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Berislav Žarnic* Gabriela Bašic**
Metanormativna načela i normama vodeno društveno medudjelovanje
Kritičko čitanje Alchourronove i Bulyginove skupovnoteorijske defnicije normativnoga sustava pokazuje da njegova deduktivna zatvorenost nije neizbježno svojstvo. Slijedeci von Wrightovu pretpostavku da aksiomi standardne deontične logike opisuju svojstva savršenoga normativnog sustava, uvodi se algoritam za prevodenje iz modalnoga u sku-povnoteorijski jezik. Prijevod nam otkriva da plauzibilnost pojedinih metanormativnih načela leži na različitim osnovama. Koristeci se metodološkim pristupom koji prepo-znaje različite aktere u normama upravljanome medudjelovanju, pokazuje se da su me-tanormativna načela obveze drugoga reda upucene različitim ulogama. Poseban slučaj jest zahtjev koji se odnosi na deduktivnu zatvorenost jer se pokazuje da je upucen ulozi koja primjenjuje, a ne onoj koja izdaje norme. Pristup je primijenjen i na slučaj čiste de-rogacije, što dovodi do novoga rezultata; svojstvo neovisnosti biva svojstvom savršenoga normativnog sustava u odnosu na mogucu derogaciju. Ovaj članak na polemički način dodiruje nekoliko točaka iznesenih u Kristanovome nedavnom članku.
Ključne riječi: normativni sustav, standardna deontična logika, metanormativna načela, derogacija, G. H. von Wright
1 NORMATIVNI SUSTAV KAO SUSTAV NORMI
U svome je nedavnom radu o razrješenju normativnih sukoba Andrej Kristan (2014) prihvatio teorijski pristup normativnosti uveden u Alchourron i Bulygin (1998). Prema skupovnoteorijskome pristupu iznesenome u Alchourron i Bulygin (1998), bilo koja rečenica p koja opisuje izvediva stanja stvari jest normativna rečenica: obavezna ako p pripada skupu logičkih posljedica 'eksplicitno zapovjedenih propozicija', dopuštena ako njezin nijek -p ne pripada skupu te zabranjena ako nije dopuštena.1 Metafora preskriptivne uporabe jezika jest ona smještanja čega u spremnik (propozicije u normativni sustav), no metafora se ne bi smjela prenapregnuti s obzirom da skupovi, za razliku od spremnika, ne-
* bzarnic@ffst.hr I Redoviti profesor u znanstvenome polju filozofije na Filozofskome fakultetu Sveučilišta u Splitu.
** gbasic@ffst.hr I Asistentica u znanstvenome polju filozofije na Filozofskome fakultetu Sveuči-lišta u Splitu.
1 Termin 'izvediva stanja stvari' preuzet je iz von Wright (1999) i označava 'stanja stvari koja mogu nastupiti kao rezultat ljudskoga djelovanja'.
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maju identiteta osim onoga koji im je zadan njihovim članstvom. Iz toga slijedi da dodavanje novih rečenica u postojeci normativni sustav stvara novi skup.
Širok je raspon obilježja koja skup propozicija može imati te postoje dva načina na koja ih se može defnirati: deskriptivno (opisujuci) i preskriptivno (pro-pisujuci). Kristan, slijedeci Alchourrona i Bulygina, defnira normativni sustav N na deskriptivan način kao skup logičkih posljedica eksplicitno zapovjedenih propozicija: A: N = Cn(A). Medutim, u ovoj se definiciji javlja nekoliko problema koje ce trebati razriješiti.
1.1 Konzistentnost i deduktivna zatvorenost
U klasičnoj je logici skup rečenica T deduktivno zatvoren samo ako je mo-guce konzistentno dodati nijek bilo koje rečenice koja nije član skupa u izvorni skup.2 Ova je činjenica simbolički prikazana formulom (1.1).3
T = Cn(T) akko ± i Cn(Tu {-p}) za svaki p i T	(1.1)
Pojmove je slijeda i konzistentnosti moguce definirati jedan drugim i oba se odnose na poželjna svojstva skupa. Postoji li razlog da se deduktivna zatvorenost promatra kao fundamentalno svojstvo u odnosu na konzistentnost? U ovome članku pokušat ce se pokazati da takav odnos prednosti jednoga svojstva pred drugim ne postoji. U analizi polazimo od skupa sadržaja eksplicitnih za-povijedi lišenog svih inherentnih logičkih svojstava, čiji je nastanak empirijska činjenica ostvarena jezičnom uporabom.4
Primjer 1. Goble (2009: 484-485) i Broome (2013: 121-122) razilaze se kod pitanja mora li normativni sustav koji sadrži eksplicitno zapovjedenu pro-poziciju (i) 'Ne smije se kampirati na javnim cestama ni u koje doba' takoder uključivati i propoziciju (ii) 'Ne smije se kampirati na javnim cestama četvrt-kom navečer'. Samo ako je normativni sustav definiran kao skup svih logičkih posljedica eksplicitnih zapovijedi, odgovor na pitanje mora biti potvrdan, no protiv takve definicije postoje uvjerljivi razlozi, kako Broome i pokazuje. S dru-
2	Slijeva nadesno, pretpostavimo u svrhu reductio ad absurdum da je nijek nečlana skupa nemoguce konzistentno dodati u deduktivno zatvoren skup. Ako je tomu tako, onda je nečlan skupa posljedica toga skupa, što je nemoguce s obzirom da je skup deduktivno zatvoren. Zde-sna na lijevo, pretpostavimo takoder u svrhu reductio ad absurdum da je proizvoljna rečenica posljedica skupa, no nije ujedno i njegov član. Ako je tako, nijek rečenice može se konzistentno dodati tome skupu, što je pak nemoguce ako je rečenica posljedica skupa.
3	Formula ± e X kaže da je falsum ± element skupa X ili drugim riječima, da je X inkonzisten-tan. Nijek prethodne formule glasi ± i X i njime se tvrdi da je skup X konzistentan.
4	U Broomeovoj teoriji zahtjeva (Broome 2013) kodeks ispostavlja skup propozicija zatvorenih pod kongruencijom, što znači da ako propozicija pripada skupu, pripada mu i svaka druga njoj ekvivalentna. Pristup u ovome članku apstrahira od svih svojstava skupova uključujuci i kongruenciju.
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ge strane, kako primjecuje Goble, odnos se izmedu (i) i (ii) tiče "zaključivanja s treba-izjavama". Pokušat cemo pokazati da su oba iznesena stajališta ispravna.
1.2 Savršena svojstva i skupovi normi
Nastavljajuci se na von Wrightov pravac razmišljanja, pokazat cemo da se deduktivna zatvorenost može razumjeti kao jedno od više savršenih svojstava skupa:
[...] klasična deontična logika u deskriptivnoj interpretaciji svojih formula prikazuje sustav normi lišen praznina i proturječja. Činjenični normativni poretci mogu imati ta svojstva te se može smatrati poželjnim da ih trebaju imati. No može li biti logička istina da normativni poredak ima ("mora imati") ta "savršena" svojstva? (von Wrigh, 1999: 32)
Standardna ili klasična KD deontična logika prihvaca mogucnost uzaja-mnoga definiranja modalnih operatora obligacije O, prohibicije F i permisije P, kao što je iskazano u (Def.) i grafički prikazano na slici 1.5 KD deontična logika proširuje propozicijsku logiku pravilom necesitacije (RN) i aksiomskim shemama (K) i (D).
Pp akko -O-p akko -Fp. Ako h p, onda h Op. O(p * q) * (Op * Oq) Op * Pp
(Def.) (RN) (K) (D)
Slika 1: Sesterokut logičkih odnosa koji su na snazi u standardnoj KD deon-tičnoj logici. Točkasta crta predstavlja odnos suprotnosti, isprekidana proturječ-je, puna crta podsuprotnost, a strelice predstavljaju podredenost (implikaciju). Deontični su pojmovi iskazani u prirodnome jeziku u lijevome šesterokutu dok onaj desni donosi odgovarajuce formule.
5 Wolenski (2008) nudi opci prikaz kvadrata, šesterokuta i osmerokuta logičkih odnosa zajedno s njihovim primjenama u različitim područjima filozofske analize.
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1.2.1	Prevodenje modalnoga u skupovnoteorijski jezik
U Žarnic (2010) definiran je prijevod iz jezika standardne deontične logike bez opetovanih operatora i dokazano je da prevedeni uvjeti standardne deontične logike opisuju "praznina lišen", deduktivno zatvoren i konzistentan tip skupa normi A/". Naši su osnovni prijevodi slični onima u Alchourron i Bulygin (1998), s razlikom u definientia gdje je 'članstvo u Cn(A/")' sad zamijenjeno 'članstvom u (moguce deduktivno otvorenome) skupu normi A/'": 'p e AT za 'Op', p £ AT za 'Pp, '-p e A/* za 'Fp'. Ukratko, veza je izmedu skupovnoteorijskoga pojma skupa normi i modalnoga pojma obligacije dana u obliku jednostavne jednadžbe: A/* = {p I Op}. Važe sljedeče korespondencije:
1.	Iz prijevoda načela (Def.) o uzajamnoj definiciji deontičnih pojmova sli-jedi da je bilo koji skup normi bez praznina (potpun), Pp v O-p, što čini svako izvedivo stanje stvari dopuštenim ili zabranjenim. Kad se prevede u skupovnoteorijski jezik, definicija (Def.) izražava logičku istinu: -p £A/* ili -p e M.6
2.	Prijevod pravila necesitacije (RN) daje tvrdnju da su logičke istine uklju-čene u skup normi, Cn(0) £ A/*, dok prijevod aksiomske sheme (K) zahtijeva zatvaranje pod modus ponens: Ako p q e M
no ova su dva uvjeta ispunjena akko je skup normi deduktivno zatvoren: N = Cn(M).
3.	Prijevod aksiomske sheme (D) daje Ako p e M, onda -p £ M. Ovaj je uvjet ispunjen ako je skup normi lišen proturječja, tj. ako je konzistentan: _L £ Cn (A/*).
Deskriptivno promatrajuči, empirijski je očito da je oprimjerenje bilo kojega od ovih svojstava kontingentne naravi. Normativno promatrajuči, potrebna su nam metanormativna načela ili intuicije kako bismo procijenili treba li neko svojstvo biti prisutno u skupu normi.
