Completeness Preservation
Carlos Caleiro 1
Walter Carnielli 2
Marelo E.Coniglio 2
Amilar Sernadas 1
CristinaSernadas 1
1
CMA,Departamento deMatematia,IST,Portugal
E-mail:fal,as,ssgmath.ist.utl.pt
2
DepartamentodeFilosoa,IFCH,UNICAMP, Brazil
E-mail:farniell,onigliogle.uniamp.br
Abstrat
Fibringhasbeenshowntobeusefulforombininglogisendowedwith
truth-funtional semantis. Onewondersifbringanbeextended inor-
der to ope with logisendowed with non-truth-funtional semantisas,
for example,paraonsistentlogis. Therstmainontribution ofthepa-
perisapositiveanswertothisquestion. Furthermore,itisshownthatthis
extendednotionofbringpreservesompletenessunderertainreasonable
onditions. This ompletenesstransferresult, theseond main ontribu-
tion of the paper, generalizesthe oneestablished byZanardo et al. and
is obtainedusing a newtehniqueexploiting the properties ofthe meta-
logiwherethe(possiblynon-truth-funtional)valuationsaredened. The
modalparaonsistentlogiofdaCostaandCarnielliisobtainedbybring
and itsompletenessissoestablished.
1 Introdution
In reent years, the problem of ombining logis has gained the attention of
many researhers in mathematial logi. Besides leading to very interesting
appliationswheneverit isneessary to work withdierent logis at the same
time, ombinations of logis are also of great interest on purely theoretial
grounds[3 ℄.
Thepratialimpatoftheproblemislear,at leastfromthepointofview
ofthoseworkinginknowledge representation(withinartiialintelligene)and
in formal speiation and veriation (within software engineering). Indeed,
intheseelds,theneedforworkingwithseveralformalismsatthesame timeis
theruleratherthantheexeption. Forinstane,inaknowledge representation
problemit maybe neessaryto work withbothtemporaland deonti aspets.
This work isa naturalsequelof theworkpresented in[5 ℄, whih startedduring avisit
by the third author to CMA at IST, entirely supportedby Funda~ao para a Ci^enia e a
Tenologia,Portugal. A subsequentvisittoCMAbytheseondandthird authorswasalso
supportedbyCoordena~ao deAperfeioamento doPessoaldo EnsinoSuperior,Brazil.
equationaland temporalspeiations.
>From a theoretial point of view, one might be tempted, for instane, to
lookatprediatetemporallogiasresultingfromtheombinationofrstorder
logiand propositionaltemporallogi. Buttheapproahwillbeofsigniane
onlyifgeneral preservation resultsareavailableaboutthemehanismused for
ombiningthelogis. Forexample,ifithadbeenestablishedthatompleteness
is preserved by the ombination mehanism Æ and it is known that logi L is
given byL 0
ÆL 00
,thentheompletenessoftheombinationLwouldfollowfrom
theompleteness ofL 0
and L 00
. No wonder that somuh theoretial eorthas
beendediated to establishingpreservationresultsand/or ndingpreservation
ounterexamples withintheommunity oflogiians workingin theproblemof
ombininglogis.
Among thedierent tehniquesforombininglogis,bring [11 ,12 ,21 ,22 ,
23 ℄deserves loseattention. But what is bring? The answeran be given in
a few paragraphs for the speial ase of logis with a propositional base, that
is,withpropositional variablesand onnetives ofarbitrary arity.
The language of the bring is obtained by the free use of the language
onstrutors (atomi symbols and onnetives) from the given logis. So, for
example,whenbringatemporallogiandadeontilogi,mixedformulae,like
((G)(O(F))), appearintheresultinglogi. Naturally,inmanyases one
wants to share some of the symbols. The previous example would involve the
onstrainedform ofbringimposedbysharing aommon propositionalpart.
At the dedutive system level, provided that the two given logis are en-
dowed with dedutive systems of the same type, the dedutive system of the
bringwillbeobtainedbythefreeuseoftheinferenerulesfromboth. Thisap-
proahwillbe ofinterestonlyifthetwogivendedutivesystems areshemati
inthesensethattheirinferenerulesareopenforappliationto formulaewith
foreignsymbols. For instane,when one representsModus Ponens by therule
MP, f(
1
2 );
1 g `
2
, in some Hilbert system, one may impliitly assume
that theinstantiation of the shema variables
1
;
2
by any formulae, possibly
withsymbolsfrom bothlogis,isallowedwhen applyingMP inthebring.
Althoughbringatthesyntatilevelisquitesimpleasdesribedabove,the
semantis of bring is muh more ompliated and it is advisable to onsider
only the speial ase where both logis have semantis with similar models.
Following [21 , 23 ℄, a onvenient, but quite general, model for a wide lass of
logiswith propositionalbase isprovidedbya triplehU;B;i whereU isa set
(of points, worlds, states, whatever), B }U, and () : B n
! B for eah
languageonstrutor ofarityn0. We lookat thepair hB;i asanalgebra
oftruth-values. In thissense, suh modelsaresaid to betruth-funtional.
Given two logis L 0
;L 00
with modelsof thistype, what is the semantis of
their bring? As rst shown in [21 ℄, it is a lass of models of the same type,
suhthatat eahpointu2U itispossibleto extratamodelfromL 0
and one
fromL 00
. Clearly,ifsymbolsareshared,thetwoextratedmodelsshouldagree
on them. In order to visualize the semantis of bring, onsider the bring
ofa propositional linear temporal logi witha propositionallinear spae logi.
Eah model of thebring willbe a loud U of points suh that at eah point
the point hBerlin,10h15m 25 Marh 2000i one knows the time line (of past,
present and future) of Berlin and the spae line(the universe taken as a line
forthesake of the example)at thattime. However, sofar, asskethed above,
thoseworkingon thesemantisofbringhaveassumedthatthelogisat hand
are truth-funtional and, therefore, have not onsidered non-truth-funtional
logislikeparaonsistent logis.
Paraonsistent logis were introdued in [6 ℄ and sine then have been the
objetof ontinuedattention, beause of theirtheoretial and pratial signif-
iane. In partiular, the paraonsistent systems C
n
of [6℄ are subsystems of
propositional lassiallogi inwhih thepriniple of Pseudo Sotus ; :
`Æ
doesnothold. Somesystemsofparaonsistent modallogiwereinvestigatedin
[9 ,10 , 8 ,20 ℄. Onemight wonder ifwe shouldtry to obtainsuh a mixedlogi
asthebringoftheunderlyingmodal andparaonsistent logis.
Therstmainontributionofthispaperisapositiveanswertothisquestion.
To this end, previous work on the semantis of bring is extended to a more
general notion of logi system enompassing the ase of non-truth-funtional
valuations. Furthermore, it is shown that thisextended notion of bring pre-
serves ompleteness under reasonable onditions. This ompleteness transfer
result, the seond main ontribution of this paper, generalizes the one estab-
lishedin[23 ℄andisobtainedusinganewtehniqueexploitingthepropertiesof
themeta-logiwherethe(possiblynon-truth-funtional)valuationsaredened.
In Setion 2, the notion of interpretation system presentation as a spe-
iation of the intended valuations within a suitable meta-logi of equational
natureisintrodued. Theinterpretation strutures appearasmodels(algebras)
of the speiation. We adopt CEQ (onditional equational logi [13, 18℄) as
themeta-logi. Setion3 denes thenotions of unonstrained and onstrained
bring of interpretation system presentations. The mainexample, bring the
paraonsistent system C
1
and the modal system KD, is disussed inSetion 4
at the semanti level. Setion 5 ontains a brief aount of the appropriate
proof-theoretinotionsand returnstothemainexampleatthededutionlevel.
