Axiomatic extensions of

n

-valued Goguen propositional logic system is first studied in this paper

which is denoted as

~

.Using the randomization method of valuation sets

the definition of -

k

randomized truth degree of formula relative to local finite theory under the

k

conjunction is given.The MP rule

HS rule

and some related properties are also discussed.Then

the definitions of -

k

randomized similarity degree and -

k

randomized pseudo-metric in

~

between two formulas are given

and some desir

able properties about -

k

randomized similarity degree and -

k

randomized pseudo-metric relative to local finite theory under conjunction are obtained.Finally

the paper introduces the concepts of randomized relative divergence degree and relative randomized consistency degree of theory relative to the fixed theory

0

in

~

and obtains the relationship between relative randomized divergence degree and relative randomized consistency degree.