1.2.2	Mogučnost stvaranja sustava normi promulgacijom normi
Po samoj je definiciji teorija Tdeduktivno zatvoren skup ili simbolički: T= Cn(T). Bilo koji deduktivno zatvoren skup Tbeskonačan je zahvaljujuči uklju-čenosti logičkih istina, čiji broj nije konačan ili simbolički: Cn(0) £ Ti |N| < ICn(0)|. Posljedično, pristajanje uz definiciju normativnoga sustava kao skupa zapovjedenih sadržaja nosi sa sobom ontološku obvezu na tvrdnju da postoje beskonačni predmeti, s obzirom da su normativni sustavi beskonačni skupovi, kao i epistemološku obvezu na istraživanje njihove spoznatljivosti. Ako se logičke istine oduzmu od skupa posljedica Cn*(T) = Cn(T) - Cn(0), nastali skup Cn*(T) uz T sadržavat ce samo relevantne posljedice skupa T, tj. one elemente čija dedukcija počiva na sadržajima iz T.
6 Ovaj se uvjet može napisati i u svojemu opcem obliku kao p £ N ili p e M. revus časopis za ustavnu teoriju i filozofiju prava
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Primjer 2. Zamislimo da je normativni sustav M. stvoren jednom jedinom zapovijedi 'Zabranjeno je činiti da nešto bude slučaj onomu tko želi da to nešto ne bude slučaj' upucen jednomu jedinom akteru i.7 Može li akter i postati svje-stan svake pojedine norme iz skupa Cn*(M)? Modalnologički prijevod izražava uvjetnu prohibiciju 'Akteru je zabranjeno činiti da bude slučaj nešto (Fii:stit p) za što želi da to nešto ne bude slučaj (Di-p)' ili simbolički: (1.2). U skup ovnote-orijskome pristupu deontični operator zamijenjen je odnosom članstva izmedu sadržaja zapovijedi i njegovoga skupa normi. Sadržaj zapovijedi glasi Ako akter i želi da -p, onda i ne čini da p bude slučaj' ili simbolički: (1.3). Sadržaj opisuje kako izgleda povinovanje normi i stoga pripada jednočlanome skupu normi M., ili simbolički: (1.4).8
Ako se varijabla p odnosi na rečenice beskonačnoga jezika £, pruža se be-skonačno mnogo rečenica koje mogu zamijeniti p u (1.4). Dakle, broj ce rečeni-ca u skupu Cn*(M) biti beskonačan.
Očito je da ni jedan normativni izvor ne može dovršiti sintaktičko stvara-nje beskonačnoga skupa sadržaja zapovijedi. Je li nužno pretpostaviti postoja-nje logičkih predmeta kao što su beskonačni, deduktivno zatvoreni skupovi? Dodatni se problem u vezi deduktivne zatvorenosti javlja s logičke strane: je li odnos posljedice koji se nalazi u definiciji teorije identičan odnosu posljedice koji definira deduktivnu zatvorenost skupa normi? Postavka o postojanju sui generis odnosa posljedice u imperativnoj jezičnoj uporabi (Žarnic 2011: 95) podupire odbacivanje redukcije odnosa posljedice u skupu normi na 'logiku pokoravanja' u indikativnome jeziku, baš kao što je Hans Kelsen tvrdio (Kelsen 1973: 254).
Ontološko se obvezivanje na postojanje beskonačnih skupova normi ili lo-gičko obvezivanje na redukciju imperativne logike na indikativnu dade lako izbjeci usvajanjem definicije skupa normi tek kao skupa sadržaja eksplicitnih zapovijedi, s jedne strane i postavkom da je deduktivna zatvorenost skupa normi pod nekom logikom kontingentno svojstvo, s druge strane.
7	Ovaj se normativni sustav može protumačiti kao utemeljen na Nietzscheovoj maksimi "Budi ono što jesi!".
8	'Quineovi navodnici' koriste se za oblikovanje imena nekoga izraza. Mogu se ispustiti ako ne postoji mogucnost zabune imena izraza za izraz, no u slučajevima gdje se ista formula i koristi i spominje, Quineovi ce se navodnici koristiti.
Di-p A Fii:stit p Di-p A -i:stitp r Di-p A -i:stit pn e M
(1.2)
(1.3)
(1.4)
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1.3 Metanormativna načela
Von Wright (1999) uvodi pojam "normativnih zahtjeva za normativne su-stave" ili pojam 'metanormativnoga načela', kako ce odsad nadalje biti zvano:
[drugi način] ... jest promatrati same ideje potpunosti i lišenosti proturječja kao normativne ideje, kao normativne zahtjeve za normativne sustave. Mogle bi se nazvati
meta-normativnim načelima. One su norme višega reda. (von Wright 1999: 33)
Na prvi pogled, čini se mogucim shvatiti metanormativna načela kao tvrd-nje o tome da skup normi treba imati odredeno svojstvo te izraziti te tvrdnje u formalnome jeziku uvodenjem ugniježdenih KD modaliteta. U svrhu analize metanormativnih načela, izražajna ce moc formalnoga jezika biti obogacena uvodenjem S5 aletičnih modaliteta nužnosti • i mogucnosti 0.9 Aletični se mo-daliteti mogu protumačiti na različite načine; kao logičke, nomološke ili histo-rijske mogucnosti.10 Pojmovi su poredani po inkluziji: historijska je mogucnost nomološka mogucnost, a nomološka je mogucnost logička.
Sljedeci popis sadrži neka prima facie plauzibilna metanormativna načela, označena njihovim dolje ponudenim formalnim prijevodima:
-	Skup normi treba biti bez praznina: (O.def).
-	Skup normi treba biti deduktivno zatvoren: (O.RN) zajedno s (O.K).11
-	Skup normi treba biti konzistentan: (O.D).
-	Skup normi treba biti ostvariv: (O.OO).12
-	Skup normi treba biti ostvaren: (O.T).
Sljedeci su formalni modalni izrazi dobiveni uporabom novoga simbola O
za obligaciju drugoga	reda:
v O-p)	(O.def)
*	O p )	(O.RN) p * q) * (Op * Oq))	(O.K)
*	Pp)	(O.D)
*	op)	(O.OO)
*	p)	(O.T)
9	S5 logika može se aksiomatizirati pravilom necesitacije: Ako I- p, onda I- Dp , aksiomskim shemama: (K) a(p • q) • (Dp • Dq), (T) Dp • p, (4) Dp • ddp, (5) •p • aOp, i definicijom: •p A - d - p.
10	Logička je mogucnost svijet u kojem vrijede zakoni logike, nomološka mogucnost svijet u kojem vrijede logički i prirodni zakoni, a historijska mogucnost je nomološka mogucnost koja leži u buducnosti druge nomološke mogucnosti.
11	Ako se modalitet • tumači kao logička nužnost, onda metanačelo kaže da zadani skup normi treba uključivati sve logičke istine.
12	Modalitet • može se u ostatku teksta tumačiti kao historijska mogucnost.
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Budua da je u skupovnoteorijskome pristupu moguce ponuditi samo prije-vode prvoga reda, on se mora proširiti kako bi mogao iskazati i metanormativ-ne izraze. Jedno od rješenja koje se namece jest postupiti s metanormativnim izrazima kao tvrdnjama da zadani skup normi pripada odredenoj klasi tipo-va skupova normi. Prijevod za obligaciju prvoga reda Op jest 'p je član skupa normi A/*", tj. p e M. Gotovo analogno tomu, izgleda da izjava 'svojstvo p jest savršeno svojstvo' i njezina ekstenzionalna reformulacija 'skup skupova normi koje ispunjavaju uvjet p član je savršenoga skupa' pružaju dostupan prijevod za tvrdnju o obligaciji drugoga reda Op. Nazovimo Savršenim skup skupova onih skupova normi koji dijele odredena savršena svojstva. Reci da je svojstvo p skupova normi savršeno svojstvo ne znači drugo doli reci da 'skup skupova normi koji ispunjavaju uvjet p jest element Savršenoga' ili simbolički, '{Af \ M ispunjava uvjet p} e Savršeno'.
Primjedba 3. Ako se prihvati Godelova pretpostavka kako svojstvo drugoga reda bivanja pozitivnim svojstvom stvara ultrafilter, onda skup skupova normi koji imaju sva savršena svojstva mora biti neprazan. Nazovimo ga Idealnim. Neka a bude skup skupova normi koje imaju odredeno savršeno svojstvo. Onda izraz 'a e Savršeno' znači isto što i 'Idealno £ a. Ultrafilter zadanoga skupa jest skup njegovih podskupova zatvorenih pod presjekom i odnosom nadskupa, pri čemu prazan skup nije njegov element i za bilo koji skup vrijedi da je ili on sam, ili njegov komplement član ultrafiltra.13
Primjedba 4. Može li skup normi imati svojstvo da svako izvedivo stanje stvari čini obvezujučim ili zabranjenim? Nazovimo ovo svojstvo svojstvom neopcionalnosti s obzirom na to da ne ostavlja mjesta za opcionalne radnje i suzdržavanje od djelovanja. Ako je normativni sustav zamišljen kao proizašao dedukcijom iz skupa normi, onda valja zapaziti da Godelov poučak nepotpuno-sti implicira neispunjivost tog uvjeta; p e Cn(N) v -p e Cn(M) za svaki skup normi formuliran u jeziku dovoljno bogatom da iskaže vlastitu sintaksu (pri-mjerice, prirodni jezik). Buduči da ni jedan normativni sustav ne može ispuniti ovaj zahtjev, svojstvo neopcionalnosti ne može unutar godelovske ontologije pozitivnih svojstava biti savršeno svojstvo.
Dalje nastavljamo s algoritmom prevodenja za formule u kojima opetovani deontični modaliteti istoga tipa nisu dopušteni, no dopušteno je da se deontični modaliteti prvoga reda pojavljuju unutar dosega onih drugoga reda.
