Real and imaginary truth in complex fuzzy implication

This paper examines the use of a parameterized family of implication operators based on Schweizer-Sklar T-norms. Removing a restriction imposed by Schweizer and Sklar when the parameter p is negative leads to improper implication operators for which the truth value can take values outside the unit interval, including complex values. Inferences using increasingly positive p produce inferences increasingly biased toward binary switching (“selectivity”), while inferences based on improper implication operators with increasingly negative p produce inferences increasingly biased to unweighted averaging (“consensus”).