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Journal articles on the topic 'Relevant logic'

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Published: 4 June 2021

Last updated: 25 January 2023

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1

Standefer, Shawn. "Tracking reasons with extensions of relevant logics." Logic Journal of the IGPL 27, no. 4 (June 25, 2019): 543–69. http://dx.doi.org/10.1093/jigpal/jzz018.

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Abstract 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 of frames for these logics. For the first kind of frame, I show soundness and highlight a difficulty in proving completeness. This motivates two alternative kinds of frames, with respect to which completeness results are obtained. Axioms to strengthen the justification logic portions of these logics are considered. I close by developing an analogy between the dot operator of justification logic and theory fusion in relevant logics.

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2

Ciuni, Roberto, Damian Szmuc, and Thomas Macaulay Ferguson. "Relevant Logics Obeying Component Homogeneity." Australasian Journal of Logic 15, no. 2 (July 4, 2018): 301. http://dx.doi.org/10.26686/ajl.v15i2.4864.

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This paper discusses three relevant logics (S*fde , dS*fde , crossS*fde) 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 for S*fde , dS*fde , crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define (a given family of) containment logics, we explore the single-premise/single-conclusion fragment of S*fde , dS*fde , crossS*fde and the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues.

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3

FRANCEZ, NISSIM. "BILATERAL RELEVANT LOGIC." Review of Symbolic Logic 7, no. 2 (April 16, 2014): 250–72. http://dx.doi.org/10.1017/s1755020314000082.

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4

Bollen, A. W. "Relevant logic programming." Journal of Automated Reasoning 7, no. 4 (December 1991): 563–85. http://dx.doi.org/10.1007/bf01880329.

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5

Strong, Gary W. "Phase logic is biologically relevant logic." Behavioral and Brain Sciences 16, no. 3 (September 1993): 472–73. http://dx.doi.org/10.1017/s0140525x00031174.

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6

Swirydowicz, Kazimierz. "There exist exactly two maximal strictly relevant extensions of the relevant logic R." Journal of Symbolic Logic 64, no. 3 (September 1999): 1125–54. http://dx.doi.org/10.2307/2586622.

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AbstractIn [60] N. Belnap presented an 8-element matrix for the relevant logic R with the following property: if in an implication A → B the formulas A and B do not have a common variable then there exists a valuation v such that v (A → B) does not belong to the set of designated elements of this matrix. A 6-element matrix of this kind can be found in: R. Routley, R.K. Meyer, V. Plumwood and R.T. Brady [82], Below we prove that the logics generated by these two matrices are the only maximal extensions of the relevant logic R which have the relevance property: if A → B is provable in such a logic then A and B have a common propositional variable.

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7

Orlowska, Ewa. "Relational proof system for relevant logics." Journal of Symbolic Logic 57, no. 4 (December 1992): 1425–40. http://dx.doi.org/10.2307/2275375.

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AbstractA method is presented for constructing natural deduction-style systems for propositional relevant logics. The method consists in first translating formulas of relevant logics into ternary relations, and then defining deduction rules for a corresponding logic of ternary relations. Proof systems of that form are given for various relevant logics. A class of algebras of ternary relations is introduced that provides a relation-algebraic semantics for relevant logics.

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8

Garson, James. "Modularity and relevant logic." Notre Dame Journal of Formal Logic 30, no. 2 (March 1989): 207–23. http://dx.doi.org/10.1305/ndjfl/1093635079.

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9

Brady, Ross T. "Gentzenizations of relevant logics with distribution." Journal of Symbolic Logic 61, no. 2 (June 1996): 402–20. http://dx.doi.org/10.2307/2275668.

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We establish cut-free left-handed Gentzenizations for a range of major relevant logics from B through to R, all with distribution. B is the basic system of the Routley-Meyer semantics (see [15], pp. 287–300) and R is the logic of relevant implication (see [1], p. 341). Previously, the contractionless logics DW, TW, EW, RW and RWK were Gentzenized in [3], [4] and [5], and also the distributionless logics LBQ, LDWQ, LTWQo, LEWQot, LRWQ, LRWKQ and LRQ in [6] and [7]. This paper provides Gentzenizations for the logics DJ, TJ, T and R, with various levels of contraction, and for the contractionless logic B, which could not be included in [4] using the technique developed there. We also include the Gentzenization of TW in order to compare it with that in [4]. The Gentzenizations that we obtain here for DW and RW are inferior to those already obtained in [4], but they are included for reference when constructing other systems. The logics EW and E present a difficulty for our method and are omitted. For background to the Gentzenization of relevant logics, see [6], and for motivation behind the logics involved, see [6], [1] and [15]. Because of the number of properties that are brought to bear in obtaining these systems, we prefer to consider Gentzenizations for particular logics rather than for arbitrary bunches of axioms.