Definicija 5. Neka £api bude jezik aletične modalne logike. Funkcija T1 prevodi formule s deontičnim modalitetima prvoga reda:
T1 (p) = p ako p e COPI
Ti(Op) = rT1 (p)"1 e N
13 Za uvid u Godelovu ontologiju svojstava vidi Kovač (2003).
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T1(Pp) = rT1 (".p)"i i N
Ti(-p) = -Ti (p)
Ti((p • q)) = (T1 (p) • T1(q))
Definicija 6. Funkcija T2 prevodi one formule čiji je glavni operator deontič-ni modalitet drugoga reda:
T2 (Op) = {N I T1(p)} e Savršeno
T2(Pp) = {N I - T1(p)} i Savršeno
Primjer 7. Neka p bude rečenica bez pojave modaliteta prvoga ili drugoga reda:
T2(O(Op • Op)) = {Af I T!(Op • Op)} e Savršeno
= {N I THOp) • T1(Op)} e Savršeno = {A/" I T1(p) e N • Op} e Savršeno = {M I rpn e A/* • Op} e Savršeno
Prijevod za uvjet (O.OO) kaže da je zahtijevanje samo onoga što je moguce savršeno svojstvo skupa normi.
Primjer 8. Prijevod za uvjet (O.T) mnogo je manje plauzibilan.
T2(O(Op • p)) = {N I rpn e M • p} e Savršeno
Ovo nam govori kako je zahtijevanje samo onoga što je slučaj savršeno svoj-stvo skupova normi, ali to očito nije željena interpretacija za načelo da skup normi treba biti ostvaren.
Nejednaka plauzibilnost prijevoda u primjerima (7) i (8) pokazuje da obligacije drugoga reda označene homonimnim izrazom — 'treba biti' u 'skup normi treba biti ostvariv' i u 'skup normi treba biti ostvaren' — ne pripadaju istoj kategoriji.
1.3.1 Načelo rimskoga prava kao norma za davatelja normi
Cilj nam je povua pojmovnu distinkciju izmedu tipova obligacija drugoga reda s obzirom na uloge aktera uključenih u promulgaciju (proglašenje), reali-zaciju (ostvarivanje) i aplikaciju (primjenu) normi. Obratimo prvo pozornost na prvi tip, točnije normativni kontekst promulgacije normi, odnosno obligacije za davatelja normi. Takozvano 'Načelo rimskoga prava' zabranjuje davatelju normi da zahtijeva neizvedive radnje s obzirom na to da nitko ne može biti obvezan učiniti nemoguce. Pokazat ce se kako je sa stajališta standardne deontične
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logike uporaba termina 'načelo' neopravdana zbog toga što če 'Načelo rimskoga prava' biti zadovoljeno normativnim sustavom čije norme konzistentno izabiru samo ono što je moguče.
Sadržaj Op — Op metanormativnoga načela (O.OO) odigrao je važnu ulogu u teoriji normativnosti. Aristotelova tvrdnja da u promišljanju "Ako se ljudi su-sretnu s nemogučim, oni odustaju" (Aristotel, Nikomahova etika, 1112b) može se razumjeti kao protupostavna formulacija srodnoga načela. U metanonorma-tivnome tumačenju aristotelovsko načelo promišljanja iskazuje da ono nemo-guče ne smije biti sadržaj namjere. Bliže načelu (O.OO) jest načelo iz rimskoga prava ultra posse nemo obligatur (ad impossibilia nemo tenetur, impossibilium nulla obligatio), samo po sebi prethodnik načela da 'treba' implicira 'može' koje je Kant formulirao i za koje se čini da Op — Op daje izravan prijevod.14 Ipak, logika se akterove sposobnosti djelovanja razlikuje od logike aletične moguč-nosti. Neki teoremi aletične logike ne vrijede u logici sposobnosti djelovanja. Primjerice, postavka da Ako je nešto slučaj, onda je to moguče, p — Op, valjana je u aletičnoj logici, no njezin parnjak u logici sposobnosti djelovanja nije: postavka Ako je nešto učinjeno, onda to može biti učinjeno ne važi u logici sposobnosti djelovanja.15 Metanormativno načelo s aletičnim modalitetom predstavlja po-opčavanje ovih načela: što god je zabranjeno zbog aletične nemogučnosti, takoder je zabranjeno i načelom da 'treba' implicira 'može', no obratno ne vrijedi.
Terminološki govoreči, uporaba termina 'načelo' nije ispravna u kontekstu načela (O.D) i (O.RN) s obzirom na to da je Op — Op poučak koji slijedi iz •p — Op u konjunkciji s Op — Pp, tj. iz sadržaja (O.D) i (O.RN). U prilog ovoj činjenici bit če dana dva stilom različita dokaza.
Poučak 9. Op — Op e Cn({D p — Op, Op — Pp})
Dokaz. Izvedimo prvo dokaz dedukcijom koji se oslanja na sintaksu jezika. Iz mp — Op te definicija deontičnih i aletičnih modaliteta dobivamo korolarij: Ako je dopušteno da neko stanje stvari bude slučaj, onda je moguče da ono bude slučaj, Pp — Op. Pretpostavimo da je p obvezatno, Op. Onda je p dopušteno, Pp, u skladu s aksiomom D. Iz korolarija slijedi da je p moguče, Op. Stoga, ako je stanje stvari obvezatno, onda je ono moguče, Op — Op. Q.E.D.
Dokaz. Kao drugo, ponudimo dokaz u semantičkim terminima! Osnovna semantička ideja modalne logike jest da je istinitosna vrijednost formule u točki vrednovanja ovisna o istinitosnim vrijednostima formule u drugim točkama vrednovanja dostupnima putem odgovarajučega odnosa. Deontični odnos
14	Brojni su odlomci iz Kantovih djela koji se bave ovim načelom. Primjerice, u Religiji unutar granica čistoga uma (1793) dana je sažeta formulacija u obliku "dužnost ne zapovijeda ništa osim onoga što možemo učiniti" (Kant: 68).
15	Izabiranje kraljice srca iz špila karata ne implicira sposobnost da se to i učini; vidi Brown (1992).
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dostupnosti, Dwv, povezuje svijet w s njegovim normativnim alternativama v u kojima je skup normi ostvaren, M c v za sve v e {v I Dwv}. Slično tomu, ale-tični odnos dostupnosti protumačen kao recimo nomološka mogucnost, Nwv, povezuje svijet w bilo s kojom od njegovih alternativa v u kojima vrijede logički i prirodni zakoni. Za neke je modalne formule (Sahlqvistove formule) moguce izračunati odgovarajuce svojstvo prvoga reda odnosa dostupnosti koristeci se Sahlqvist-van Benthemovim algoritmom.16 Poznato je da Op — Pp karakteri-zira svojstvo serijalnosti deontičnoga odnosa, Vx3y Dxy. Kao što je prethodno rečeno, to znači da je zadani skup normi konzistentan. Koristeci se algoritmom, moguce je izračunati sljedeca meduodnosna svojstva:
•	Op — 0p karakterizira Vx3y (Dxy A Nxy) meduodnosno svojstvo. Može ga se nazvati 'svojstvom konvergirajuce serijalnosti odnosnoga para' i ono kaže da uvijek postoji deontički dostupna situacija koja je takoder i nomološki mo-guca. Ili da se poslužimo metaforom profesora Segerberga, nema tragičnih dilema (Segerberg, 2003). Skup normi koji oprimjeruje ovo svojstvo pruža moguc i zakonit izlaz iz bilo koje situacije.
•	• p — Op karakterizira podredenost deontičnoga odnosa nomološkome VxVy(Dxy — Nxy). Skup normi može se ostvariti samo u nomološki mogucim situacijama: ako postoji zakonit izlaz iz situacije, on je takoder i moguc izlaz.17
Lako je uvidjeti da ako je deontični odnos serijalan i podreden nomološko-me, on uvijek mora s njim konvergirati u nekoj točki, točki u kojoj su norme ostvarene u nomološki mogucemu svijetu.18 Stoga,'treba' implicira 'može' nije samoopravdavajuce načelo, vec posljedica drugih načela. Q.E.D.
2 NORME I DRUŠTVENO MEDUDJELOVANJE
U komunikaciji se obično prepoznaju dvije akterske uloge: uloga pošiljatelja i uloga primatelja poruke, no u normama vodenome društvenom medudjelova-nju, osim uloga izdavatelja normi i podložnika normama postoji i dodatna ulo-ga, ona primjenitelja normi. Komunikacija je vrsta čina, a to prema Parsonsovoj
16	Van Benthem definira skup formula algoritamski prevodivih u njihove ekvivalente prvoga reda u sljedecemu počku: "Poučak 19. Postoji efektivan algoritam koji prevodi sve modalne aksiome oblika A — B u odgovarajuca svojstva prvoga reda, gdje je A sastavljeno od osnovnih formula • Dp koristeci se samo A, V, 0, B, je 'pozitivno': sastavljeno od propozicijskih slova samo pomocu A, V, 0, •" (van Benthem 2010: 106).
17	Ilustracije radi upotrijebimo Sahlqvist-van Benthemov algoritam za odredivanje korespon-dencija. Započinjemo s (i) Dp — Op i primjenjujemo standardni prijevod u dva koraka: (ii) VP STx(dp — Op); (iii) VP(Vy(RNxy — Py) — Vy(ROxy — Py)). Potom odredujemo minimalno vrednovanje (iv) Pu := Rnxu i izvodimo supstituciju: (v) Vy(RNxy — RNxy) — Vy(ROxy — RNxy). Pojednostavljenjem dobivamo (vi) T— Vy(ROxy — RNxy) i konačno (vii) VxVy(ROxy — RNxy).
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definiciji znači da pošiljatelj poruke ima neki cilj u situaciji čiji su uvjeti i sredstva podredeni normativnim zahtjevima.19 Posljednji uvjet u Parsonsovoj definiciji djelovanja upozorava na njegovu normativnu dimenziju. Slično tomu, Habermas izjednačava društveni svijet s normativnim kontekstom.20 Djelovanja koje se odnose na norme (proglašenje, pokoravanje, primjena) kao djelovanja i kao društvene činjenice moraju imati svoje vlastite normativne kontekste koji su, u skladu s našom pretpostavkom, eksplicirani u njihovim metanormativnim načelima.