Setion6 establishes the ompletenesspreservation theorem and appliesit for
proving the ompleteness of the modal paraonsistent logi in [8 ℄. Setion 7
disussesappliationsof self-bring,namely inthe ontext ofthe C
n
hierarhy
ofparaonsistentsystems. Setion8onludeswithan assessment ofwhatwas
ahievedand what lays ahead.
2 Speifying valuation semantis
Sinetheobjetlogisunderinvestigationarepropositional-based,thefollowing
notionofsignaturesuÆes forourpurposes:
Denition2.1 Anobjetsignature isafamilyC=fC
k g
k2N
whereeah C
k is
aset (of onnetives of arityk).
Inpartiular, thesetof propositional symbolsis inludedinC
0 .
1 2
shemavariables,to beusedininferenerules,suhthat \C
0
=;.
We denote byL(C ;) thesetof shema formulae indutivelybuiltfromC
and. Forexample,
1
(p_ :
2
)isa shemaformula, ifp2C
0 ,
:
2C
1 and
;_2C
2 .
Our next step willbeto denean equational signatureinduedbya given
objetsignatureC asa meta-linguistideviewhihpermitsto talkaboutthe
semantis of logis based on C. For this purpose, it is onvenient to onsider
two new sorts, (for formulae) and (for truth-values). As usual, denotes
theemptystring over S=f;g and, if w2S
and s2 S,then O
ws
denotes
thesetof operationswithdomainw and odomain s.
Denition2.2 GivenanobjetsignatureC,theindued meta-signatureisthe
2-sortedequational signature(C ;)=hS;Oi where S=f;g and:
O
=C
0 [;
O
k
=C
k
fork>0;
O
=fvg;
O
=f>;?g;
O
=f g;
O
=fu;t;)g;
O
!s
=;in theother ases.
Weshalluse(C)todenotethesubsignature(C ;;),thatis,whereO
=C
0 .
Inthisontext, and are thesortsof shemaformulae and truth-values,
respetively. The symbols >, ?, , u, t and ) are used as generators of
truth-values. Thesymbolv willbeinterpretedas avaluationmap.
We onsider the following sets of variables for (C) and (C ;): X
=
fx
1
;x
2
;:::g and X
= fy
1
;y
2
;:::g. Reall that a term t is alled a ground
term if it does not ontain variables, and that a substitution is said to be
groundifit replaesevery variable byaground term.
We want to write valuation speiations (within the adopted meta-logi
CEQ) over (C) and X. Reall thata CEQ-speiation is omposedof on-
ditionalequationsof thegeneralform:
(equation
1
& ::: &equation
n
! equation)
withn0. Conditionalequations areuniversallyquantied,although,for the
sake of simpliity, we omit the quantier, ontrarily to the notation used in
[18 ℄. Forexample, (!v(y
1
^y
2
)=u(v(y
1 );v(y
2
)))isaonditionalequationof
sort , supposingthat ^2 C
2
. It is lear, from this and also theforthoming
examples, that we only need to onsider speiations ontaining exlusively
onditionalequations(ormeta-axioms) ofsort. Suh speiationsarealled
-speiations inthesequel.
itmight seemthatwe have two dierent ways to represent arbitrary formulae:
by meansof propositional shema variables(i.e.,
1
;
2
, et.) and bymeans of
variablesofsort(i.e.,y
1
;y
2
,et.). Theformershallindeedrepresentarbitrary
formulaebutonlyintheontext ofHilbertCaluli(tobedenedinSetion5).
The latter represent arbitrary formulae inthe meta-language of CEQ. In this
meta-language,propositionalshema variablesappearasonstants.
Denition2.3 Aninterpretationsystempresentation(isp)isapairS =hC ;Si
whereC is anobjetsignatureand S is a-speiationover(C).
Denition2.4 Given an isp S, the lass Int (S) of interpretation strutures
presented by S is the lass of all Heyting algebras over (C ;) satisfying the
speiationS.
We denote by S
thespeiationomposed of themeta-axioms inS plus
-equations over (C) speifyingthe lass of all Heyting algebras. Note that
Int(S), that is, the lass of all algebras over (C ;) satisfying S
, is always
non-empty. Indeed, the trivialalgebra with singleton arrier sets for all sorts
satisesanysetof onditional equations.
Forthesakeofeonomyofpresentation,weintroduethefollowingabbrevi-
ations: x
1 x
2
foru(x
1
;x
2 )=x
1
, and ,(x
1
;x
2
) foru()(x
1
;x
2 );)(x
2
;x
1 )).
The relation symbol denotes a partialorder on truth-values. Furthermore,
thepartialorderisaboundedlattiewithmeet u,joint,top>andbottom?
(f. [2 ℄). Asexpeted,givenan algebraA,a
1
A a
2
and,
A (a
1
;a
2
)areabbre-
viationsof u
A (a
1
;a
2 )=a
1 andu
A ()
A (a
1
;a
2 );)
A (a
2
;a
1
)), respetively. It is
also well known that the Heyting algebra axioms further entail the following
result:
Proposition 2.5 Let S be an isp, t
1 and t
2
terms of sort and A 2 Int(S).
Then,forevery assignment overA:
[[t
1
℄℄
A
A [[t
2
℄℄
A
i )
A ([[t
1
℄℄
A
;[[t
2
℄℄
A )=>
A
and
[[t
1
℄℄
A
=[[t
2
℄℄
A
i ,
A ([[t
1
℄℄
A
;[[t
2
℄℄
A )=>
A :
As explained, our framework is intended to study properties of bring of
non-truth-funtionallogisingeneral. We nowillustratethenotionofispwith
two examplesthat willbe usedthroughout therest ofthe paper.
Example2.6 Paraonsistent systemC
1 [6 ℄:
Objet signature- C:
{ C
0
=fp
n
:n2Ng[ft;fg;
{ C
1
=f:g;
{ C
2
=f^;_;g.
{ Truth-valuesaxioms{furtheraxiomsinorder to obtainaspeia-
tion of thelassof all Boolean algebras,e.g.,adding theequation:
( ! ( (x
1 ))=x
1 ).
{ Valuation axioms:
( !v(t)=>);
( !v(f)=?);
( !v(y
1
^y
2
)=u(v(y
1 );v(y
2 )));
( !v(y
1 _y
2
)=t(v(y
1 );v(y
2 )));
( !v(y
1 y
2
)=)(v(y
1 );v(y
2 )));
(v(y
1
)=?! v(:
y
1
)=>);
(v(::y
1
)=>!v(y
1
)=>);
(v(y Æ
2
)=>&v(y
1 y
2
)=>&v(y
1 :y
2
)=>!
v(y
1
)=?);
(v(y Æ
1
)=>&v(y Æ
2
)=>!v((y
1
^y
2 )
Æ
)=>);
(v(y Æ
1
)=>&v(y Æ
2
)=>!v((y
1 _y
2 )
Æ
)=>);
(v(y Æ
1
)=>&v(y Æ
2
)=>!v((y
1 y
2 )
Æ
)=>).
As usualin theC
n
systems, Æ
isan abbreviationof:(^:).
