Rend. Mat. Acc. Lincei s. 9, v. 16143-157 2005) Matematica.(6)

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

148A.BRESSAN

(c)In1962prof.Bressan,consideringonlymass-pointsystems,inadeepworkofwellknowndifficulty,reachedanimpeccablesolutioninthesenseofmathematicallogicandhewasthefirsttodoso.(d)Inhismorerecentstudieshetreatsclassicalmechanicsinauniversethatcontainsmathematicalobjectscalled`observers',and(d')headdsasuitableexistenceaxiomthatwaslacking(5).(e)Inaddition,heconstructsaformalconceptofphysicalpossibilitywhichabstractlyrepresentsaphysicalexperiment(6).(f)Presentlyprof.Bressaniscompletinghislogicalaxiomatizationofmechanicswithsomeofhispupils(Montanaro,Pitteri,etc.),andinawaycapableofincludingthemechanicsofcontinuousmediainthesenseofNollandhisfollowers.(g)AfteradigressionTruesdellrecommendsthepublicationofaBressan'spaper,theonethatappearedin1987as[10],because`itcarriesoutaremarkablestep'intheprogrammentionedin(f),themaincircumstancesrelevantforithavingbeensketchedin(a)to(e).(h)LastlyTruesdelladdsthatBressanistheuniqueperson(asfarasheknows)capableofmasteringmathematicallogic,rationalmechanics,electromagnetismandspecialrelativity;inhisopinionItalyandtheAccademiadeiLinceicanbeproudofhim.

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Here,andaswellasinhisbookTruesdell1984,Truesdellmentionsfieldsoutsidephysics,likelogicandphilosophyofscience.HisremarksaboutBressan'shighstandinginthesefieldsgetaveryauthoritativesupportbyearlierassertionsofthephilosopherofscienceN.BelnapinhisForewordtoGIMC,[...](7).

(5)ThelackingaxiomisAmmissione10.2inMet,p.106(alsorecalledinGIMC,footnote3onpp.110-111).Thisisframedasapossibilitycondition,but(d'),whereexistenceaxiomsubstantiallystandsforpossibilityaxiom,iscorrectbecauseofsomepeculiartechnicalfeaturesofMet;andthisshowsthatTruesdellhadgraspedthemaintechnicalsemanticfeaturesofthatpaper.

(6)Thisconstruction,performedinMetandrefinedlaterinGIMCandBressan1985,agreeswiththeviewsofHamel1908andHamel1927.

(7)Inhisforeword,pp.xiiitoxxvofGIMC,BelnappreliminarysaysthatGIMC`isthemostimportantcontributiontodateconcerningtheintroductionofquantifiersintomodallogic.Itsurpassesanyarticleorbookinthegeneralityofitsconceptions,thedegreeoftheirdevelopmentandtheprofundityoftheiranalysis'(p.xiii,line3).BelnapespeciallypointsoutBressan's`newanalysisofpredication'(p.xiv,line10).

Inthesubsequentsections1to8Belnapdetailsandmotivateshispreliminaryassertions.Forinstancehesays(p.xiv,sect.1,line3):`...that[Bressan'smodallanguage]MLyis-uniquelyamongmodallogics-acompletetypetheorywithnoupperlimitonitstypes,isextremelyimportant'.Indeed,asisbroadlymentionedonp.xvii,line-9,bythelackofthislimitGIMCsolvespositivelytheproblemconsideredinCarnap1954-see[19]-,pp.195to196.AmongBressan's`distinctivesemanticfeatures'(pp.xviiitoxxiii)Belnapmentionsattributes,lambdaabstraction,anddefinitedescriptions.Furthermore,inNewDirections(p.xxiv,lines1to6),BelnapespeciallypointsoutBressan'snotionofabsoluteattribute,andsays:`ThearticulationanddeploymentofthisnotionisextremelyimportanttoBressan'senterprise...'.Thisis