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10

Zaverucha, Gerson. "Relevant logic as a basis for paraconsistent epistemic logics." Journal of Applied Non-Classical Logics 2, no. 2 (January 1992): 225–41. http://dx.doi.org/10.1080/11663081.1992.10510783.

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11

Verdée, Peter, Inge De Bal, and Aleksandra Samonek. "A non-transitive relevant implication corresponding to classical logic consequence." Australasian Journal of Logic 16, no. 2 (February 3, 2019): 10. http://dx.doi.org/10.26686/ajl.v16i2.5273.

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In this paper we first develop a logic independent account of relevant implication. We propose a stipulative denition of what it means for a multiset of premises to relevantly L-imply a multiset of conclusions, where L is a Tarskian consequence relation: the premises relevantly imply the conclusions iff there is an abstraction of the pair <premises, conclusions> such that the abstracted premises L-imply the abstracted conclusions and none of the abstracted premises or the abstracted conclusions can be omitted while still maintaining valid L-consequence. Subsequently we apply this denition to the classical logic (CL) consequence relation to obtain NTR-consequence, i.e. the relevant CL-consequence relation in our sense, and develop a sequent calculus that is sound and complete with regard to relevant CL-consequence. We present a sound and complete sequent calculus for NTR. In a next step we add rules for an object language relevant implication to thesequent calculus. The object language implication reflects exactly the NTR-consequence relation. One can see the resulting logic NTR-> as a relevant logic in the traditional sense of the word. By means of a translation to the relevant logic R, we show that the presented logic NTR is very close to relevance logics in the Anderson-Belnap-Dunn-Routley-Meyer tradition. However, unlike usual relevant logics, NTR is decidable for the full language, Disjunctive Syllogism (A and ~AvB relevantly imply B) and Adjunction (A and B relevantly imply A&B) are valid, and neither Modus Ponens nor the Cut rule are admissible.

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12

Jago, M. "Recent Work in Relevant Logic." Analysis 73, no. 3 (July 1, 2013): 526–41. http://dx.doi.org/10.1093/analys/ant043.

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13

SHRAMKO, YAROSLAV V. "Relevant Variants of Intuitionistic Logic." Logic Journal of IGPL 2, no. 1 (1994): 47–53. http://dx.doi.org/10.1093/jigpal/2.1.47.

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14

Goble, Lou. "An Incomplete Relevant Modal Logic." Journal of Philosophical Logic 29, no. 1 (February 2000): 103–19. http://dx.doi.org/10.1023/a:1004774422224.

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15

Jago, Mark. "Truthmaker Semantics for Relevant Logic." Journal of Philosophical Logic 49, no. 4 (January 7, 2020): 681–702. http://dx.doi.org/10.1007/s10992-019-09533-9.

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AbstractI 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 investigate ways to add conjunction, disjunction, and negation; and I discuss how to justify contraposition and excluded middle within a truthmaker semantics.

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16

Punčochář, Vít. "A Relevant Logic of Questions." Journal of Philosophical Logic 49, no. 5 (January 21, 2020): 905–39. http://dx.doi.org/10.1007/s10992-019-09541-9.

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17

Mares, Edwin D. "General information in relevant logic." Synthese 167, no. 2 (October 22, 2008): 343–62. http://dx.doi.org/10.1007/s11229-008-9412-9.

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18

Brady, Ross T. "Normalized natural deduction systems for some relevant logics I: The logic DW." Journal of Symbolic Logic 71, no. 1 (March 2006): 35–66. http://dx.doi.org/10.2178/jsl/1140641162.

