International Journal of Computational Intelligence Systems

In Press, Uncorrected Proof, Available Online: 14 February 2020

On the Use of Conjunctors With a Neutral Element in the Modus Ponens Inequality

Authors
Ana Pradera1, *, Sebastia Massanet2, 3, Daniel Ruiz2, 3, Joan Torrens2, 3
1 Departamento de Ciencias de la Computación, Arquitectura de Computadores, Lenguajes y Sistemas Informáticos y Estadística e Investigación Operativa, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
2 Soft Computing, Image Processing and Aggregation (SCOPIA) Research Group, Department of Mathematics and Computer Science, University of the Balearic Islands, 07122 Palma, Spain
3 Balearic Islands Health Research Institute (IdISBa), 07010 Palma, Spain
*Corresponding author. Email: ana.pradera@urjc.es
Corresponding Author
Ana Pradera
Received 14 May 2019, Accepted 4 February 2020, Available Online 14 February 2020.
DOI
https://doi.org/10.2991/ijcis.d.200205.002How to use a DOI?
Keywords
Modus ponens, Fuzzy implication function, Conjunctor, Neutral element, Semi-copula, T-norm, Uninorm
Abstract

The inference rule of Modus Ponens has been extensively investigated in the framework of approximate reasoning, especially for the case of t-norms. Recently, more general kinds of conjunctors have also been considered, like semi-copulas, copulas, and conjunctive uninorms. A common feature of all these kinds of conjunctors is the fact that they have a neutral element e[0,1]. This paper is devoted to the study of Modus Ponens for conjunctors with a neutral element with no additional conditions. Many properties are proved to be necessary for a fuzzy implication function I to satisfy the Modus Ponens with respect to a conjunctor with neutral element e[0,1]. Although the most usual families of fuzzy implication functions do not satisfy all these properties, other possibilities for I are presented showing many new examples and generalizing some already known results on this topic. Moreover, all fuzzy implication functions satisfying the Modus Ponens with respect to the least (and with respect to the greatest) conjunctor with neutral element e[0,1] are characterized. The particular case of e=1, that provides semi-copulas, is studied separately, retrieving many known results that can be easily derived from the current study.

Copyright
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The Modus Ponens inequality [15] is a well-known functional inequality that comes out when using the classical Modus Ponens rule in order to implement forward fuzzy inference processes. The latter are approximate reasoning schemes that allow to infer a conclusion of the form “y is Q” from two premises: a fuzzy proposition “x is P” and a fuzzy conditional statement “If x is P, then y is Q”. The inequality, which involves a fuzzy implication function [1,69] I:[0,1]2[0,1] (used to model the fuzzy conditional) and a bivariate aggregation function[1013] C:[0,1]2[0,1] (needed to aggregate the premises), is written as C(a,I(a,b))b for any a,b[0,1].

The aggregation of the premises in these inference processes has traditionally been performed by means of triangular norms[4,14,15], even though lately other functions such as overlap functions [16,17] and conjunctive uninorms [18,19] have also been investigated for this purpose. Another recent paper, Ref. [20], shows that conjunctors (aggregation functions having zero as annihilator element) are the only aggregation functions that may be able to solve the Non-Contradiction principle and hence also the Modus Ponens inequality, since the satisfaction of the former (with respect to the natural negation of the fuzzy implication function) is a necessary condition for the satisfaction of the latter.

This paper explores the generalization of the aggregation function used in the Modus Ponens inequality to the class of conjunctors with a neutral element e[0,1]1 a broad family of aggregation functions that includes semi-copulas (when the neutral element is equal to one) such as triangular norms, copulas or representable aggregation functions, as well as conjunctive uninorms or continuous generated functions (otherwise). Our main goal is to investigate the relationships that exist between the properties of conjunctors with a neutral element and those of the fuzzy implication functions that may satisfy the Modus Ponens inequality with respect to them. We will then apply such relationships to the most important families of fuzzy implication functions in order to study whether they satisfy or not the Modus Ponens inequality with respect to this kind of conjunctors, allowing many new examples and general results.

The paper is organized as follows. Section 2 recalls the main issues related to negation, aggregation, and fuzzy implication functions, as well as the most important available results regarding the Modus Ponens inequality. Section 3 provides several necessary and/or sufficient conditions for the satisfaction of the Modus Ponens inequality with respect to a conjunctor with a neutral element. Next, Sections 4 and 5 focus, respectively, on the specific cases of conjunctors with a neutral element e1 and those with neutral element e=1 (semi-copulas). Finally, Section 6 ends with some conclusions and pointers to future work.

2. PRELIMINARIES

In this section we recall the definitions and main properties of negation functions, conjunctors with a neutral element and fuzzy implication functions, as well as the main results related to the Modus Ponens inequality.

2.1. Negation Functions

Negation functions constitute a well-known tool allowing to represent fuzzy negations.

Definition 1.

[2123] A function N:[0,1][0,1] is called a negation function if it is decreasing and satisfies N(0)=1 and N(1)=0. Moreover,

  • N is vanishing when N(x)=0 for some x1,

  • N is filling when N(x)=1 for some x0.

Typical examples of negation functions are the classical negation Nc, defined as Nc(x)=1x for all x[0,1], as well as the least and the greatest negation functions, given, respectively, by

N(x)=1if x=0,0otherwise,N(x)=0if x=1,1otherwise.

