# A theorem on implication functions defined from triangular norms

Didier Dubois, Henri Prade

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## To cite this version:

Didier Dubois, Henri Prade. A theorem on implication functions defined from triangular norms. Bulletin pour les sous-ensembles flous et leurs applications, 1984, 18, pp.33-41. hal-04583381

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# A theorem on implication functions defined from triangular norms

Didier Dubois, Henri Prade

<figure><img src="image_1.png" /></figure>

## To cite this version:

Didier Dubois, Henri Prade. A theorem on implication functions defined from triangular norms. Bulletin pour les sous-ensembles flous et leurs applications, 1984, 18, pp.33-41. hal-04583381

HAL Id: hal-04583381

https://hal.science/hal-04583381v1

Submitted on 22 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
A THEOREM ON IMPLICATION FUNCTIONS
DEFINED FROM TRIANGULAR NORMS

Didier DUBOIS Henri PRADE

Laboratoire Langages et Systèmes Informatiques

Université P. SABATIER

31062 - TOULOUSE CEDEX

**ABSTRACT** : Several transformations which enable implication functions in multi-valued logics to be generated from conjunctions have been proposed in the literature. It is proved that for a rather general class of conjunctions modeled by triangular norms, the generation process is closed, thus shedding some light on the relationships between seemingly independent classes of implication functions.

0 - INTRODUCTION

It is now well-known (Alsina et al [1], Dubois [2], Prade [7]) that a good model of a fuzzy set-theoretic intersection, or equivalently of a conjunction function in multivalued logics, is a triangular norm (Menger [6], Schweizer and Sklar [8]). Using a De-Morgan-like transformation yields the corresponding model of a fuzzy set-theoretic union or multivalued disjunction function. Let * be a triangular norm, modeling a conjunction, and n a negation function. If P, Q ... are propositions, whose degrees of truth are v(P), v(Q) ... then

$$v(P \cap Q) = v(P) * v(Q)$$

$$v(\neg P) = n(v(P))$$

Several techniques for deriving implication functions from conjunctions have been proposed, in the past, namely (e.g. Valverde [10])

$$a) v(P \to Q) = v(\neg (P \land \neg Q)) = n(v(P) * n(v(Q)))$$

$$b) v(P \to Q) = \sup\{s, s \in [0,1], v(P) * s \le v(Q)\}$$

The motivation of a) is clear. Transformation b) defines a pseudo-complement in a Brouwerian lattice, and is related to the following identity in set theory :
A THEOREM ON IMPLICATION FUNCTIONS
DEFINED FROM TRIANGULAR NORMS

Didier DUBOIS Henri PRADE

Laboratoire Langages et Systèmes Informatiques

Université P. SABATIER

31062 - TOULOUSE CEDEX

**ABSTRACT** : Several transformations which enable implication functions in multi-valued logics to be generated from conjunctions have been proposed in the literature. It is proved that for a rather general class of conjunctions modeled by triangular norms, the generation process is closed, thus shedding some light on the relationships between seemingly independent classes of implication functions.

O - INTRODUCTION

It is now well-known (Alsina et al [1], Dubois [2], Prade [7]) that a good model of a fuzzy set-theoretic intersection, or equivalently of a conjunction function in multivalued logics, is a triangular norm (Menger [6], Schweizer and Sklar [8]). Using a De-Morgan-like transformation yields the corresponding model of a fuzzy set-theoretic union or multivalued disjunction function. Let * be a triangular norm, modeling a conjunction, and n a negation function. If P, Q ... are propositions, whose degrees of truth are v(P), v(Q) ... then

$$v(P \cap Q) = v(P) * v(Q)$$

$$v(\neg P) = n(v(P))$$

Several techniques for deriving implication functions from conjunctions have been proposed, in the past, namely (e.g. Valverde [10])

$$a) v(P \to Q) = v(\neg(P \wedge \neg Q)) = n(v(P) * n(v(Q)))$$

$$b) v(P \to Q) = \sup\{s, s \in [0,1], v(P) * s \le v(Q)\}$$

The motivation of a) is clear. Transformation b) defines a pseudo-complement in a Brouwerian lattice, and is related to the following identity in set theory :
$$ \bar{A} \cup B = \overline{A \setminus B} = U_S, A \cap S \subseteq B $$

where $A \setminus B$ is a set-difference and the bar stands for complementation. Although apparently unrelated, these two classes of implication functions, and a third one obtained by contraposition can be generated by the same processes if the class of conjunction operations from which they stem is enlarged. This is the topic of this paper. After a background on triangular norms and their functional representation is recalled, the main theorem is expressed and proved. The introduced transformations are applied to the basic t-norms.