ThereadershouldbewarnedthatweareusingBooleanalgebrashereasa
metamathematial environment suÆient to arryout the omputations
of truth-valuesfortheformulaeinC
1
. Speiallywe arenotintroduing
anyunaryoperatorintheBoolean algebrasorrespondingto paraonsis-
tent negation, but we are omputing the values of formulae of the form
:
bymeansofonditional equationsinthe algebras. Inother words, :
doesnotorrespondtotheBooleanalgebraomplement . Thereforewe
are notattahing anyalgebraimeaning to C
1 .
1
4
Intheontext ofanispS,atruth-funtionalderived onnetive ofarityk is
a-termy
1 :::y
k
:Æwherethesetofshemavariablesourringintheshema
formula Æ isfy
1
;:::;y
k
g andsuhthat some equationof theform
v(Æ)=t v(y)
x
is derived from S
(within the meta-logi CEQ), where t is some -term in
whihonlythevariablesx
1
;:::;x
k
mayour,and v(y)
x
isthesubstitutionsuh
that v(y)
x (x
n
)=v(y
n
) forevery n1.
In C
1
,lassialnegation := y
1 :
:
y
1
^y Æ
1
(take tas (x
1
)) and equiva-
lene := y
1 y
2 :(y
1 y
2 )^(y
2 y
1
) (take t as ,(x
1
;x
2
)) are bothtruth-
funtionalderivedonnetives. And,ofourse,soaretheprimitiveonjuntion
y
1 y
2 :y
1
^y
2
,disjuntiony
1 y
2 :y
1 _y
2
,and impliationy
1 y
2 :y
1 y
2 . But,
paraonsistent negationy
1 :
:
y
1
is nottruth-funtional.
1
The question of algebraizing paraonsistent logi is a separate issue and we refer the
interestedreaderto[19 ,16℄.
modality y
1 :Ly
1
would requirein (C) the extra generator inO
satis-
fying:
( !(>)=>);
( !(u(x
1
;x
2
))=u((x
1 );(x
2 )));
( !u((x
1
); (( (x
1
)))=(x
1 ));
( !v(Ly
1
)=(v(y
1 ))).
Example2.7 Modal systemKD [14 , 15 ℄:
Objet signature- C:
{ C
0
=fp
n
:n2Ng[ft;fg;
{ C
1
=f:;Lg;
{ C
2
=f^;_;g.
Meta-axioms - S:
{ Truth-valuesaxioms:
Further axiomsinorder to obtaina speiationof the lassof
all Boolean algebras asinthepreviousexample.
{ Valuation axioms:
( !v(t)=>);
( !v(f)=?);
( ! v(:
y
1
)= (v(y
1 )));
( !v(y
1
^y
2
)=u(v(y
1 );v(y
2 )));
( !v(y
1 _y
2
)=t(v(y
1 );v(y
2 )));
( !v(y
1 y
2
)=)(v(y
1 );v(y
2 )));
( !v(Lt)=>);
( !v(L(y
1
^y
2
))=u(v(Ly
1
);v(Ly
2 )));
( !u(v(Ly
1
);v(:L:y
1
))=v(Ly
1 );
(v(y
1
)=v(y
2
)!v(Ly
1
)=v(Ly
2 )).
Note that the last four axioms are related to the axioms on . Their
inlusionallowsusto speifytheintendedmodalalgebraswhileavoiding
the useof thethatis notinthesignature(C). 4
Returning to the rst example (paraonsistent system C
1
), it is straight-
forward to verify that every paraonsistent valuation introdued in [7 ℄ has a
ounterpart, up to isomorphismbeause of the built-in freedom in the truth-
values, inInt(S). Furthermore,the additionalinterpretation strutures do not
hange thesemanti entailment (asdened below). Notethat it iseasy to ex-
tend this example in order to set up the isp's for the whole hierarhy C
n by
speifyingtheparaonsistent n-valuationsintroduedin[17 ℄.
forwardtoverifythateveryKripkemodelhasaounterpartinInt(S): onsider
thealgebraoftruth-valuesgiven bythepowersetofthesetofworlds. Further-
more,every generalmodelin[23 ℄ also has aounterpart inInt(S): take hB;i
as the algebra of the truth-values. Again, the extra interpretation strutures
donothangethe semanti entailment.
We are now ready to dene the (global and loal) semanti entailments.
To this end, we need to refer to the denotation [[t℄℄
A
of a meta-term t given
an assignment over an algebra A. As expeted, an assignment maps eah
variable to an element inthearrierset of thesort ofthe variable. In thease
ofa groundterm t,as usual,we justwrite[[t℄℄
A
forits denotation inA.
Denition2.8 Given an isp S, a set of shema formulae and a shema
formula Æ, we saythat:
S
g
Æ ( globally entails Æ) i, for every A 2 Int(S), v
A ([[Æ℄℄
A ) = >
A
wheneverv
A ([[℄℄
A )=>
A
forevery 2 ;
S
l
Æ ( loally entails Æ)i,foreveryA2Int(S)andb2A
,v
A (b)
A
v
A ([[Æ℄℄
A
) wheneverv
A (b)
A v
A ([[℄℄
A
) forevery 2 .
Observe that S
l
Æ implies that S
g
Æ, provided that v
A
(b) = >
A for
some b 2 A
. On the other hand, if either = ; or A
= f?
A
;>
A g with
?
A
A
>
A
,then S
l
Æ i
S
g Æ.
Beforeturning ourattention to theproblemof bringisp's, we rst dene
theategories Sig and Isp. The objets of theategory Sig areobjet signa-
tures. A morphismh :C ! C 0
inSig is of the form h =fh
k :C
k
!C 0
k g
k2N ,
eahh
k
beingamap. TheobjetsoftheategoryIspareisp's. Theappropriate
notionofmorphismintheategory Ispisasfollows: eah h:hC ;Si!hC 0
;S 0
i
isamorphismh:C !C 0
inSigsuh thatforeahs2S,
^
h(s)is inS 0
. Here,
^
h is the unique map from the meta-language over (C) to the meta-language
over (C 0
) that extends the map h. Note that suh a morphism imposes the
onditionthatforevery A 0
2Int(S 0
)its redutto(C), viah andtheidentity
onthe generators,is inInt(S).
In what follows, we shall also use the forgetful funtor N from Isp to the
ategorySig ofobjet signatures. Thisfuntor mapseah S to theunderlying
objet signature C and eah morphism h to the underlying objet signature
morphism.
3 Fibring non-truth-funtional logis
We assume that we are given two isp's S 0
and S 00
. Following the method
proposed in [21 ℄, we start by onsidering the notion of unonstrained bring
thatorrespondstoombiningthetwoisp'swithoutsharinganyofthesymbols
of the objet signatures C 0
and C 00
. That is, if we so ombine C
1
and KD we
shall obtain inthe result of the bring two dierent symbols for onjuntion,
disjuntion, et. This onstrution appears as the oprodut of S and S in
theategory Isp(G). Therefore,
S 0
S 00
=hC ;Si
where:
C =C 0
C 00
withinSig withtheinjetionsi 0
andi 00
;
S =
^
i 0
(S 0
)[
^
i 00
(S 00
).
Proposition 3.1 GivenS 0
andS 00
asabove,a(C 0
C 00
;)-algebraAbelongs
to Int(S 0
S 00
) i:
Aj i
0
(C 0
;)
2Int(S 0
);
Aj i
00
(C 00
;)
2Int (S 00
);
whereAj i
0
(C 0
;)
and Aj i
00
(C 00
;)
arethe reduts of A to thesignatures (C 0
;)
and(C 00
;),respetively,viatheindiatedinlusionsand theidentityon the
generators.