2.2. Conjunctors With a Neutral Element

Conjunctors with a neutral element constitute a special class of aggregation functions (see Refs. [1013]). In this paper we will only be interested in the bivariate case (where aggregation functions are increasing functions A:[0,1]2[0,1] verifying A(0,0)=0 and A(1,1)=1) and in the following properties.

Definition 2.

[1013] Let A be an aggregation function and let N be a negation function.

  • A is a conjunctor when it satisfies A(1,0)=A(0,1)=0.

  • A has a neutral element e[0,1] (NE(e)) when A(x,e)=A(e,x)=x for all x[0,1].

  • A has zero divisors (0Div) when there exist a,b[0,1] such that A(a,b)=0.

  • A satisfies the Non-Contradiction principle with respect to N (NC(N)) when A(x,N(x))=0 for all x[0,1].

Note that when A is a conjunctor, due to its increasingness, A has zero as annihilator element.

Conjunctors with a neutral element may be classified into the two following categories:

  1. Conjunctors with neutral element e=1, also known as semi-copulas.2 Semi-copulas are conjunctive (Cmin) and may or may not have zero divisors. Some distinguished families of semi-copulas are the following:

    • Triangular norms (t-norms for short) [21,22], which are associative and commutative semi-copulas.

    • Copulas[27], which are semi-copulas C satisfying C(0,x)=C(x,0)=0 for all x[0,1] and the so-called 2-increasing property: C(x1,y1)C(x1,y2)C(x2,y1) + C(x2,y2)0 for all x1,y1,x2,y2[0,1] such that x1x2 and y1y2.

    The least and the greatest semi-copulas are, respectively, the drastic product t-norm TD (given by TD(x,y)=0 if (x,y)[0,1]2 and TD(x,y)=min(x,y) otherwise) and the minimum t-norm TM (TM(x,y)=min(x,y)). Other well-known families of t-norms are the so-called strict t-norms (or t-norms in the Product family), defined as (TP)φ(x,y)=φ1(φ(x)φ(y)), and nilpotent t-norms (or t-norms in the Łukasiewicz family), (TLK)φ(x,y)=φ1(max(0,φ(x) + φ(y) − 1)), where φ:[0,1][0,1] is an increasing bijection [21,22].

  2. Conjunctors with a neutral element e[0,1], which are conjunctive (Cmin) on the square [0,e]2, disjunctive (Cmax) on [e,1]2 and averaging (minCmax) elsewhere. These conjunctors are averaging on the whole unit square only when the limiting functions min and max are chosen in the regions [0,e]2 and [e,1]2, respectively. They have zero divisors if and only if C|[0,e]2 has zero divisors. One distinguished family of aggregation functions in this category is the class of conjunctive uninorms, which are associative and commutative aggregation functions U:[0,1]2[0,1] with a neutral element e[0,1] satisfying U(0,1)=0 that behave as scaled t-norms/t-conorms on [0,e]2/[e,1]2 (see Ref. [28] for a recent survey on these functions).

    Figure 1 depicts the structure of Ce and Ce, the least and the greatest conjunctors with a neutral element e1 (note that the first one is also the least uninorm and belongs to the class Umin).

Figure 1

The least (Ce, left) and the greatest (Ce, right) conjunctors with a neutral element e[0,1].

2.3. Fuzzy Implication Functions

The most accepted definition of fuzzy implication function is the following.

Definition 3.

[1,9,29] A fuzzy implication function is a function I:[0,1]2[0,1] verifying the following properties:

  1. I is decreasing in the first variable.

  2. I is increasing in the second variable.

  3. I(0,0)=I(1,1)=1 and I(1,0)=0.

The papers in Refs. [8,30,31] provide recent compilations regarding these functions. The least and the greatest implication functions are given, respectively, by

I(x,y)=1if x=0 or y=1,0otherwise.
and
I(x,y)=0if x=1,y=0,1otherwise.

Other popular fuzzy implication functions are recalled below:

  • The Gödel implication

    IGD(x,y)=1if xy,yotherwise.

  • The Goguen implication

    IGG(x,y)=1if xy,yxotherwise.

  • The Łukasiewicz implication

    ILK(x,y)=1if xy,1x+yotherwise.

  • The Weber implication

    IWB(x,y)=1if x1,yotherwise.

Recall also (see e.g., Ref. [1]) that, similarly to what happens with aggregation functions, any fuzzy implication function I may be transformed by means of an automorphism φ:[0,1][0,1] into a new fuzzy implication function denoted as Iφ and given by Iφ(x,y)=φ1(I(φ(x),φ(y))). Remember finally that if I is a fuzzy implication function, then NI:[0,1][0,1], defined as NI(x)=I(x,0) for any x[0,1], is a negation function, called the natural negation[1] of the fuzzy implication function. Besides, the following additional properties of fuzzy implication functions will be needed in the remainder of this paper.

Definition 4.

[1,32] Let I be a fuzzy implication function. Then:

  • I satisfies the left neutrality principle (NP) when

    y[0,1]:I(1,y)=y(NP)

  • I satisfies the identity principle when

    x[0,1]:I(x,x)=1(IP)

  • satisfies the ordering property when

    x,y[0,1]:xyI(x,y)=1(OP)

  • I satisfies the consequent boundary property when

    x,y[0,1]:I(x,y)y(CP)

Recall that on the one hand, (OP) implies (IP) and on the other hand, (NP) implies (CB) (see e.g., Ref. [32]).

2.4. The Modus Ponens Inequality

The Modus Ponens scheme, a well-known classical inference rule allowing to perform forward reasoning, has usually been translated to the fuzzy framework as follows:

Definition 5.

