This defines the

unary operation x [right arrow] [bar.x] on each epigroup; we call this operation pseudoinversion.

Let M be a ([logical not], A, V, [right arrow])-type algebra, where [logical not] is a

unary operation and [conjuncton], [disjunction], and [right arrow] are binary operations.

An MV-algebra M = (M; 0, [logical not], [direct sum]) is an algebra where [direct sum] is an associative and commutative binary operation on M having 0 as the neutral element, a

unary operation [logical not] is involutive with a [direct sum] [logical not]0 = [logical not]0 for all a [member of] M, and moreover the identity a [direct sum] [logical not](a [direct sum] [logical not]b) = b [direct sum] [logical not](b [direct sum] [logical not]a) is satisfied for all a,b [member of] M.

Intensional algebra for the intensional FOL in Definition 1 is a structure [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with binary operations [conj.sub.s] : [D.sub.j] x[D.sub.I] [right arrow] [D.sub.i],

unary operation neg : [D.sub.I] [right arrow] [D.sub.I],

unary operations [exists.sub.n] : [D.sub.I] [right arrow] [D.sub.I], such that for any extensionalization function he W, and u [member of] [D.sub.k], [upsilon] [member of] [D.sub.j], k, j [greater than or equal to] 0,

* that is a

unary operation interpreted so that('[s.sub.A]') denotes the class of models determined by the [L.sub.A]-sentence [s.sub.A].

Moreover, the

unary operation [sup.-1] of <Q, [[direct sum].sub.k]> will be denoted by [[??].sub.k] and the element [[??].sub.k] x coincides with k/x.

Then the meaning of the rules of constructing well-formed formulas P[1],..., P[10] can be explained as follows: for each k from 1 to 10, the rule P[k] determines a partial

unary operation Op[k] on the set Degr(B) with the value being an element of Ls(B).

An algebra (A, [conjunction], [disjunction],[(-).sup.~]) where A is non-empty set with 1, [conjunction], [disjunction] are binary operations and [(-).sup.-] is a

unary operation satisfying

and a

unary operation *: S [right arrow] S satisfying the following three axioms: