Modus tollens with respect to uninorms: U -Modus Tollens International Journal of Approximate Reasoning

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Contents lists available atScienceDirect

International Journal of Approximate Reasoning

www.elsevier.com/locate/ijar

Modus tollens with respect to uninorms: U -Modus Tollens

Isabel Aguiló

^a

^,

^b

, Juan Vicente Riera

^a

^,

^b

^,∗ , Jaume Suñer

^a

^,

^b

, Joan Torrens

^a

^,

^b

aSoftComputing,ImageProcessingandAggregation(SCOPIA)ResearchGroup,Dept.ofMathematicsandComputerScience,Universityof the BalearicIslands,Ctra.deValldemossa,Km.7.5,07122Palma,Spain

bBalearicIslandsHealthResearchInstitute(IdISBa),07010Palma,Spain

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received16January2020

Receivedinrevisedform6October2020 Accepted8October2020

Availableonline19October2020

Keywords:

ModusTollens Uninorm

Implicationfunction RU -implication

Infuzzylogicandapproximatereasoning theinferencerulegivenbytheModusTollens usually derives into an inequality involving three logical operators: a conjunction, an implicationfunctionandanegation.Untilnow,inthisscenariotheconjunctionhasbeen commonly modeled by a t-norm, but recently the possibility of using a more general conjunctionhasbeenpointedout.Inthiswork,wewanttogeneralizetheModusTollens inequality byusingaconjunctiveuninorm insteadofat-norm,leading tothe so-called U -ModusTollens.First,wegiveastudyofthisnewpropertyforimplicationfunctionsin generalandthenwespeciallyfocusonresidualimplicationsderivedfromuninorms.Inall cases,weprovethatthere arealotofsolutionsofthe U -ModusTollensand wegivea characterizationofallthesolutionsinsomeparticularcases.

©²⁰²⁰^TheÂuthors.^Published^byÊlsevierÎnc.^Thisîsânôpenâccessârticleûnder^the^CC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Implicationfunctionsarecommonlyusednotonlytomodelfuzzyconditionals, butalsotocarry outall inferencepro- cessesinanyfuzzyrulebasedsystem.Thus,theimportanceofimplicationfunctionsmainlyliesintheirapplications,which cover awide rangeof differentﬁelds,inparticular, all thosewhere fuzzycontrol andapproximatereasoningcan be ap- plied(seeforinstance[3,5,18]).Thisisalsothemainreasonforwhichitisimportanttohaveawiderangeofimplication functionstobeusedindifferentcontexts(see[29]).

Fromthetheoreticalpointofview,oneofthemaininterestingtopicsonimplicationfunctionsisthestudyandthechar- acterization ofthoseimplicationsthatsatisfycertainadditionalpropertiesthatusuallyarisefromtheconcreteapplications.

Therearemanyoftheseproperties,butwecanhighlightamongthemthosederivedfromthebasicinferencerules:Modus PonensandModusTollens(see [9,10,12]).Intheframeworkoffuzzylogic,theseinferencerulesareguaranteedwhenthe involvedlogicaloperatorssatisfythefollowingfunctionalinequalities,respectively:

T

(

x

,

I

(

x

,

y

)) ≤

^y ^{for all x}

,

y

∈ [

⁰

,

1

],

⁽¹⁾

and

T

(

N

(

y

),

I

(

x

,

y

)) ≤

N

(

x

)

for all x

,

y

∈ [

0

,

1

],

(2)

*

Correspondingauthorat:SoftComputing,ImageProcessingandAggregation(SCOPIA)ResearchGroup,Dept.ofMathematicsandComputerScience, Universityofthe BalearicIslands,Ctra.deValldemossa,Km.7.5,07122Palma,Spain.

E-mailaddresses:[email protected](I. Aguiló),[email protected](J.V. Riera),[email protected](J. Torrens).

https://doi.org/10.1016/j.ijar.2020.10.003

0888-613X/©²⁰²⁰^TheÂuthors.^Published^byÊlsevierÎnc.^Thisîsânôpenâccessârticleûnder^the^CC^BY-NC-ND^license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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whereT isat-norm, I isanimplicationfunctionandN isanegation.

When implicationfunctionshaveto be usedin managing inferenceprocesses, one ofthesetwo propertiesorboth of them are essential.Thus, dueto their signiﬁcance,both inequalities havebeenlargely studied inthe literature formany authors(seeforinstance[2,3,16,18,26–28,30]).Firststudieswerecarriedoutforimplicationfunctionsderivedfromt-norms andt-conorms.Speciﬁcally,residuatedimplicationfunctionsand

(

S

,

N

)

-implicationswerestudiedin[2,26,27],whereasthe caseof Q L and D-implicationswas studiedin[28].Asageneralizationoftheseclassesofimplications,theircounterparts derived from more general kinds ofaggregation functions havebeen introduced, specially those derived fromuninorms like RU -implicationsand

(

U

,

N

)

-implications(seeforinstance[1,4,6,17,22–24]).Recently,boththeModusPonensandthe Modus Tollens have been studied also for these two kinds of implications derived from uninorms, (see [16] and [15], respectively).

Infact,althoughuninormswereinitiallyintroducedintheframeworkofaggregationfunctions(see[8,31]),theyhavealso beenstudiedandusedaslogicaloperatorsduetothefactthatthey arealwaysconjunctive(theirvalue inthepoint

(

0

,

1

)

is0)ordisjunctive(theirvaluein

(

0

,

1

)

is1).Inparticular,conjunctiveuninormsarecommonlyusedasfuzzyconjunctions.

Thus, the idea of substituting the t-norm T by a conjunctive uninorm U in the Modus Ponensand the Modus Tollens inference rules becomes naturaland it turns out to be an interesting line of research. Note that, in the Modus Ponens case,thisideawasalreadycarriedout in[19] leadingtotheso-calledU -Modus Ponens(orequivalently U -conditionality).

Initial resultsintheseworksprovethat theusualimplications derivedfromt-normsandt-conorms,aswell asgenerated Yager implications, never satisfy U -Modus Ponens, whereas implications derived fromuninorms appear to be the most suitable onesto satisfyit. Inthisdirection, thecaseof RU -implications hasalreadybeensolved in[19], andthe caseof

(

U

,

N

)

-implicationshasbeenrecentlystudiedin[20].

Onthecontrary,asimilarstudyforthecaseoftheModusTollensisstillmissing.Inthispaper,wewanttodealprecisely withthistaskandwewanttostudy thesamegeneralizationoftheModusTollens.Thatis,wewanttostudytheproperty obtained from Equation (2) by substituting the t-norm T by a conjunctive uninorm U , that we will call the U -Modus Tollens:

U

(

N

(

y

),

I

(

x

,

y

)) ≤

^N

(

x

)

for all x

,

y

∈ [

⁰

,

1

],

⁽³⁾

where U is aconjunctive uninorm, I is an implicationfunction andN is afuzzy negation.We wanttohighlight that in uninorm-basedfuzzylogics(see[14])apropositionP isconsideredtruewhenitsassignedvalueisgreaterthanorequalto e (wheree denotestheneutralelementofuninormU ).Inthiscontext,therulegivenbyEquation(3) couldbeinterpreted asfollows:“IfthevalueofnotQ isgreaterthanorequaltoe andthevalue of P

→

^{Q is}^greater^thanôrêqual^toê,^then thevalueofnotP isalsogreaterthanorequaltoe”.

Thispaperisorganized asfollows.Afterthisintroduction,Section 2isdevotedtosome preliminariescompilingall the notationsandconceptsnecessarytofollowthearticle.InSection3theU -ModusTollensisintroducedanditisanalyzedfor generalimplicationfunctions.Section4isdevotedtothecaseofRU -implicationsderivedfromthethreemostusualclasses ofconjunctiveuninorms:uninormsin

U

min,representableuninorms andidempotentuninorms.Finally,Section 5includes someconclusionsandsomefutureworkproposals.

2. Preliminaries

Werecallsomeconceptsthatwewillusealongthepaper.

Deﬁnition2.1.([11]) A function N

: [

⁰

,

1

] → [

⁰

,

1

]

^is ^said ^to ^be ^a ^fuzzynegation if it is decreasing with N

(

0

) =

^{1 and} N

(

0

) =

^1.^A^fuzzy^negation^{N is}^said^to^be

•

Strict whenitisstrictlydecreasingandcontinuous.

•

Strong whenitisaninvolution,i.e.,N

(

N

(

x

)) =

^{x for}^all^x

∈ [

⁰

,

1

]

^.

Deﬁnition2.2.([3,7]) AbinaryoperatorI

: [

⁰

,

1

] × [

⁰

,

1

] → [

⁰

,

1

]

îs^said^to^beânimplicationfunction,oranimplication,ifit satisfies:

(I1) I

(

x

,

z

) ≥

^I

(

y

,

z

)

whenx

≤

^y,^for^all^z

∈ [

⁰

,

1

]

^. (I2) I

(

x

,

y

) ≤

^I

(

x

,

z

)

when y

≤

^z,^for^all^x

∈ [

⁰

,

1

]

^. (I3) I

(

0

,

0

) =

^I

(

1

,

1

) =

^{1 and}^I

(

1

,

0

) =

^0.

Notethatitfollowsfromthedeﬁnition,that I

(

0

,

x

) =

^{1 and} ^I

(

x

,

1

) =

^{1 for}^all^x

∈ [

⁰

,

1

]

^whereas^thesymmetricalvalues I

(

x

,

0

)

andI

(

1

,

x

)

arenotderivedfromthedeﬁnition.

Specialinterestingpropertiesforimplicationfunctionsare:

•

^The^neutrality^property,

I

(

1

,

y

) =

y for all y

∈ [

⁰

,

1

].

^{(N P )}

(3)

•

^The^identity^principle,

I

(

x

,

x

) =

1 for all x

∈ [

⁰

,

1

].

^{(I P )}

•

^Thecontrapositiveproperty withrespecttoanegationN

I

(

x

,

y

) =

I

(

N

(

y

),

N

(

x

))

for all x

,

y

∈ [

0

,

1

].

(C P

(

N

)

Deﬁnition2.3.([3,7]) Let I be a fuzzyimplication function. The function N_I givenby N_I

(

x

) =

^I

(

x

,

0

)

forall x

∈ [

⁰

,

1

]

^is alwaysafuzzynegation,whichiscalledthenaturalnegation of I.

Deﬁnition2.4.([8,31]) Auninorm isatwo-placefunctionU

: [

⁰

,

1

]

²

−→ [

⁰

,

1

]

^which^isassociative,commutative,increasing ineachplace,andsuchthatthereexistssomeelemente

∈ [

⁰

,

1

]

^,^called^neutral^element,^such^that^U

(

e

,

x

) =

^{x for}^all^x

∈ [

⁰

,

1

]

^.

Itisclearthatuninormsgeneralizebotht-normsandt-conorms,sincetheyareretrievedfromuninormsjusttakinge

=

¹ ande

=

^0,respectively.Foranyother valuee

∈]

⁰

,

1

[

^,^theôperation^worksâsâ^t-normⁱⁿ^the^square

[

⁰

,

e

]

²^,^as^a^t-conorm in

[

^e

,

1

]

²^,ândîts^valuesâre^between^minimumând^maximumⁱⁿ^the^setôf^points Â

(

e

)

givenby

A

(

e

) = [

⁰

,

e

[ × ]

^e

,

1

] ∪ ]

^e

,

1

] × [

⁰

,

e

[.

We will usually denote a uninorm with neutralelement e and underlying t-norm T and t-conorm S, by U

≡

^T

,

e

,

S

^. Moreover, forauninorm U ,itisalways U

(

1

,

0

) ∈ {

⁰

,

1

}

^and^{U is}^said ^to^be conjunctive whenU

(

1

,

0

) =

^0,^anddisjunctive whenU

(

1

,

0

) =

1.

Therearemanydifferentkindsofuninorms.Themostcommononescanbefoundin[13].Letusrecallherethestructure ofthethreemost-usedclassesofconjunctiveuninormsthatwillbeusedalongthepaper.

Proposition2.1.([8,13])LetU

: [

⁰

,

1

]

²

→ [

⁰

,

1

]

^beâûninorm^with^neutralêlementê

∈ ]

⁰

,

1

[

^.^If^U

(

0

,

1

) =

^0,^then^the^function x

→

^U

(

x

,

1

)

iscontinuousexceptforx

=

^{e if}ândônlyîf^{U is}^given^by

U

(

x

,

y

) =

⎧ ⎪

⎨

⎪ ⎩

eT

_x

e

,

^y_e

if

(

x

,

y

) ∈ [

⁰

,

e

]

²

,

e

+ (

1

−

e

)

S

x−^e 1−^e

,

₁^y⁻₋_e^e

if

(

x

,

y

) ∈ [

e

,

1

]

²

,

min

(

x

,

y

)

if

(

x

,

y

) ∈

A

(

e

),

whereT isat-normandS isat-conorm.

Wewilldenoteby

U

minthefamilyoftheseuninormsandbyU

≡

^T

,

e

,

S

mintheuninormin

U

minwithneutralelement e andunderlyingt-norm T andt-conormS.

Idempotentuninormswerecompletelycharacterizedforthegeneralcasein[25] inthefollowingway.

Proposition2.2.([25,13])AbinaryoperatorU

: [

⁰

,

1

]

²

→ [

⁰

,

1

]

îsânîdempotentûninorm^with^neutralêlementê

∈ [

⁰

,

1

]

îfândônly ifthereexists adecreasingfunctiong

: [

⁰

,

1

] → [

⁰

,

1

]

^,^symmetric^with^respect^to^the^main^diagonal,^with^g

(

e

) =

^e,^such^that

U

(

x

,

y

) =

⎧ ⎪

⎨

⎪ ⎩

min

(

x

,

y

)

if y

<

g

(

x

)

or

(

y

=

^g

(

x

)

and x

<

g²

(

x

)),

max

(

x

,

y

)

if y

>

g

(

x

)

or

(

y

=

g

(

x

)

and x

>

g²

(

x

)),

x or y if y

=

g

(

x

)

and x

=

g²

(

x

),

whereg²

(

x

) =

^g

(

g

(

x

))

,beingU commutativeinthepoints

(

x

,

y

)

suchthaty

=

^g

(

x

)

withx

=

^g²

(

x

)

.

AnyidempotentuninormU withneutralelemente andassociatedfunction g willbedenotedbyU

≡

^g

,

e

ide,andthe class ofidempotent uninormswillbe denotedby Uide.Obviously,foranyoftheseuninormsthe underlyingt-norm TU is theminimumandtheunderlyingt-conormS_U isthemaximum.

Anotherwellknownclassofuninormsistheonegivenbyalluninormsthatcanberepresentedbyunaryfunctions.

Deﬁnition2.5.([8,13]) Auninorm U with neutralelement e

∈

^]0

,

1[ is said to be representable ifthere exists a strictly increasingandcontinuousfunctionh

: [

⁰

,

1

] → [−∞, +∞]

^with^h

(

0

) = −∞

^,^h

(

e

) =

^{0 and}^h

(

1

) = +∞

^such^that ^{U is}^given by

U

(

x

,

y

) =

h⁻¹

(

h

(

x

) +

^h

(

y

))

if

(

x

,

y

) / ∈ {(

⁰

,

1

), (

1

,

0

)},

0

(

or 1

)

otherwise

.

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Functionh iscalledanadditivegeneratorofU .Thusitisclearthattherearetwodifferentrepresentableuninormswith thesamegeneratorh,aconjunctiveone(which isthenleft-continuous)andadisjunctiveone(whichisright-continuous).

Any representable uninorm U withneutralelement e andadditive generator h willbe denotedby U

≡

^e

,

h

rep, andthe classofrepresentableuninormswillbedenotedbyUrep.Moreover,foranyrepresentableuninormU

≡

^e

,

h

repthefunction

N_h

(

x

) =

^h⁻¹

( −

^h

(

x

))

for all x

∈ [

⁰

,

1

]

isalwaysastrongnegationassociatedtotheuninormU .

Ontheother hand,differentclassesofimplications derivedfromuninorms havealsobeenstudied.Amongthem,only RU -implicationswillbeusedalongthepaperandforthisreasontheyarerecalledhere.

Deﬁnition2.6.([6]) LetU beauninorm.Theresidualoperation derivedfromU isthebinaryoperationgivenby

I_U

(

x

,

y

) =

^sup

{

^z

∈ [

⁰

,

1

] |

^U

(

x

,

z

) ≤

^y

}

⁽⁴⁾

forallx

,

y

∈ [

⁰

,

1

]

^.