[1,9,29] Let I be a fuzzy implication function and let C be a conjunctor. Then I satisfies the Modus Ponens inequality with respect to C when

x,y[0,1]:C(x,I(x,y))y(MP(C))

As it has been mentioned in the introduction, the Modus Ponens inequality (MP(C)) has mostly been studied when the conjunctor C is taken as a t-norm (usually a continuous one) and the fuzzy implication function belongs to one of the most important families of implications: see e.g., Ref. [1] Section 7.4. for the main results related to (S,N)-implications, R-implications, and QL-implications, Ref. [15] for RU and (U,N)-implications and Ref. [14] for probabilistic implications and S-implications. Recently, some authors [18,19] have also dealt with the use of conjunctive uninorms instead of t-norms, studying the case of RU-implications, and others (see Ref. [16]) have considered the use of overlap functions, dealing with the so-called O-conditionality. Since t-norms and conjunctive uninorms are special cases of conjunctors with a neutral element, in the current paper we will recover and generalize some of these results.

Remark 1

Recall also (see e.g., Proposition 7.4.3. in Ref. [1] for the case of t-norms) that when studying the Modus Ponens inequality, the monotonicity of the conjunctor clearly allows one to focus on the greatest functions, since:

  • If I satisfies (MP(C)), then I satisfies (MP(C*)) for any conjunctor C* such that C* C.

  • If I satisfies (MP(C)), then any fuzzy implication function I such that II does also satisfy (MP(C)).

An interesting necessary condition for the satisfaction of the Modus Ponens inequality (which is in some cases also sufficient) comes from considering residuated functions, which are functions obtained from the residuation scheme pq{t:ptq} as

IC(x,y)=sup{t[0,1]:C(x,t)y},x,y[0,1],
where C is a binary function (see e.g., Refs. [33,34]). Depending on the properties of the underlying function C, residuated functions may qualify as implication functions (according to Definition 3) and/or may satisfy the so-called residuation property (see e.g., Refs. [24,35]), given by
x,y,z[0,1]:C(x,y)zyIC(x,z)((RP))

The study of the properties of the residuated function IC and the satisfaction of (RP) was first undertaken for t-norms but was later on generalized to other functions. In the case of conjunctors the following can be stated.

Proposition 1.

[24,25,35] Let C be a conjunctor and let IC be its corresponding residuated function.

  1. IC is a fuzzy implication function if and only if C(1,t)0 for all t[0,1].

  2. If C is left-continuous, then (RP) is satisfied, and the supremum of IC can be replaced by a maximum.

Note that condition (i) in the previous proposition is satisfied, in particular, if C has a neutral element e, since C(1,t)C(e,t)=t. On the other hand, as noted for example in Ref. [4], the residuation property may be used to study the satisfaction of the Modus Ponens inequality. Indeed, the following result was proved in Ref. [4] when dealing with t-norms (and in Ref. [36] for conjunctive uninorms), but it can be adapted for conjunctors in general.

Proposition 2.

[4,36] Let C be a conjunctor, let IC be its corresponding residuated function and let I be a fuzzy implication function.

  • If I satisfies (MP(C)), then IIC.

  • If C is left-continuous, then I satisfies (MP(C)) if and only if IIC.

Proof.

The first item is direct from the definition of IC. To prove the second one, taking x=a,y=I(a,b), and z=b in Equation (RP) we obtain

C(a,I(a,b))bI(a,b)IC(a,b)a,b[0,1],
which ends the proof.

3. SOME GENERAL CONDITIONS FOR THE SATISFACTION OF THE MODUS PONENS INEQUALITY WITH RESPECT TO CONJUNCTORS WITH A NEUTRAL ELEMENT

In this section we analyze the satisfaction of the Modus Ponens inequality with respect to conjunctors with a neutral element e[0,1], studying the relationships that exist between the properties of the conjunctor and those of the fuzzy implication function involved.

We begin by recalling that the paper in Ref. [20], which deals with the Non-Contradiction principle NC(N) (see Subsection 2.2), points out that the fulfillment of this principle with respect to the natural negation of a fuzzy implication function is a necessary condition for the satisfaction of the Modus Ponens inequality: indeed, it suffices to take the value y=0 in (MP(C)) to obtain C(x,I(x,0))=C(x,NI(x))=0. As a consequence, all the results presented in Ref. [20] become necessary conditions for the satisfaction of the Modus Ponens inequality. The following Proposition encompasses the main ones related to conjunctors with a neutral element (Propositions 13, 15, 17, and 18 in Ref. [20]).

Proposition 3.

[20] Let C be a conjunctor with a neutral element e[0,1], let I be an implication function satisfying the Modus Ponens inequality with respect to C and let NI be the natural negation of the fuzzy implication function (NI(x)=I(x,0)). Then:

  1. C(x,y)=0 for all x,y[0,1] such that yNI(x), which means that C (C|[0,e]2 when e1) must satisfy (0Div) unless it is NI=N.

  2. [xe:NI(x)=0] and [x<e,x0:NI(x)<e] (note that this implies, in particular, that NI is non-filling and, whenever e1, NI is in addition vanishing and non-continuous at least on x=0).

  3. If e=1, C=(aj,bj,Tj)jJ is a non-trivial ordinal sum t-norm and NIN, then:

    1. There exists iJ such that ai=0 and bi1.

    2. [xbi:NI(x)=0] and [x<bi,x0:NI(x)<bi] (note that this implies that NI must be vanishing and non-continuous at least on x=0).

