Modus Ponens Generating Function in the Class of ^-valuations of Plausibility

We discuss the problem of construction of inference procedures which can manipulate with uncertainties measured in ordinal scales and fulfil to the property of strict monotonic­ ity of conclusion. The class of A-valuations of plausibility is considered where operations based only on information about linear or­ dering of plausibility values are used. In this class the modus ponens generating function fulfiling to the property of strict monotonic­ ity of conclusions is introduced. 1 STABILITY OF DECISIONS IN INFERENCE PROCEDURES Human judgements about plausibility, truth, certainty values of premises, rules and facts are usually qualita­ tive and measured in ordinal scales. Representation of these judgements by numbers from interval L = [0, lJ or L = [0, 100] and using over these numbers quanti­ tative operations such as multiplication, addition and so on is not always correct. Let's consider example. Let R1 and R2 are two rules of some expert system: Rl: If At then H1,pv(RI), (1) R2: If A2 then H2,pv(R2), (2) where pv(RI) and pv(R2) are the plausibility, cer­ tainty, truth values of rules measured in some linearly ordered scale L, for example L = [0, 1]. Often plausi­ bilities of conclusions are calculated by: pv(Ht) = pv(Rl) * pv(At), (3) pv(H2) = pv(R2) * pv(A2), (4) where pv(At) and pv(A2} are the plausibilities of premises and * is some T-norm, for example multi­ plication operation (Godo, Lopez de Mantaras et al. 1988; Hall1990; Trillas, Valverde 1985; Valverde, Tril­ las 1985; Forsyth 1984). Generally the plausibility of conclusion can be calculated by means of a modus po­ nens generating function mpgf: pv(HI) = mpgf(p·v(At),pv(Rt)). Let in (1)-( 4) the qualitative information about plau­ sibility values is the next: pv(At) < pv(A2} < pv(R2) < pv(RI), (5) that is the plausibility values of premises are less than the plausibility values of rules, the plausibility value of A1 is less than the plausibility value of A2 and the plausibility value of rule R2 is less than the plausibility value of rule Rl. Let these plausibility values are inter­ preted as the next quantitative values from L = (0, 1]: pv(A1) = 0.3 < pv(A2) = 0. 4 < pv(R2} = 0.6 <