Logic in Asia: Studia Logica Library Series Editors: Fenrong Liu · Hiroakira Ono · Kamal Lodaya Shier Ju Alessandra Palmigiano Minghui Ma Editors Nonclassical Logics and Their Applications Post-proceedings of the 8th International Workshop on Logic and Cognition Logic in Asia: Studia Logica Library Editors-in-Chief Fenrong Liu, Tsinghua University and University of Amsterdam, Beijing, China Hiroakira Ono, Japan Advanced Institute of Science and Technology (JAIST), Ishikawa, Japan Kamal Lodaya, Bengaluru, India Editorial Board Natasha Alechina, University of Nottingham, Nottingham, UK Toshiyasu Arai, Chiba University, Chiba Shi, Inage-ku, Japan Sergei Artemov, City University of New York, New York, NY, USA Mattias Baaz, Technical university of Vienna, Austria, Vietnam Lev Beklemishev, Institute of Russian Academy of Science, Russia Mihir Chakraborty, Jadavpur University, Kolkata, India Phan Minh Dung, Asian Institute of Technology, Thailand Amitabha Gupta, Indian Institute of Technology Bombay, Mumbai, India Christoph Harbsmeier, University of Oslo, Oslo, Norway Shier Ju, Sun Yat-sen University, Guangzhou, China Makoto Kanazawa, National Institute of Informatics, Tokyo, Japan Fangzhen Lin, Hong Kong University of Science and Technology, Hong Kong Jacek Malinowski, Polish Academy of Sciences, Warsaw, Poland Ram Ramanujam, Institute of Mathematical Sciences, Chennai, India Jeremy Seligman, University of Auckland, Auckland, New Zealand Kaile Su, Peking University and Griffith University, Peking, China Johan van Benthem, University of Amsterdam and Stanford University, The Netherlands Hans van Ditmarsch, Laboratoire Lorrain de Recherche en Informatique et ses Applications, France Dag Westerstahl, Stockholm University, Stockholm, Sweden Yue Yang, Singapore National University, Singapore Syraya Chin-Mu Yang, National Taiwan University, Taipei, China Logic in Asia: Studia Logica Library Thisbookseriespromotestheadvanceofscientificresearchwithinthefieldoflogic in Asian countries. It strengthens the collaboration between researchers based in Asia with researchers across the international scientific community and offers a platform for presenting the results of their collaborations. One of the most prominent features of contemporary logic is its interdisciplinary character, combining mathematics, philosophy, modern computer science, and even the cognitiveandsocialsciences.Theaimofthisbookseriesistoprovideaforumfor current logic research, reflecting this trend in the field’s development. The series accepts books on any topic concerning logic in the broadest sense, i.e., books on contemporary formal logic, its applications and its relations to other disciplines. It accepts monographs and thematically coherent volumes addressing important developments in logic and presenting significant contributions to logical research. In addition, research works on the history of logical ideas, especially on the traditions in China and India, are welcome contributions. The scope of the book series includes but is not limited to the following: (cid:129) Monographs written by researchers in Asian countries. (cid:129) Proceedings of conferences held in Asia, or edited by Asian researchers. (cid:129) Anthologies edited by researchers in Asia. (cid:129) Research works by scholars from other regions of the world, which fit the goal of “Logic in Asia”. The series discourages the submission of manuscripts that contain reprints of previously published material and/or manuscripts that are less than 165 pages/ 90,000 words in length. Please also visit our webpage: http://tsinghualogic.net/logic-in-asia/background/ Relation with Studia Logica Library ThisseriesispartoftheStudiaLogicaLibrary,andisalsoconnectedtothejournal Studia Logica. This connection does not imply any dependence on the Editorial OfficeofStudiaLogicaintermsofeditorialoperations,thoughtheseriesmaintains