Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 19 studies the combination of various methods of designing for reliability, availability, maintainability and safety, as well as the latest techniques in probability and possibility modelling, mathematical algorithmic modelling, evolutionary algorithmic modelling, symbolic logic modelling, artificial intelligence modelling, and object-oriented computer modelling, in a logically structured approach to determining the integrity of engineering design. . | Analytic Development of Reliability and Performance in Engineering Design 163 Instead of min and max the product and algebraic sum for intersection and union may be respectively used. The two equations can be verified by substituting 1 for true and 0 for false. Table Truth table applied to propositions p q p A q p v q p q p q p T T T T T T F T F F T F F F F T F T T F T F F F F T T T In traditional propositional logic there are two very important inference rules associated with implication and proposition specifically the inferences modusponens and modus tollens. Modus ponens Premise 1 x is A Premise 2 IF x is A THEN y is B Consequence y isB . Modus ponens is associated with the implication A implies B . In terms of propositions p and q modus ponens is expressed as P A p q q Modus tollens Premise 1 y is not B Premise 2 IF x is A THEN y is B Consequence x is not A . In terms of propositions p and q modus tollens is expressed as q A p q p Modus ponens plays a central role in engineering applications such as control logic largely due to its basic consideration of cause and effect. Modus tollens has in the past not featured in engineering applications and has only recently been applied to engineering analysis logic such as in engineering design analysis with the application of FMEA and FMECA. Although traditional fuzzy logic borrows notions from crisp logic it is not adequate for engineering applications offuzzy control logic because cause and effect is the cornerstone of modelling in engineering control systems whereas in traditional propositional logic it is not. Ultimately this has prompted redefinition of fuzzy implication operators for engineering applications of fuzzy control logic. An understanding of why the traditional approach fails in engineering is essential. The extension of crisp logic to fuzzy logic is made by replacing the bivalent membership functions of crisp logic with fuzzy membership functions. 164 3 Reliability and Performance
