I think there are two [ETA: three] distinct claims about apparently circular preferences that need to be (but are perhaps not always) adequately distinguished.
One is that apparently circular preferences are not able to be represented by a utility function. As Tim rightly points out, much of the time this isn’t really true: if you extend your state-descriptions sufficiently, they usually can be.
A different claim is that, even if they can be represented by a utility function, such preferences are irrational. Usually, the (implicit or explicit) argument here is that, while you could augment your state description to make the resulting preferences transitive, you shouldn’t do so, because the additional factors are irrelevant to the decision. Whether this is a reasonable argument or not depends on the context.
ETA:
Yet another claim is that circular preferences prevent you from building, out of a set of binary preferences, a utility function that could be expected to predict choice in non-binary contexts. If you prefer Spain from the set {Spain,Greece}, Greece from the set {Greece,Turkey}, and Turkey from the set {Turkey,Spain}, then there’s no telling what you’ll do if presented with the choice set {Spain,Greece,Turkey}. If you instead preferred Spain from the final set {Spain,Turkey} (while maintaining your other preferences), then it’s a pretty good shot you’ll also prefer Spain from {Spain,Greece,Turkey}.
I think there are two [ETA: three] distinct claims about apparently circular preferences that need to be (but are perhaps not always) adequately distinguished.
One is that apparently circular preferences are not able to be represented by a utility function. As Tim rightly points out, much of the time this isn’t really true: if you extend your state-descriptions sufficiently, they usually can be.
A different claim is that, even if they can be represented by a utility function, such preferences are irrational. Usually, the (implicit or explicit) argument here is that, while you could augment your state description to make the resulting preferences transitive, you shouldn’t do so, because the additional factors are irrelevant to the decision. Whether this is a reasonable argument or not depends on the context.
ETA:
Yet another claim is that circular preferences prevent you from building, out of a set of binary preferences, a utility function that could be expected to predict choice in non-binary contexts. If you prefer Spain from the set {Spain,Greece}, Greece from the set {Greece,Turkey}, and Turkey from the set {Turkey,Spain}, then there’s no telling what you’ll do if presented with the choice set {Spain,Greece,Turkey}. If you instead preferred Spain from the final set {Spain,Turkey} (while maintaining your other preferences), then it’s a pretty good shot you’ll also prefer Spain from {Spain,Greece,Turkey}.