    3. Ti(xbi,ybi)=0 for all x[0,bi],y[0,NI(x)].

In the following we present some additional necessary conditions for fuzzy implication functions satisfying the Modus Ponens inequality with respect to conjunctors with a neutral element, and we analyze their consequences.

Proposition 4.

Let C be a conjunctor with a neutral element e[0,1] and let I be an implication function satisfying the Modus Ponens inequality with respect to C. Then

  1. I(x,y)<ex,y[0,1] such that x>y.

  2. I(x,y)yx,y[0,1] such that xe.

Proof.

We prove the items step by step.

  • Suppose there exist x0,y0[0,1] such that x0>y0 and I(x0,y0)e. In this case, from the monotonicity of C and the fact that e is the neutral element of C, we deduce

    C(x0,I(x0,y0))C(x0,e)=x0>y0,
    obtaining a contradiction.

  • Note first that choosing x=e in (MP(C)) provides C(e,I(e,y))=I(e,y)y for any y[0,1]. Thus, by monotonicity of I we have

    I(x,y)I(e,y)y
    for any x,y[0,1] such that xe.

Remark 2

When e=1 the first item in the proposition above may be written as I(x,y)1 for all x>y and hence, it is a generalization of Proposition 7.4.2. in Ref. [1]. Note also that, if C=min, the stronger condition I(x,y)y when x>y is obtained. Similarly, the second item when e=1 can be written as I(1,y)y for all y[0,1] and hence, it is a generalization of the result given in Ref. [4] for continuous t-norms.

On the other hand, when e1, Proposition 4 recovers some weaker statements related to uninorms proved in Ref. [19].

Next result involves several additional properties of fuzzy implication functions.

Proposition 5.

Let C be a conjunctor with a neutral element e[0,1] and let I be an implication function satisfying the Modus Ponens inequality with respect to C. Then:

  1. NI(x)<e for any x0 and NI(x)=0 for any xe.

  2. If I satisfies (IP), then it necessarily satisfies (OP).

  3. If I satisfies (CB), then necessarily e=1 and I satisfies (NP).

  4. If e1, then

    • I(x,y)<y for any x>ye, in particular I(1,y)<y for any 1>ye (i.e., I does not satisfy (CB) either (NP)).

    • I does not satisfy (IP) either (OP).

Proof.

Again we give the proof step by step.

  1. It suffices to choose y=0 in Proposition 4 (note that this is also a consequence of item 2 in Proposition 3).

  2. Directly comes from the fact that (OP) is equivalent to (IP) and [I(x,y)1 if x>y].

  3. Suppose on the contrary that I satisfies (CB) and e1. Then choosing x>ye we get I(x,y)<ey by Proposition 4. This contradicts (CB) and hence e=1. Now (NP) follows directly by taking x=1 in the second item of Proposition 4.

  4. The first part of this item is a direct consequence of the previous one. To prove the second part just take x=ye with x1, then Proposition 4 provides I(x,x)x1.

We finally deal with some sufficient conditions that apply to a broad class of conjunctors that includes, as a particular case, conjunctors with a neutral element.

Proposition 6.

Let C be a conjunctor, let e[0,1] and let I be a fuzzy implication. Then:

  1. Suppose C(x,e)x for any x[0,1] and I(x,y)e for any x,y[0,1] such that xy,x0,y1. If C(x,I(x,y))y for all x,y[0,1] such that x>y then I satisfies (MP(C)).

  2. Suppose C(e,y)y for any y[0,1] and I(x,y)y for any x,y[0,1] such that xe,x0. If C(x,I(x,y))y for all x,y[0,1] such that x>e then I satisfies (MP(C)).

Proof.

  1. It suffices to check that (MP(C)) is true for any x,y[0,1] such that xy. This is obvious if x=0 or y=1, and otherwise we have, by monotonicity of C, C(x,I(x,y))C(x,e)xy.

  2. We now have to prove that (MP(C)) is true for any x,y[0,1] such that xe. If x=0, C(x,I(x,y))=0y for any y[0,1]. Otherwise, using the monotonicity of C it is C(x,I(x,y))C(e,I(x,y))I(x,y)y.

In the limit case where e=1 we get the following results, which are valid, in particular, for any conjunctive aggregation function and hence for any semi-copula:

Corollary 7.

Let C be a conjunctor and let I be a fuzzy implication function.

  1. Suppose C(x,1)x for any x[0,1]. If C(x,I(x,y))y for all x,y[0,1] such that x>y then I satisfies (MP(C)) (i.e., the values of I when xy are indifferent).

  2. Suppose C(1,y)y for any y[0,1] and I(x,y)y for any x,y[0,1] such that x0. Then I satisfies (MP(C)).

  3. Suppose Cmin (i.e., C is conjunctive) and I satisfies I(x,y)y for any x,y[0,1] such that x>y. Then I satisfies (MP(C)).

Proof.

The two first items come directly from Proposition 6, and the third one is a combination of them (indeed, because of monotonicity, the conditions C(x,1)x and C(1,y)y are equivalent to Cmin).

4. THE MODUS PONENS INEQUALITY WITH RESPECT TO CONJUNCTORS WITH A NEUTRAL ELEMENT e[0,1]

The present section is specifically devoted to the use of conjunctors with a neutral element e[0,1]. First of all, we summarize in Figure 2 some of the conditions found in Section 3 when they are restricted to this kind of conjunctors. Note that the fact that I can not satisfy (NP) allows to discard many important families of fuzzy implication functions.