# 1 - BACKGROUND

A triangular co-norm $\perp$ (t-co-norm for short) is a two-place function from $I \times I$ to $I$ (where $I$ denotes the real interval $[0,1]$) such that

$$ i) \forall (a, b) \in I^2, a \perp b = b \perp a $$

$$ ii) \forall (a,b,c) \in I^3, (a \perp b) \perp c = a \perp (b \perp c) $$

$$ iii) \text{ if } a \le b \text{ and } c \le d, \text{ then } a \perp c \le b \perp d $$

$$ iv) \forall a \in I, 0 \perp a = a $$

$$ v) 1 \perp 1 = 1 $$

A continuous triangular co-norm $\perp$ such that

$$ vi) \forall a \in ]0,1[, a \perp a > a \text{ (Archimedean property)} $$

can be expressed in terms of a generator $\varphi$ which is a continuous strictly increasing function from $I$ to $[0, +\infty]$, with $\varphi(0) = 0$, under the form (Ling [5]):

$$ a \perp b = \varphi^*(\varphi(a) + \varphi(b)) $$

(1)

where $\varphi^*$ is the pseudo-inverse of $\varphi$, defined by

$$ \varphi^*(a) = \begin{cases} \varphi^{-1}(a) & \text{if } a \in [0, \varphi(1)] \\ 1 & \text{if } a \in [\varphi(1), +\infty) \end{cases} $$

(2)

if $\varphi(1) < +\infty$, the t-co-norm $\perp$ is said to be nilpotent; otherwise the t-norm is said to be strict.

Note that

$$ \varphi^* \circ \varphi = \text{id.} $$

(3)

while $\varphi \circ \varphi^* \ne \text{id.}$ Moreover $\varphi$ is defined up to a positive multiplicative constant $\lambda$, i.e. $\lambda$ and $\lambda\varphi$, $\lambda > 0$, generate same the co-norm.

By definition, a strong negation function $n$ is a continuous strictly decreasing function from $I$ to $I$, such that

$$ vii) \forall a \in I, n(n(a)) = a $$

$$ viii) n(0) = 1 $$

Trillas [9] proved that any strong negation function $n$ can be generated from a continuous strictly increasing function $\varphi$ from $I$ to $[0, +\infty)$, such that $\varphi(0) = 0$ and $\varphi(1) < +\infty$, under the form

$$ n(a) = \varphi^{-1}(\varphi(1) - \varphi(a)) $$

(4)

To a t-co-norm $\perp$ and a strong negation function $n$, is associated a two-place function from $I^2$ to $I$ defined by

$$ a * b = n(n(a) \perp n(b)) $$

(5)

(5) expresses a n-duality since then $a \perp b = n(n(a) * n(b))$; * is called a triangular norm; it satisfies, i, ii, iii, and

$$ ix) \forall a, 1 * a = a $$

$$ x) 0 * 0 = 0 $$

instead of $iv$ and $v$ respectively; $\perp$ satisfies $vi$ if and only if $*$ satisfies

$$ xi) \forall a \in ]0,1[, a * a < a $$

Then $*$ and $\perp$ are said to be Archimedean.

Any t-norm $*$ is such that

$$ \forall (a, b) \in I^2, T_w(a, b) = \begin{cases} a & \text{if } b = 1 \\ b & \text{if } a = 1 \\ 0 & \text{otherwise} \end{cases} \le a * b \le \min(a, b) $$

(6)

while, for any t-co-norm $\perp$,

$$ \forall (a, b) \in I^2, \max(a, b) \le a \perp b \le T_w^*(a, b) = \begin{cases} a & \text{if } b = 0 \\ b & \text{if } a = 0 \\ 1 & \text{otherwise} \end{cases} $$

(7)

Note that $\min(\text{resp. } \max)$ is a continuous t-norm (resp. t-co-norm) which does not satisfy $\text{xi}$ (resp. $\text{vi}$), and $T_w^*$ (resp. $T_w$) is a non-continuous t-norm (resp. t-co-norm).
$$ \bar{A} \cup B = \overline{A \setminus B} = U_S, A \cap S \subseteq B $$

where $A \setminus B$ is a set-difference and the bar stands for complementation. Although apparently unrelated, these two classes of implication functions, and a third one obtained by contraposition can be generated by the same processes if the class of conjunction operations from which they stem is enlarged. This is the topic of this paper. After a background on triangular norms and their functional representation is recalled, the main theorem is expressed and proved. The introduced transformations are applied to the basic t-norms.

# 1 - BACKGROUND

A triangular co-norm $\perp$ (t-co-norm for short) is a two-place function from $I \times I$ to $I$ (where $I$ denotes the real interval $[0,1]$) such that

$$ i) \forall (a, b) \in I^2, a \perp b = b \perp a $$

$$ ii) \forall (a,b,c) \in I^3, (a \perp b) \perp c = a \perp (b \perp c) $$

$$ iii) \text{ if } a \le b \text{ and } c \le d, \text{ then } a \perp c \le b \perp d $$

$$ iv) \forall a \in I, 0 \perp a = a $$

$$ v) 1 \perp 1 = 1 $$

A continuous triangular co-norm $\perp$ such that

$$ vi) \forall a \in ]0,1[, a \perp a > a \text{ (Archimedean property)} $$

can be expressed in terms of a generator $\varphi$ which is a continuous strictly increasing function from $I$ to $[0, +\infty]$, with $\varphi(0) = 0$, under the form (Ling [5]):

$$ a \perp b = \varphi^*(\varphi(a) + \varphi(b)) $$

(1)

where $\varphi^*$ is the pseudo-inverse of $\varphi$, defined by

$$ \varphi^*(a) = \begin{cases} \varphi^{-1}(a) & \text{if } a \in [0, \varphi(1)] \\ 1 & \text{if } a \in [\varphi(1), +\infty) \end{cases} $$

(2)

if $\varphi(1) < +\infty$, the t-co-norm $\perp$ is said to be nilpotent; otherwise the t-norm is said to be strict.

Note that

$$ \varphi^* \circ \varphi = \text{id.} $$

(3)

while $\varphi \circ \varphi^* \ne \text{id.}$ Moreover $\varphi$ is defined up to a positive multiplicative constant $\lambda$, i.e. $\lambda$ and $\lambda\varphi$, $\lambda > 0$, generate same the co-norm.

By definition, a strong negation function $n$ is a continuous strictly decreasing function from $I$ to $I$, such that

$$ vii) \forall a \in I, n(n(a)) = a $$

$$ viii) n(0) = 1 $$

Trillas [9] proved that any strong negation function $n$ can be generated from a continuous strictly increasing function $\varphi$ from $I$ to $[0, +\infty)$, such that $\varphi(0) = 0$ and $\varphi(1) < +\infty$, under the form

$$ n(a) = \varphi^{-1}(\varphi(1) - \varphi(a)) $$

(4)

To a t-co-norm $\perp$ and a strong negation function $n$, is associated a two-place function from $I^2$ to $I$ defined by

$$ a * b = n(n(a) \perp n(b)) $$

(5)

(5) expresses a n-duality since then $a \perp b = n(n(a) * n(b))$; * is called a triangular norm; it satisfies, i, ii, iii, and

$$ ix) \forall a, 1 * a = a $$

$$ x) 0 * 0 = 0 $$

instead of $iv$ and $v$ respectively; $\perp$ satisfies $vi$ if and only if $*$ satisfies

$$ xi) \forall a \in ]0,1[, a * a < a $$

Then $*$ and $\perp$ are said to be Archimedean.

Any t-norm $*$ is such that

$$ \forall (a, b) \in I^2, T_w(a, b) = \begin{cases} a & \text{if } b = 1 \\ b & \text{if } a = 1 \\ 0 & \text{otherwise} \end{cases} \le a * b \le \min(a, b) $$

(6)

while, for any t-co-norm $\perp$,

$$ \forall (a, b) \in I^2, \max(a, b) \le a \perp b \le T_w^*(a, b) = \begin{cases} a & \text{if } b = 0 \\ b & \text{if } a = 0 \\ 1 & \text{otherwise} \end{cases} $$

(7)

Note that $\min(\text{resp. } \max)$ is a continuous t-norm (resp. t-co-norm) which does not satisfy $\text{xi}$ (resp. $\text{vi}$), and $T_w^*$ (resp. $T_w$) is a non-continuous t-norm (resp. t-co-norm).
Let us check that the condition $n(s) \le a$ is satisfied for this choice of $s$,
i.e. $n(\varphi^*(\varphi(n(a)) + \varphi(b))) \le a$.

This inequality is equivalent to $\varphi^*(\varphi(n(a)) + \varphi(b)) \ge n(a)$ which obviously
holds since $\varphi(n(a)) + \varphi(b) \ge \varphi(n(a))$ and $\varphi^* \circ \varphi = \text{id}$.