Proposition2.3.([6])LetU beauninormandI_Uitsresidualoperation.ThenI_Uisanimplication,calledRU -implication,ifandonly ifthefollowingconditionholds

U

(

x

,

0

) =

0 for all x

<

1

.

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Thispropertyincludesallconjunctiveuninormsbutalsomanydisjunctiveones,forinstancetheclassesofrepresentable andidempotentuninormswith g

(

0

) =

1 (see[6] and[22]).However,whenwedealwithleft-continuousuninormsU ,we clearly havethat U satisﬁescondition (5) if andonly ifit is conjunctive. There are some properties of RU -implications derived fromuninorm,that canbe deduceddirectlyfromthedeﬁnition,orcan beproved inthe similarwaythat itwas doneforthosederivedfromt-norms.Formoredetailssee[6].

3. Modustollenswithrespecttouninorms:U -ModusTollens

Aswehavealreadymentioned,themainobjectiveofthisworkisthestudy ofthegeneralization oftheModusTollens property obtainedby changingthet-norm T bya uninorm U ,ina similarwayasit was doneforthe ModusPonensin [19].ForobviousreasonswewillrefertothisgeneralizationastheU -ModusTollens.Thus,letusbeginbygivingtheformal deﬁnitionintheframeworkoffuzzylogic.

Deﬁnition3.1.LetI beafuzzyimplicationfunction,U auninormandN afuzzynegation.ItissaidthatI andN satisfythe U -ModusTollens propertywithrespecttoU when

U

(

N

(

y

),

I

(

x

,

y

)) ≤

^N

(

x

)

for all x

,

y

∈ [

⁰

,

1

].

⁽⁶⁾

Note that the deﬁnition has been given for a uninorm in generaland not for a conjunctive one.¹ This is because this propertyonU isdirectlyderivedfromthedeﬁnitionofU -ModusTollensasitisprovedinthefollowingproposition.

Proposition3.1.LetI beafuzzyimplicationfunctionandN anegationthatsatisfytheU -ModusTollenswithrespecttoauninormU . ThentheuninormU mustnecessarilybeconjunctive.

Proof. If I and N satisfy the U -Modus Tollens with respect to U , just take x

=

^{1 and} ^y

=

^{0 in} ^Equation ^{(6) to} ^obtain U

(

N

(

0

),

I

(

1

,

0

)) ≤

^N

(

1

)

,thatis,U

(

1

,

0

) =

^0.

Inviewofthepropositionabove,fromnowon,wewillconsideronlyconjunctiveuninormsintheU -ModusTollensprop- erty. By Deﬁnition 3.1,in general, U -Modus Tollensdepends on the uninorm U , the fuzzy implication I,and the fuzzy negation N. Example3.1showsthat,foranyconjunctiveuninorm U ,ifwe requireU -Modus Tollenstobevalidforsome particularnegationN,the admissibleimplicationsI dependonthechoiceofN.

Example3.1.Letusseehowthe U -Modus Tollensbehaves inthecases wheretheextremefuzzynegations are involved, thatis,whenweconsiderthesmallestandlargestfuzzynegations.

1 Recallthatat-normT isusedtomodelaconjunctioninfuzzylogic.Thus,thenormalgeneralizationwouldbetotakeaconjunctiveuninorm.

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i) Letusconsiderthesmallestfuzzynegation N

=

^ND1,givenby ND1

(

x

) =

1 if x

=

⁰

,

0 if x

>

0

.

Note that,if we take anyconjunctiveuninorm U andthe negation N_D₁,any implicationfunction I trivially satisﬁes Equation (6) forall y

>

0 (becausethen ND1

(

y

) =

⁰⁾ ^and^also ^when ^x

=

^{0 (because}^then ^ND1

(

x

) =

^1). ^On ^the^other hand,when y

=

^{0 and}^x

>

0 Equation(6) holdsifandonlyif

U

(

N

(

0

),

I

(

x

,

0

)) =

^U

(

1

,

N_I

(

x

)) ≤

^N

(

x

) =

⁰

,

which impliesNI

(

x

) =

^{0 for}^all ^x

>

0.Thus, we getthefollowing conclusion:“AnimplicationfunctionI satisﬁestheU - ModusTollenswithrespecttoaconjunctiveuninormU andthefuzzynegationND1ifandonlyif NI

=

^ND1”.

ii) Letusconsidernowthegreatestfuzzynegation N

=

^ND2,givenby ND₂

(

x

) =

1 if x

<

1

,

0 if x

=

¹

.

Asimilarargumentshowsthefollowingconclusion:“AnimplicationfunctionI fulﬁlls theU -ModusTollenswithrespect toaconjunctiveuninormU andthefuzzynegationND2 ifandonlyifI

(

1

,

y

) =

^{0 for}^all ^y

<

1”.

Similarly tothe caseoft-norms, whenthe uninorm U isleft-continuous, wecan easily derivea generalresultabout U - ModusTollensasfollows.

Proposition3.2.LetU beaconjunctiveleft-continuousuninorm,I_U itsresidualimplicationfunctionandN afuzzynegation.Then, animplicationfunctionI satisﬁestheU -ModusTollenswithrespecttoU andN ifandonlyif

I

(

x

,

y

) ≤

^IU

(

N

(

y

),

N

(

x

))

for all x

,

y

∈ [

⁰

,

1

],

Proof. JustrecallthatwhenU isleft-continuoustheresiduationpropertyholds(see[6])andthen U

(

N

(

y

),

I

(

x

,

y

)) ≤

^N

(

x

) ⇐⇒

^I

(

x

,

y

) ≤

^IU

(

N

(

y

),

N

(

x

)).

However, the abovecondition doesnot holdwhen U is not left-continuousandmoreover, evenwhen itholds, itcan be difficult to check itin manycases. Thus,we want to deeplystudy the U -Modus Tollenscondition to give more spe- cific results. Afirst analysisof the U -Modus Tollensallows usto give some generalresults that we listin thefollowing proposition.

Proposition3.3.LetI beafuzzyimplicationfunctionandN anegationthatsatisfytheU -ModusTollenswithrespecttoaconjunctive uninormU .Lete betheneutralelementofU ,thenthefollowingpropertiesaresatisﬁed

1. U

(

N

(

y

),

I

(

1

,

y

)) =

^{0 for}^all^y

∈ [

⁰

,

1

]

^.

2. NI

(

x

) ≤

^N

(

x

)

forallx

∈ [

⁰

,

1

]

^.^Also,^for^all^x

∈ [

⁰

,

1

]

^such^that^NI

(

x

) ≥

^{e it}^must^be^N

(

x

) =

^1.

3. Take

α

N

=

^sup

{

^x

∈ [

⁰

,

1

] |

^N

(

x

) ≥

^e

}

^.^Then a) If

α

N

=

^{0 then}^N

(

x

) <

e forallx

>

0.

b) If

α

N

>

0 thenitmustbeI

(

1

,

y

) =

^{0 for}^all^y

< α

N. 4. IfI satisﬁes(N P),then

α

N

=

^0.

5. IfI satisﬁes(I P) andN isnonﬁlling(N

(

x

) <

1 forx

>

0),itmustbe

α

N

=

^0.

Proof. Letusprovethemstepbystep.

1. ThisﬁrstitemisadirectconsequenceoftheU -ModusTollensproperty,justtakingx

=

^{1 in}^Equation^(6).

2. Consider NI

(

x

) =

^I

(

x

,

0

)

,thenwehave

N_I

(

x

) =

^U

(

e

,

I

(

x

,

0

)) ≤

^U

(

1

,

I

(

x

,

0

)) ≤

^N

(

x

),

wherethelastinequalityfollowsbytakingy

=

^{0 in}^Equation^(6).

Ontheotherhand,ifN_I

(

x

) ≥

^e,^taking^again^y

=

^{0 in}^Equation^(6),^we^obtain N

(

x

) ≥

^U

(

N

(

0

),

I

(

x

,

0

)) =

^U

(

1

,

N_I

(

x

)) ≥

^U

(

1

,

e

) =

¹

.

3. Letusconsidernow

α

N

=

^sup

{

^x

∈ [

⁰

,

1

] |

^N

(

x

) ≥

^e

}

^.

(6)

a) If

α

N

=

^0,^we^directly^derive^from^the^deﬁnition^of

α

N that N

(

x

) <

e forallx

>

0.

b) If

α

N

>

0 wegetN

(

y

) ≥

^{e for}^all ^y

< α

N andconsequently, I

(

1

,

y

) =

^U

(

e

,

I

(

1

,

y

)) ≤

^U

(

N

(

y

),

I

(

1

,

y

)) ≤

^N

(

1

) =

⁰

,

thatis,I

(

1

,

y

) =

^{0 for}^all ^y

< α

N.

4. If

α

N

>

0,wegetI

(

1

,

y

) =

^{0 for}^all ^y

< α

N andconsequentlyI doesnotsatisfy(N P).

5. Suppose inthiscasethat I

(

x

,

x

) =

^{1 for}^all ^x

∈ [

⁰

,

1

]

^and^N

(

x

) <

1 for x

>

0.Inthiscase, supposeagainthatthere is somex

>

0 withN

(

x

) ≥

^e.^Taking^x

=

^{y in}^Expression^{(6) we}^will^get

N

(

x

) ≥

^U

(

N

(

x

),

I

(

x

,

x

)) ≥

^U

(

e

,

1

) =

¹

,

whichisacontradiction.Thus,itmustbeagainN

(

x

) <

e forallx

>

0 andconsequently,

α

N

=

^0.

Remark3.1.SomeinterestingconsequencescanbederivedfromProposition3.3asfollows.

(i) IfwewanttodealwithcontinuousfuzzynegationsintheU -ModusPonens,thenwewillhave

α

N

>

0.FromItem4in Proposition3.3wegetthat anypossiblesolutionoftheU -ModusPonensfailsto satisfy(N P).Consequently,allusual implicationsderivedfromt-normsandt-conorms(thatis,R,

(

S

,

N

)

,Q L andD-implications),aswellasYagergenerated implications,never satisfy U -ModusTollens(sinceall ofthemsatisfy (N P)).Fortunately, we willseeinnext sections thatmanysolutionscanbefoundamongthoseimplicationsderivedfromuninorms,especiallyamongRU -implications.

(ii) On the other hand, if we want to consider implication functions satisfying (N P), the same argument leads to the necessityofconsideringnegationswith

α

N

=

^0,^that^is,^negations ^{N such}^that ^N

(

x

) <

e forallx

>

0 andconsequently, non-continuous.

Intherestofthissectionwewilldealwiththecase

α

N

=

^{0 and}^also^when^the^involvedimplications satisfy(N P).We willcharacterizeinparticularall solutionsinthecasewhenthenegation N isstrictlydecreasingintheinterval

]

⁰

,

e

[

^and U isauninormin

U

min.Westartbyprovingthatforthiskindofuninorms,U-ModusTollensalwaysholdsforallx

≤

^{y and} soitissuﬃcienttoproveitfor

(

x

,

y

)

suchthat y

<

x.