Figure 2

Some conditions for the satisfaction of (MP(C)) when C is a conjunctor with a neutral element e[0,1] (conditions do not apply to dashed lines).

Proposition 8.

Let C be a conjunctor with a neutral element e[0,1] and let I be a fuzzy implication function. If I is an (S,N)-implication, an R-implication, a QL-implication, an f, g, h or hg-generated implication, a probabilistic implication or an S-implication, then I does not satisfy (MP(C)).

Proof.

All the abovementioned families of fuzzy implication functions satisfy (NP) (see e.g., Refs. [8,14,30,31]) and hence Proposition 5.4 can be applied.

Nevertheless, there are still many fuzzy implication functions that could satisfy the Modus Ponens inequality with respect to a conjunctor with a neutral element e[0,1]. Observe, as it was recalled in Section 2.2, that the behavior of these conjunctors depends on the region of the unit square which is considered, since they are below min on [0,e]2, above max on [e,1]2 and averaging otherwise. The next result analyzes the consequences of choosing the least possible values on each of these regions.

Proposition 9.

Let C be a conjunctor with neutral element e[0,1] and let I be an implication function.

  1. Suppose C|[0,e]2=0 and let R1={(x,y)[0,1]2:x>y, x<e}. Then I satisfies (MP(C)) on R1 if and only if I(x,y)<e for any (x,y)R1.

  2. Suppose C|[e,1]2=max and let R2={(x,y)[0,1]2:xy<1,xe}. Then I satisfies (MP(C)) on R2 if and only if I(x,y)y for any (x,y)R2.

  3. Suppose C|[e,1]×[0,e]=min and let R3={(x,y)[0,1]2:x>y,xe}. Then I satisfies (MP(C)) on R3 if and only if I(x,y)<e for any (x,y)R3 such that ye and I(x,y)y for any (x,y)R3 such that y<e.

  4. Suppose C|[0,e]×[e,1]=min and let R4={(x,y)[0,1]2:0<xy<1,x<e}. Then I satisfies (MP(C)) on R4.

Proof.

The necessity part of any of the three first items is given by Proposition 4. The sufficiency is proved below:

  1. If (x,y)R1 and I(x,y)<e then we have C(x,I(x,y))=0y.

  2. Suppose that (x,y)R2 and I(x,y)y. If it were I(x,y)e then Proposition 6 proves the conclusion. Otherwise, if I(x,y)>e then C(x,I(x,y))=max(x,I(x,y))y since both x and I(x,y) are smaller than or equal to y.

  3. Suppose now that (x,y)R3 and we distinguish two cases. If ye and I(x,y)<e we have (x,I(x,y))[e,1]×[0,e] and consequently

    C(x,I(x,y))=min(x,I(x,y))=I(x,y)<ey.

    On the other hand, if y<e and I(x,y)y we again have (x,I(x,y))[e,1]×[0,e] and

    C(x,I(x,y))=min(x,I(x,y))=I(x,y)y.

  4. Finally, if (x,y)R4 we again distinguish two cases. If it were I(x,y)<e this would obviously entail C(x,I(x,y))C(x,e)=xy, and otherwise it would be (x,I(x,y))[0,e]×[e,1] and then C(x,I(x,y))=min(x,I(x,y))=xy.

Note that the combination of the results given in Proposition 9 allows us to characterize the capacity of the least conjunctor with a neutral element e1 (the least uninorm with neutral element e, depicted on the left part of Figure 1) for solving the Modus Ponens inequality.

Proposition 10.

Let Ce be the least conjunctor with a neutral element e[0,1] and let I be a fuzzy implication function. Then I satisfies (MP(C)) with C=Ce if and only if the two following conditions hold:

  • I(x,y)y for any x,y[0,1] such that xe and either ye or xy.

  • I(x,y)<e for any x,y[0,1] such that x>y and either ye or x<e.

Remark 3

The two following remarks regarding the above characterization are worth noting:

  • I satisfies (MP(Ce)) is not equivalent to IICe, where ICe is the residuated implication built from Ce and given (see e.g., Ref. [37]) by

    ICe(x,y)=1if xy,x<e,eif x>y,(x<e or ye),yotherwise.

    Indeed, as stated in Proposition 2, if I satisfies (MP(Ce)) then clearly IICe, but the converse is not necessarily true if C is not left-continuous, as it happens with Ce. Note in particular that ICe itself does not satisfy (MP(Ce)), since taking e.g., (x0,y0) such that x0>y0,x0<e provides Ce(x0,ICe(x0,y0))=Ce(x0,e)=x0>y0.

  • Proposition 10 is actually a characterization of fuzzy implication functions satisfying (MP(C)) with respect to at least some conjunctor with a neutral element e[0,1], since Ce is the least of them.

The regions R1 to R4 considered in Proposition 9 are depicted in the top of Figure 3, whereas the possible values of fuzzy implication functions I satisfying (MP(C)) with C=Ce proved in Proposition 10 can be viewed in the bottom of the figure.

Figure 3

Regions R1 to R4 considered in Proposition 9 can be viewed in the top of the figure. Dashed segments do not belong to the corresponding region. The bottom of the figure corresponds to the possible values of the fuzzy implication functions satisfying (MP(Ce)) characterized in Proposition 10.

Example 1.

Fixed some e[0,1], let us show that there are many examples of fuzzy implication functions satisfying conditions in Proposition 10 and, consequently, satisfying (MP(Ce)).