Now, if $\varphi(s) \le \varphi(n(a)) + \varphi(0)$, then
$s \le \varphi^*(\varphi(n(a) + \varphi(0))) \le \varphi^*(\varphi(n(a) + \varphi(b)))$, $\forall b \in [0,1]$. Thus we get

$$
\begin{align}
a \dashv\dashv b &= \max[n(a), \varphi^*(\varphi(n(a)) + \varphi(b))] \\
&= \varphi^*(\varphi(n(a)) + \varphi(b)) \tag*{(17)}
\end{align}
$$

$$
= a[\mathcal{J}_n(*)]b
$$

Note that $\mathcal{V}_n(\ast) = \ast$.

$$
\underline{Q} \cdot \underline{E} \cdot \underline{D} \cdot
$$

3.2 - *Proof of (13)* (Continuous Archimedean t-norms)

We proceed with the same relaxed assumptions as the proof of (12). It is

easy to figure out that

$$
a[\mathcal{A} \circ \mathcal{J}_n \circ \mathcal{L} (*)]b = a[\mathcal{A}' \circ b = \begin{cases} \varphi^*(\varphi(a) - \varphi(n(b))) & \text{if } n(b) \le a \\ 0 & \text{if } n(b) \ge a \end{cases} \tag{18}
$$

Then

$$
\begin{align*}
a[\mathcal{L} \circ \mathcal{R} \circ \mathcal{J}_n \circ \mathcal{L} (*)]b &= a \dashv\dashv b = \sup\{s, s \in [0,1], a[\mathcal{A}'] s \le b\} \\
&= \max[n(a), \sup\{s, s \in [0,1], n(s) \le a, \varphi^*(\varphi(a) - \varphi(n(s))) \le b\}]
\end{align*}
$$

\text{Note that } \varphi^*(\varphi(a) - \varphi(n(s))) \le b
\begin{align*}
\Rightarrow \varphi(a) - \varphi(n(s)) &\le \varphi(b) \text{ if } \varphi(a) - \varphi(0) \ge \varphi(n(s)) \\
\Rightarrow s &\le n[\varphi^*(\varphi(a) - \varphi(b))] \text{ if } a \ge b
\end{align*}

If $a \le b$, $\forall s$, $\varphi(n(s)) \ge 0 \ge \varphi(a) - \varphi(b)$;

We can check that $n[n[\varphi^*(\varphi(a) - \varphi(b))]] \le a$, if $a \ge b$.

Indeed this is equivalent to $\varphi^*(\varphi(a) - \varphi(b)) \le a$
which obviously holds since $\varphi(a) - \varphi(b) \le \varphi(a)$ and $\varphi^* \circ \varphi = \text{id}$;
Now, if $\varphi(a) - \varphi(0) < \varphi(n(s))$, then
$s \le n \cdot \varphi(\varphi(a) - \varphi(0)) \le n \cdot \varphi^*(\varphi(a) - \varphi(b))$, $\forall b \in [0, 1]$.

$$
\text{Thus we get } a^{**+} b = \begin{cases} \max[n(a), n[\varphi^*(\varphi(a) - \varphi(b))]] & \text{if } a \ge b \\ 1 & \text{if } a \le b \end{cases}
$$

but $a \ge \varphi^*(\varphi(a) - \varphi(b))$ and finally

$$
\begin{align*}
a^{**+} b &= \begin{cases} n[\varphi^*(\varphi(a) - \varphi(b))] & \text{if } a \ge b \\ 1 & \text{if } a \le b \end{cases} \tag{19} \\
&= n(b) \dashv n(a) = a[\mathcal{V}_n \circ \mathcal{E} (*)] b
\end{align*}
$$

$$
\underline{Q} \cdot \underline{E} \cdot \underline{D} \cdot
$$

3.3 - Particular case : nilpotent t-norms

It is assumed that (⊥, *, n) are based on the same generator φ, i.e.
n(a) = φ*(φ(1) - φ(a)), a ⊥ b = φ*(φ(a) + φ(b)), a * b = n(n(a) ⊥ n(b)). It is easy
to figure out that φ ⋉ n(a) = φ(1) - φ(a), ∀ a. Hence
a * b = φ*(φ(n(a)) + φ(b)) = φ*(φ(1) - φ(a) + φ(b)).

$$
\begin{align*}
\text{Besides, if } a \ge b \\
\qquad n[\varphi^*(\varphi(n(b)) - \varphi(n(a)))] &= n[\varphi^*(\varphi(a) - \varphi(b))] \\
&= \varphi^*(\varphi(1) + \varphi(b) - \varphi(a))
\end{align*}
$$

since $a \ge b \Rightarrow \varphi(a) - \varphi(b) \le \varphi(1)$

and $n \circ \varphi^*(k) = \varphi^*(\varphi(1) - k)$ if $k \le \varphi(1)$

if a ≤ b then φ(1) - φ(a) + φ(b) ≥ φ(1) and a * b = 1

We have proved that a * b = a * b.

Now

$$
\ast\ast\dashv = \mathcal{V}_n(\ast\ast) = \mathcal{V}_n(\ast\ast) = \ast\ast
$$

$$
\mathcal{A}' = \mathcal{J}_n(\ast\ast) = \mathcal{J}_n(\ast\ast) = \ast
$$

$$
\mathcal{A}' = \mathcal{J}_n (\mathbf{\ast\ast}) = \mathcal{J}_n (\mathbf{\ast}) = \mathbf{\ast}
$$

3.4 - min_and_Tw

The proof in this case can be obtained by straight-forward check for min.
The theorem does not hold for $T_w$.

Examples

We provide the results of the transformations for the four basic t-norms
in the appended table 1, where n(a) = 1 - a in any case.
Let us check that the condition $n(s) \le a$ is satisfied for this choice of $s$,
i.e. $n(\varphi^*(\varphi(n(a)) + \varphi(b))) \le a$.

This inequality is equivalent to $\varphi^*(\varphi(n(a)) + \varphi(b)) \ge n(a)$ which obviously
holds since $\varphi(n(a)) + \varphi(b) \ge \varphi(n(a))$ and $\varphi^* \circ \varphi = \text{id}$.

Now, if $\varphi(s) \le \varphi(n(a)) + \varphi(0)$, then
$s \le \varphi^*(\varphi(n(a) + \varphi(0))) \le \varphi^*(\varphi(n(a) + \varphi(b)))$, $\forall b \in [0,1]$. Thus we get

$$
\begin{align}
a \dashv b &= \max[n(a), \varphi^*(\varphi(n(a)) + \varphi(b))] \\
&= \varphi^*(\varphi(n(a)) + \varphi(b)) \tag*{(17)}
\end{align}
$$

$$
= a[\mathcal{J}_n(*)]b
$$

Note that $\mathcal{V}_n(\ast) = \ast$.

$\underline{Q} \cdot \underline{E} \cdot \underline{D} \cdot$

3.2 - *Proof of (13)* (Continuous Archimedean t-norms)

We proceed with the same relaxed assumptions as the proof of (12). It is

easy to figure out that

$$
a[\mathcal{A} \circ \mathcal{J}_n \circ \mathcal{L} (*)]b = a[\mathcal{A}' \circ b = \begin{cases} \varphi^*(\varphi(a) - \varphi(n(b))) & \text{if } n(b) \le a \\ 0 & \text{if } n(b) \ge a \end{cases} \tag{18}
$$

Then

$$
\begin{align*}
a[\mathcal{L} \circ \mathcal{R} \circ \mathcal{J}_n \circ \mathcal{L} (*)]b &= a \dashv b = \sup\{s, s \in [0,1], a[\mathcal{A}'] s \le b\} \\
&= \max[n(a), \sup\{s, s \in [0,1], n(s) \le a, \varphi^*(\varphi(a) - \varphi(n(s))) \le b\}]
\end{align*}
$$

$$
\begin{align*}
& \text{Note that } \varphi^*(\varphi(a) - \varphi(n(s))) \le b \\
& \quad \Rightarrow \varphi(a) - \varphi(n(s)) \le \varphi(b) \text{ if } \varphi(a) - \varphi(0) \ge \varphi(n(s)) \\
& \quad \Rightarrow s \le n[\varphi^*(\varphi(a) - \varphi(b))] \text{ if } a \ge b
\end{align*}
$$

If $a \le b$, $\forall s$, $\varphi(n(s)) \ge 0 \ge \varphi(a) - \varphi(b)$;

We can check that $n[n[\varphi^*(\varphi(a) - \varphi(b))]] \le a$, if $a \ge b$.

Indeed this is equivalent to $\varphi^*(\varphi(a) - \varphi(b)) \le a$

which obviously holds since $\varphi(a) - \varphi(b) \le \varphi(a)$ and $\varphi^* \circ \varphi = \text{id}$;

Now, if $\varphi(a) - \varphi(0) < \varphi(n(s))$, then
$s \le n \cdot \varphi(\varphi(a) - \varphi(0)) \le n \cdot \varphi^*(\varphi(a) - \varphi(b))$, $\forall b \in [0,1]$.

