In quantitative judgments we may think x = y, or, as Boolero oses x = v ° p p y = - ° y, or, as Jevons proposes, x = xy, or, as Venn proposes, x which is not y=o; and equational symbolic logic is useful whenever we think in this quantitative way.

We look at denotational semantics, using sets and functions, the substitution lemma and equational theory.

It is proved using a direct axiomatization of Kleisli categories of equational lifting monads.

Our framework allows equational reasoning of non-deterministic functions, even tho it is possible to specify non-determinism.

- (i.) The statement of problems in equational form should precede the solution of equations.

In quantitative judgments we may think x = y, or, as Boolero oses x = v ° p p y = - ° y, or, as Jevons proposes, x = xy, or, as Venn proposes, x which is not y=o; and equational symbolic logic is useful whenever we think in this quantitative way.

Our framework allows equational reasoning of non-deterministic functions, even tho it is possible to specify non-determinism.