Again following the method in [21 ℄, it is easy to introdue the notion of
onstrained bring by sharing onnetives and/or propositional symbols that
orrespondstoombiningthetwoisp'swhilesharingsomeofthesymbolsofthe
objetsignatures C 0
and C 00
: theonstrution appearasa o-Cartesian lifting
by thefuntor N along the signatureoequalizer for the envisaged pushoutof
the signatures. We refrain from dwellingfurther on this onstrution sine it
doesnotbringanyinsighttothemainissueofthispaper(thatis,thebringof
logispossiblywithnon-truth-funtionalsemantis). Foraninterestingexample
ofonstrained bringsee thenext setion.
4 Appliation I: modal paraonsistent logi
The idea is to ombine C
1
and KD by bring them while sharing the propo-
sitional symbols, onjuntion, disjuntion, impliation, true and false. Let
S 0
=hC 0
;S 0
i be theisp for C
1
as desribed inExample2.6 and S 00
=hC 00
;S 00
i
theispforKD asdesribed inExample2.7.
We workrst inthe ategory Sig. Consider thepropositional-basedsigna-
tureB asfollows:
B
0
=fp
n
:n2Ng[ft;fg;
B
2
=f^;_;g;
B
k
=; fortheother valuesofk.
Thepropositional-basedsignatureC 0
fBf
C
00
oftheenvisagedonstrained
bringis obtained by the pushout of C 0
f 0
B f
00
! C 00
where f 0
and f 00
are the
obvious inlusions.
The uniquemorphismz fromthe oprodutC 0
C 00
to C 0
f 0
Bf 00
C
00
is now
usedto obtaintheenvisaged ispasbrieydesribed attheendof theprevious
setion. Theisp we want,
S 0
f 0
Bf 00
S
00
;
orrespondingtotheonstrainedsharingofS 0
andS 00
whilesharingthesymbols
in B, appears as the o-Cartesian lifting by the funtor N along z on the isp
S 0
S 00
. That is,
S 0
f 0
Bf 00
S
00
=hC 0
f 0
Bf 00
C
00
;z(^
^
i 0
(S 0
)[
^
i 00
(S 00
))i:
Asexpeted,theinterpretationstruturespresentedbythisisparepreisely
thosealgebras inthe unonstrainedbringthatagree on thesharedsymbols.
Note that we end up having two negations: z(i 0
(:)) oming from C 0
and
z(i 00
(:))
oming from C 00
. The former is a paraonsistent negation and the
latteristhelassialnegationinheritedfromKD. Clearly,the(lassial)strong
negationy
1 ::y
1
^y Æ
1
inheritedfromC
1
ollapses into z(i 00
(:)).
One partiularlyinteresting appliation of modal paraonsistent logi is in
oping with ertain problems of deonti logi having to do with deonti para-
doxes and moral dilemmas. In [8 ℄ a paraonsistent deonti logi alled C D
1 is
introdued by ombining the paraonsistent system C
1
with modal KD inter-
preting the modal operator L as \obligatory". In order to reover C D
1
at the
semanti level in our setting we have to add one additional meta-axiom for
valuations:
(v(y Æ
1
)=>!v((Ly
1 )
Æ
)=>):
Using the terminology introdued in [4℄, this proedure an be seen as a
splittingof C D
1
intheomponentsKD and C
1 .
5 Logi systems
This setion is devoted to extending the bring tehniques to the syntatial
ounterpart of isp's. As we have hinted before,we shall use the propositional
shemavariablesin=f
1
;
2
;:::g to writeinferenerules.
Denition5.1 A Hilbert alulus over is a triple hC ;P;Di in whih (1) C
is an objet signature, (2) P is a subset of }
n
L(C ;)L(C ;), (3) D is a
subsetof(}
n
L(C ;)n;)L(C ;),and (4)DP.
Givenanyr=h ;iinP,the(nite)set isthesetofpremisesofr and
is theonlusion; we willoften writer =hPrem (r);Con (r)i. If Prem (r)=;,
then r is said to be an axiom shema; otherwise, it is said to be a proof rule
shema. Eah r inD issaid to bea derivation rule shema.
P =fh;;
1 (
2
1 )i;
h;;(
1
(
2
3
))((
1
2 )(
1
3 ))i;
h;;(:
1 :
2 )(
2
1 )i;
h;;L(
1
2
)(L
1 L
2 )i;
h;;L
1
:
L :
1 i;
h;;(
1 _
2
)(:
1
2 )i;
h;;(
1
^
2
):(:
1 _:
2 )i;
h;;t(
1
1 )i;
h;;f (
1
^ :
1 )i;
hf
1
;
1
2 g;
2 i;
hf
1 g;L
1 ig;
D=fhf
1
;
1
2 g;
2 ig.
Example5.3 TheHilbertalulusC
1
iseasilydenedadaptingthewellknown
axiomatispresentedin[7 ℄.
Denition5.4 A shema formula Æ 2 L(C ;) is provable from the set of
shema formulae L(C ;) in the Hilbert alulus hC ;P;Di, denoted by
` PD
p
Æ, i there is a sequene
1
;:::;
m
2 L(C ;) +
suh that
m
=Æ and,
fori=1 to m,either
(1)
i
2 ,or
(2)there existaruler2P and ashemavariablesubstitution :!L(C ;)
suhthat Con(r)=
i
and Prem (r)f
1
;:::;
i 1 g.
Denition5.5 A shema formula Æ 2 L(C ;) is derivable from the set of
shema formulae L(C ;) in the Hilbert alulus hC ;P;Di, denoted by
` PD
d
Æ, i there is a sequene
1
;:::;
m
2 L(C ;) +
suh that
m
=Æ and,
fori=1 to m,either
(1)
i
2 ,or
(2)
i
isprovable from theemptysetof formulae,or
(3) there exist a rule r 2 D and a shema variable substitution suh that
Con(r)=
i
andPrem (r)f
1
;:::;
i 1 g.
Clearly,if ` PD
d
Æ then also ` PD
p
Æ. Furthermore,if ` PD
p
Æ then` PD
d Æ.
The following struturality properties are also immediate: for every shema
variable substitution , if ` PD
p
Æ then ` PD
p
Æ, and if ` PD
d
Æ then
` PD
d Æ.
Denition5.6 The unonstrained bring of the Hilbert aluli hC 0
;P 0
;D 0
i
andhC 00
;P 00
;D 00
i istheHilbertalulus
hC 0
;P 0
;D 0
ihC 00
;P 00
;D 00
i = hC 0
C 00
;
^
i 0
(P 0
)[
^
i 00
(P 00
);
^
i 0
(D 0
)[
^
i 00
(D 00
)i:
Denition5.7 The onstrained bring of the Hilbert alulihC ;P ;Di and
hC 00
;P 00
;D 00
i sharing C aording to the injetive morphismsf 0
:C ! C 0
and
f 00
:C ! C 00
is the Hilbert alulushC 0
;P 0
;D 0
i f
0
Cf 00
hC
00
;P 00
;D 00
i dened as
follows:
hC 0
f 0
Cf 00
C
00
;z(^
^
i 0
(P 0
))[z(^
^
i 00
(P 00
));z(^
^
i 0
(D 0
))[z(^
^
i 00
(D 00
))i:
ByadoptingthenotionofHilbertalulusmorphismproposedin[21 ℄,both
formsof bringabove again appearasuniversal onstrutions(oprodut and
o-Cartesianlifting). ItfollowsthatthereisamorphismfromeahgivenHilbert
alulustothe bring,e.g., h 0
fromhC 0
;P 0
;D 0
ito hC ;P;Di. Therefore:
if ` P
0
D 0
p
Æ then
^
h 0
( )` PD
p
^
h 0
(Æ);
if ` P
0
D 0
d
Æ then
^
h 0
( )` PD
d
^
h 0
(Æ).
ThebringofHilbertaluliforC
1
and KD,sharingthepropositionalsym-
bols,onjuntion,disjuntion,impliation,trueandfalse,istheHilbertalulus
wherewe have alltheproofand derivation rulesforbothC
1
andKD. In order
to getthedeonti paraonsistentsystem C D
1
of [8℄,at theproof-theoreti level,
we needto introduethefollowingproof rule:
h;;
Æ
1
(L
1 )
Æ
i:
Denition5.8 AlogisystemisatupleL=hC ;S;P;DiwherethepairhC ;Si
onstitutes anisp and thetriplehC ;P;Di onstitutes a Hilbertalulus.
As expeted, the (unonstrained and onstrained) bring of logi systems
is obtained by the orresponding bring of the underlying isp's and Hilbert
aluli.
Example5.9 The logi systems for C
1
and KD will be denoted by L
C
1 and
L
KD
,respetively,andtheirbringwhilesharingB willbedenotedbyL
C
1 KD
.
Denition5.10 GivenalogisystemL=hC ;S;P;Di,wesaythatthededu-
tionsystemhC ;P;Di issound w.r.t. the isp hC ;Si, orsimplythatL is sound,
iforeveryset ofshema formulae andevery shema formulaÆ:
`
PD
p
Æ implies S
g Æ;
`
PD
d
Æ implies S
l Æ.
We say that hC ;P;Di is adequate w.r.t. hC ;Si, or simply that L is adequate,
iforeveryset ofshema formulae andevery shema formulaÆ:
`
PD
p
Æ is impliedby S
g Æ;
`
PD
d
Æ is impliedby S
l Æ.
Furthermore,we saythat hC ;P;Di isomplete w.r.t. hC ;Si,or simplythat L
isomplete, iit isbothsoundand adequate.
Example5.11 The logisystemsL
C
1
and L
KD
areomplete.
Themaingoalistoestablishthepreservationofompletenessbybring. Tothis
end, it is onvenient to take advantage of the ompleteness of CEQ asproved
forinstanein[13 ,18 ℄byenodingthededutionsystemofthemeta-logiCEQ
intheobjetHilbertalulus.
Inorderto dealwithloalreasoningatthemeta-level,weshalltake advan-
tageof thefollowingtwo shema variablesubstitutions:
+1
suh that +1
(
i )=
i+1
forevery i1;
1
suh that 1
(
1 )=
1 and
1
(
i )=
i 1
forevery i2.
Notethatif isashemaformulathen +1
isavariantof where
1 does
notour. Furthermore, easily, +1
1
=.
6.1 Enoding
Firstwe analyse what an be obtainedproof-theoretially withinCEQ. Given
an isp S = hC ;Si, we adopt the following abbreviations, where [fÆg
L(C ;):
`
S
g
Æ for S
[f(!v()=>): 2 g` CEQ
(C ;)
v(Æ)=>;
`
S
l
Æ for S
[f(!v(
1
)v(
+1
)): 2 g` CEQ
(C ;) v(
1
)v(Æ +1
).
Theorem 6.1 Given an ispS =hC ;Si and [fÆgL(C ;),we have:
S
g
Æ i `
S
g Æ;
S
l
Æ i `
S
l Æ.
Proof: This is an immediate onsequene of the ompleteness of the meta-
logiCEQ. Inthe loalase it isessential to notethat, sineshema variables
annot our in S
, we an freely hange the denotation of shema variables
given by an algebra A 2 Int (S) (namely aording to +1
or 1
) and still
obtainan algebrainInt (S). Thefatthat
1
annot our inshemaformulae
instantiatedby +1
doestherest. QED
Fortheenvisaged enoding,itis neessaryto assume thatthelogisystem
at handis suÆientlyexpressive:
Denition6.2 Alogi systemL=hC ;S;P;Di issaid to berih i:
1. t;f 2C
0
and ^;_;2C
2
;
2. S
` CEQ
(C ;)
v(t)=>;
3. S
` CEQ
(C ;)
v(f)=?;
4. S
` CEQ
(C ;) v(y
1
^y
2
)=u(v(y
1 );v(y
2 ));
5. S
`
(C ;) v(y
1 _y
2
)=t(v(y
1 );v(y
2 ));
6. S
` CEQ
(C ;) v(y
1 y
2
)=)(v(y
1 );v(y
2 ));
7. hf
1
;
1
2 g;
2 i2D.
Example6.3 BothlogisystemsL
C
1 andL
KD
,aswellasmanyotherommon
logis,arerih.
Within a rih logi system it is possible to translate from the meta-logi
levelto theobjetlogilevel. Aground termofsort over(C ;) ismapped
to aformulainL(C ;) aordingto the followingrules:
v()
is
>
ist;
?
isf;
(t)
is t
f;
u(t
1
;t
2 )
ist
1
^t
2
;
t(t
1
;t
2 )
ist
1 _t
2
;
)(t
1
;t
2 )
ist
1 t
2 .
Moreover, a ground -equation (t
1
= t
2
) is translated to (t
1
= t
2 )
given by
t
1 t
2
. Finally,ifE isasetofground-equations,thenE
willdenotetheset
feq
: eq2Eg.
Lemma6.4 LetLbearihlogisystemandt aground-termover(C ;).
Then:
S
` CEQ
(C ;) v(t
)=t:
Proof: Immediatebydenition, takinginto aount the requirementsof rih-
nessand theompleteness ofCEQ. QED
Lemma6.5 LetLbearihlogisystem,t
1 andt
2
ground-termsover(C ;)
andA2Int (S). Then:
[[t
1
℄℄
A
A [[t
2
℄℄
A
i v
A ([[t
1 t
2
℄℄
A )=>
A
and
[[t
1
℄℄
A
=[[t
2
℄℄
A
i v
A ([[t
1 t
2
℄℄
A )=>
A :
Proof: DiretorollaryofProposition2.5usingthepreviousLemmaandtaking
into aount theompleteness ofCEQ. QED
In a rih logi system, under ertain onditions (f. Denition 6.6 below),
oneanenodetherelevantpartofthemeta-reasoningintotheobjetalulus.
foreveryonditionalequation(eq
1
& ::: &eq
n
! eq)inS
andeveryground
substitution:
f(eq
1 )
;:::;(eq
n )
g` PD
p
(eq)
:
Finally, we obtain the main results of this Setion relating adequay to
equationalappropriateness. Suhresultsareimportantbeauseitismuheasier
to analyze the preservation by bring of equational appropriateness than of
adequaydiretly.
Theorem 6.7 Every rihand adequate logisystemis equationally appropri-
ate.
Proof: Assume that L is a rih and adequate logi system and A 2 Int(S),
and let (t
1
=s
1
& ::: &t
n
= s
n
! t= s) be a onditional equation in S
and a groundsubstitution.