2.1 Normativni konteksti s normama odnosnih djelovanja
Kao što je gore navedeno, u normama vodenome medudjelovanju postoje tri aktera: uloga izdavatelja normi, ona podložnika normama i ona primjenitelja normi; tako postoje i tri tipa djelovanja koja se odnose na norme: promulgacija normi, normama upravljano djelovanje i na normama zasnovano prosudivanje. U ovoj vrsti medudjelovanja izdavatelj normi njihovom promulgacijom upravlja djelovanjima podložnika normama, o čijemu pokoravanju njima prosuduje primjenitelj normi.
Prvo se okrecemo normativnome kontekstu radnje promulgacije normi. Prema našemu tumačenju, jezik KD logike jest deskriptivan jezik čiji aksiomi opisuju svojstva skupova normi: aksiom K definira posljedičnost, a aksiom D definira konzistentnost. Ako je promulgacija skupa normi djelovanje (u Parsonsovome smislu) ili društvena činjenica (u Habermasovome smislu), onda barem jedno od njegovih svojstva jest ili dopušteno, ili zabranjeno. Naprimjer, ako se ne smatra poželjnim da promulgirani skup normi bude inkonzistentan, onda poželjnost svojstva konzistentnosti utemeljuje normativni kontekst za promulgaciju normi. Ovo se poželjno svojstvo može protumačiti kao obligacija drugoga reda i može se izraziti tvrdnjom da skup normi treba biti konzistentan, kao što je iskazano u (O.D) gore. Što se tiče pitanja je li poželjno da skup normi sadrži sve svoje deduktivne posljedice, čini se da je niječan odgovor neizbježan jer proizvodenje beskonačnoga teksta nije izvedivo djelovanje. Stoga poželjnost
19	Parsonsova definicija čina: "...'čin' uključuje logički sljedece: (1) Podrazumijeva činitelja, 'aktera.' (2) U svrhu definicije čin mora imati 'cilj,' buduce stanje stvari prema kojemu je process djelovanja usmjeren. (3) Mora započeti u 'situaciji' čiji se razvojni smjerovi razlikuju u jed-nome ili više pogleda od stanja stvari kojemu je djelovanje usmjereno, tj. cilja. Ovu je pak situaciju moguce analizirati na dvije skupine elemenata: one nad kojima akter nema kontrole, tj. one koje ne može u skladu sa svojim ciljem izmijeniti ili spriječiti da budu izmijenjeni, te one nad kojima ima kontrolu. Prvospomenuti se elementi mogu nazvati 'okolnostima' djelo-vanja, a drugospomenuti 'sredstvima.' Konačno, (4) inherentan je poimanju ovoga jedinstva, u njegovoj analitičkoj uporabi, odredeni način povezanosti izmedu ovih elemenata. To znači da u izboru alternativnih sredstava za neki cilj, u mjeri u kojoj situacija dopušta alternative, postoji 'normativno usmjeravanje 'čina' (Parson, 1937: 44).
20	Habermas piše: "Društveni se svijet sastoji u normativnome kontekstu koji nalaže koja medudjelovanja pripadaju cjeloukupnosti zakonitih medusobnih odnosa" (Haberma, 1984: 88).
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konzistentnosti pripada kategoriji različitoj od poželjnosti posljedičnosti ili de-duktivne zatvorenosti.
Drugo, istražimo normativni kontekst pokoravanja normama. Poseban se tip poželjnosti javlja u metanormativnoj postavci (O.T), postavci koja se može plauzibilno protumačiti kao Pokoravanje je normama poželjno, Dužnost se mora ispuniti, Norme se trebaju ostvariti itd.21 Kao što je gore primijeceno, nije medu-tim plauzibilno tumačiti ovu postavku kao tvrdnju o poželjnome svojstvu skupa normi s obzirom na to da tvrdnja Poželjno je da norme zahtijevaju samo ono što je slučaj završava u svojevrsnome normativnom kolapsu. Postavku prije treba razumjeti kao načelo pokoravanja s obzirom na to da pokazuje kako je norma ono čemu se treba pokoriti. Iz ove perspektive gledano, postoji važna razlika izmedu dviju metanormi: za razliku od izdavatelja normi, podložnik normama nema obveza koje se tiču svojstava skupova normi, a za razliku od podložnika normama, izdavatelj normi nema obveza koje se tiču pokoravanja njima.
Trece, pozabavimo se normativnim kontekstom primjene normi. Primjenitelj normi ili sudac odlučuje o deontičnome statusu stanja stvari uspostavljenoga djelovanjem podložnika normama. Pretpostavimo da je podložnik uspostavio stanje da p. Primjenitelj normi treba odrediti deontični status p s obzirom na neki skup normi J\f, a to može učiniti dvjema logički ekvivalentnim metodama: bilo dodavanjem p u N i provjeravanjem konzistentnosti proširenoga skupa M u {p}, bilo ispitivanjem je li -p posljedica skupa N. Prema prvoj metodi, ako M u {p} nije konzistentno, onda je p zabranjeno, a ako je konzistentno, p je dopušteno, kao što je pokazano u (2.5) i (2.6). Sličan slučaj vrijedi i za drugu metodu, kao što je pokazano u (2.7) i (2.8).
Primjenitelj normi izvodi dedukciju, no nema nikakvoga "normativnoga su-stava", tj. deduktivno zatvorenoga skupa Cn(M) koji bi morao prethoditi ili bi mogao nastati iz tako dobivenoga odredenja deontičnoga statusa stanja stvari koje je uspostavio podložnik normama svojim djelovanjem ili suzdržavanjem od djelovanja. Iako zahtjev drugoga reda za deduktivnom zatvorenošcu ili načelo posljedičnosti ne definira savršeno svojstvo empirijskoga skupa normi, on
21 B. Chellas odobrava uporabu obligacija drugoga reda unutar postavke OU. O(OA • A) ili O.T u našoj notaciji: "Primijetimo da je OU poučak deontične S5 ... Shema izražava postavku da treba biti slučaj da što god treba biti slučaj, bude slučaj. Radi se o često raspravljanome načelu u deontičnoj logici jer je jedan od rijetkih plauzibilnih slučajeva poučka oblika OA u kojemu A nije trivijalno." (Chellas 1980: 193).
Ako ± e (N u {p}), onda Fp. Ako ± i (J\f u {p}), onda Pp. Ako -p e Cn(A0, onda Fp. Ako -p i Cn(M), onda Pp
(2.5)
(2.6)
(2.7)
(2.8)
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definira metanormativni kontekst za primjenitelja normi. Načelo posljedičnosti pokazuje da se normativne prosudbe trebaju podvrgavati zakonima logike.
Usmjerene obligacije drugoga reda Različita metanormativna načela prila-žu se različitim ulogama u normama vodenome medudjelovanju. Dok su norme uvijek upucene podložnicima normama, obligacije drugoga reda mogu se razlikovati po njihovim adresatima, kao što je pokazano u tablici 1:
uloge u normama upravljanome Njihove obligacije drugoga reda: medudjelovanju:
izdavatelj normi g	Treba stvarati skupove normi
Tablica 1. Različite uloge u normama upravljanome medudjelovanju i njihove obligacije drugoga reda
Ova činjenica upozorava na potrebu reformulacije metanormativnih načela razmatranih u Odsječku 1.3: obligacije drugoga reda O moraju se indeksirati imenima uloga za koje vrijede.22 Ako se uloga izdavatelja normi označi indeksom G, uloga podložnika normama sa s te uloga primjenitelja normi (suca) sa J, izborom iz metanormativnih načela dobiva se sljedeča reformulacija:
Og(Osp A Ps p)	(Og.D)
Os(Osp p)	(Os.T)
Oj(Os(p q) a (Osp a Os q))	(Oj.K)
Čitanje se reformuliaranih metanormativnih načela može iskazati u termi-nima modalne semantike. Naprimjer, (Oj.K) čitamo 'Logičke posljedice obligacija podložnika normama jesu njegove obligacije u svim svjetovima u kojima su zadovoljene obligacije primjenitelja normi'.
2.2 Savršeno svojstvo u odnosu na derogaciju
Dinamični fenomen revizije teorije prepoznat je najprije i najistaknutije unutar pravne tradicije. Uspostavljeno je nekoliko načela za razrješenje nor-
22 Na isto je upozorio i Yamada (2011: 63): "Formula oblika O,f znači da je za djelatnika i obve-zatno učiniti da bude slučaj da 9. Iako indeksiranje deontičnih operatora skupom djelatnika nije standardno u deontičnoj logici, moramo moči razlikovati djelatnike kojima su zapovijedi upucene od ostalih ako želimo rabiti deontičnu logiku za razmišljanje o tome kako činovi zapovijedi mijenjaju situacije".
SUDACj
PODLOŽNIK NORMAMA s
savršenih svojstava
Treba se pokoravati normama
Treba primjenjivati norme
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mativne inkonzistentnosti zahvaljujuci odredenju hijerarhijskih odnosa medu skupovima normi na temelju njihove razine opcenitosti (lex specialis derogat legi generali), vremenskoga prethodenja (lex posterior derogat legi priori) i pravne podredenosti (lex superior derogat legi inferiori).23 Prema Kristan (2014), načela su normativnih sukoba "pravila o pravilima" koja nastaju promulgacijom i stoga pripadaju skupu normi. Gledano iz perspektive normama upravljanoga medudjelovanja, ova se pravila obracaju ulozi primjenitelja zakona dajuci mu metodu za ponovno uspostavljanje konzistentnosti. Postojanje normativnoga sukoba pokazuje da davatelj normi nije uspio zadovoljiti načelo višega reda, načelo konzistentnosti ili preciznije rečeno, vanjske konzistentnosti izmedu skupova normi, medutim zahtjev za konzistentnošcu još uvijek vrijedi za pri-mjenitelja normi.
U najjednostavnijemu slučaju čiste derogacije "važenje jedne zakonske norme ukida se i ni jedna nova ne zauzima njezino mjesto," da upotrijebimo Kelsenov opis (1973: 269). Kristan (2014) tvrdi da u ovome slučaju gdje je jedna jedina norma x normativnoga sustava derogirana "novi skupovi A i Cn(A) sa-stavljeni su od svih elemenata prijašnjih skupova, osim norme x" (i posljedica ovisnih o x). Ova tvrdnja nije opcevaljana.
Najjednostavnija derogacija odgovara operaciji kontrakcije u AGM teoriji (Alchourron, Gärdenfors i Makinson, 1985). Primjenjujuci pojam AGM kontrakcije u normativnome kontekstu, dobivena je sljedeca definicija za operaciju čiste derogacije: sadržaj norme p skupa normi N derogiran je akko operacijom nastaje novi skup N+p koji je najveci moguci podskup skupa M koji ne povlači za sobom p. Operacija čiste derogacije pododredena je s obzirom da ce u tipič-nome slučaju biti više od jednoga najvecega moguceg podskupa M koji ne po-vlači za sobom p. Skup takvih skupova može se nazvati preostatkom skupa M oduzimanjem p, J\f _L p. On sadrži sve i samo one skupove M. koji ispunjavaju sljedece uvjete:
1.	Uvjet očuvanja: novi skup normi nastao derogacijom podskup je izvornoga skupa, M. £ M.
2.	Uvjet nepovlačenja: novi skup ne povlači za sobom derogiranu normu, p ž Cn(M).
3.	Uvjet najvece moguce veličine: novi skup zadržava najveci moguci broj normi iz izvornoga skupa, ne postoji skup M. 'takav da M. c M. £ N i p ž Cn( M).
Analogno operaciji kontrakcije, operacija N+p čiste derogatcije treba do-datnu operaciju izbora y za izabiranje člana iz preostatka: N+p = y(A/" _L p). Poseban i uredan slučaj čiste derogacije javlja se kad su norme početnoga skupa
23 Aksiomi za složene hijerarhijske odnose nastale kombinacijom temelja dani su u Malec (2001).
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normi medusobno neovisne, tj. kad ni jedna norma nije proizašla iz drugih, tj. p £ Cn(M - {p}) za svaki p e M. Samo u ovome posebnom slučaju vrijedi da čista derogacija ne zahtijeva odabir člana s obzirom na to da postoji točno jedan član preostatka skupa, naime M - {p}, (2.9).
Ako p £ Cn(M - {p}) za sve p e M, onda A/V p = M - {p}	(2.9)
U svjetlu moguce derogacije, neovisnost normi biva jednim od savršenih svojstava skupa normi, ono koje lišavanjem primjenitelja normi tereta izbora omogucuje "uniformnost sudske prakse". Ako skup normi nema svojstvo neovi-snosti, onda bi čista derogacija mogla dovesti do zamjene uloga; bivajuči prisiljenim izabrati izmedu elemenata preostatka skupa, primjenitelj normi zapravo bi postao njihovim izdavateljem.
— Zahvala.— Priprema ovoga članka potpomognuta je subvencijam Filozofskoga fakulteta Sveučilišta u Splitu. Autori zahvaljuju anonimnim recenzentima na kritičkim čitanjima ruko-pisa i konstruktivnim primjedbama.
S engleskog prevela Gabriela Basic.
Bibliografija
Carlos E. ALCHOURRON & Eugenio BULYGIN, 1998: The expressive conception of norms. Normativity and Norms: Critical Perspectives on Kelsenian Themes. Ur. L. Paulson and B. Litschewski-Paulson. New York: Oxford University Press. 383-410
Carlos E. ALCHOURRON, Peter GARDENFORS & David MAKINSON, 1985: On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic 50 (1985) 2. 510-530.
ARISTOTEL, 2000: Nicomachean Ethics. Preveo Roger Crisp. Cambridge: Cambridge University Press.
Johan van BENTHEM, 2010: Modal Logicfor Open Minds. Chicago: University of Chicago Press.
John BROOME, 2013: Rationality Through Reasoning. Malden: Wiley-Blackwell.
Mark A. BROWN, 1992: Normal bimodal logics of ability and action. Studia Logica: An International Journal for Symbolic Logic 51 (1992) 3-4. 519-532.
Brian F. CHELLAS, 1980: Modal Logic: An Introduction. Cambridge: Cambridge University Press.
Lou GOBLE, 2009: Normative conflicts and the logic of 'ought'. Noüs 43(2009) 3. 450-489.
Jürgen HABERMAS, 1984: The Theory of Communicative Action: Reason and the Rationalization of Society. Boston: Beacon Press.
Immanuel KANT, 1998 [1793]: Religion Within the Boundaries of Mere Reason: And Other Writings. Preveo i uredio Allen Woodand & G. Di Giovanni. Cambridge: Cambridge University Press.
Hans KELSEN, 1973: Essays in Legal and Moral Philosophy. Dordrecht: D. Reidel Publishing Company.
Srečko KOVAČ, 2003: Some weakened Gödelian ontological systems. Journal of Philosophical Logic 32 (2003) 6. 565-588.
Andrej KRISTAN, 2014: V bran izraznemu pojmovanju pravil. Revus. Časopis za ustavnu teoriju i filozofiju prava (2014) 22.
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Andrej KRISTAN, u tisku: In Defence of the Expressive Conception of Norms. Problems of Normativity, Rules and Rule-following. Ur. Michal Araszkiewicz, Pawel Banas, Tomasz Gizbert-Studnicki i Krzysztof Pleszka. Dordrecht: Springer.
Andrzej MALEC, 2001: Legal reasoning and logic. Studies in Logic, Grammar and Rhetoric 4 (2001) 17. 97-101.
Talcott PARSONS, 1937: The Structure of Social Action: A Study in Social Theory with Special Reference to a Group ofRecent European Writers. New York: McGraw Hill Book Company.
Krister SEGERBERG, 2003: Some Meinong/ Chisholm theses. Logic, Law, Morality (Thirtheen essays in practical philosophy in honour of Lennart Aqvist). Ur. Krister Segerberg i Rysiek Sliwinski. Uppsala: Uppsala University. 67-77.
Jan WOLENSKI, 2008: Applications of squares of oppositions and their generalizations in philosophical analysis. Logica Universalis 2 (2008). 13-29.
Georg Henrik von WRIGHT, 1999: Deontic logic: a personal view. Ratio Juris 12 (1999) 1. 26-38.
Tomoyuki YAMADA, 2011: Acts of requesting in dynamic logic of knowledge and obligation. European Journal of Analytic Philosophy 7 (2011) 2. 59-82.
Berislav ŽARNIC, 2010: A logical typology of normative systems. Journal of Applied Ethics and Philosophy 2 (2010) 1. 30-40.
Berislav ŽARNIC, 2011: Dynamic models in imperative logic. Theory of Imperatives from Different Points of View. Ur. Anna Brožek, Jacek Jadacki i Berislav Žarnic. Warsaw: Wydawnictwo Naukowe Semper. 60-119.
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Berislav Žarnic* Gabriela Bašic**
Metanormative Principles and Norm Governed Social Interaction
Critical examination of Alchourron and Bulygin's set-theoretic definition of normative system shows that deductive closure is not an inevitable property. Following von Wright's conjecture that axioms of standard deontic logic describe perfection-properties of a norm-set, a translation algorithm from the modal to the set-theoretic language is introduced. The translations reveal that the plausibility of metanormative principles rests on different grounds. Using a methodological approach that distinguishes the actor roles in a norm governed interaction, it has been shown that metanormative principles are directed second-order obligations and, in particular, that the requirement related to deductive closure is directed to the norm-applier role rather than to the norm-giver role. The approach has been applied to the case of pure derogation yielding a new result, namely, that an independence property is a perfection-property of a norm-set in view of possible derogation. This paper in a polemical way touches upon several points raised by Kristan in his recent paper.
Keywords: normative system, standard deontic logic, metanormative principles, derogation, G. H. von Wright
1 THE NORMATIVE SYSTEM AS A SET OF NORMS
In his recent work on normative conflict resolution, Andrej Kristan (forthcoming) adopted the theoretical approach to normativity introduced by Alchourron and Bulygin (1998). According to the set-theoretic approach presented in Alchourron and Bulygin (1998), any sentence p describing "doable" states of affairs is a normative sentence: obligatory if p belongs to the set of logical consequences of "explicitly commanded propositions", permitted if its negation -p does not belong to the set, and prohibited if not permitted.1 The metaphor for prescriptive use of language is that of putting something into a container (a proposition into the norm-set). The metaphor should not be stretched too far since sets unlike containers have no identity other than what is
* bzarnic@ffst.hr I Professor in Philosophy at the University of Split, Faculty of Humanities and Social Sciences.
** gbasic@ffst.hr I Assistant in Philosophy at the University of Split, Faculty of Humanities and Social Sciences.
1 The term 'doable states of affairs' is taken from von Wright (1999) and denotes 'states of affairs which can come to obtain as the result of human action'.
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given to them by their membership and, consequently, adding sentences to an existing norm-set creates a new set.
The range of properties that a set of propositions can have is vast and there are two ways to define them: descriptively and normatively. For example, on the descriptive side, a norm-set need not exemplify consistency as a matter of fact but, on the normative side, it may be subordinated to the consistency requirement as a matter of value. Kristan, following Alchourron and Bulygin, defines the normative system W in a descriptive way as a set of logical consequences of explicitly commanded propositions A: M = Cn(A). There are, however, some problems with this definition which will need to be resolved.
1.1 Consistency and deductive closure
In classical logic, a set T of propositions is deductively closed just in case the negation of any non-member of the original set can be consistently added to it.2 This fact is symbolically represented by the formula (1.1).3
T= Cn(T) iff ± £ Cn(T u {-p}) for all p £ T	(1.1)
The notions of consequence and consistency are interdefinable and are both about desirable properties. Is there a reason to regard deductive closure as a property more fundamental than consistency? In this paper we will try to show that there is no order of precedence between these properties. Let the set of contents of explicit commands be the starting point of our analysis. This set is devoid of any inherent logical properties and its creation is an empirical fact brought about by the use of language.4
Example 1. Goble (2009: 484-5) and Broome (2013: 121-2) disagree on the question whether a normative system containing the explicitly commanded proposition (i) 'There shall be no camping at any time on public streets' must also include the proposition (ii) 'There shall be no camping on public streets on Thursday night'. Only if the normative system is defined as a set of all logical consequences of explicit commands, must the answer be in the affirmative, but
2	For the left-to-right direction, suppose, for the purpose of a reductio ad absurdum, that the negation of a non-member cannot be consistently added to the deductively closed set. If so, then the non-member of the set is a consequence of it, which is impossible since it is deductively closed. For the right-to-left direction, suppose, for the purpose of a reductio ad absurdum, that an arbitrary sentence is a consequence of the set but not its member. If so, the negation of the sentence can be consistently added to the set, which is impossible if the sentence is a consequence of the set.
