Varieties of contextuality emphasizing (non)embeddability

The notion of "contextuality" in terms of nonclassical (paradoxical) properties of probability distributions is refined by concentrating on the nonclassical logico-algebraic structures of the observables supporting such probabilities. These structures often, but not always, induce and are reflected by nonclassical properties of the probability distributions. Two types of contextuality are introduced, originating either in nonclassical probabilities on classical logics, or in genuine nonclassical logics of observables which cannot be faithfully (homomorphically) embedded into (extended) Boolean algebras. It is suggested to substitute the term "contextual" by "indeterminate" or "value indefinite" in the spirit of partial functions of theoretical computer sciences.