$$
\text{Thus we get } a^{**+} b = \begin{cases} \max[n(a), n[\varphi^*(\varphi(a) - \varphi(b))]] & \text{if } a \ge b \\ 1 & \text{if } a \le b \end{cases}
$$

but $a \ge \varphi^*(\varphi(a) - \varphi(b))$ and finally

$$
\begin{align*}
& a^{**+} b = \begin{cases} n[\varphi^*(\varphi(a) - \varphi(b))] & \text{if } a \ge b \\ 1 & \text{if } a \le b \end{cases} \tag{19} \\
& = n(b) \dashv n(a) = a[\mathcal{V}_n \circ \mathcal{E} (*)] b
\end{align*}
$$

$\underline{Q} \cdot \underline{E} \cdot \underline{D} \cdot$

3.3 - Particular case : nilpotent t-norms

It is assumed that (⊥, *, n) are based on the same generator φ, i.e.
n(a) = φ*(φ(1) - φ(a)), a ⊥ b = φ*(φ(a) + φ(b)), a * b = n(n(a) ⊥ n(b)). It is easy
to figure out that φ ⋉ n(a) = φ(1) - φ(a), ∀ a. Hence
a * b = φ*(φ(n(a)) + φ(b)) = φ*(φ(1) - φ(a) + φ(b)).

$$
\begin{align*}
\text{Besides, if } a \ge b \\
\qquad n[\varphi^*(\varphi(n(b)) - \varphi(n(a)))] &= n[\varphi^*(\varphi(a) - \varphi(b))] \\
&= \varphi^*(\varphi(1) + \varphi(b) - \varphi(a))
\end{align*}
$$

since $a \ge b \Rightarrow \varphi(a) - \varphi(b) \le \varphi(1)$

and $n \circ \varphi^*(k) = \varphi^*(\varphi(1) - k)$ if $k \le \varphi(1)$

if a ≤ b then φ(1) - φ(a) + φ(b) ≥ φ(1) and a * b = 1

We have proved that a * b = a * b.

$$
\begin{align*}
\text{Now } \ast\ast\to &= \mathcal{V}_n(\ast\to) = \mathcal{V}_n(\ast\to) = \ast\to \\
\mathcal{A}' &= \mathcal{J}_n(\ast\to) = \mathcal{J}_n(\ast\to) = \ast \\
\mathcal{A}'' &= \mathcal{J}_n(\ast\ast\to) = \mathcal{J}_n(\ast\to) = \ast.
\end{align*}
$$

3.4 - min_and_Tw

The proof in this case can be obtained by straight-forward check for min.
The theorem does not hold for $T_w$.

Examples

We provide the results of the transformations for the four basic t-norms
in the appended table 1, where n(a) = 1 - a in any case.
<table><caption>Table 1 - Conjunctions and implications</caption><thead><tr><th>t-norm *</th><th>T<sub>W</sub>(a, b)</th><th>max(a + b - 1, 0)</th><th>a, b</th><th>min(a, b)</th></tr></thead><tbody><tr><td>nature</td><td>discontinuous</td><td>nilpotent</td><td>strict</td><td>non-Archimedean</td></tr><tr><td>co-norm</td><td>T<sup>*</sup><sub>W</sub>(a, b)</td><td>min(a + b, 1)</td><td>a + b - ab</td><td>max(a, b)</td></tr><tr><td rowspan="2">a *→ b</td><td>1 - a if b = 0</td><td></td><td></td><td></td></tr><tr><td>b if a = 1<br>1 otherwise</td><td>min(1, 1 - a + b)</td><td>1 - a + ab</td><td>max(1 - a, b)</td></tr><tr><td rowspan="2">a *→ b</td><td>1 if a &lt; 1</td><td>min(1, 1 - a + b)</td><td>min(1, b/a)<br>(1 if a = 0)</td><td>1 if a ≤ b<br>b if a &gt; b</td></tr><tr><td>b if a = 1</td><td></td><td></td><td></td></tr><tr><td rowspan="2">a *→ b</td><td>1 if b &gt; 0</td><td>min(1, 1 - a + b)</td><td>min(1, 1 - a)<br>(1 if b = 1)</td><td>1 if a ≤ b<br>1 - a if a &gt; b</td></tr><tr><td>1 - a if b = 0</td><td></td><td></td><td></td></tr><tr><td rowspan="2">a *<sup>1</sup>→ b</td><td>0 if b &lt; 1</td><td>max(a + b - 1, 0)</td><td>max(0, <sup>a+b-1</sup>&frasl;<sub>b</sub>)<br>(0 if b = 0)</td><td>0 if a + b ≤ 1<br>a if a + b &gt; 1</td></tr><tr><td>a if b = 1</td><td></td><td></td><td></td></tr><tr><td rowspan="2">a *<sup>1</sup>→ b</td><td>0 if a &lt; 1</td><td>max(a + b - 1, 0)</td><td>max(0, <sup>a+b-1</sup>&frasl;<sub>a</sub>)<br>(0 if a = 0)</td><td>0 if a + b ≤ 1<br>b if a + b &gt; 1</td></tr><tr><td>b if a = 1</td><td></td><td></td><td></td></tr></tbody></table>

