Look at the Wikipedia definition, which is actually quite good. It says (I paraphrase slightly) it is a tone that continually ascends in pitch, yet which ultimately seems to get no higher. There are two interesting words in that definition, namely 'seems' and 'ultimately', and applying or removing them gives four possible combinations.
(1) The Shepard tone ascends in pitch but does not ascend in pitchDefinition (1) is clearly absurd. A definition needs to give us a way of distinguishing one thing from another, but this gives us nothing, since it includes pitches which do ascend and those which do not ascend, i.e. includes every kind of pitch whatsoever. Definition (2) is better, but is it correct? Surely not. If we take the first two or three tones as they occur in order, they clearly are ascending, and it is not that they just seem to ascend, at least in the initial phases of the sequence. Both definitions (3) and (4) incorporate the term 'ultimately', and here we are getting somewhere. What seems paradoxical about the tone is the way that after a full octave has been ascended, we seem to be back where we were. It's like one of those Sisyphean nightmares where we seem to be climbing forever, and find ourselves back in the same place. Or the Blair Witch Project (for those who remember that).
(2) The Shepard tone ascends in pitch but does not seem to ascend in pitch
(3) The Shepard tone ascends in pitch but does not ultimately ascend in pitch
(4) The Shepard tone ascends in pitch but does not ultimately seem to have ascended in pitch
Yet, in this case at least, is there really any contradiction between appearance and reality? The whole point of the octave interval is the strong resemblance between the two tones of the interval (say, middle C and top C). And where there is resemblance or similarity there is (formal) identity. So it is no paradox to say we are ultimately back in the same place. All the Shepard tone does is to eliminate the respect in which the tones of the interval are different, i.e. eliminates the respect in which C and C' are different, while retaining the similarity.
Why should we find the 'paradox' any more paradoxical than angular movement or modulo change? If you keep on turning around long enough, you will be facing the same direction again. It is midnight, then time passes for each successive hour until it is midnight again. Definition (3) captures the Shepard tone best. It is a tone which ascends in pitch but does not ultimately ascend in pitch, just as an orbit is a movement which changes place but which does not ultimately change in place. What's the problem?
More about the principle of contradiction later.