This paper deals with the joint verification of the inequalities of Modus Ponens and Modus Tollens in Fuzzy Logic. Although the case of Modus Ponens was partially considered before, the case of Modus Tollens and the joint satisfaction of both inequalities are studied by the first time. The cases of R-implications, Simplications, Q-implications and MamdaniLarsen implications are separatedly cons...

This paper deals with numerical functions able to functionally express operators defined as , and verifying either Modus Ponens or Modus Tollens, or both. The concrete goal of the paper is to search for continuous t-norms and strong-negation functions for which it is either (Modus Ponens), or (Modus Tollens), or both, for all in and a given . Functions are taken among those in the most usual fa...

A fuzzy rough set approach has been proposed without using any fuzzy logical connectives. By this approach, gradual decision rules are induced from a given decision table. Fuzzy-rough modus ponens and fuzzy-rough modus tollens have been formulated based on the gradual decision rules. The equivalence condition between fuzzy-rough modus ponens and fuzzy-rough modus tollens is discussed. The condi...

It is introduced an approach to the problem of "building logics" over sets of facts with only structured by a fuzzy relation of "consequence". Notions of "Truth" and "Falsehood" are introduced by means of two generalizations for classical Modus Ponens and Modus Tollens. Studying this concepts a theoretical base to explain the mode of reasoning in expert systems is obtained. By studying fuzzy tr...

We establish a probabilized version of modus tollens, deriving from p(E jH)1⁄4 a and p( E)1⁄4 b the best possible bounds on p( H). In particular, we show that p( H)! 1 as a, b ! 1, and also as a, b ! 0.

This paper defends a counterexample to Modus Tollens, and uses it to draw some conclusions about the logic and semantics of indicative conditionals and probability operators in natural language. Along the way we investigate some of the interactions of these expressions with knows, and we call into question the thesis that all knowledge ascriptions have truthconditions. A probabilistic dynamic s...