If it is the ase that v
A ([[(t
i )
(s
i )
℄℄
A ) = >
A
for i = 1;:::;n, then,
aordingto thepreviousLemma,thismeanspreiselythat[[t
i
℄℄
A
=[[s
i
℄℄
A for
i = 1;:::;n. Consider the assignment = [[ ℄℄
A
Æ. It is straightforward to
verifythat[[r℄℄
A
=[[r℄℄
A
,forevery-termrover(C ;). So,sinebydenition
ofInt(S)weknowthatAisamodeloftheonditionalequation,itimmediately
followsthat also [[t℄℄
A
=[[s℄℄
A
, orequivalently,v
A ([[(t)
(s)
℄℄
A )=>
A .
Therefore, wehave f(t
1 )
(s
1 )
;:::;(t
n )
(s
n )
g S
g (t)
(s)
.
Now, equational appropriateness follows easily sine, from adequay, we must
have f(t
1 )
(s
1 )
;:::;(t
n )
(s
n )
g` PD
p (t)
(s)
. QED
Before proving the onverse of this theorem, we need to establish some
tehniallemmas.
Lemma6.8 LetLbeanequationallyappropriatelogisystemand [fÆgbea
setofshemaformulaewhere
1
doesnotour. Iff
1
: 2 g` PD
p
1 Æ
then ` PD
d Æ.
Proof: First of all we note that, sine S
must ontain a speiation of the
lassof all Heyting algebras,equational appropriatenessimpliesthat every in-
tuitionistitheorem written with t;f;^;_; must be provable in the Hilbert
alulus. Inthesequel, we shallusethisfat withoutfurthernotie.
Letusassumethatf
1
: 2 g` PD
p
1
Æ. Immediately,bythenite
harater ofderivations,f
1
1
;:::;
1
n g`
PD
p
1
Æ,whereeah
i 2 .
Let now be the shema formula
1
^:::^
n
and take the shema variable
substitution suh that (
1
) = and (
i ) =
i
for every i 2. Sine
1
doesnotourin orÆ,thestruturalityofproofseasilyimpliesthatwemust
alsohave f
1
;:::;
n g`
PD
p
Æ. Butitislearbyeasy intuitionisti
reasoningthat` PD
p
i
for i=1;:::;n. So, itfollows that ` PD
p
Æ, and
sinebyfurtherintuitionistireasoningwehave` PD
p
1
(:::(
n
):::),
thederivation ruleof Modus Ponens immediatelyimpliesthat ` PD
d
Æ. QED
ground-equations over(C ;) and a ground substitution. If
S
[f(!eq):eq2Eg` CEQ
(C ;) t
1
=t
2
theneithert
1
;t
2
arethesame term ofsort , ort
1
;t
2
areof sort and
E
` PD
p (t
1 )
(t
2 )
:
Proof: TherulesofCEQ[18 ℄aretheusualReexivity,Symmetry,Transitivity,
Congruene and a form of Modus Ponens that allows us to obtain eq from
eq
1
;:::;eq
n
foranygiven onditionalequation (eq
1
& ::: &eq
n
! eq ).
Given a ground substitution ,we show byindution on the lengthn of a
proofof S
[f(!eq):eq2Eg` CEQ
(C ;) t
1
=t
2
that E
` PD
p (t
1 )
(t
2 )
.
Base: n=1.
(i) t
1 is s
1
0
, t
2 is s
2
0
and t
1
= t
2
is obtained by CEQ Modus Ponens from
(!s
1
=s
2 )2S
.
Obviously t
1 and t
2
are -terms. Immediately,byequational appropriateness,
we have ` PD
p (s
1
0
)
(s
2
0
)
and therefore E
` PD
p (t
1 )
(t
2 )
by the
monotoniityofprovability.
(ii) t
1 is s
1
0
, t
2 is s
2
0
and t
1
=t
2
is obtained by CEQ Modus Ponens from
(!s
1
=s
2
) withs
1
=s
2 2E.
Obviouslyt
1 and t
2
arelosed-terms,s
1
0
iss
1 ands
2
0
iss
2
. Thus,(t
1 )
(t
2 )
2 E
, i.e., t
1 t
2 2 E
and E
` PD
p t
1 t
2
by the extensiveness of
provability.
(iii)t
1 and t
2
are thesame term,of either sort or,and t
1
=t
2
isobtained
byReexivity.
If the sort is we are done. Otherwise, obviously, (t
1 )
and (t
2 )
are the
same formula and trivial intuitionisti reasoning allows us to onlude that
` PD
p
1
1
. Therefore, E
` PD
p (t
1 )
(t
2 )
by the struturality and
monotoniityofprovability.
Step: n>1.
(i)t
1
=t
2
isobtainedfrom S
[f(!eq):eq2Eg` CEQ
(C ;) t
2
=t
1
bySymme-
try.
Ifthetermshavesort,byindutionhypothesis,theyoinide. Otherwise,also
by indution hypothesis, we know that E
` PD
p (t
2 )
(t
1 )
. Elementary
intuitionistireasoning allows us to onlude that ` PD
p (
1
2 ) (
2
1 ),
andtherefore also E
` PD
p (t
1 )
(t
2 )
.
(ii)t
1
=t
2
isobtainedfrom S
[f(!eq):eq2Eg` CEQ
(C ;) t
1
=t
3
;t
3
=t
2 by
Transitivity.
Ifthetermshavesort byindutionhypothesistheyoinide. Otherwise,also
byindutionhypothesis,weknowthatE
` PD
p (t
1 )
(t
3 )
;(t
3 )
(t
2 )
.
Simpleintuitionistireasoningallowsustoonludethatf
1
2
;
2
3 g`
PD
p
1
3
,and therefore alsoE
` PD
p (t
1 )
(t
2 )
.
(iii) t
1
is f(t
11
;:::;t
1k ), t
2
is f(t
21
;:::;t
2k
) and t
1
= t
2
is obtained from
S
[f(!eq):eq2Eg` CEQ
(C ;) t
11
=t
21
;:::;t
1k
=t
2k
byCongruene.
It t
1 and t
2
have sort then f 2 C
k
and therefore all the terms t
ij
are also
1j 2j 1
oinideswitht
2
. Otherwise,f aneitherbevorageneratoramong ;u;t;).
Intherst ase k =1 and byindutionhypothesis t
11
oinideswith t
21 sine
they must have sort . Therefore, t
1 and t
2
also oinide and we repeat step
(iii) of the Base to obtain E
` PD
p (t
1 )
(t
2 )
. Finally, if f is a gen-
erator then all the terms t
ij
are of sort . Thus, by indution hypothesis,
E
` PD
p (t
11 )
(t
21 )
;:::;(t
1k )
(t
2k )
. Usingagain intuitionistirea-
soningwe have that f
1
2
;
3
4 g `
PD
p (
1
^
3 ) (
2
^
4 ); (
1 _
3 )
(
2 _
4 );(
1
3 )(
2
4
). Given that istranslated usingf and this
isenoughto guarantee thatE
` PD
p (t
1 )
(t
2 )
.
(iv)t
1 is s
1
0
,t
2 is s
2
0
and t
1
=t
2
is obtainedusing theonditional equation
(s
11
= s
21
& ::: & s
1k
= s
2k
! s
1
= s
2 ) 2 S
from S
[f( ! eq) : eq 2
Eg` CEQ
(C ;) s
11
0
=s
21
0
;:::;s
1k
0
=s
2k
0
byCEQModus Ponens.