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Fitch-style natural deduction was first introduced into relevant logic by Anderson in [1960], for the sentential logic E of entailment and its quantincational extension EQ. This was extended by Anderson and Belnap to the sentential relevant logics R and T and some of their fragments in [ENT1], and further extended to a wide range of sentential and quantified relevant logics by Brady in [1984]. This was done by putting conditions on the elimination rules, →E, ~E, ⋁E and ∃E, pertaining to the set of dependent hypotheses for formulae used in the application of the rule. Each of these rules were subjected to the same condition, this condition varying from logic to logic. These conditions, together with the set of natural deduction rules, precisely determine the particular relevant logic. Generally, this is a simpler representation of a relevant logic than the standard Routley-Meyer semantics, with its existential modelling conditions stated in terms of an otherwise arbitrary 3-place relation R, which is defined over a possibly infinite set of worlds. Readers are urged to refer to Brady [1984], if unfamiliar with the above natural deduction systems, but we will introduce in §2 a modified version in full detail.Natural deduction for classical logic was invented by Jaskowski and Gentzen, but it was Prawitz in [1965] who normalized natural deduction, streamlining its rules so as to allow a subformula property to be proved. (This key property ensures that each formula in the proof of a theorem is indeed a subformula of that theorem.)

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19

Yoshimitsu, Akihiro. "Anomalies of Classical Logic in View of Relevant Logic." Kagaku tetsugaku 45, no. 2 (2012): 65–81. http://dx.doi.org/10.4216/jpssj.45.2-65.

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20

Vidal-Rosset, Joseph. "Why Intuitionistic Relevant Logic Cannot Be a Core Logic." Notre Dame Journal of Formal Logic 58, no. 2 (2017): 241–48. http://dx.doi.org/10.1215/00294527-3839326.

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21

Brady, Ross T. "Gentzenizations of relevant logics without distribution. I." Journal of Symbolic Logic 61, no. 2 (June 1996): 353–78. http://dx.doi.org/10.2307/2275666.

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The history of the Gentzenization of relevant logics goes back to Kripke [17], who in 1959 Gentzenized R→ and went on to prove its decidability. Formulae were separated by commas on the left side of the turnstile, the commas just representing nested implications. Kripke employed just a singleton formula to the right of the turnstile. He also considered adding negation, as well as other connectives, but it was not until 1961 that Belnap and Wallace, in [5], Gentzenized and proved its decidability, though their Gentzenization employed commas on both sides of the turnstile. Subsequently, in 1966, the logic R without distribution, now called LR (for lattice R), was Gentzenized in a similar style by Meyer in [20]. He also went on to show decidability for LR by extending Kripke's argument. Later, in 1969, Dunn Gentzenized R+ (published in [1], pp. 381–391) using two structural connectives (commas and semicolons) to the left of the turnstile, and with a single formula to the right. Here, the commas represent conjunction and the semicolons represent an intensional conjunction, called “fusion”. This is all nicely set out in McRobbie [19], where he also introduces left-handed Gentzenizations and analytic tableaux for a number of fragments of relevant logics. In 1979, further work on distributionless logic was done by Grishin, in a series of papers, including [16], in which he produced a Gentzenization of quantified RW without distribution (which we will call LRWQ), and used it to prove the decidability of this quantified logic.

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22

Schurz, Gerhard. "Relevant deduction." Erkenntnis 35, no. 1-3 (July 1991): 391–437. http://dx.doi.org/10.1007/bf00388295.

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23

Avron, Arnon. "On purely relevant logics." Notre Dame Journal of Formal Logic 27, no. 2 (April 1986): 180–94. http://dx.doi.org/10.1305/ndjfl/1093636610.

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24

Kashima, Ryo. "On semilattice relevant logics." MLQ 49, no. 4 (July 2003): 401–14. http://dx.doi.org/10.1002/malq.200310043.

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25

BADIA, GUILLERMO. "INFINITARY PROPOSITIONAL RELEVANT LANGUAGES WITH ABSURDITY." Review of Symbolic Logic 10, no. 4 (August 1, 2017): 663–81. http://dx.doi.org/10.1017/s1755020317000132.

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AbstractAnalogues of Scott’s isomorphism theorem, Karp’s theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An “interpolation theorem” (of a particular sort introduced by Barwise and van Benthem) for the infinitary quantificational boolean logic L∞ω holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definability property.

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26

Васюков, В. Л. "Game Theoretical Semantic for Relevant Logic." Logical Investigations 21, no. 2 (September 28, 2015): 42–52. http://dx.doi.org/10.21146/2074-1472-2015-21-2-42-52.

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In 1979 D.E. Over proposed game theoretical semantics for first-degree entailment formulated by Anderson and Belnap. In order to extend this approach to include other systems of relevant logc (e.g., $\boldsymbol{R}$) we have two promoting facts. Firstly, there is Routley- Meyer’s situational semantic for system$\boldsymbol{R}$ of relevant logic. Secondly, this semantics shows some resemblance with W__ojcicki’s situational semantic of non-fregean logic for which the situational game semantics was developed by author exploiting essentially the notion of non-fregean games. In the paper an attempt is done to give a partial account of these results and some conception of situational games developed which laid down into foundation of the game theoretical semantics of relevant logic $\boldsymbol{R}$.

