We develop alternative semantics for Lukasiewicz logic and for cancellative hoop logic according to the following idea. We formalize statements reflecting an inexact knowledge of certain (sharp) properties; we assume that all what can be known about a property is its expressive strength. To this end, we consider a Boolean algebra endowed with an automorphism group or, alternatively, with a measure. The Boolean algebra is meant to model a collection of properties; and the additional structure is used to identify pairs of properties which, although possibly distinct, are equally strong. Propositions are defined as subsets of the algebra containing with any element also those identified with it in this way. We show that then, the set of all propositions carries the structure of an MV-algebra or of a cancellative hoop.
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