3	The formula ± e X says that falsum ± is an element in X or, in other words, that X is incon-
sistent. The negation of the former formula is ± £ X and it says that X is consistent.
4	In Broome's (2013) theory of requirements, a code delivers a set of propositions closed under congruence, i.e., if a proposition belongs to the set, then so does any proposition equivalent to it. In our approach, all properties, including congruence, are abstracted away.
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there are compelling reasons against it, as Broome shows. On the other hand, as Goble notes, the relation between (i) and (ii) concerns "one's reasoning with ought-statements". We will try to show here that both positions are correct.
1.2 Perfection properties and norm-sets
Extending von Wright's line of thought, we will show that deductive closure can be understood as one among other perfection-properties.
[C]lassic deontic logic, on the descriptive interpretation of its formulas, pictures a ga-pless and contradiction-free system of norms. A factual normative order may have these properties, and it may be thought desirable that it should have them. But can it be a truth of logic that a normative order has ("must have") these "perfection"-proper-ties? Von Wright (1999: 32)
Standard or classical KD deontic logic accepts the interdefinability of the modal operators of obligation O, prohibition F and permission P as stated in (Def.) and graphically represented in Figure 1.5 KD deontic logic extends propo-sitional logic with the necessitation rule (RN) and axiom schemata (K) and (D).
Pp iff -O-p iff -Fp. If I- p, then I- Op. O(p A q) a (Op A Oq) Op A Pp
(Def.) (RN) (K) (D)
Figure 1: The hexagon of logical relations holding in standard KD deontic logic. The dotted line represents the contrariety relation, the dashed line represents contradiction, the full line represents subcontrariety, and the arrows represent subalterna-tion (implication). Deontic concepts are expressed in natural language in the left hexagon while the right hexagon presents the corresponding formulas.
Wolenski (2008) has given a generalized account of squares, hexagons and octagons of logical relations, along with their applications to different domains of philosophical analysis.
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1.2.1	Translating modal language to set-theoretic language
In Žarnic (2010), translation from the language of standard deontic logic without iterated operators is defined and it is proved that translated conditions of standard deontic logic describe the gapless, deductively closed and consistent type of norm-set AT. Our basic translations are similar to Alchourron and Bulygin (1998) but with a slight difference in the definientia since 'membership in Cn(A/")' is now replaced by 'membership in the (possibly deductively unclosed) norm-set A/*": 'p e A/" for 'Op', -p g A/* for 'Pp', —p e A/" for 'Fp'. In short, the connection between the set-theoretic notion of norm-set and the modal notion of obligation is given by the simple equation: N = {p I Op}. The following correspondences hold:
1.	It follows from the translation of principle (Def.) on the interdefinability of deontic notions that any norm-set is gapless (complete), Pp v O-p, making each doable state of affairs either permitted or forbidden. When translated to set-theoretic language the definition (Def.) expresses a logical truth: -p g A/* or -p e NI
2.	The translation of the necessitation rule (RN) gives the claim that logical truths are included in a norm-set, Cn(0) £ A/*, while the translation of (K) axiom schema requires closure under modus ponens: If p • q e N and p e M, then q e M. Taken together, these two conditions are fulfilled iff a norm-set is deductively closed: A/* = Cn(A/*).
3.	The translation of the (D) axiom schema gives: If p e M, then -p g M. This condition is fulfilled if a norm-set is free of contradiction, i.e. if it is consistent: ± g Cn(M).
On the descriptive side, it is empirically evident that the exemplification of any of these properties is a contingent matter. On the normative side, we must employ meta-normative principles or intuitions in order to evaluate whether some property ought to be encoded by a norm-set.
1.2.2	On the possibility of creating a norm-system by norm-promulgation
By its own definition, theory T is a deductively closed set, or symbolically: T = Cn(T). Any deductively closed set T is infinite thanks to the inclusion of logical truths, whose number is not finite, or symbolically: Cn(0) £ T and INI < ICn(0)|. Consequently, one who agrees to define a normative system as a set of command contents has an ontological obligation to concede that infinite objects exist, since normative systems are infinite sets, and also an epistemolo-
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gical obligation to investigate their knowability. If logical truths are subtracted from the set of consequences Cn*(T) = Cn(7~) - Cn(0), the resulting set Cn*(T) in addition to T will contain relevant consequences, i.e., those elements whose deduction relies on a content from T.
Example 2. Imagine that the normative system M is created by the single command 'It is forbidden to see to it that something is the case if one desires it not to be the case' directed to a single actor i.7 Can actor i become aware of all and each norm from the set Cn*(W? Modal logic translation yields a conditional prohibition 'An actor is forbidden to see to it that something is the case (Fii:stit p) if she desires it not to be the case (Di-p)', or symbolically: (1.2). In the set-theoretic approach, the deontic operator is replaced by the membership relation relation between a command content and its norm-set. The content of the command is: 'If actor i desires that -p then i does not see to it that p is the case', or symbolically: (1.3). The content describes what conformation with the norm looks like and so it belongs to the single command norm-set M, or symbolically: (1.4)8
If the variable p ranges over sentences of an infinite language £, then it provides infinitely many sentences that can replace p in (1.4). So, the number of sentences in the set Cn*(M) will be infinite.
It is obvious that no normative source can complete the syntactic creation of an infinite set of command contents. Is it necessary to assume the existence of logical objects as infinite, deductively closed sets? The additional problem of deductive closure arises on the side of logic: is the consequence relation which defines a theory identical to the relation that defines the deductive closure of a norm-set? The thesis on the existence of a sui generis consequence relation in imperative language use (Žarnic 2011: 95) supports the rejection of the reduction of the consequence relation to the 'logic of observance' in the language of indicatives, just as Hans Kelsen claimed (Kelsen 1973: 254).
One can easily avoid ontological commitment to the existence of infinite norm-sets or logical commitment to the reduction of imperative-logic to indicative-logic by adopting the definition that a norm-set is merely a set of con-
7	This normative system can be interpreted as founded on Nietzsche's maxim "Be thyself!".
8	'Quine quotes' are used for forming the name of an expression. Their use can be omitted if there is no possibility of confusion, but in cases where the same formula is both used and mentioned, Quine quotes will be used.
Di-p A Fii:stit p Di-p a -i:stitp Di-p a -i:stit p e M
(1.2)
(1.3)
(1.4)
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tents of explicit commands, and the thesis that the deductive closure of a normset under some logic is a contingent property.
1.3 Metanormative principles
Von Wright (1999: 33) introduced the notion of "normative demands on normative systems" or the notion of 'the metanormative principle', as it will be called hereafter
[another way] /.../ is to view the ideas of completeness and freedom of contradiction
as themselves normative ideas, as normative demands on normative systems. They
could be called meta-normative principles. They are norms of higher order.
At first sight, it seems possible to understand meta-normative principles as claims that a norm-set ought to have a certain property and to express these claims in formal language by allowing embedded KD modalities. For the purpose of analysis of metanormative principles, the expressive power of formal language will be enriched by introducing S5 alethic modalities of necessity • and possibility O.9 Alethic modalities can be interpreted in different ways: as logical, as nomological, and as historical possibilities.10 The concepts are ordered by inclusion: historical possibility is a nomological possibility and nomo-logical possibility is a logical possibility.
The following list contains some prima facie plausible metanormative principles, denoted by tags of their formal translations given below:
-	A norm-set ought to be gapless: (O.def.).
-	A norm-set ought to be deductively closed: (O.RN) with (O.K).11
-	A norm-set ought to be consistent: (O.D).
-	A norm-set ought to be realizable: (O.OO).12
-	A norm-set ought to be realized: (O.T).
The following formal modal expressions for the listed metanormative principles are obtained using new symbol O for the second-order obligation:
O(Pp vO-p)	(O.def.)
9	S5 logic can be axiomatized by rule of necessity: If I- p, then I- Dp, axiom schemata: (K) m(p • q) • (Dp • Dq), (T) Dp • p, (4) Dp • • Dp, (5) Op • dOp, and the definition: Op « -d-p.
10	Logical possibility is a world where laws of logic hold, nomological possibility is a world where logical and natural laws hold, and historical possibility is a nomological possibility that lies in the future of another nomological possibility.
11	If modality • is interpreted as logical necessity, then the meta-principle says that a given norm-set ought to include all logical truths.
12	Modality O can be interpreted as historical possibility in the remainder of the text.
O(dp * Op)
O(O(p * q) * (Op * Oq))
(O.RN) (O.K)
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0(0p * Pp) o(op * op) 0(0p * p)
(O.D) (o.oo) (O.T)
Since in the set-theoretic approach only first-order translations can be given, the approach will have to be extended in order to accommodate metanormative expressions. One suggestive solution is to treat metanormative expressions as claims that a given norm-set type belongs to a certain class of norm-set types. The translation for the first-order obligation Op is 'p is a member of the normset A/*. Almost analogously, the statement 'property p is a perfection property' and its extensional reformulation 'the set of norm-sets satisfying condition p is a member of the perfection-set' seem to provide a viable translation for the second-order obligation claim Op. Let's call Perfect the set of sets of normsets sharing certain perfection properties. To say that a property p of normsets is a perfection property means to say that 'the set of norm-sets that satisfy condition p is an element in Perfect', or symbolically '{A/" I M satisfies condition p} e Perfect'.
Remark 3. If one accepts Godel's assumption that the second order property of being a positive property creates an ultrafilter, then a set of norm-sets having all perfection properties must be non-empty. Let's call it Ideal. Let a be the set of norm-sets having a certain perfection property. Then the expression 'a e Perfect' means the same as 'Ideal £ a'. An ultrafilter of a given set is a set of its subsets that is closed under intersection and superset relation, the empty set is not its element and for any set either the set or its complement is a member of the ultrafilter.13
Remark 4. Can a norm-set have the property of making each doable state of affairs either obligatory or forbidden? Let's call this property the property of non-optionality since it leaves no place for optional acts and forbearances. If a normative system is conceived as generated by deduction from a normset, then it should be noted that Godel's incompleteness theorem implies the unsatisfiability of the condition p e Cn(M) v -p e Cn(N) for a norm-set formulated in a language that is rich enough to express its own syntax (e.g., natural language). Since no normative system can satisfy this requirement, the property of non-optionality cannot be a perfection property under the Godelian ontology of positive properties.
Next we proceed to the translation algorithm for the formulas where iterated deontic modalities of the same type are not allowed while first-order deontic modalities are allowed to occur within the scope of second-order ones.
13 For an investigation into Godel's ontology of properties, see Kovac (2003).
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Definition 5. Let £• pl be the language of alethic modal logic. Function T1 translates formulas with first-order deontic modalities:
T1(p) = p if p e£apL
T1(Op) = rT1(py e M
Ti(Pp) = rT1(-pr £ M
T1(—p) = —T1(p)
T1((p * q)) = (T1(p) T1A))
Definition 6. Function T2 translates those formulas whose main operator is a second-order deontic modality:
T2(Op) = {M I T1(p)} e Perfect
A(Pp) = {M I - T1(p)} £ Perfect
Example 7. Let p be a sentence with no occurrence of first-order or second-order modalities.
T2(O(Op Op)) = {Af I T1(Op Op)} e Perfect
= {Af I T1(Op) T1(Op)} e Perfect = {AA I T1(p) e AT Op} e Perfect = {Af I rpn e AT Op} e Perfect
The translation for the condition (O.OO) says that requiring only that which is possible is a perfection property of norm-sets.
Example 8. The translation for the condition (O.T) is much less plausible.
T2(O(Op p)) = {Af I p e N p} e Perfect
This says that requiring only that which is the case is a perfection property of norm-sets and that is obviously not the intended translation for the principle a norm-set ought to be realized.
The unequal plausibility of the translations in examples (7) and (8) shows that second order obligations designated by the homonymous expression — 'ought to be' in 'a norm-set ought to be realizable' and in 'a norm-set ought to be realized' — do not belong to the same category.
1.3.1 Roman Law principle as a norm for the norm-giver
We aim to draw a conceptual distinction between types of second order obligations with respect to the roles of actors involved in norm promulgation, norm realization and norm application. First, let our attention be drawn to the first
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type, namely to the normative context of norm promulgation, the obligations for the norm-giver. The so-called 'Roman Law principle' forbids the norm-giver to require non-doable acts since no-one can be obliged to do the impossible. It will be shown that from the standpoint of standard deontic logic the use of the term 'principle' is unjustified because the Roman Law principle will be satisfied by a normative system whose norms consistently select only that which is possible.
The content Op — Op of the metanormative principle (O.OO) has played an important role in normativity theory. Aristotle's claim that in deliberation "If people meet with an impossibility, they give up" (Aristotle, Nicomachean Ethics, 1112b) can be understood as a contrapositive formulation of a related principle. In metanormative interpretation, the Aristotelian deliberation principle states that the impossible ought not to be the content of an intention. Closer to the (O.OO) principle comes the Roman Law principle ultra posse nemo obligatur (ad impossibilia nemo tenetur, impossibilium nulla obligatio), itself a predecessor of the 'ought' implies 'can' principle that Kant formulated and for which Op — Op seems to be the direct translation.14 Nevertheless, the logic of the actor's ability differs from the logic of alethic possibility. Some theorems of alethic logic fail in the logic of ability. For example, the thesis If something is the case, then it is possible, p — Op, is valid in alethic logic but its ability counterpart is not: the thesis If something is done, then it can be done fails in the logic of ability.15 The metanormative principle with alethic modality is an over-generalization of these principles: whatever is forbidden because of alethic impossibility is also forbidden by the 'ought' implies 'can' principle, but the converse does not hold.
Terminologically speaking, the use of the term 'principle' is not correct in the context of principles (O.D) and (O.RN) since Op — Op is a theorem that follows from • p — Op in conjunction with Op — Pp, i.e. from the contents of (O.D) and (O.RN). Two proofs, different in style, will be given for the fact.
Theorem 9. Op — Op e Cn({D p — Op, Op — Pp})
Proof. First, let us give a deduction proof relying on the syntax of the language. From Dp — Op and the definitions of deontic and alethic modalities we obtain the corollary: If a state of affairs is permitted to be the case, then it is possible for it to be the case, Pp — Op. Assume that p is obligatory, Op. Then p is permitted, Pp, according to axiom D. From the corollary it follows that p is possible, Op. Therefore, if a state of affairs is obligatory, then it is possible, Op — Op. Q.E.D.
14	There are numerous passages in Kant's works dealing with the principle. For example, in Religion Within the Boundaries of Mere Reason (1793), a succinct formulation is given as "duty commands nothing but what we can do" Kant (1998: 68).
15	Picking the queen of hearts out of a card deck does not imply the ability to do so; see Brown (1992).
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Proof. Second, let us give a proof in semantic terms! The basic semantic idea of modal logic is that the truth value of a formula at a point of valuation depends on the formula's truth values at other valuation points accessible via an appropriate relation. The deontic accessibility relation, Dwv, connects the world w to its normative alternatives v in which the norm-set is realized, M Q v for all v e {v | Dwx). Similarly, the alethic accessibility relation interpreted, say, as nomological possibility, Nwv, connects the world w to any of its alternatives v in which all logical and natural laws hold. For some modal formulas (Sahlqvist formulas), the corresponding first-order property of the accessibility relation can be computed using the Sahlqvist-van Benthem algorithm.16 It is known that Op — Pp determines the seriality property of the deontic relation,Vx3y Dxy. As stated above, this means that the given norm-set is consistent. Using the algorithm the following interrelation properties can be computed:
•	Op — 0p determines Vx3y (Dxy A Nxy) interrelation property. It could be termed as the 'convergent seriality property of a relation pair' and it says that there is always a deontically accessible situation which is also nomologically possible. Or to use Professor's Segerberg's metaphor, there are no tragic dilemmas (Segerberg, 2003). A set of norms which exemplifies this property provides a possible and legal way out of any situation.
•	• p — Op determines the subordination of the deontic relation under nomological VxVy(Dxy — Nxy). A set of norms can be realized only in nomo-logically possible situations: if there is a legal way out of a situation, then this is also a possible way out.17
It is easy to see that if the deontic relation is serial and subordinated to the nomological, it must always have a point of convergence with it, a point where norms are realized in a nomologically possible world.18 Therefore, 'ought' implies 'can' is not a self-justifying principle but a consequence of other principles. Q.E.D.
16	Van Benthem defines the set of formulas algorithmically translatable to their first order equivalents in the following theorem: "Theorem 19. There exists an effective algorithm which translates all modal axioms of the form A — B into corresponding first-order properties, where A is constructed from basic formulas • •» Dp using only A, V, 0, B is 'positive': constructed from proposition letters with only A, V, 0, •," (van Benthem 2010: 106).
17	For the purpose of illustration let us use the Sahlqvist-van Benthem algorithm to determine correspondences. We start with (i) Op — Op and apply standard translation in two steps: (ii) VP STx(dp — Op); (iii) VP(Vy(RNxy — Py) — Vy(ROxy — Py)). Then we determine the minimal valuation (iv) Pu := Rnxu and perform substitution: (v) Vy(RN xy — RNxy) — Vy(ROxy — RNxy). By simplification we get (vi) T — Vy(ROxy — RNxy) and, finally, (vii) VxVy(ROxy — RNxy).
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2 NORMS AND SOCIAL INTERACTION
Two actor roles in communication are commonly recognized: the role of sender and the role of receiver, but, in a norm governed social interaction, besides the roles of norm-giver and norm-subject there is an additional role, the role of norm-applier. Communication is a kind of action, and that, according to Parsons' (1937) definition, means that the sender has an aim in a situation whose conditions and means are subordinated to normative requirements.19 The last condition in Parsons' definition of action points to its normative dimension. Similarly, Habermas equates the social world with the normative context.20 The acts related to norms (promulgation, observance, application), as acts and social facts, must have their own normative contexts which, according to our hypothesis, are made explicit in their metanormative principles.
2.1 Normative contexts for norm related acts
As noted above, in a norm governed interaction there are three actor roles: the norm-giver, the norm-subject and the norm-applier role; and there are three types of norm related actions: norm-promulgation, norm-regulated action, norm-based judgement. In this kind of interaction, a norm-giver by norm-promulgation regulates the actions of a norm-subject whose observance of the norms is judged by a norm-applier.
First, we turn to the normative context of the norm-promulgation act. According to our interpretation, the language of KD logic is a description language and its axioms describe properties of norm-sets: the axiom K defines consequentiality and the axiom D defines consistency. If the promulgation of a norm-set is an act (in Parsons' sense) or a social fact (in Habermas' sense), then at least one of its properties is either permitted or forbidden. For example, if it is not considered desirable that a promulgated norm-set is inconsistent, then desirability of the consistency property constitutes the normative context for norm promulgation. This desirable property can be interpreted as a second-
19	Parsons' definition of action: "...an 'act' involves logically the following: (1) It implies an agent, an 'actor.' (2) For purposes of definition the act must have an 'end,' a future state of affairs toward which the process of action is oriented. (3) It must be initiated in a 'situation' of which the trends of development differ in one or more important respects from the state of affairs to which the action is oriented, the end. This situation is in turn analyzable into two elements: those over which the actor has no control, that is which he cannot alter, or prevent from being altered, in conformity with his end, and those over which he has such control. The former may be termed the 'conditions' of action, the latter the 'means.' Finally (4) there is inherent in the conception of this unit, in its analytical uses, a certain mode of relationship between these elements. That is, in the choice of alternative means to the end, in so far as the situation allows alternatives, there is a 'normative orientation' of action" Parsons (1937: 44).
20	Habermas writes: "A social world consists of a normative context that lays down which interactions belong to the totality of legitimate interpersonal relations" Habermas (1984: 88).
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order obligation and can be expressed by the claim that a norm-set ought to be consistent, as stated in (O.D) above. As regards the question whether it is desirable that a norm-set has all of its deductive consequences, a negative answer seems inevitable since the production of an infinite text is not a doable act. Therefore, the desirability of consistency belongs to a category different from the desirability of consequentiality or deductive closure.