## 4 - CONCLUSION

An important consequence of the theorem is the closure of the generation processes of implication functions from a triangular norm based on transformations a) and b). Iterating these processes does not yield an uncontrollable birth of implication families. This closure property is a strong point in favor of methods a) and b). Moreover we have generated a new kind of conjunction operations A which coincide with classical conjunctions of the proposition calculus, but are not commutative when non-extreme truth values are combined. This lack of commutativity may be natural in some contexts when propositions which do not play the same role have to be combined

in a conjunctive manner. For instance when P is to be combined with P → Q, for the purpose of detaching Q; or if the concept of time is accounted for, and P ∧ Q underlies the assumption that P was known before Q. So we believe that this new class of conjunctions (and dual disjunctions) does not challenge our intuition. Unfortunately, many of these operations are not associative either.

However, we do not claim that the closure theorem is a complete answer to the problem of generating multivalued implication functions. Several families fall outside our framework: for instance those derived from a conjunction and a negation by means of the identity P → Q = P ∧ (P ∧ Q); or the implication proposed by Yager [11], based on the power operation: a → b = bª!

## REFERENCES

[1] - Alsina, C., Trillas, E., Valverde, L. (1980). On non-distributive logical connectives for fuzzy sets theory. BUSEFAL n° 3, LSI, Univ. P. Sabatier, Toulouse, 18-29.

[2] - Dubois, D. (1980). Triangular norms for fuzzy sets. Proc. 2nd Int. Seminar on Fuzzy Set Theory, Johannes Kepler Univ., Linz, Austria, Sept. 15-20, (E.P. Klement, Ed.), 39-68.

[3] - Dubois, D., Prade, H. (1983). Fuzzy set-theoretic differences and inclusions and their use in fuzzy arithmetics and analysis. Proc. 5th Int. Seminar on Fuzzy Set Theory, (E.P. Klement, Ed.), J. Kepler University, Linz, Austria.

[4] - Dubois, D., Prade, H. (1984). Fuzzy set-theoretic differences and inclusions and their use in the analysis of fuzzy equations. To appear in Control and Cybernetics (Poland).

[5] - Ling, C.H. (1965). Representation of associative functions. Publ. Math. Debrecen 12, 189-212.

[6] - Menger, K. (1942). Statistical Metrics. Proc. Nat. Acad. Sci. USA, 28, 535-537.

[7] - Prade, H. (1980). Unions et intersections d'ensembles flous. BUSEFAL n° 3, LSI, Univ. P. Sabatier, Toulouse, 58-62.

[8] - Schweizer, B., Sklar, A. (1963). Associative functions and abstract semigroups. Publ. Math. Debrecen, 10, 69-81.

[9] - Trillas, E. (1979). Sobre funciones de negacion en la teoria de conjuntos difusos. Stochastica, III, n° 1, 47-59.

[10] - Valverde, L. (1982). Indistinguishability relations with implicative part. Proc. 4th Int. Seminar on Fuzzy Set Theory, J. Kepler Universität, Linz, Austria.