cooperative ties to the journal. This book series is also a sister series to Trends in Logic and Outstanding Contributions to Logic. For inquiries and to submit proposals, authors can contact the editors-in-chief Fenrong Liu at [email protected] or Hiroakira Ono at [email protected]. More information about this series at http://www.springer.com/series/13080 Shier Ju Alessandra Palmigiano Minghui Ma (cid:129) (cid:129) Editors Nonclassical Logics and Their Applications Post-proceedings of the 8th International Workshop on Logic and Cognition 123 Editors ShierJu Alessandra Palmigiano Institute of Logic andCognition Schoolof Business andEconomics SunYat-sen University Vrije Universiteit Amsterdam Guangzhou, Guangdong,China Amsterdam, TheNetherlands Department ofMathematics and Minghui Ma AppliedMathematics SunYat-Sen University University of Johannesburg Guangzhou, Guangdong,China Johannesburg, SouthAfrica ISSN 2364-4613 ISSN 2364-4621 (electronic) Logicin Asia: Studia Logica Library ISBN978-981-15-1341-1 ISBN978-981-15-1342-8 (eBook) https://doi.org/10.1007/978-981-15-1342-8 ©SpringerNatureSingaporePteLtd.2020 TheChapter“TheCategoryofNode-and-ChoiceForms,withSubcategoriesforChoice-SequenceForms and Choice-Set Forms” is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/). For further details see licence informationinthechapter. Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. Theuse ofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc. inthis publi- cationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromthe relevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface The8thinstalmentoftheInternationalConferenceonLogicandCognition(WOLC 2016)tookplaceon5–9December2016attheInstituteofLogicandCognitionof Sun Yat-Sen University in Guangzhou. This instalment focused on non-classical logics and their applications. The conference provided a very interactive environ- ment in which experts from disciplines ranging from formal linguistics and theo- retical computer science to analytic philosophy, game theory, social choice, and managementsciencecouldnotonlydiscusshowtoadvanceformaltheoriessoasto meet the main challenges in their own disciplines, but could also find a common ground based on the tools, insights and techniques developed in the study of non-classical logics. The discussions and research directions with which the conference participants engaged reverberate in their research and the research of colleagues in their net- work. This volume collects a—necessarily non-exhaustive—sample of ways in which advancements in (the mathematical theory of) non-classical logics are obtained in response to challenges arising in other scientific fields. Amsterdam, The Netherlands Alessandra Palmigiano Guangzhou, China Shier Ju Guangzhou, China Minghui Ma v Contents Hyperstates of Involutive MTL-Algebras that Satisfy ð2xÞ2 ¼2ðx2Þ. . . . 1 Tommaso Flaminio and Sara Ugolini The Category of Node-and-Choice Forms, with Subcategories for Choice-Sequence Forms and Choice-Set Forms . . . . . . . . . . . . . . . . 15 Peter A. Streufert About the Temporal Logic of the Lexicographic Products of Unbounded Dense Linear Orders: A New Study of Its Computability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Philippe Balbiani Contact Logic is Finitary for Unification with Constants . . . . . . . . . . . . 85 Philippe Balbiani and Çiğdem Gencer A Multi-agent Default Theory of Permission . . . . . . . . . . . . . . . . . . . . . 105 Huimin Dong Algebraic Semantics for Hybrid Logics . . . . . . . . . . . . . . . . . . . . . . . . . 