Second, let us investigate the normative context of norm observance. A specific type of desirability appears in the metanormative thesis (O.T), a thesis which can be plausibly interpreted as Conformation to norms is desirable, Duty must be done, Norms ought to be realized, and so on.21 As noted above, it is not plausible, however, to interpret the thesis as a claim about a desirable property of a norm-set since the claim It is desirable that norms require only what is the case results in a kind of normative collapse. Rather, the thesis can be understood as an observance principle since it shows that a norm is that which ought to be observed. From this perspective, there is an important difference between the two meta-norms: unlike the norm-giver, the norm-subject has no obligations with respect to the properties of norm-sets, and unlike the norm-subject, the norm-giver has no obligations with respect to the observance of norms.
Third, let us discuss the normative context of norm application. The norm-applier or judge decides on the deontic status of a state of affairs brought about by the norm-subject's act. Suppose that a norm-subject has brought about that p. The norm-applier has to determine the deontic status of p with respect to some norm-set N and can do so by two logically equivalent methods: either by adding p to M and testing the consistency of the extended set A/" u {p} or by examining whether -p is a consequence of N. According to the first method, if H u {p} is not consistent, then p is forbidden, and if it is consistent, then p is permitted, as shown in (2.5) and (2.6). A similar case holds for the second method, as shown in (2.7) and (2.8).
The norm-applier performs deduction but there is no "normative system", no deductively closed set Cn(N) that needs to precede or can result from the thus
21 B. Chellas approves the use of the second order obligation within the thesis OU. O(OA • A) or O.T in our notation: "Note that OU is a theorem of deontic S5 /.../ The schema expresses the thesis that it ought to be the case that whatever ought to be the case be the case. It is a much discussed principle in deontic logic, because it is one of the few plausible cases of a theorem of the form OA in which A is non-trivial ..." (Chellas 1980: 193).
If ± e (Af u {p}), then Fp. If ± g (N u {p}), then Pp. If -p e Cn(A/"), then Fp. If -p g Cn(A0, then Pp
(2.5)
(2.6)
(2.7)
(2.8)
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obtained determination of the deontic status of the state of affairs brought about by a norm-subject act or by forbearance. Although the second-order requirement of deductive closure or the consequentiality principle does not define the perfection-property of an empirical norm-set, it does define the metanormative context for the norm-applier. The consequentiality principle shows that normative judgements ought to obey the laws of logic.
Directed second-order obligations. Different metanormative principles are attached to different roles in norm-governed interaction. While norms are always directed to norm-subjects, second-order obligations can be differentiated by their addressees as shown in Table 1. This fact indicates the need to reformulate the metanormative principles discussed in Section 1.3: second-order obligations O must be indexed by their holders' names.22 If the norm-giver role is denoted by the index g, the norm-subject role by s and the norm-applier (judge) role by j, the selection of metanormative principles obtains the following reformulation:
Og (Osp * Ps p) Os (Osp p) Oj (Os(p q) (Osp
Osq))
(Og.D) (Os.T) (Oj.K)
The reading of reformulated metanormative principles can be given in terms of modal semantics. For example, (Oj.K) reads 'The logical consequences of the norm-subject's obligations are norm-subject obligations in all the worlds where the norm-applier's obligations are satisfied'.
Table 1: The different roles in norm governed interaction and their second-order obligations.
RoLES in norm governed interaction
NORM-GIVER g
NORM-SUBJECT s JUDGE j
Their second-order obligations:
ought to create norm-sets with perfection properties ought to observe norms ought to apply norms
22 The same point has been made by Yamada (2011: 63): "The formula of the form Oif means that it is obligatory upon agent i to see to it that f. Although indexing of deontic operators with a set of agents is not standard in deontic logic, we need to be able to distinguish agents to whom commands are given from other agents if we are to use deontic logic to reason about how acts of commanding change situations".
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2.2 A perfection-property related to derogation
The dynamic phenomenon of theory revision has been first and foremost recognized within the legal tradition. Several principles for the resolution of normative inconsistency have been established thanks to the determination of hierarchical relations between norm-sets on the grounds of their generality level (lex specialis derogat legi generali), temporal precedence (lex posterior de-rogat legi priori) and legal subordination (lex superior derogat legi inferiori).23 According to Kristan (forthcoming), the principles of normative conflict are "rules about rules" which are generated by promulgation and thus belong to a norm-set. Viewed from the perspective of norm-governed interaction, these rules address the role of the norm-applier, giving a method for consistency restoration. Normative conflict shows that the norm-giver has failed to satisfy the higher-order principle of consistency or, more precisely, external consistency between norm-sets, but the requirement of consistency still holds for the norm-applier.
In the simplest case of pure derogation "the validity of a legal norm is repealed and no new one takes its place," to use Kelsen's (1973: 269) description. Kristan claims that in this case where a single norm x of a normative system is derogated "the new sets A and Cn(A) are composed of all the elements of the previous ones, except x" (and the consequences depending on x). This claim is not generally valid.
The simplest derogation corresponds to the contraction operation in AGM theory (Alchourron, Gardenfors, and Makison, 1985). Applying the notion of AGM contraction to the normative context, the following definition for the operation of pure derogation is obtained: a norm-content p of a norm-set M is derogated iff the operation results in a new set M+p which is a maximal subset of M that does not entail p. The operation of pure derogation is sub-determined since, typically, there will be more than one maximal subset of M not entailing p. The set of such sets can be called the remainder set of M by p, M _L p. It contains all and only those sets M. that satisfy the following conditions:
1.	The preservation condition: a new norm-set resulting from derogation is a subset of the original set, M. £ M.
2.	The non-entailment condition: a new set does not entail the derogated norm, p i Cn(M).
3.	The maximality condition: a new set retains the maximal number of norms from the original set, there is no M'such that M. c A4' £ M and p i Cn(M.').
23 The axioms for complex hierarchical relations resulting from combinations of grounds are given in Malec (2001).
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Analogously to the contraction operation, the operation A/Vp of pure derogation needs an additional choice operation y to pick a member of the remainder set: A/Vp = y(A/" _L p). The special and neat case of pure derogation arises when the norms of the initial norm-set are independent, i. e. when no norm from the set is entailed by the rest, i. e. p £ Cn(N - {p}) for all p e Af. Only in this special case does it hold that pure derogation imposes no need to choose since there is exactly one member in the remainder set, namely N - {p}, (2.9).
If p £ Cn(M - {p}) for all p e M, then M+p = M - {p}	(2.9)
In view of possible derogation, independence turns out to be another perfection property of a norm-set, one that by relieving the burden of choice from the norm-applier enables "uniformity of judicial practice". If a norm-set does not have the independence property, then pure derogation could lead to the switching of roles: by being forced to choose between the elements of the remainder set, the norm-applier actually becomes the norm-giver.
— Acknowledgments.— The preparation of this paper has been sup-ported by a grantfrom the Faculty of Humanities and Social Sciences-University of Split. The authors would like to thank the anonymous referees for critical reading of the manuscript and constructive suggestions.
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SYNOPSES, KEYWORDS, AND BIOGRAPHICAL NOTES 148
Synopsis
Berislav Žarnic Gabriela Bašic
Metanormative Principles and Norm Governed Social Interaction
SLOV. | Metanormativna načela in z normami urejano družbeno življenje. Če kritično preverimo opredelitev normativnega sistema, ki sta jo z uporabo teorije množic podala Alchourron in Bulygin, ugotovimo, da deduktivna zaprtost ni njegova neizbežna lastnost. Sledeč von Wrightovemu sklepu, da aksiomi standardne deontične logike opisujejo popolnostne lastnosti množice norm, avtorja predstavita algoritem, ki modal-ni jezik prevaja v jezik teorije množic. Prevodi nam razkrijejo, da ima verodostojnost metanormativnih načel različne temelje. Ob ločitvi igralnih vlog, ki jih imajo udeleženci z normami urejane interakcije, se izkaže, da so metanormativna načela usmerjene obveznosti drugega reda, predvsem pa, da so zahteve, povezane z deduktivno zaprtostjo, naslovljene na vlogo uporabnika norm in ne na vlogo njihovega ustvarjalca. Če ločevanje vlog uporabimo tudi na primeru čiste derogacije, pridemo do novih zaključkov. Ugotovimo namreč, da je neodvisnost iz vidika morebitne derogacije popolnostna lastnost dane množice norm. Avtorja se tako polemično dotikata nekaterih točk, ki jih je nedavno v svojem članku izpostavil Kristan.
Ključne besede: normativni sistem, standardna deontična logika, metanormativna načela, derogacija, G. H. von Wright
ENG. I Critical examination of Alchourron and Bulygin's set-theoretic definition of normative system shows that deductive closure is not an inevitable property. Following Von Wright's conjecture that axioms of standard deontic logic describe perfection-properties of a norm-set, a translation algorithm from the modal to the set-theoretic language is introduced. The translations reveal that the plausibility of metanormative principles rests on different grounds. Using a methodological approach that distinguishes the actor roles in a norm governed interaction, it has been shown that metanormative principles are directed second-order obligations and, in particular, that the requirement related to deductive closure is directed to the norm-applier role rather than to the norm-giver role. The approach has been applied to the case of pure derogation yielding a new result, namely, that an independence property is a perfection-property of a norm-set in view of possible derogation. This paper in a polemical way touches upon several points raised by Kristan in his recent paper.
Keywords: normative system, standard deontic logic, metanormative principles, derogation, G. H. von Wright
Summary: 1. The Normative System as a Set of Norms. — 1.1. Consistency and Deductive Closure. — 1.2. Perfection Properties and Norm-sets. — 1.2.1. Translating Modal Language to Set-theoretic Language. — 1.2.2. On the Possibility of Creating a Norm-system by Norm-promulgation. — 1.3. Metanormative Principles. — 1.3.1. Roman Law Principle as a Norm for the Norm-giver. — 2. Norms and Social Interaction. — 2.1. Normative Contexts for Norm Related Acts. — 2.2. A perfection-property to derogation.
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Berislav Žarnic is a Professor in Philosophy at the University of Split, Faculty of Humanities and Social Sciences. I Address: Sveučilište u Splitu, Filozofski fakultet, Sinjska 2, 21000 Split. E-mail: bzarnic@ffst.hr.
Gabriela Bašic is an Assistant in Philosophy at University of Split, Faculty of Humanities and Social Sciences. I Address: Sveučilište u Splitu, Filozofski fakultet, Sinjska 2, 21000 Split. E-mail: gbasic@ffst.hr.
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