Various fuzzy implications Each of these is a fuzzy logic generalization of the classical implication operation.
fimplication_minimal(x, y)fimplication_maximal(x, y)
fimplication_kleene(x, y)
fimplication_lukasiewicz(x, y)
fimplication_reichenbach(x, y)
fimplication_fodor(x, y)
fimplication_goguen(x, y)
fimplication_goedel(x, y)
fimplication_rescher(x, y)
fimplication_weber(x, y)
fimplication_yager(x, y)
Numeric vector of the same length as x and y.
The ith element of the resulting vector gives the result
of calculating I(x[i], y[i]).
numeric vector with elements in \([0,1]\)
numeric vector of the same length as x,
with elements in \([0,1]\)
A function \(I: [0,1]\times [0,1]\to [0,1]\) is a fuzzy implication if for all \(x,y,x',y'\in [0,1]\) it holds: (a) if \(x\le x'\), then \(I(x, y)\ge I(x', y)\); (b) if \(y\le y'\), then \(I(x, y)\le I(x, y')\); (c) \(I(1, 1)=1\); (d) \(I(0, 0)=1\); (e) \(I(1, 0)=0\).
The minimal fuzzy implication is given by \(I_0(x, y)=1\) iff \(x=0\) or \(y=1\), and 0 otherwise.
The maximal fuzzy implication is given by \(I_1(x, y)=0\) iff \(x=1\) and \(y=0\), and 1 otherwise.
The Kleene-Dienes fuzzy implication is given by \(I_{KD}(x, y)=max(1-x, y)\).
The Lukasiewicz fuzzy implication is given by \(I_{L}(x, y)=min(1-x+y, 1)\).
The Reichenbach fuzzy implication is given by \(I_{RB}(x, y)=1-x+xy\).
The Fodor fuzzy implication is given by \(I_F(x, y)=1\) iff \(x\le y\), and \(max(1-x, y)\) otherwise.
The Goguen fuzzy implication is given by \(I_{GG}(x, y)=1\) iff \(x\le y\), and \(y/x\) otherwise.
The Goedel fuzzy implication is given by \(I_{GD}(x, y)=1\) iff \(x\le y\), and \(y\) otherwise.
The Rescher fuzzy implication is given by \(I_{RS}(x, y)=1\) iff \(x\le y\), and \(0\) otherwise.
The Weber fuzzy implication is given by \(I_{W}(x, y)=1\) iff \(x<1\), and \(y\) otherwise.
The Yager fuzzy implication is given by \(I_{Y}(x, y)=1\) iff \(x=0\) and \(y=0\), and \(y^x\) otherwise.
Klir G.J, Yuan B., Fuzzy sets and fuzzy logic. Theory and applications, Prentice Hall PTR, New Jersey, 1995.
Gagolewski M., Data Fusion: Theory, Methods, and Applications, Institute of Computer Science, Polish Academy of Sciences, 2015, 290 pp. isbn:978-83-63159-20-7
Other fuzzy_logic:
fnegation_yager(),
tconorm_minimum(),
tnorm_minimum()