Rend. Mat. Acc. Lincei s. 9, v. 16143-157 2005) Matematica.(10)

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

152A.BRESSAN

(d)ThesemanticsforMLyisbasedontheextensionalsemanticsforELy 1,onthet),andonformula(9)onp.21(11).Inextensionalcorrespondentt th,(tP

connectionwiththenewweakestadmissibilityconditionsforELy 1,thissemanticsturnsouttobeextensional(whileinGIMC,line-1,p.18,itoughttobeessentiallymodal);InowregardasadvantageousthepossibilityoftreatingmodalandextensionalsemanticsuniformlyfollowingZanardo1981andZanardo2004(i.e.,[63]and[66])(12).

AS15InthepresentintroductionwehaveusedauniqueseriesAS1,AS2,...;tomarksomerelevantpartsofit.E.g.,ASrhasexactlyoneentrywritteninboldcharacter;inthepartmarkedbyitnotyetknownexplicationsareintroduced.ForobviousmotivesASrhassomeentriesbothafterandbeforeit.Thelattershowreaderswheretheycanfindahelptounderstandyetunclearwritings.ThesameholdsforthepossiblepartsASr;s,(s 1;2;...)ofeveryelementASrofaboveseries.

AS15,1Intheinitialpagesofthisintroductionitscontentsisincluded.Theanaloguecanbedoneinanycontributingwork.(Eachofthesehasaseriesof,e.g.,ASrstartingwithr 1).

AS15,2Anycontributingworkisdividedinsectionsmarkedby(§r)(brieflysections(§r)),and(§r)isdividedinnumbers(1),(2),...;andeachnumber(s)initems(i),(ii),...E.g.thesub-item(ii3)(ifitexists)canbeusedinsteadof(ii)forgreaterprecision.AS15,3E.g.in(§r)``see(s)(ii)''orin(§r)(s)``see(ii)''means:see(§r)(s)(ii).

AS15,4Formoreclarity,onecanbesuperabundantbothinplacing(above)marksforsections,numbers,anditems(inboldcharacter)andinreferringtothem(innormalcharacter).

AS16E.g.,by`Bressan19RS'IabbreviatetheuniqueBressan'spaperappearedintheyear19RS(providedsuchpaperexists).Ifinthatyearanumbery>0ofBressan'spapersappeared,then(beingy< I)Ilabelthemwiththeindexesa1toayandIabbreviatethemby`Bressanai19RS'(i 1;...;y).

(11)(a)TheclassQIofQIs(quasiintensions)oftypetP ty-seeN2atp.40ofGIMC-isdeterminedby 1ht)isdefinedbytherecursivedefinition(a1)QIty Qyth-seeformula(9)onp.21-wheret t,(tP

hOh (h 1);rh (h 1;r)(r 1;...;y);

hh(t1;...;tn)h (t1;...;tn;y 1);hhh(t1;...;tn:t0)h (t1;...;tn:t0):

(b)HerewementiontwocorrectionstobedoneinGIMC:yy(1st)in(b)onp.19:j1POytn3j1POt1;...;jnPOtn.(2nd)definition(8)onp.21must(obviously)bechangedinto h.

(12)Seep.47ofZanardo1981,fromline7toline10.Theyareimproved,withinZanardo2004,byDefinition1.1,Remark(e)andDefinition1.2inpart(A).ThesedefinitionsallowustoembodyextensionallogicintothelogicalcalculusMCyafterhavingweakened,followingZanardo,therequirementcardG!2usedinGIMCintocardG!1.