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27

Milne, P. "Intuitionistic relevant logic and perfect validity." Analysis 54, no. 3 (July 1, 1994): 140–42. http://dx.doi.org/10.1093/analys/54.3.140.

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28

Badia, Guillermo. "On Sahlqvist Formulas in Relevant Logic." Journal of Philosophical Logic 47, no. 4 (August 22, 2017): 673–91. http://dx.doi.org/10.1007/s10992-017-9445-y.

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29

Došen, Kosta. "The first axiomatization of relevant logic." Journal of Philosophical Logic 21, no. 4 (November 1992): 339–56. http://dx.doi.org/10.1007/bf00260740.

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30

Meyer, Robert. "Relevant Arithmetic." Australasian Journal of Logic 18, no. 5 (July 21, 2021): 150–53. http://dx.doi.org/10.26686/ajl.v18i5.6904.

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This is a republication of R.K. Meyer's "Relevant Arithmetic", which originally appeared in the Bulletin of the Section of Logic 5 (1976). It sets out the problems that Meyer was to work on for the next decade concerning his system, R#.

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31

Meyer, Robert K. "Ternary relations and relevant semantics." Annals of Pure and Applied Logic 127, no. 1-3 (June 2004): 195–217. http://dx.doi.org/10.1016/j.apal.2003.11.015.

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32

Meyer, Robert K., and Igor Urbas. "Conservative Extension in Relevant Arithmetic." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 32, no. 1-5 (1986): 45–50. http://dx.doi.org/10.1002/malq.19860320106.

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33

Bimbo, Katalin, Jon Michael Dunn, and Nicholas Ferenz. "Two Manuscripts, One by Routley, One by Meyer: The Origins of the Routley-Meyer Semantics for Relevance Logics." Australasian Journal of Logic 15, no. 2 (July 4, 2018): 171. http://dx.doi.org/10.26686/ajl.v15i2.4066.

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A ternary relation is often used nowadays to interpret an implication connective of a logic, a practice that became dominant in the semantics of relevance logics. This paper examines two early manuscripts --- one by Routley, another by Meyer --- in which they were developing set-theoretic semantics for various relevance logics. A standard presentation of a ternary relational semantics for, let us say, the logic of relevant implication R is quite illuminating, yet the invention of this semantics was fraught with false starts. Meyer's manuscript, in which he builds on some ideas from Routley's manuscript, essentially contains a relational semantics for which R^{ot} is sound and complete.

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34

Brady, Ross T. "Gentzenizations of relevant logics without distribution. II." Journal of Symbolic Logic 61, no. 2 (June 1996): 379–401. http://dx.doi.org/10.2307/2275667.

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In Part I, we produced a Gentzenization L6LBQ of the distributionless relevant logic LBQ, which contained just the one structural connective ‘:’ and no structural rules. We compared it with the corresponding “right-handed” system and then proved interpolability and decidability of LBQ. Knowledge of Part I is presupposed.In Part II of this paper, we will establish Gentzenizations, with appropriate interpolation and decidability results, for the further distributionless logics LDWQ, LTWQo, LEWQot, LRWQ, LRWKQ and LRQ, using essentially the same methods as were used for LBQ in Part I. LRWQ has been Gentzenized by Grishin [2], but the interpolation result is new and the decidability result is proved by a substantial simplification of his method. LR has been Gentzenized and shown to be decidable by Meyer in [5], by extending a method of Kripke in [3], and McRobbie has proved interpolation for it in [4], but here the Gentzenization and interpolation results are extended to quantifiers.We axiomatize these logics as follows. The primitives and formation rules are as before, except that LTWQ and LEWQ require the extra primitive ‘o’, a 2-place connective (called ‘fusion’), and LEWQ also requires ‘t’, a sentential constant.

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35

BADIA, GUILLERMO. "THE RELEVANT FRAGMENT OF FIRST ORDER LOGIC." Review of Symbolic Logic 9, no. 1 (October 20, 2015): 143–66. http://dx.doi.org/10.1017/s1755020315000313.

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AbstractUnder a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulations. Furthermore, the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation property holds for the relevant fragment of first order logic.

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36

Seki, Takahiro. "General Frames for Relevant Modal Logics." Notre Dame Journal of Formal Logic 44, no. 2 (April 2003): 93–109. http://dx.doi.org/10.1305/ndjfl/1082637806.