  1. The parameterized family of fuzzy implication functions given by

    Ia,e(x,y)=1if xy,x<e,aif a<y<x,yotherwise,
    satisfies (MP(Ce)), for any a such that 0a<e.

  2. There are also examples among the residual implications IU derived from uninorms, or RU-implications for short. For instance, it can be easily viewed from Proposition 10 that all RU-implications derived from uninorms in some of the following classes3 satisfy (MP(Ce)):

    1. Uninorms in Umin with neutral element e<e whose underlying t-conorm is given by the maximum (this can be derived also from Proposition 7 in Ref. [19]).

    2. All representable uninorms with neutral element e (this can be also derived from Theorem 5 in Ref. [19]).

    3. All idempotent uninorms with g(e)=0 where g is the Id-symmetrical function associated to the uninorm (see also Proposition 10 in Ref. [19]).

The next Proposition deals with the cases where the greatest values are chosen in each of the four different regions R1 to R4 (note that in this case there are some small differences in the location of the border values to each region) of the unit square.

Proposition 11.

Let C be a conjunctor with neutral element e[0,1] and let I be an implication function.

  1. Suppose C|[0,e]2=min and let R1={(x,y)[0,1]2:x>y, xe}. Then I satisfies (MP(C)) on R1 if and only if I(x,y)y for any (x,y)R1.

  2. Suppose C|[e,1]2=1 and let R2={(x,y)[0,1]2:xy, x>e,y1}. Then I satisfies (MP(C)) on R2 if and only if I(x,y)e for any (x,y)R2.

  3. Suppose C|[e,1]×[0,e]=max and let R3={(x,y)[0,1]2: x>y,x>e}. Then I satisfies (MP(C)) on R3 if and only if I(x,y)=0 for any (x,y)R3.

  4. Suppose C|[0,e]×[e,1]=max and let R4={(x,y)[0,1]2: xy,xe,x0,y1}. Then I satisfies (MP(C)) on R4 if and only if I(x,y)max(e,y) for any (x,y)R4.

Proof.

The sufficiency part of items 1, 2, and 4 is given by Proposition 6, whereas the sufficiency of item 3 is obvious since I(x,y)=0 implies C(x,I(x,y))=0y. The necessity is proved below:

  1. Take (x,y)R1. Since x>y we have I(x,y)<e by Proposition 4, and consequently

    C(x,I(x,y))=min(x,I(x,y)).

    Now, if it were I(x,y)>y we would deduce C(x,I(x,y))>y obtaining a contradiction with the Modus Ponens, thus it must be I(x,y)y.

  2. Consider now (x,y)R2 and suppose that I(x,y)>e. Since x>e, y1 and C|[e,1]2=1 we obtain C(x,I(x,y))=1y contradicting again the Modus Ponens. Thus, we have I(x,y)e in this case.

  3. Let (x,y)R3 and suppose that I(x,y)0. In this case, Proposition 4 provides I(x,y)<e, and since x>e we get the contradiction

    C(x,I(x,y))=max(x,I(x,y))=x>y.

  4. Finally, consider (x,y)R4 and let us distinguish two cases. If I(x,y)e then obviously I(x,y)max(e,y). On the other hand, if I(x,y)>e and taking into account that xe we obtain by the Modus Ponens that

    I(x,y)=max(x,I(x,y))=C(x,I(x,y))y,
    and hence I(x,y)max(e,y).

Note that the first item in Proposition 11 forces implications functions to verify NI=N (see Proposition 3, item 1) and it may be applied, in particular, to the greatest conjunctor with a neutral element e1, Ce. Moreover, the combination of the four items of this Proposition characterizes the family of implications functions that satisfy the Modus Ponens inequality with respect to Ce.

Proposition 12.

Let Ce be the greatest conjunctor with a neutral element e[0,1] and let I be a fuzzy implication function. Then I satisfies (MP(C)) with C=Ce if and only if the three following conditions hold:

  • I(x,y)=0 for any x,y[0,1] such that x>y,x>e.

  • I(x,y)y for any x,y[0,1] such that (x>y,xe) or (xe,ye,x0).

  • I(x,y)e for any x,y[0,1] such that (xy,x>e,y1) or (xy,y<e,x0).

The regions R1 to R4 slightly modified in Proposition 11 are depicted in the top of Figure 4, whereas the possible values of implications I satisfying (MP(C)) with C=Ce proved in Proposition 12 can be viewed in the bottom of the figure.

Figure 4

Regions R1 to R4 considered in Proposition 11 can be viewed in the top of the figure. Dashed segments do not belong to the corresponding region. The bottom of the figure corresponds to the possible values of the fuzzy implication functions satisfying (MP(Ce)) characterized in Proposition 12.

Remark 4

Since Ce is left-continuous, Proposition 2 shows that the characterization of the satisfaction of (MP(Ce)), given in the previous proposition, is equivalent to the inequality IICe, where the latter is the residuated implication of Ce, given by

ICe(x,y)=1if x=0 or y=1,0if x>y,x>e,yif (y<xe) or (xey,x0),eotherwise.

Observe finally that the fact that Ce is the greatest conjunctor with neutral element e, along with Remarks 1 and 4 and Proposition 12, allows for the following result.

Proposition 13.

Let I be a fuzzy implication function and let Ce be the greatest conjunctor with a neutral element e[0,1]. The following items are equivalent:

  1. I satisfies (MP(Ce)).

  2. I satisfies (MP(C)) with respect to any conjunctor C such that CCe, in particular any conjunctor with a neutral element e[0,1].

