Anderson and Belnap’s selection criteria for an adequate theory of entailment. The criteria are grouped into three categories: criteria pertaining to modality, those pertaining to relevance, and those related to expressive strength. The leitmotif of both this paper and its prequel is the relevant legitimacy of disjunctive syllogism. Relevant logics are commonly held to be paraconsistent logics. It is shown in this paper, however, that both E and R can be extended to explosive logics (...)
Epistemology of Logic in Logic and Philosophy of Logic

Logical Consequence and Entailment in Logic and Philosophy of Logic

Logical Pluralism in Logic and Philosophy of Logic

Modal Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Boolean Negation and Non-Conservativity III: The Ackermann Constant.Tore Fjetland Øgaard - 2021 - Logic Journal of the IGPL 29 (3):370-384.details
It is known that many relevant logics can be conservatively extended by the truth constant known as the Ackermann constant. It is also known that many relevant logics can be conservatively extended by Boolean negation. This essay, however, shows that a range of relevant logics with the Ackermann constant cannot be conservatively extended by a Boolean negation.
Logical Consequence and Entailment in Logic and Philosophy of Logic

Logical Constants in Logic and Philosophy of Logic

Negation in Philosophy of Language

Relevance Logic in Logic and Philosophy of Logic

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Boolean Negation and Non-Conservativity I: Relevant Modal Logics.Tore Fjetland Øgaard - 2021 - Logic Journal of the IGPL 29 (3):340-362.details
Many relevant logics can be conservatively extended by Boolean negation. Mares showed, however, that E is a notable exception. Mares’ proof is by and large a rather involved model-theoretic one. This paper presents a much easier proof-theoretic proof which not only covers E but also generalizes so as to also cover relevant logics with a primitive modal operator added. It is shown that from even very weak relevant logics augmented by a weak K-ish modal operator, and up to the strong (...)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Logical Constants in Logic and Philosophy of Logic

Modal and Intensional Logic in Logic and Philosophy of Logic

Negation in Philosophy of Language

Relevance Logic in Logic and Philosophy of Logic

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Boolean Negation and Non-Conservativity II: The Variable-Sharing Property.Tore Fjetland Øgaard - 2021 - Logic Journal of the IGPL 29 (3):363-369.details
Many relevant logics are conservatively extended by Boolean negation. Not all, however. This paper shows an acute form of non-conservativeness, namely that the Boolean-free fragment of the Boolean extension of a relevant logic need not always satisfy the variable-sharing property. In fact, it is shown that such an extension can in fact yield classical logic. For a vast range of relevant logic, however, it is shown that the variable-sharing property, restricted to the Boolean-free fragment, still holds for the Boolean extended (...) logic. (shrink)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Logical Constants in Logic and Philosophy of Logic

Negation in Philosophy of Language

Relevance Logic in Logic and Philosophy of Logic

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From Hilbert Proofs to Consecutions and Back.Tore Fjetland Øgaard - 2021 - Australasian Journal of Logic 18 (2):51-72.details
Restall set forth a "consecution" calculus in his "An Introduction to Substructural Logics." This is a natural deduction type sequent calculus where the structural rules play an important role. This paper looks at different ways of extending Restall's calculus. It is shown that Restall's weak soundness and completeness result with regards to a Hilbert calculus can be extended to a strong one so as to encompass what Restall calls proofs from assumptions. It is also shown how to extend the calculus (...)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Proof Theory in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Relevance for the Classical Logician.Ethan Brauer - 2020 - Review of Symbolic Logic 13 (2):436-457.details
Although much technical and philosophical attention has been given to relevance logics, the notion of relevance itself is generally left at an intuitive level. It is difficult to find in the literature an explicit account of relevance in formal reasoning. In this article I offer a formal explication of the notion of relevance in deductive logic and argue that this notion has an interesting place in the study of classical logic. The main idea is that a premise is relevant to (...)
Relevance Logic in Logic and Philosophy of Logic