Proposition3.4.LetI beanimplicationfunction,N afuzzynegationwith

α

N

=

^0,^and^{U a}^uninormⁱⁿ

U

minwithneutralelement e

∈]

⁰

,

1

[

^.^Then^I

,

N satisfytheU -ModusTollenswithrespecttoU ifandonlyif

U

(

N

(

y

),

I

(

x

,

y

)) ≤

N

(

x

)

for all y

<

x

.

Proof. WeonlyneedtoprovethattheU -ModusTollensisalwayssatisﬁedinpointswherex

≤

^y.^Sinceîtîsôbvious^when x

=

^0,^we^deal^with^values⁰

<

x

≤

^y.^In^this^case^we^have^N

(

y

) ≤

^N

(

x

)

andsince

α

N

=

^0,^it^is^N

(

x

) <

e forall x

>

0.Then wehavetwocases:

a) IfI

(

x

,

y

) >

e,then

U

(

N

(

y

),

I

(

x

,

y

)) ≤

^U

(

N

(

x

),

I

(

x

,

y

)) =

^min

{

^N

(

x

),

I

(

x

,

y

) } =

^N

(

x

).

b) IfI

(

x

,

y

) ≤

^e,^then

U

(

N

(

y

),

I

(

x

,

y

)) ≤

^U

(

N

(

x

),

I

(

x

,

y

)) =

^eTU

N

(

x

)

e

,

^I

(

x

,

y

)

e

≤

^e ^N

(

x

)

e

=

^N

(

x

).

Wenowstudythecaseinwhichtheimplication I satisﬁes(N P),whichisthecaseofmostoftheimplicationsderived fromt-normsandt-conorms.

Proposition3.5.LetI beanimplicationfunctionthatsatisﬁes(N P),N afuzzynegation,andU aconjunctiveuninormwithneutral elemente

∈]

⁰

,

1

[

^.^If^I

,

N satisfytheU -ModusTollenswithrespecttoU ,then:

1. U

(

N

(

y

),

y

) =

^{0 for}^all^y

∈ [

⁰

,

1

]

^.

2. N

(

x

) =

^{0 for}^all^x

≥

^{e and}^N

(

x

) <

e forallx

>

0.

3. U

(

N

(

y

),

I

(

e

,

y

)) =

^{0 for}^all^y

∈ [

⁰

,

1

]

^.

4. IfN isstrictlydecreasingintheinterval

]

⁰

,

e

[

^,^then^I

(

x

,

y

) <

e forally

<

x

<

e.

Proof. Letusproveallitemsonebyone.

1. U

(

N

(

y

),

y

) =

^{0 for}^all ^y

∈ [

⁰

,

1

]

^follows^directly^by^taking^x

=

^{1 in}^Equation^{(6) since} ^{I satisﬁes}^{(N P}^).

(7)

2. AccordingtoProposition3.3weknowthatifI satisﬁes(N P),then

α

N

=

^{0 and}^N

(

x

) <

e forallx

>

0.Moreover,justby putting y

=

^{e in}^the^previous ^condition,^we ^obtain ^N

(

e

) =

^U

(

N

(

e

),

e

) =

^{0 and}consequently, N

(

x

) =

^{0 for}^all ^x

≥

^e,^by decreasingness.

3. Now,takingx

=

^{e in}^Equation^{(6) we}^get^U

(

N

(

y

),

I

(

e

,

y

)) ≤

^N

(

e

) =

^0,^and^so^U

(

N

(

y

),

I

(

e

,

y

)) =

^{0 for}^all ^y

∈ [

⁰

,

1

]

^. 4. To prove thisitem letusconsider some x

,

y such that y

<

x

<

e and supposethat I

(

x

,

y

) >

e.First, note that y

>

0

becauseI

(

x

,

0

) =

^NI

(

x

) ≤

^N

(

x

) <

e.Consequently,wehaveN

(

y

) <

e andso,sinceN isstrictlydecreasingin

]

⁰

,

e

[

^,

U

(

N

(

y

),

I

(

x

,

y

)) ≥

^min

{

^N

(

y

),

I

(

x

,

y

) } =

^N

(

y

) >

N

(

x

),

leadingtoacontradiction.Thus,weobtainI

(

x

,

y

) <

e forall y

<

x

<

e.

WhenN isanegationstrictlydecreasingintheinterval

]

⁰

,

e

[

^and^theimplicationI satisﬁes(N P),wecangiveacharac- terizationofthoseuninormsin

U

minthatsatisfytheU -ModusTollens,asfollows.

Theorem3.1.LetI beanimplicationfunctionthatsatisﬁes(N P),U

≡

^TU

,

e

,

SU

minauninormin

U

minwithneutralelemente

∈ ]

⁰

,

1

[

^,^and^{N a}^fuzzy^negation^strictly^decreasingⁱⁿ

]

⁰

,

e

[

^.^Then^I

,

N satisfytheU -ModusTollenswithrespecttoU ifandonlyifthe followingpropertieshold:

1. U

(

N

(

y

),

I

(

e

,

y

)) =

^{0 for}^all^y

∈ [

⁰

,

1

]

^.

2. N

(

x

) =

^{0 for}^all^x

≥

^{e and}^N

(

x

) <

e forallx

>

0.

3. I

(

x

,

y

) <

e forally

<

x

<

e.

4. IandNverifytheModusTollenswithrespecttoT_Uforally

<

x,whereIandNaregivenby

I

(

x

,

y

) =

1 if x

≤

y

,

I(ex,ey)

e if y

<

x

.