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37

Slaney, John K., and Robert K. Meyer. "A structurally complete fragment of relevant logic." Notre Dame Journal of Formal Logic 33, no. 4 (September 1992): 561–66. http://dx.doi.org/10.1305/ndjfl/1093634487.

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38

Keene, G. B., and Stephen Read. "Relevant Logic: A Philosophical Examination of Inference." Philosophical Quarterly 40, no. 159 (April 1990): 259. http://dx.doi.org/10.2307/2219818.

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39

Mares, Edwin D., and Robert Goldblatt. "An alternative semantics for quantified relevant logic." Journal of Symbolic Logic 71, no. 1 (March 2006): 163–87. http://dx.doi.org/10.2178/jsl/1140641167.

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AbstractThe quantified relevant logic RQ is given a new semantics in which a formula ∀xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of ‘extensional confinement’: ∀x(A ⋁ B) → (A ⋁ ∀xB), with x not free in A. Validity of EC requires an additional model condition involving the boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation.

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40

Wavering, Michael James. "Piaget's Logic of Meanings: Still Relevant Today." School Science and Mathematics 111, no. 5 (May 2011): 249–52. http://dx.doi.org/10.1111/j.1949-8594.2011.00083.x.

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41

Mares, Edwin. "Relevant Logic and the Philosophy of Mathematics." Philosophy Compass 7, no. 7 (June 14, 2012): 481–94. http://dx.doi.org/10.1111/j.1747-9991.2012.00496.x.

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42

Brady, Ross T. "Rules in relevant logic - I: Semantic classification." Journal of Philosophical Logic 23, no. 2 (April 1994): 111–37. http://dx.doi.org/10.1007/bf01050340.

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43

Lycke, Hans. "An adaptive logic for relevant classical deduction." Journal of Applied Logic 5, no. 4 (December 2007): 602–12. http://dx.doi.org/10.1016/j.jal.2006.03.011.

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44

Brady, Ross T. "Rules in relevant logic ? II: Formula representation." Studia Logica 52, no. 4 (1993): 565–85. http://dx.doi.org/10.1007/bf01053260.

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45

Brink, Chris. "A comment on relevant truth table logic." Journal of Applied Non-Classical Logics 2, no. 2 (January 1992): 243–46. http://dx.doi.org/10.1080/11663081.1992.10510784.

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46

Leslie, Neil, and Edwin D. Mares. "CHR: A Constructive Relevant Natural-deduction Logic." Electronic Notes in Theoretical Computer Science 91 (February 2004): 158–70. http://dx.doi.org/10.1016/j.entcs.2003.12.011.

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47

Mares, Edwin D. "Relevant logic and the theory of information." Synthese 109, no. 3 (December 1996): 345–60. http://dx.doi.org/10.1007/bf00413865.

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48

Meyer, Robert. "Arithmetic Formulated Relevantly." Australasian Journal of Logic 18, no. 5 (July 21, 2021): 154–288. http://dx.doi.org/10.26686/ajl.v18i5.6905.

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The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that (1) it is trivial that relevant arithmetic is absolutely consistent, but (2) classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under (1), I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, of course, is exactly the formula whose unprovability was sought in the Hilbert program for proving arithmetic consistent. Under (2), I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment.

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49

Szmuc, Damián. "Inferentialism and Relevance." Análisis Filosófico 41, no. 2 (November 1, 2021): 317–36. http://dx.doi.org/10.36446/af.2021.458.

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This paper provides an inferentialist motivation for a logic belonging in the connexive family, by borrowing elements from the bilateralist interpretation for Classical Logic without the Cut rule, proposed by David Ripley. The paper focuses on the relation between inferentialism and relevance, through the exploration of what we call relevant assertion and denial, showing that a connexive system emerges as a symptom of this interesting link. With the present attempt we hope to broaden the available interpretations for connexive logics, showing they can be rightfully motivated in terms of certain relevantist constraints imposed on assertion and denial.

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50

Mortensen, Chris. "Remark on Relevant Arithmetic." Australasian Journal of Logic 18, no. 5 (July 21, 2021): 426–27. http://dx.doi.org/10.26686/ajl.v18i5.6915.

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Abstract:

This is a brief note about the history of the analysis of the collection of theories, RM3modn, in Meyer and Mortensen "Inconsistent Models for Relevant Arithmetics" Journal of Symbolic Logic 49 (1984).

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