123 Willem Conradie and Claudette Robinson vii Hyperstates of Involutive MTL-Algebras that Satisfy (2x)2 = 2(x2) TommasoFlaminioandSaraUgolini Abstract StatesofMV-algebrashavebeentheobjectofintensivestudyandattempts ofgeneralizations.Theaimofthiscontributionistoprovideapreliminaryinvesti- gation for states of prelinear semihoops and hyperstates of algebras in the variety generatedbyperfectandinvolutiveMTL-algebras(IBP -algebrasforshort).Ground- 0 ingonarecentresultshowingthatIBP -algebrascanbeconstructedfromaBoolean 0 algebra,aprelinearsemihoopandasuitablydefinedoperatorbetweenthem,ourfirst investigationonstatesofprelinearsemihoopswillsupportandjustifythenotionof hyperstateforIBP -algebrasandwillactuallyshowthateachsuchmapcanberep- 0 resentedbyaprobabilitymeasureonitsBooleanskeleton,andastateonasuitably definedabelian(cid:2)-group. · · · Keywords IBP -algebras Abelian(cid:2)-groups Prelinearsemihoop Statesof 0 · prelinearsemihoop Hyperstates 1 Motivation States of MV-algebras have been introduced by Daniele Mundici (1995) as averagingprocessesfortruth-valuesofŁukasiewiczformulas.Thesearemappings ofanyMV-algebraintherealunitinterval[0,1]satisfyinganormalizationcondition andageneralizedversionoftheusualadditivitylaw(seeFlaminioandKroupa2015; T.Flaminio IIIA-CSIC,CampusdelaUniversidadAutònomadeBarcelonaS/n,08193 Bellaterra,Spain e-mail:[email protected] B S.Ugolini( ) DepartmentofComputerScience,UniversityofPisa,Pisa,Italy e-mail:[email protected] ©SpringerNatureSingaporePteLtd.2020 1 S.Juetal.(eds.),NonclassicalLogicsandTheirApplications, LogicinAsia:StudiaLogicaLibrary, https://doi.org/10.1007/978-981-15-1342-8_1 2 T.FlaminioandS.Ugolini Mundici2011formoredetails).ThestatesofMV-algebrasarestronglyconnected tostatesofabelian(cid:2)-groups(Goodearl1986)throughMundici’scategoricalequiva- lencebetweenthecategoryofMV-algebraswithhomomorphismsandthecategory ofabelian(cid:2)-groupswithstrongorderunit(unital(cid:2)-groups)andunit-preserving(cid:2)- group homomorphisms (Mundici 1986). Indeed, if A is an MV-algebra and G is A itscorrespondingunital(cid:2)-group,thenthestatesofAandthestatesofG arein1-1 A correspondence. MV-algebraicstateshavebeenwidelystudiedinthelastyears(cf.Flaminioand Kroupa 2015 and Mundici 2011 for a brief survey), and many attempts have been made to define states to alternative algebraic structures. In particular, the task of definingstatesonperfectMV-algebrashasbeentheobjectofseveralproposals(Di Nola et al. 2000; Diaconescu et al. 2014a,b) since the very notion of state given inMundici(1995)trivializeswhenappliedtothesestructures.Indeed,everyperfect MV-algebrahasonlyonestate:thefunctionsmappingitsradical,i.e.theintersection ofitsmaximalfilters, Rad(A)in1anditsco-radicalcoRad(A)in0(seeDiNola etal.2000andSect.2belowforfurtherdetails). In this contribution, mimicking the insights provided by Mundici’s categorical equivalencetothestudyofstatetheory,weshalldefineanotionofhyperstate(i.e., hyperreal-valued state) on a wider class of algebras called IBP -algebras which 0 properlycontainsperfectMV-algebrasandforwhichwerecentlyprovidedacate- goricalequivalencewithrespecttoacategorywhoseobjectsareprelinear-semihoop triples, that is, systems (B,H,∨ ) where B is a Boolean algebra, H is a prelinear e semihoopand∨ : B×H → H isasuitablydefinedmap,intuitivelyrepresenting e thenaturaljoinbetweenelementsofBandH.If(B,H,∨ )and(B(cid:4),H(cid:4),∨(cid:4))aretwo e e triples, a morphism between them is a pair (f,g) where f :B→B(cid:4) is a Boolean homomorphism,g :H→H(cid:4)isaprelinearsemihoophomomorphism,andforevery (b,c)∈ B×H,g(b∨ c)= f(b)∨(cid:4) g(c). e e Thedefinitionofhyperstatethatwewillpresentandstudyinthefollowingsections is grounded on the fact that Boolean algebras already possess a well-established notionofstate:probabilityfunctions.Asforprelinearsemihoops,wewillshowinthe nextsectionthateachofthemhasahomomorphicimage(asabelian(cid:2)-monoid)inan abelian(cid:2)-group.Thus,takingintoaccountthecategoricalequivalencebetweenIBP - 0 algebrasandprelinearsemihooptriples,ourmainresultwillprovethatanyhyperstate onanIBP -algebrasplitsintoaprobabilitymeasureoftheBooleanskeletonanda 