⁽⁷⁾

N

(

x

) =

1 if x

=

⁰

,

N(ex)

e otherwise

.

⁽⁸⁾

Proof. Let ussupposeﬁrstthat I and N satisfythe U -Modus Tollenswithrespectto U .Properties 1,2and3holdfrom Proposition3.5andthus,weonlyneedtoprovethat IandN satisfytheModusTollenswithrespectto TU forall y

<

x.

Wewilldistinguishtwocases:

•

^If ^y

=

^{0 then,}

TU

(

N

(

y

),

I

(

x

,

0

)) =

TU

1

,

^I

(

ex

,

0

)

e

=

^I

(

ex

,

0

)

e

=

^N^I

(

ex

)

e

≤

^N

(

ex

)

e

=

N

(

x

),

wherethelastinequalityisduetoProposition3.3.

•

^If ^y

>

0,wehave

T_U

(

N

(

y

),

I

(

x

,

y

)) =

^TU

N

(

ey

)

e

,

^I

(

ex

,

ey

)

e

=

^U

(

N

(

ey

),

I

(

ex

,

ey

))

e

≤

^N

(

ex

)

e

=

^N

(

x

),

wherethelastinequalityholdsbecauseI

,

N satisfyU -ModusTollenswithrespecttoU .

Conversely, supposethat conditions1-4holdandwewill provethat I

,

N satisfy theU -Modus Tollenswithrespectto U . According to Proposition3.4 the U -Modus Tollensneeds to be checkedonly inpoints

(

x

,

y

)

where y

<

x. Thus, we will considerthreecases:

•

^If ^y

=

^0,^then

U

(

N

(

y

),

I

(

x

,

y

)) =

^U

(

1

,

I

(

x

,

0

)) =

^U

(

1

,

N_I

(

x

)) =

^NI

(

x

),

wherethelastequalityholdsbecauseU

∈ U

min.ThentheresultfollowsfromProposition3.3.

•

^If⁰

<

y

<

x

<

e thenI

(

x

,

y

) <

e and N

(

y

) <

e byconditions3and2,respectively.Inthiscase, U

(

N

(

y

),

I

(

x

,

y

)) =

^e

·

^TU

N

(

y

)

e

,

^I

(

x

,

y

)

e

=

^e

·

^TU

N

_y e

,

I

_x e

,

^y

e

≤

^e

·

^N

_x e

=

^N

(

x

),

wherethelastinequalityholdsfromcondition4.

(8)

Fig. 1. Plot of the fuzzy negation Ne(left) and structure of the fuzzy implication Ie (right) given in Example3.2.

•

^If^x

≥

^e,^then^I

(

x

,

y

) ≤

^I

(

e

,

y

)

andhence,

U

(

N

(

y

),

I

(

x

,

y

)) ≤

^U

(

N

(

y

),

I

(

e

,

y

)) =

⁰

,

whereinthiscasethelastequalityisduetocondition1.

Remark3.2.Letuspointouttwointerestingfacts.

i) According to the theorem above, the proof that I

,

N satisfy the U -Modus Tollens with respect to a uninorm U

≡

^TU

,

e

,

S_U

min in

U

min, requires in particular to check that I and N satisfy the Modus Tollens withrespect to T_U. However, thisquestion was alreadysolved in[26,27] and consequently, our characterizationtheorem iscomplete. In fact,manyexamplescanbederivedfromconditionsinTheorem3.1(seeExample3.2below).

ii) NotethatinTheorem3.1,thet-conormSU playsnoroleinthesatisfactionoftheU -ModusTollensandso,SU canbe anyt-conorm.Moreover,thesamepropertyholdswhenthevaluesoftheimplicationI areoutoftherectangle

[

⁰

,

e

[

²^, withtheuniqueconditionthatitsatisﬁes(N P).

Example3.2.Letusconsidertheuninormin

U

min,U

≡

^TU

,

e

,

SU

minwhereTU

=

^TLK istheŁukasiewiczt-normandSU is anyt-conorm.LetNe bethenegationgivenby

Ne

(

x

) =

⎧ ⎪

⎨

⎪ ⎩

1 if x

=

⁰

,

e

−

^x ^{if 0}

<

x

<

e

,

0 if x

≥

^e

.

Ontheotherhand,weconsiderthefollowingfuzzyimplication

I_e

(

x

,

y

) =

⎧ ⎪

⎨

⎪ ⎩

1 if x

=

^{0 or y}

=

¹

,

max

(

e

−

^x

,

y

)

if 0

<

x

≤

^{e and 0}

≤

^y

≤

^e

,

y otherwise

.

It is easy to seethat these operators satisfy all the conditionsof Theorem 3.1and therefore Ie

,

Ne satisfy the U -Modus TollenswithrespecttotheuninormU (Fig.1).

4. ModusTollensforR U -implications

Inthissection,givenaconjunctiveuninormU andafuzzynegationN,wewillinvestigatewhichRU -implicationssatisfy theU -ModusTollensinequality.WewilldoitforRU -implicationsderivedfromthethreeclassesofuninormsrecalledinthe preliminaries.Notethatintheprevioussectionwehaveseenthat,forfuzzynegationsN with

α

N

>

0 (casethatincludesall continuousnegations),onlyimplications I suchthat I

(

1

,

y

) =

^{0 for}^all ^y

< α

N cansatisfytheU -ModusTollens.Fortunately, thisconditionissatisﬁedby RU -implicationsderivedfromrepresentableuninormsandforsomeidempotentsonesandso, inthesecaseswewillseethatcontinuous(evenstrictandstrong)negationscanbeconsidered.

