The half-open real unit interval (0,1] is closed under the ordinary
multiplication and its residuum. The corresponding infinite-valued
propositional logic has as its equivalent algebraic semantics the equational
class of cancellative hoops. Fixing a strong unit in a cancellative hoop
-equivalently, in the enveloping lattice-ordered abelian group- amounts to
fixing a gauge scale for falsity. In this paper we show that any strong unit in
a finitely presented cancellative hoop H induces naturally (i.e., in a
representation-independent way) an automorphism-invariant positive normalized
linear functional on H. Since H is representable as a uniformly dense set of
continuous functions on its maximal spectrum, such functionals -in this context
usually called states- amount to automorphism-invariant finite Borel measures
on the spectrum. Different choices for the unit may be algebraically unrelated
(e.g., they may lie in different orbits under the automorphism group of H), but
our second main result shows that the corresponding measures are always
absolutely continuous w.r.t. each other, and provides an explicit expression
for the reciprocal density.Comment: 24 pages, 1 figure. Revised version according to the referee's
suggestions. Examples added, proof of Lemma 2.6 simplified, Section 7
expanded. To appear in the Journal of Symbolic Logi