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Truthmaker Semantics for Relevant Logic.Mark Jago - 2020 - Journal of Philosophical Logic 49 (4):681-702.details
I develop and defend a truthmaker semantics for the relevant logic R. The approach begins with a simple philosophical idea and develops it in various directions, so as to build a technically adequate relevant semantics. The central philosophical idea is that truths are true in virtue of specific states. Developing the idea formally results in a semantics on which truthmakers are relevant to what they make true. A very natural notion of conditionality is added, giving us relevant implication. I then (...)
Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Truthmaker Semantics in Philosophy of Language

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Putting the Stars in Their Places.Shay Allen Logan - 2020 - Thought: A Journal of Philosophy 9 (3):188-197.details
This paper presents a new semantics for the weak relevant logic DW that makes the role of the infamous Routley star more explicable. Central to this rewriting is combining aspects of both the American and Australian plan for understanding negations in relevance logics.
Logical Consequence and Entailment in Logic and Philosophy of Logic

Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Actual Issues for Relevant Logics.Shawn Standefer - 2020 - Ergo: An Open Access Journal of Philosophy 7.details
In this paper, I motivate the addition of an actuality operator to relevant logics. Straightforward ways of doing this are in tension with standard motivations for relevant logics, but I show how to add the operator in a way that permits one to maintain the intuitions behind relevant logics. I close by exploring some of the philosophical consequences of the addition.
Relevance Logic in Logic and Philosophy of Logic

Two-Dimensional Semantics in Philosophy of Language

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Cut and Gamma I: Propositional and Constant Domain R.Yale Weiss - 2020 - Review of Symbolic Logic 13 (4):887-909.details
The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain quantifiers). The status of (...)
Proof Theory in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Farewell to Suppression-Freedom.Tore Fjetland Øgaard - 2020 - Logica Universalis 14 (3):297-330.details
Val Plumwood and Richard Sylvan argued from their joint paper The Semantics of First Degree Entailment and onward that the variable sharing property is but a mere consequence of a good entailment relation, indeed they viewed it as a mere negative test of adequacy of such a relation, the property itself being a rather philosophically barren concept. Such a relation is rather to be analyzed as a sufficiency relation free of any form of premise suppression. Suppression of premises, therefore, gained (...)
20th Century Logic in 20th Century Philosophy

History of Logic in Logic and Philosophy of Logic

Logical Consequence and Entailment in Logic and Philosophy of Logic

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Mathematical Explanation by Law.Sam Baron - 2019 - British Journal for the Philosophy of Science 70 (3):683-717.details
Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...)
Deductive-Nomological Explanation in General Philosophy of Science

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Negation on the Australian Plan.Francesco Berto & Greg Restall - 2019 - Journal of Philosophical Logic 48 (6):1119-1144.details
We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. (...)
Many-Valued Logic in Logic and Philosophy of Logic

Modal Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Stoic Sequent Logic and Proof Theory.Susanne Bobzien - 2019 - History and Philosophy of Logic 40 (3):234-265.details
This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic (...)
Ancient Greek and Roman Logic in Ancient Greek and Roman Philosophy

Proof Theory in Logic and Philosophy of Logic

Propositional Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Notes on Stratified Semantics.Shay Logan - 2019 - Journal of Philosophical Logic 48 (4):749-786.details
In 1988, Kit Fine published a semantic theory for quantified relevant logics. He referred to this theory as stratified semantics. While it has received some attention in the literature, 1–20, 1992; Mares & Goldblatt, Journal of Symbolic Logic 71, 163–187, 2006), stratified semantics has overall received much less attention than it deserves. There are two plausible reasons for this. First, the only two dedicated treatments of stratified semantics available are, 27–59, 1988; Mares, Studia Logica 51, 1–20, 1992), both of which (...)
Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Tracking Reasons with Extensions of Relevant Logics.Shawn Standefer - 2019 - Logic Journal of the IGPL 27 (4):543-569.details
In relevant logics, necessary truths need not imply each other. In justification logic, necessary truths need not all be justified by the same reason. There is an affinity to these two approaches that suggests their pairing will provide good logics for tracking reasons in a fine-grained way. In this paper, I will show how to extend relevant logics with some of the basic operators of justification logic in order to track justifications or reasons. I will define and study three kinds (...)
Modal Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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A Note on the Relevance of Semilattice Relevance Logic.Yale Weiss - 2019 - Australasian Journal of Logic 16 (6):177-185.details
A propositional logic has the variable sharing property if φ → ψ is a theorem only if φ and ψ share some propositional variable. In this note, I prove that positive semilattice relevance logic and its extension with an involution negation have the variable sharing property. Typical proofs of the variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing property for (...) the systems discussed. (shrink)
Relevance Logic in Logic and Philosophy of Logic