  3. I satisfies the three conditions given in Proposition 12 (bottom of Figure 4).

  4. IICe, where ICe is given in Remark 4.

Example 2.

Fixed some e[0,1], let us give some examples of fuzzy implication functions satisfying conditions in Proposition 12, i.e., satisfying (MP(Ce)), and consequently (MP(C)) with respect to any conjunctor C with neutral element e. Of course ICe itself is one of them, as it has been proved before. The following parameterized family of fuzzy implication functions presents other possibilities.

Ia,b(x,y)=1if x=0 or y=1,0if x>y,x>e,aif (aye) and (0<xe),bif (by<1) and (0<xe),eif e<x;y<1,yotherwise,
where a,b are such that 0aeb1.

Results in Propositions 9 and 11 can be also used to characterize those fuzzy implication functions satisfying (MP(C)) with C=Ue, where Ue is the least idempotent uninorm with neutral element e and with C=Ue, where Ue is the greatest conjunctive idempotent uninorm with neutral element e. These uninorms are respectively given by (see for instance Ref. [28]):

Ue(x,y)=max(x,y)if x,ye,min(x,y)otherwise,
and
Ue(x,y)=0if x=0 or y=1,min(x,y)if 0<x,ye,max(x,y)otherwise,
and the results are presented in the next proposition.

Proposition 14.

Let Ue be the least idempotent uninorm with neutral element e, Ue the greatest conjunctive idempotent uninorm with neutral element e and let I be a fuzzy implication function. Then

  1. I satisfies (MP(C)) with C=Ue if and only if the following two conditions hold:

    • I(x,y)y for any x,y[0,1] such that [x>y,y<e] or [exy<1].

    • I(x,y)<e for any x,y[0,1] such that ey<x.

  2. I satisfies (MP(C)) with C=Ue if and only if the following three conditions hold:

    • I(x,y)=0 for any x,y[0,1] such that e,y<x.

    • I(x,y)y for any x,y[0,1] such that [y<xe] or [0<xy,ye].

    • I(x,y)e for any x,y[0,1] such that xy<e.

Note that Ue is an example of a uninorm in Umin which is not left-continuous and whose residual implication IUe does not satisfy the Modus Ponens with Ue. On the contrary, Ue is a left-continuous uninorm and consequently the characterization given in the second item of the previous proposition is equivalent to say IIUe.

5. THE MODUS PONENS INEQUALITY WITH RESPECT TO SEMI-COPULAS

This section analyzes the satisfaction of (MP(C)) when C is a conjunctor with neutral element e=1, i.e., a semi-copula. Figure 5 encompasses the conditions mentioned in Section 3 when they are specifically applied to semi-copulas.

Figure 5

Some conditions for the satisfaction of (MP(C)) when C is a semi-copula (conditions do not apply to the dashed line).

The least and the greatest semi-copulas are two well-known t-norms, TD and TM, whose definition was recalled in Section 2.2, along with other important families of t-norms. The next Proposition characterizes their use in the Modus Ponens inequality.

Proposition 15.

Let I be a fuzzy implication function. The following statements are true:

  • I satisfies (MP( TD)) if and only if I(1,y)y for any y[0,1] and I(x,y)1 for all x,y[0,1] such that x>y.

  • I satisfies (MP((TLK)φ)) if and only if I(x,y)φ1(φ(y) +1φ(x)) for all x,y[0,1] such that x>y, i.e., IILKφ.

  • I satisfies (MP((TP)φ)) if and only if I(x,y)φ1(φ(y)φ(x)) for all x,y[0,1] such that x>y, i.e., IIGGφ.

  • I satisfies (MP(TM)) if and only if I(x,y)y for all x,y[0,1] such that x>y, i.e., IIGD.

Proof.

The first item is a matter of calculation (note that TD is not left-continuous and the satisfaction of (MP(TD)) implies but is not equivalent to IIWB-the Weber implication is the residuated implication associated to TD-). The following three items are obtained from Proposition 2 since it is well-known (see e.g., Ref. [1]) that ILK,IGG, and IGD are, respectively, the residuated implication functions associated to the continuous t-norms TLK,TP, and TM.

Similarly to what was noticed in the previous section, the facts that TD and TM are, respectively, the least and the greatest semi-copulas, along with Proposition 15 and Remark 1, allow for the following equivalences:

Proposition 16.

A fuzzy implication function I satisfies (MP(C)) with respect to at least some semi-copula C if and only if I(1,y)y for any y[0,1] and I(x,y)1 for all x,y[0,1] such that x>y.

Proposition 17.

Let I be a fuzzy implication function. The following items are equivalent:

  1. I satisfies (MP( TM)).

  2. I satisfies (MP(C)) with respect to any C such that CTM (in particular any semi-copula).

  3. I satisfies I(x,y)y for all x,y[0,1] such that x>y.

  4. IIGD.

It is worth noting that the above results allow to easily recover some already existing results concerning the satisfaction of the Modus Ponens, as shown in the example below.

Example 3.

Probabilistic implications and probabilistic S-implications [38] are two recently introduced classes of fuzzy implication functions which are defined, respectively, as

IC(x,y)=1if x=0,C(x,y)xif x0,
and
I~C(x,y)=C(x,y)x+1,
where C is a copula.4 The paper in Ref. [14] investigates, among other properties, the T-conditionality (Modus Ponens with respect to t-norms) of these functions, with the following results:
  1. “A probabilistic implication IC satisfies (MP(TM)), and consequently (MP(T)) for any t-norm T, if and only if C(x,y)xy for all x,y[0,1] such that x>y” (see Ref. [14], Theorem 4.2). This result could have been directly obtained from Proposition 17.