Obviously, all the terms are -terms and the indution hypothesis implies
that E
` PD
p (s
11
0
)
(s
21
0
)
;:::;(s
1k
0
)
(s
2k
0
)
. Moreover, by
equational appropriateness, we have f(s
11
0
)
(s
21
0
)
;:::;(s
1k
0
)
(s
2k
0
)
g` PD
p (s
1
0
)
(s
2
0
)
. Therefore, E
` PD
p (t
1 )
(t
2 )
. QED
Theorem 6.10 Every equationally appropriatelogi systemisadequate.
Proof: Let L be an equationally appropriate logi system and [fÆg
L(C ;).
If
S
g
Æ then also ` S
g
Æ, i.e., S
[f( ! v() = >) : 2 g ` CEQ
(C ;)
v(Æ) = >. Using the previous Lemma, we have that f tg ` PD
p
Æ t.
Trivial intuitionistireasoning allows us to onlude that ` PD
p
1
(
1 t),
andtherefore itfollows that ` PD
p Æ.
If
S
l
Æthenalso ` S
l
Æ,i.e., S
[f(!v(
1
)v(
+1
)): 2 g` CEQ
(C ;)
v(
1
) v(Æ +1
). Using the previous Lemma, we have that f(
1
^ +1
)
1
: 2 g ` PD
p (
1
^Æ +1
)
1
. Trivial intuitionisti reasoning allows us
to onlude that ` PD
p (
1
2
) ((
1
^
2
)
1
), and therefore it follows
that f
1
+1
: 2 g ` PD
p
1
Æ
+1
. Thus, we already know that
+1
` PD
d Æ
+1
and bystruturality,using 1
,we have ` PD
d
Æ. QED
The equivalenebetweenadequay and equational appropriatenessforrih
systemswillbeusedbelowforshowingthatadequayispreservedbybringrih
systems,butthisequivalenemayalso be usefulforestablishing theadequay
oflogisendowedwithasemantispresentedbyonditionalequations. Indeed,
it is a muh easier taskto verify equational appropriateness than to establish
adequaydiretly.
6.2 Preservation of ompleteness by bring
We onsiderinturnthepreservation ofsoundness and ofadequay bybring.
Theorem 6.11 Soundnessis preserved bybring.
Proof: Let L be the bring of two sound logi systems L and L . It is
enough to prove the following: ` PD
p
Æ implies that ` S
g
Æ, and ` PD
d Æ
implies that ` S
l
Æ, by Theorem 6.1. Moreover, it is enough to prove that
Prem (r)` S
g
Con(r)foreveryr 2P,andPrem(r)` S
l
Con (r)forevery r2D.
Let r 2 P. Assume, without loss of generality, that r =
^
h 0
(r 0
). Then, by
denitionof proof,Prem (r) 0
` P
0
D 0
p
Con(r) 0
and, so,Prem(r) 0
` S
0
g
Con(r) 0
,by
thesoundnessof L 0
. This meansthat
S 0
[f( !v(
0
)=>): 0
2Prem(r) 0
g` CEQ
(C 0
;)
v(Con (r) 0
)=>
andthen,bytheuniformnessof CEQunderhangeofnotationby
^
h 0
(f. [18 ℄),
we obtain
^
h 0
(S 0
)
[f(!v(
^
h 0
( 0
))=>): 0
2Prem (r) 0
g` CEQ
(C ;) v(
^
h 0
(Con (r) 0
))=>:
This immediately implies
^
h 0
(Prem (r) 0
) ` S
g
^
h 0
(Con (r) 0
), that is, Prem(r) ` S
g
Con(r). Theproofforderivationsissimilar. QED
We, nally, onsider the problem of preservation of adequay by bring,
takingadvantage of thetehnial mahinery presentedbefore on theenoding
ofthemeta-logi in theobjetHilbertalulus.
Lemma6.12 Rihnessis preserved by bringprovidedthatonjuntion, dis-
juntion,impliation,true and falseare shared.
Proof: Itistrivialthatthesignatureandvaluationrequirementsarepreserved
sinewearesharingonjuntion,disjuntion,impliation,trueandfalse. More-
over, itislearthatModusPonens isstilladerivationruleinthebring.QED
Lemma6.13 Equationalappropriatenessispreservedbybringprovidedthat
onjuntion,disjuntion,impliation,trueand falseareshared.
Proof: Let L 0
and L 00
be equationally appropriate logi systems, and L their
bringby sharing onjuntion, disjuntion,impliation, true and false. From
thepreviousLemmawealready knowthatL isrih.
Now, let eq be (t
1
=s
1
& ::: &t
n
=s
n
! t= s)2 S
,and a ground
substitution. Clearly,bydenitionof bring, eq must bethe translationof a
onditional equation in some of the omponents. Let us assume, without loss
of generality, that eq omes from L 0
, i.e., eq is
^
h 0
(eq 0
), where eq 0
is the
onditionalequation (t 0
1
=s 0
1
& ::: &t 0
n
=s 0
n
!t 0
=s 0
)2S 0
. Sine we know
thatL 0
isequationally appropriate, itfollows that
f(t 0
1
0
)
(s 0
1
0
)
;:::;(t 0
n
0
)
(s 0
n
0
)
g` P
0
D 0
p (t
0
0
)
(s 0
0
)
;
where 0
isthefollowingground substitution:
forevery i1, 0
(x
i
)=v(
2i 1
) and 0
(y
i )=
2i .
f
^
h 0
((t 0
1
0
)
(s 0
1
0
)
);:::;
^
h 0
((t 0
n
0
)
(s 0
n
0
)
)g` PD
p
^
h 0
((t 0
0
)
(s 0
0
)
):
Considernowthesubstitution onshema variablesdened by:
(
2i 1
)=(x
i )
;
(
2i
)=(y
i ).
Using and thestruturalityofthe Hilbertalulus,now, we must alsohave
f
^
h 0
((t 0
1
0
)
(s 0
1
0
)
);:::;
^
h 0
((t 0
n
0
)
(s 0
n
0
)
)g` PD
p
^
h 0
((t 0
0
)
(s 0
0
)
):
But, in fat, a straightforward indutive proof allows us to onlude that
^
h 0
((u 0
0
)
)=(
^
h 0
(u 0
))
foreverytermu 0
ofsort over(C 0
;),andtherefore
f(t
1 )
(s
1 )
;:::;(t
n )
(s
n )
g` PD
p (t)
(s)
:
Thus, Lis equationallyappropriate. QED
Theorem 6.14 Giventwo rih andompletelogisystems,theirbringwhile
sharingonjuntion,disjuntion,impliation,true andfalseis alsoomplete.
Proof: The preservation of adequay is animmediate onsequeneof the two
previouslemmas and the equivalenebetween equational appropriateness and
adequayforrih systems. QED
Example6.15 Bybringwhilesharingonjuntion,disjuntion,impliation,
true and false the logi systems L
C1
and L
KD
we obtain a new modal para-
onsistent logi system L
C
1 KD
that is omplete. Observe that if we add to
L
C
1 KD
:
(v(y Æ
1
)=>!v((Ly
1 )
Æ
)=>)asa meta-valuation axiom;
h;;
Æ
1
(L
1 )
Æ
i asan axiominthe Hilbertalulus;
we obtain a omplete logi system that is equivalent to the system C D
1 of [8 ℄
bothat theproof-theoretiand thesemanti levels.
7 Appliation II: self-bring of truth-funtional and
non-truth-funtional logis
It is not diÆult to see that, if S is a truth-funtional isp (that is, all the
onnetivesaretruth-funtionalderivedonnetives)thentheself-bringSS
ofS withitself, withoutsharing of onnetives(just sharingthe propositional
symbolsinC
0
),produesaopyofS whereeahonnetiveappearsdupliate.