WewillstudytheU -ModusTollens forRU -implicationsderivedfromuninormsintheclassof

U

min,fromrepresentable uninormsandfromidempotentuninorms,andwewilldevoteasubsectiontoeachofthesecases.

(9)

4.1. CaseofRU -implications derivedfromuninormsin

U

min

In this subsection, we will deal with RU -implications derived from uninorms in

U

min, that is, uninorms U0

≡

^T0

,

e₀

,

S₀

min withneutral element e₀

∈]

⁰

,

1

[

ând ûnderlying^t-norm ând^t-conorm ^T0 and S₀, respectively. Recall that forthiskindofuninorms,RU -implicationsaregivenbythefollowingstructure.

Proposition4.1.(Theorem5.4.7in[3])LetU

≡

^TU

,

e

,

SU

minbeauninormin

U

minandIU itsresidualimplication.Then

IU

(

x

,

y

) =

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

1 if x

≤

^y

<

e

,

eIT_U

(

_e^x

,

_e^y

)

if y

<

x

≤

^e

,

e

+ (

1

−

e

)

ISU

(

^x₁⁻₋^e_e

,

₁^y⁻₋^e_e

)

if e

≤

x

≤

y

,

e if e

≤

^y

<

x

,

y otherwise

.

(9)

Forthiskindof RU -implicationswe againgetthat the U -Modus Tollenscan besatisﬁed onlywithnegations N with

α

N

=

^{0 and}consequently,withnon-continuousnegations.Moreover,forthiskindofRU -implicationswehavethefollowing result.

Proposition4.2.LetIU0 beafuzzyimplicationderivedfromU0

≡

^T0

,

e0

,

S0

min,N afuzzynegation,andU

≡

^TU

,

e

,

SU

^a^con- junctiveuninorm.IfIU0

,

N satisfy theU -ModusTollenswithrespecttoU ,thenthefollowingpropertieshold:

1. U

(

N

(

y

),

y

) =

^{0 for}^all^y

≤

^e0. 2. U

(

N

(

y

),

1

) =

^N

(

y

)

forally

<

e0. 3.

α

N

=

^{0 (that}^is,^N

(

x

) <

e forallx

>

0).

4. Ife

≤

^e0N

(

x

) =

^{0 for}^all^x

≥

^e.

5. Ife0

<

e,TUisacontinuoust-normandU

(

e0

,

e0

) =

^e0,thenN

(

x

) <

e0forallx

>

0 andN

(

x

) =

^{0 for}^all^x

≥

^e0.

Proof. 1. FollowsdirectlyfromEquation(6) byputtingx

=

^{1 and}^takingîntoâccount^thatÎU₀

(

1

,

y

) =

^{y for}^all ^y

≤

^e0. 2. Foranyx

<

e0,wehaveIU0

(

x

,

x

) =

^{1 and}^if^we^take^x

=

^{y in}^Equation^(6),^we^have

U

(

N

(

y

),

IU₀

(

y

,

y

)) =

^U

(

N

(

y

),

1

) ≤

^N

(

y

).

Theresultfollowsfromthefactthat U

(

N

(

y

),

1

) ≥

^N

(

y

)

holdsalways.

3. ItisadirectconsequenceofProposition3.3,becauseitis I_U₀

(

1

,

y

) >

0 forall y

>

0.

4. Ifwetake y

=

^{e in}^Property^1,^we^have^U

(

N

(

e

),

e

) =

^N

(

e

) =

^{0 and}^bydecreasingness,N

(

x

) =

^{0 for}^all^x

≥

^e.

5. First,letusseethat N

(

x

) <

e0 forall0

<

x

<

e0.Indeed,since TU iscontinuous ande0

<

e isan idempotentelement, ifwehavethat N

(

x

) ≥

^e0withe₀

>

x

>

0,then

U

(

e₀

,

x

) =

^x ^{for all} ⁰

≤

^x

≤

^e0 (10)

andthen,applyingProperty1weget

0

=

^U

(

N

(

x

),

x

) ≥

^U

(

e₀

,

x

) =

^x

leadingtoacontradiction.Now,wehavethatN

(

x

) <

e0forallx

>

0 bydecreasingness,andweonlyneedtoprovethat N

(

e0

) =

^{0 to}^conclude^the^proof.^But^this^is^clear^because^N

(

e0

) <

e0andthen

0

=

^U

(

N

(

e0

),

e0

) =

^N

(

e0

),

wheretheﬁrstequalityisduetoProperty1andthesecondisduetoEquation(10).

Inordertocharacterizethesolutionsinthiscase,wewilldistinguishthreedifferentpossibilitiesaccordingtotheorder oftheneutralelementse ande0.Webeginbythemostsimplecasewhenbothelementscoincide.

Theorem4.1.LetI_U₀betheRU -implicationderivedfromU₀

≡

^T0

,

e₀

,

S₀

min,N afuzzynegation,andU

≡

^TU

,

e

,

S_U

^a^conjunc- tiveuninormwithneutralelemente

=

^e0.ThenIU0andN satisfytheU -ModusTollenswithrespecttoU ,ifandonlyifthefollowing propertieshold

1. U

(

N

(

y

),

y

) =

^{0 for}^all^y

≤

ê,ândÛ

(

N

(

y

),

1

) =

^N

(

y

)

forally

<

e.

2. N

(

x

) <

e forallx

>

0,andN

(

x

) =

^{0 for}^all^x

≥

^e.

3. I_T₀andNsatisfytheModusTollenswithrespecttothet-normT_Uforally

<

x,whereI_T₀istheresidualimplicationderivedfrom thet-normT0andNisgivenbyEquation(8).