0 state of the largest prelinear semihoop contained in it. As a consequence, we will prove that if the IBP -algebra actually belongs to the variety generated by perfect 0 MV-algebras,thenitshyperstatesaregivenbyaprobabilitymeasureonitsBoolean skeletonandastateofasuitablydefinedabelian(cid:2)-group. Thepresentpaperisstructuredinthefollowingway:Sect.2isdevotedtorecalling basicnotionsandresultsaboutabelian(cid:2)-groups,MV-algebras,perfectMV-algebras andtheirstates,whileinSect.3wewillproveafirstresultwhichpartiallyextends the usual Grothendieck group construction to lattice-ordered monoids. That result and its corollary will be used in Sect.4 to introduce a suitable notion of state of a prelinear semihoop which, in turn, allows to introduce a notion of hyperstate of IBP -algebrasinSect.5.InthesameSect.5wewillprovethateveryhyperstateofan 0 HyperstatesofInvolutiveMTL-Algebras… 3 IBP -algebrasplitsintoaprobabilitymeasureonitsBooleanskeletonandastateof 0 thelargestprelinearsemihoopcontainedinit.WeendthispaperwithSect.6which isdevotedtoconcludingremarksandfutureworkonthissubject. 2 Abelian(cid:2)-Groups,MV-AlgebrasandTheirStates An abelian (cid:2)-group with strong unit (or unital (cid:2)-group for short) is a pair (G,u) whereGisanabelian(cid:2)-group(seeGoodearl1986)andu ∈Gsatisfiesthefollowing requirement:foreveryx ∈G thereisanaturalnumbernsuchthatx ≤u+···+u where,inthepreviousexpression,u+···+u isthen-timessumofu inG,and≤ denotesthelatticeorderofG. Astateofan(cid:2)-groupGisagrouphomomorphismσ totheadditivegroupRof realswhichfurthersatisfies:forallx ≥0inG,σ(x)≥0inR.If(G,u)isaunital (cid:2)-group,astateof(G,u)isanystateofGsuchthatσ(u)=1(seeGoodearl1986 forfurtherdetails). MV-algebras can be introduced as those structures A=(A,⊕,¬,0,1) of type (2,1,0,0)forwhichthereexistsaunital(cid:2)-group(G ,u)suchthat A={x ∈G | A A 0≤ x ≤u}, x ⊕y =(x +y)∧u,¬x =u−x and1=u.Furthermore,forevery MV-algebraA,theunital(cid:2)-group(G ,u)isunique.Indeed,thepreviousconstruction A inducesacategoricalequivalence,establishedbyMundici’sfunctor(cid:4),betweenthe categoriesofunital(cid:2)-groupswithunit-preserving(cid:2)-grouphomomorphismsandthat ofMV-algebraswithMV-homomorphisms(Mundici1986).Inparticular,itisworth noticing that for every morphism h in the category of unital (cid:2)-groups, (cid:4)(h) is an MV-homomorphismthatisdefinedbyrestriction. ThelatterconstructionsuggeststhatwecanspeakaboutstatesofanMV-algebra Arestrictinganystateof(G ,u)bothinitsdomain,whichthusbecomes A,andits A codomain that, since 1 is a strong unit for the (cid:2)-group R, restricts to the real unit interval [0,1]. Indeed, by a state of A we mean any map s : A→[0,1] such that s(1)=1 and s(x ⊕y)=s(x)+s(y) for all x,y ∈ A such that x +y (the group sum)coincideswithx ⊕y (theMV-sum)(Mundici1995). AlthoughstatesofMV-algebrasresemblefinitelyadditiveprobabilitymeasures onBooleanalgebras,theyareintimatelyrelatedwithBorel(andhenceσ-additive) regular measures. Indeed, by the Kroupa-Panti Theorem (Kroupa 2006 and Panti 2009), for every MV-algebra A the set of its states S(A) is in 1-1 correspondence withthesetofBorelregularmeasuresonthecompactandHausdorffspaceMax(A) of maximal MV-filters of A. Precisely, for every state s of A there exists a unique BorelregularmeasureμofMax(A)suchthatsistheLebesgueintegralw.r.t.μ(see alsoFlaminioandKroupa2015;Mundici2011formoredetails). EveryMV-algebraadmitsatleastonestate.However,therearerelevantexamples of MV-algebras whose unique state is trivial, i.e., it only takes Boolean values, 0 and 1. This is the case, for instance, of perfect MV-algebras (Di Nola and Lettieri 1994).MimickingthewayweusedtointroduceMV-algebrasingeneral,perfectMV- algebrasare,uptoisomorphisms,thoseMV-algebrasoftheform(cid:4)(Z×G,(1,0)) where (cid:4) is Mundici’s functor, Z is the (cid:2)-group of integers, G is any (cid:2)-group, ×