[11] - Yager, R. (1980). An approach to inference in approximate reasoning. Int. J. Man-Machine Studies 13, 323-338.
<table><caption>Table 1 - Conjunctions and implications</caption><thead><tr><th>t-norm *</th><th>T<sub>W</sub>(a, b)</th><th>max(a + b - 1, 0)</th><th>a, b</th><th>min(a, b)</th></tr></thead><tbody><tr><td>nature</td><td>discontinuous</td><td>nilpotent</td><td>strict</td><td>non-Archimedean</td></tr><tr><td>co-norm</td><td>T<sub>W</sub><sup>*</sup>(a, b)</td><td>min(a + b, 1)</td><td>a + b - ab</td><td>max(a, b)</td></tr><tr><td rowspan="2">a *→ b</td><td>1 - a if b = 0</td><td></td><td></td><td></td></tr><tr><td>b if a = 1</td><td>min(1, 1 - a + b)</td><td>1 - a + ab</td><td>max(1 - a, b)</td></tr><tr><td></td><td>1 otherwise</td><td></td><td></td><td></td></tr><tr><td rowspan="2">a *→ b</td><td>1 if a &lt; 1</td><td>min(1, 1 - a + b)</td><td>min(1, b/a)</td><td>1 if a ≤ b</td></tr><tr><td>b if a = 1</td><td></td><td>(1 if a = 0)</td><td>b if a &gt; b</td></tr><tr><td rowspan="2">a **→ b</td><td>1 if b &gt; 0</td><td>min(1, 1 - a + b)</td><td>min(1, 1 - a)</td><td>1 if a ≤ b</td></tr><tr><td>1 - a if b = 0</td><td></td><td>(1 if b = 1)</td><td>1 - a if a &gt; b</td></tr><tr><td rowspan="2">a *<sup>†</sup> b</td><td>0 if b &lt; 1</td><td>max(a + b - 1, 0)</td><td>max(0, <sup>a+b-1</sup>&frasl;<sub>b</sub>)</td><td>0 if a + b ≤ 1</td></tr><tr><td>a if b = 1</td><td></td><td>(0 if b = 0)</td><td>a if a + b &gt; 1</td></tr><tr><td rowspan="2">a *<sup>†</sup> b</td><td>0 if a &lt; 1</td><td>max(a + b - 1, 0)</td><td>max(0, <sup>a+b-1</sup>&frasl;<sub>a</sub>)</td><td>0 if a + b ≤ 1</td></tr><tr><td>b if a = 1</td><td></td><td>(0 if a = 0)</td><td>b if a + b &gt; 1</td></tr></tbody></table>

## 4 - CONCLUSION

An important consequence of the theorem is the closure of the generation processes of implication functions from a triangular norm based on transformations a) and b). Iterating these processes does not yield an uncontrollable birth of implication families. This closure property is a strong point in favor of methods a) and b). Moreover we have generated a new kind of conjunction operations A which coincide with classical conjunctions of the proposition calculus, but are not commutative when non-extreme truth values are combined. This lack of commutativity may be natural in some contexts when propositions which do not play the same role have to be combined

in a conjunctive manner. For instance when P is to be combined with P → Q, for the purpose of detaching Q; or if the concept of time is accounted for, and P ∧ Q underlies the assumption that P was known before Q. So we believe that this new class of conjunctions (and dual disjunctions) does not challenge our intuition. Unfortunately, many of these operations are not associative either.

However, we do not claim that the closure theorem is a complete answer to the problem of generating multivalued implication functions. Several families fall outside our framework: for instance those derived from a conjunction and a negation by means of the identity P → Q = P ∧ (P ∧ Q); or the implication proposed by Yager [11], based on the power operation: a → b = bª!

## REFERENCES

[1] - Alsina, C., Trillas, E., Valverde, L. (1980). On non-distributive logical connectives for fuzzy sets theory. BUSEFAL n° 3, LSI, Univ. P. Sabatier, Toulouse, 18-29.

[2] - Dubois, D. (1980). Triangular norms for fuzzy sets. Proc. 2nd Int. Seminar on Fuzzy Set Theory, Johannes Kepler Univ., Linz, Austria, Sept. 15-20, (E.P. Klement, Ed.), 39-68.

[3] - Dubois, D., Prade, H. (1983). Fuzzy set-theoretic differences and inclusions and their use in fuzzy arithmetics and analysis. Proc. 5th Int. Seminar on Fuzzy Set Theory, (E.P. Klement, Ed.), J. Kepler University, Linz, Austria.

[4] - Dubois, D., Prade, H. (1984). Fuzzy set-theoretic differences and inclusions and their use in the analysis of fuzzy equations. To appear in Control and Cybernetics (Poland).

[5] - Ling, C.H. (1965). Representation of associative functions. Publ. Math. Debrecen 12, 189-212.

[6] - Menger, K. (1942). Statistical Metrics. Proc. Nat. Acad. Sci. USA, 28, 535-537.

[7] - Prade, H. (1980). Unions et intersections d'ensembles flous. BUSEFAL n° 3, LSI, Univ. P. Sabatier, Toulouse, 58-62.

[8] - Schweizer, B., Sklar, A. (1963). Associative functions and abstract semi-groups. Publ. Math. Debrecen, 10, 69-81.

[9] - Trillas, E. (1979). Sobre funciones de negacion en la teoria de conjuntos difusos. Stochastica, III, n° 1, 47-59.

[10] - Valverde, L. (1982). Indistinguishability relations with implicative part. Proc. 4th Int. Seminar on Fuzzy Set Theory, J. Kepler Universität, Linz, Austria.

[11] - Yager, R. (1980). An approach to inference in approximate reasoning. Int. J. Man-Machine Studies 13, 323-338.