  2. “Any probabilistic implication satisfies (MP(C)) with C=TP (and with any weaker t-norm, in particular C=TLK)” (see Ref. [14], Proposition 4.3). This may be proved using Proposition 15 taking into account that ICIGG (see Ref. [39]).

  3. “No probabilistic S-implication satisfies (MP(C)) with C=TM” (see Ref. [14], Proposition 4.5). This result may be directly obtained from Proposition 3, item 1, since clearly NI~C=NcN and TM does not have zero divisors.

  4. “Each probabilistic S-implication satisfies (MP(C)) with C=TLK” (see Ref. [14], Proposition 4.6). Indeed, this may be obtained from Proposition 15 by just noting that since any copula C satisfies CTM, any probabilistic S-implication satisfies I~CI~TM, and the latter is nothing but ILK.

  5. “A probabilistic S-implication I~C satisfies (MP(C)) with C=TP if and only if I~CIGG” (See Ref. [14], Proposition 4.7). Again, this can be obtained directly from Proposition 15. Note nevertheless that the condition I~CIGG is never true, and hence this sentence could be better expressed as “No probabilistic S-implication satisfies (MP(C)) with C=(TP)φ”. This fact could have been directly obtained from Proposition 3, item 1, taking into account that NI~CN and that (TP)φ does not satisfy (0Div).

Finally, let us note that the above examples could be easily adapted to the classes of the so-called survival implications and survival S-implications, since it was proved in Refs. [39,40] that these classes respectively coincide with the classes of probabilistic implications and probabilistic S-implications.

6. CONCLUSIONS AND FUTURE WORK

The inference rule of Modus Ponens is a property of paramount importance in approximate reasoning which is not studied only for the case of t-norms any more. Indeed, several more general classes of conjunctors have been considered recently, most of them having a neutral element e[0,1]. Therefore, in this paper, many new results are proved in which the necessary properties for the fuzzy implication function I to satisfy the Modus Ponens with respect to a conjunctor with neutral element e[0,1] are determined. Although these properties are not fulfilled by well-known families such as (S,N), R, or QL-implications, in this paper many examples of admissible fuzzy implication functions are presented in which the common feature is that they do not satisfy the left NP. Particularly interesting are the characterizations of all fuzzy implication functions satisfying the Modus Ponens with respect the least and the greatest conjunctors with neutral element e[0,1].

As future work, we want to study with more detail the Modus Ponens property with respect to some of the families of conjunctors with neutral element e[0,1] such as ordinal sum t-norms, representable aggregation functions [4143], continuous generated functions [44], or conjunctive uninorms. The study of the latter has been made fixing the class of RU-implications [19] and some preliminary results are available for (U,N)-implications [18]. However, these results must be expanded and many other classes of fuzzy implication functions derived from uninorms have not been studied yet.

ACKNOWLEDGMENTS

This paper has been partially supported by the Spanish Grants TIN2015-66471-P and TIN2016-75404-P, AEI/FEDER, UE.

Footnotes

1

Conjunctors have annihilator zero and hence can not have zero as neutral element.

2

Note that the terms semi-copula and conjunctor are sometimes used interchangeably (e.g. Ref. [12]), whereas in this paper, as well as in Refs. [20,2426], semi-copulas constitute a proper subclass of conjunctors.

3

See Ref. [28] for the different classes of uninorms and Ref. [1] for the corresponding RU-implications.

4

In the first case, to guarantee that IC is a fuzzy implication function, the copula C must satisfy additionally the condition C(x1,y)x2C(x2,y)x1 for all x1x2.

REFERENCES

4.E. Trillas, C. Alsina, and A. Pradera, On MPT-implication functions for fuzzy logic, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A: Matemáticas (RACSAM), Vol. 98, 2004, pp. 259-271.
5.E. Trillas and L. Valverde, On mode and implication in approximate reasoning, M. Gupta (editor), Approximate Reasoning in Expert Systems, North Holland, Amsterdam, 1985, pp. 157-166.
6.M. Baczyński, Fuzzy implication functions: recent advances, Mathw. Soft Comput., Vol. 19, 2012, pp. 13-19.
17.G.P. Dimuro and B. Bedregal, Fuzzy implications and the law of O-conditionality: the case of residual implications derived from overlap functions, in Proceedings of 8th International Summer School on Aggregation Operators, 2015, pp. 97-111.
24.K. Demirli and B. De Baets, Basic properties of implicators in a residual framework, Tatra Mountains Math. Publ., Vol. 16, 1999, pp. 31-46.
43.G. Mayor and J. Torrens, On a class of binary operations: non-strict archimedean aggregation functions, in Proceedings of ISMVL’88 (Palma de Mallorca, Spain), 1988, pp. 54-59.
Journal
International Journal of Computational Intelligence Systems
Publication Date
2020/02
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
https://doi.org/10.2991/ijcis.d.200205.002How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Ana Pradera
AU  - Sebastia Massanet
AU  - Daniel Ruiz
AU  - Joan Torrens
PY  - 2020
DA  - 2020/02
TI  - On the Use of Conjunctors With a Neutral Element in the Modus Ponens Inequality
JO  - International Journal of Computational Intelligence Systems
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.200205.002
DO  - https://doi.org/10.2991/ijcis.d.200205.002
ID  - Pradera2020
ER  -