In fat, if 2 C
k
(k > 0) and 0
is its dupliate then v
A ([[(t
1
;:::;t
k )℄℄
A ) is
equal to v
A ([[(t
1
;:::;t
k )℄℄
A
) for every interpretation A, assignment over A
and termst
1
;::;t
k
of sort . Additionally,if2C
0
is a onstant symbol, suh
as t or f, then v
A ([[℄℄
A
) is also equal to v
A ([[℄℄
A
). Of ourse, this property
an be extended to every rih and omplete logi system L while sharing C
0 ,
onjuntion,disjuntionandimpliationintheself-bringLL. Insuhases
theformulae (t
1
;:::;t
k
) and 0
(t
1
;:::;t
k
) will be equivalent in LL, even if
and 0
arenot expliitlyshared.
Ontheotherhand,ifSisanispwithsomenon-truth-funtionalonnetive
,thenv
A ([[(t
1
;:::;t
k )℄℄
A
)andv
A ([[
0
(t
1
;:::;t
k )℄℄
A
)donotneessarilyoinidein
themodelsofSS and,onsequently,theformulae(t
1
;:::;t
k )and
0
(t
1
;:::;t
k )
arenotneessarily equivalentinLL,unless and 0
are expliitlyshared.
Asa onrete instane, onsidertheisp S ofexample 2.6, representing the
semantisof the paraonsistent alulus C
1
(f. [6 ℄). Ifwe performthe bring
SS of S with itself, while sharing the symbols in C
0
, then we obtain two
families of onnetives: f^;_;;:g and f^
0
;_ 0
; 0
;: 0
g. A model for S S
gives avaluation mapv
A
suhthat
v
A ([[t
1 t
2
℄℄
A )=v
A ([[t
1
0
t
2
℄℄
A
) for2f^;_;g;
beause those onnetivesare truth-funtional. On theother hand, v
A ([[:t℄℄
A )
does not oinide neessarily with v
A ([[:
0
t℄℄
A
). For example, let v
1 and v
2 be
two C
1
-bivaluationssuh that v
1 (p
1 ) =v
1 (:p
1
)=1 and v
2 (p
1
)=1, v
2 (:p
1 )=
0. Moreover, assume that v
1
(p) = v
2
(p) for every other propositional symbol
p2C
0
. We an thusdenean interpretation Aof SS asfollows:
A
=f0;1g (withits usualBoolean algebrastruture);
v
A
restrited to thefragment with: oinideswithv
1
;
v
A
restritedto thefragment with: 0
oinideswithv
2
;
v
A
extended tothemixedlanguageCC 0
isobtainedfromv
1 and v
2 by
usingthe sametehniquesusedintheproofof Proposition6.1in[4℄.
The interpretation A satises: v
A ([[:p
1
℄℄
A ) 6= v
A ([[:
0
p
1
℄℄
A
), showing that :
and : 0
do not ollapse. Considering the bring at the logi system level, we
obtain, by the ompleteness-preservation theorem 6.15, that :t
1
and : 0
t
1 are
notequivalentformulae (unlessthey are boththeorems).
Theexampleaboveshows thatthebringof C
n
;n1withitselfprodues,
for every n, two disjoint opies of C
n
(as the same argument an be applied
to the whole hierarhy of paraonsistent aluli C
n
). We exlude C
0
(whih is
justtheClassialPropositionalCalulus)sineinthisasetheself-bringwill
ollapsewithC
0
,beausethissystemistruth-funtional. Ontheotherhand,the
bringof C
n
withC
m
,with m>n,produesanew paraonsistentsystemwith
two paraonsistent negations,:
n and :
m
,whoseaxiomsorrespond to adding
theaxiomsofC
n
andofC
m
,andwhoseinterpretationsaregivenbymapswhih
are simultaneously C
n and C
m
valuations. It is an open question whether or
not there exists a formula (in the language of C
m
) whih enodes in C
m the
paraonsistentnegation:
n
,form>n(this,infat,happenswiththelassial
negation, whih is representable in every aluli C
n
). If this question has a
positive answer, then thebring of C
n and C
m
(withoutsharing the negation)
m n m
sharingthe negation) willbeequivalent to C
n
. If we generalize thisargument,
inludinginthe objetsignature C an unarysymbol :
n
forevery n0,then
theinnitebringofthewholehierarhyC
n
,withoutsharingthenegations:
n ,
willprodue a new paraonsistent system with innitelymany paraonsistent
negations. If negations are shared in the bring, the result oinides with
theClassial Propositional Calulus C
0
. It is worth remarkingthat, althought
we onentrate most of the time on paraonsistent aluli beause these are
exellent examples of interesting non-truth-funtional logisystems, it is lear
thatourtreatment is fullygeneral.
8 Conluding remarks
The rst main ontribution of this paper is a general semantis for bring
propositional-basedlogisenompassingsystemswithnon-truth-funtionalval-
uations. Thisgoalwasahievedbyreognizingthatsuhvaluationsarespeied
in some appropriate meta-logi, in this ase onditional equational logi. Al-
though restrited to systems with a nitary propositional base, the proposed
semantis deals with a wide variety of logis from paraonsistent to modal,
many-valuedand intuitionistisystems. Withinthissetting,bringappears as
a universal onstrution within the underlyingategory, generalizingprevious
resultsfortruth-funtionalsystems [21 ℄.
Theseondmainontributionofthispaperistheompletenesspreservation
theorem that generalizesthe result establishedin [23 ℄ and is obtained usinga
new tehnique exploiting the properties of onditional equational logi where
the(possiblynon-truth-funtional)valuationsaredened.
Asan example ofappliation of our tehniques,the useof bringfor om-
bining paraonsistent logis with other logis is illustrated by reovering the
modal paraonsistent logi C D
1
of da Costa and Carnielli[8 ℄ as a bring. We
believethatbringisaverynaturalwayofestablishingnewombinedsystems
involving non-truth-funtionallogis. This approahis more widelyappliable
asit appears to be: a large familyof logis (inluding many-valued and intu-
itionisti)admits bivalued non-truth-funtional semantis(f. [1 ℄). Whenever
thathappens, theompleteness preservation theoreman thenbe used fores-
tablishingtheompletenessoftheresultaslongasthegivenlogisareomplete
andfulll therequirementsof rihness.
Otherlinesofresearhareobvious,towardsrelaxingtheassumptionsofthis
paper. Forinstane,wemaywanttoworkwithmoregeneralobjetlogis(e.g.,
prediatelogis),orwithamoregeneralmeta-logi(e.g.,disjuntiveonditional
equational logi), or with an even more general universe of truth-values (e.g.,
involving less or extra generators), or with weaker rihness requirements and
stillobtainompleteness preservationbybring.
Still other lines of researh are related to more general forms of bring,
namelyheterogeneousformsofbringwherewewanttoombinetwo(ormore)
logis thatare dened inquite dierent forms(either at thededutive system
levelorat thesemantilevel). To summarize,ourapproahoersaframework
express a large variety of logi systems; suh representations of logi systems
onstitute a ategory and they an be ombined by means of bring (that is,
throughuniversalonstrutionsintheategory). We have alsoproved thatthe
ompletenessofsuhlogisystemsispreservedbybringunderertainreason-
ableassumptionson thelogisystems,guaranteeing theimportantpropertyof
non-destrutivenessof ourbringonstrutions.
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