This is a bibliography of R.K. Meyer's published articles on relevant arithmetic

Ferguson, Thomas Macaulay

Priest, Graham

English

Open Journal Systems at the Victoria University of Wellington Library

Australasian Journal of LogicRobert Meyer’s Publications onRelevant ArithmeticThomas Macaulay Ferguson and Graham PriestILLC, University of Amsterdam and Arché Research Centre, University of St. AndrewsThe Graduate Center, City University of New YorkAbstractThe bibliography appearing below collects the publications in whichMeyer’s investigations into relevant arithmetic saw print. Each bib-liographic item is accompanied by a short description of the text orother remarks. We include papers on relevant arithmetic coauthoredby Meyer, but omit both Meyer’s work on relevant logic and the workpublished independently by his collaborators.• R. K. Meyer. Relevant arithmetic. Bulletin of the Sectionof Logic, 5:133–137, 1976.◦ This abstract offers a précis of the results and motivations outlined inMeyer’s monographs Arithmetic Formulated Relevantly and The Consis-tency of Arithmetic. R] is formally defined and the relationships—both philo-sophical and formal—it bears to classical arithmetic are sketched out. Impor-tantly, several notable results—e.g., the Post and arithmetical consistenciesof R] and the admissibility of the γ rule in R]]—are first announced—althoughnot proven—in this abstract.◦ This work is reprinted in this issue.• R. K. Meyer. Arithmetic Formulated Relevantly. Unpub-lished monograph, 1976.◦ Along with The Consistency of Arithmetic, one of two extended treatmentson relevant arithmetic written by Meyer in 1976. Despite thematic overlapAustralasian Journal of Logic (18:5) 2021, Article no. 2147with its companion, this manuscript is gentler and more introductory in itstone, its case for R] made through illustrations of how relevant arithmeticcaptures mathematical intuitions more closely than classical PA. The rest ofthe text serves as a guidebook acquainting the reader with interesting resultsabout R], largely avoiding its companion’s bellicose tone.◦ This work is reprinted in this issue.• R. K. Meyer. The Consistency of Arithmetic. Unpublishedmonograph, 1976.◦ Like its companion Arithmetic Formulated Relevantly, this 1976 mono-graph is a rigorous philosophical investigation of the relevant arithmetic R].If the former work is a gentle invitation to R], this work enlists the Postconsistency of R]—along with the apparently finitary nature of its proof—todrive a sustained and vigorous series of arguments targeted at a purportedlyoverzealous reading of Gödel.◦ This work is reprinted in this issue.• R. K. Meyer and C. Mortensen. Inconsistent models for rel-evant arithmetics. Journal of Symbolic Logic, 49(3):917–929,1984.◦ This paper is notable for including the first appearance in print of a proofof a cornerstone of Meyer’s program: the Post consistency of R]. Notably,the paper demonstrates the utility of finite, inconsistent models of arithmeticas powerful tools for establishing results about relevant arithmetics.◦ This work is reprinted in this issue.•R. K. Meyer and I. Urbas. Conservative extensions in relevantarithmetic. Zeitschrift für mathematische Logic und Grundla-gen der Mathematik, 32:45–50, 1986.◦ This paper’s main result is that all positive theorems of R] have a proofin which no instances of axioms including negation ( e.g., excluded middle)occur. Defining positive subsystems of a range of other relevant arithmetic,the authors extend this conservativity result to arithmetics like E] (arithmeticunder the relevant logic E).• R. K. Meyer and C. Mortensen. Alien Intruders in Rele-vant Arithmetic. Technical Report TR-AR-9/87, AustralianNational University, 1987.Australasian Journal of Logic (18:5) 2021, Article no. 2148◦ Running with Dedekind’s description of nonstandard naturals as “alien in-truders,” this joint technical report examines the model theory of nonstandardelements in relevant arithmetic. The primary “Alien Intruder Theorem” isthat there exist models of R] in which every rational number plays the role ofa positive integer.◦ This work is reprinted in this issue.• H. Friedman and R. K. Meyer. Whither relevant arithmetic?Journal of Symbolic Logic, 57(3):824–831, 1992.◦ The central result of this joint paper—that γ is not admissible in R]—crucially affected the trajectory of Meyer’s program. The authors observethat the complex ring C serves as a model of R]. But as a countermodel to aPA theorem QRF , C witnesses that some theorems of PA are unprovable inR], which entails that γ fails for ⊃.• J. K. Slaney, R. K. Meyer, and G. Restall. Linear arithmeticdesecsed. Logique et Analyse, 39:379–388, 1996.◦ Meyer defined a secondary formula to be a ϕ for which both ϕ → 0 = 0and 0 6= 0 → ϕ are provable and showed that in R] every zeroth-degreeformula is a secondary formula, i.e., R] is secsed. This paper identifies aconnection between an arithmetic’s being secsed and validity of contractionin its underlying logic, which the authors examine in the context of the lineararithmetic LL].◦ This work also appears as ANU Technical Report TR-ARP-2/96.• R. K. Meyer. Kurt Gödel and the consistency of R]]. In P. Ha-jek, editor, Gödel ’96: Logical Foundations of Mathematics,Computer Science and Physics, pages 247–256. Cambridge Uni-versity Press, Cambridge, 1996.◦ This paper includes reworked selections from The Consistency of Arithmeticcritical of Hilbert and Gödel. A description of a finite model for R]] follows,leading to a discussion of the sense in which this model “finitarily provesthe consistency of R]].” Special attention is paid to a tension between this“finitary” proof and the nonconstructivity of Meyer’s proof of γ admissibilityin R]].• R. K. Meyer and G. Restall. ‘Strenge’ arithmetics. Logiqueet Analyse, 42:205–220, 1999.Australasian Journal of Logic (18:5) 2021, Article no. 2149◦ This paper tackles a host of topics related to modal notions in relevant arith-metic. Observing that there exist restricted senses in which R] avoids thetraditional fallacies of modality, a more traditional presentation of modalityis applied to E], on which the authors prove a number of interesting features,e.g., that E] authentically commits no fallacies of modality. Finally, apply-ing metavaluations to E]] is shown to yield a complete theory of arithmeticTE] which, the authors suggest, may have a stronger claim to being “truearithmetic” than R]].• R. K. Meyer. ⊃E is admissible in “true” relevant arithmetic.Journal of Philosophical Logic, 27(4):327–351, 2004.◦ After some interesting discussion defending the thesis that R]] character-izes true arithmetic, the paper provides a proof that γ is admissible in R]](in contrast to its inadmissibility in R]). The methods developed to prove theresult—although infinitary and highly nonconstructive—are independently in-teresting.• R. K. Meyer. Fallacies of division. In M.-H. Lee, editor,Proceedings of the XXII World Congress of Philosophy, vol-ume 13, pages 71–80. 2012.◦Meyer recalls his stock suggestion that 0 = 2 does not relevantly entail 0 = 1and considers an objection involving division. This leads to an argument thatsome classical theorems characterizing division are au fond elaborate casesof fallacies of relevance.Australasian Journal of Logic (18:5) 2021, Article no. 2

Robert Meyer's Publications on Relevant Arithmetic

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Robert Meyer's Publications on Relevant Arithmetic

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