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Substitution in Relevant Logics.Tore Fjetland Øgaard - 2019 - Review of Symbolic Logic (3):1-26.details
This essay discusses rules and semantic clauses relating to Substitution—Leibniz’s law in the conjunctive-implicational form s=t ∧ A(s) → A(t)—as these are put forward in Priest’s books "In Contradiction" and "An Introduction to Non-Classical Logic: From If to Is." The stated rules and clauses are shown to be too weak in some cases and too strong in others. New ones are presented and shown to be correct. Justification for the various rules are probed and it is argued that Substitution ought (...) to fail. (shrink)
Relevance Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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On Elimination of Quantifiers in Some Non-Classical Mathematical Theories.Guillermo Badia & Andrew Tedder - 2018 - Mathematical Logic Quarterly 64 (3):140-154.details
Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.
Model Theory in Logic and Philosophy of Logic

Nonclassical Logic, Misc in Logic and Philosophy of Logic

Quantifiers, Misc in Philosophy of Language

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Relevance in Structured Argumentation.AnneMarie Borg & Christian Straßer - 2018 - In Jérôme Lang (ed.), Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-18).details
Logical Consequence and Entailment in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Relevant Logics Obeying Component Homogeneity.Roberto Ciuni, Damian Szmuc & Thomas Macaulay Ferguson - 2018 - Australasian Journal of Logic 15 (2):301-361.details
This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi (...)
Many-Valued Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Sylvan's Jungle Volume 1: Exploring Meinong's Jungle and Beyond.Maureen Eckert - 2018 - International: Synthese Library.details
In this first volume of The Sylvan Jungle, the editors present a scholarly edition of the first chapter, "Exploring Meinong's Jungle," of Richard Routley's 1000-plus page book, Exploring Meinong's Jungle and Beyond. Going against the Quinean orthodoxy, Routley’s aim was to support Meinong’s idea that we can truthfully refer to non-existent and even impossible objects, like Superman, unicorns and the (infamous) round-square cupola on Berkeley College. The tools of non-classical logic at Routley’s disposal enabled him to update Meinong’s project for (...)
20th Century Logic in 20th Century Philosophy

History of Logic in Logic and Philosophy of Logic

Nonexistent Objects in Metaphysics

Paraconsistent Logic in Logic and Philosophy of Logic

Philosophical Traditions

Relevance Logic in Logic and Philosophy of Logic

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Reflections on Routley's Ultralogic Program.Daniel Nolan - 2018 - Australasian Journal of Logic 15 (2):407-430.details
In this paper, I take up three tasks in turn. The first is to set out what Routley thought we should demand of an all-purpose universal logic, and some of his reasons for those demands. The second is to sketch Routley's own response to those demands. The third is to explore how else we could satisfy some of the theoretical demands Routley identified, if we are not to follow him in endorsing Routleyan Ultralogic as a foundational logic. As part of (...)
Logic and Philosophy of Logic, Misc in Logic and Philosophy of Logic

Logic in Philosophy in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Science, Logic, and Mathematics

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Logics for Modelling Collective Attitudes.Daniele Porello - 2018 - Fundamenta Infromaticae 158 (1-3):239-27.details
We introduce a number of logics to reason about collective propositional attitudes that are defined by means of the majority rule. It is well known that majoritarian aggregation is subject to irrationality, as the results in social choice theory and judgment aggregation show. The proposed logics for modelling collective attitudes are based on a substructural propositional logic that allows for circumventing inconsistent outcomes. Individual and collective propositional attitudes, such as beliefs, desires, obligations, are then modelled by means of minimal modalities (...)
Attitude Ascriptions in Philosophy of Language

Judgment Aggregation in Social and Political Philosophy

Logic and Information in Philosophy of Computing and Information

Ontology, Misc in Metaphysics

Relevance Logic in Logic and Philosophy of Logic

Social Choice Theory, Misc in Social and Political Philosophy

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