This paper deals with numerical functions J : [0,1] x [0,1] → [0,1] able to functionally express operators →: [0,1]X x [0,1]Y → [0,1]XxY defined as (μ → σ)(x,y) = J(μ(x),σ(y)), and verifying either Modus Ponens or Modus Tollens, or both. The concrete goal of the paper is to search for continuous t-norms T and strong-negation functions N for which it is either T(a, J(a,b)) ≤ b (Modus Ponens) or T(N(b), J(a,b)) ≤ N(a) (Modus Tollens), or both, for all a,b in [0,1] and a given J. Functions J are taken among those in the most usual families considered in Fuzzy Logic, namely, R-implications, S-implications, Q-implications and Mamdani-Larsen implications. En passant, the cases of conditional probability and material conditional's probability are analyzed.
@article{urn:eudml:doc:41637,
title = {On MPT-implication functions for fuzzy logic.},
journal = {RACSAM},
volume = {98},
year = {2004},
pages = {259-271},
mrnumber = {MR2136170},
zbl = {1070.03014},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41637}
}
Trillas, Enric; Alsina, Claudi; Pradera, Ana. On MPT-implication functions for fuzzy logic.. RACSAM, Tome 98 (2004) pp. 259-271. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41637/