The basis of relativity, as science knows it, is the relativity of time. But we didn't need Einstein to tell us that time is relative. We could have worked that out for ourselves, long ago. All we needed to know is what just about everyone knows, that when a body is receding from us its light-spectrum is shifted towards the red (the 'ticks' of its atomic clocks appear slowed) and when it is approaching us its spectrum is shifted towards the blue (its atomic ticks appear quickened). This is the well-known and well-proven Doppler effect. From this knowledge alone we can deduce the relativity of time in a perfectly simple, logical way, as may now be demonstrated.
For instance, let O and P be two astronauts somewhere in deep space. Starting together, they move steadily apart at such a speed that the ticks of each one's clocks, as viewed by the other are twice as long as his own . Note that any precept that there is some underlying God's-eye overview that can be taken of the views of O and P together must be cancelled. If held to throughout, this merely defeats the argument by adhering dogmatically to what it seeks to refute. (We mention this, since it constitutes the characteristic fallacy of the anti-relativist dogma of the self-styled 'Dissidents'.)
Now in order to meet up again there has to be some action which ends the outward relative motion and reverses it into the opposite direction. This action may, logically, be undertaken either by O or P or both. Let us suppose, in the first instance, that this action is taken by P after a time of two years as measured by his (P's) atomic clocks. We know that due to the Doppler lengthening of the ticks of P's atomic clocks in O's view of P, O will see that action of P's after four years of his (O's) own time.
Let P, having completed that reversing action, now commence his course back towards O at the same steady speed as that with which they parted company. Neglecting the time taken for his turnaround, P meets up with O after a further time of two years, as measured by his (P's) own atomic clocks. Meanwhile, during that backwards journey of P, after the four years already elapsed in O's time, O will observe the ticks of P's atoms from then on at the faster (blued) rate of half their standard length, which means that in the two years that P's atoms will have registered in re-uniting with O, O's atoms will have registered a half of that time, which is, of course, one year.
It follows, then, that while the total time registered by P's clocks between their leaving and meeting up again is just four years, for O it is the four years registered by his own atomic clocks plus another year, which is five years. So when they get back together, if they are identical twins, the result of the motion is that twin O will have aged one year more than twin P, and twin P will have aged one year less than twin O.
Clearly, if it had been O rather than P who had taken the reversing action, then the result would have been the other way around, with O having registered the four years and P registering the five. Of course, if they had both taken the same reversing action at the same pre-arranged times, then there would be no resulting difference in their ages. This sufficiently demonstrates how motion can affect the durations of bodies relatively to one another in what Einstein's theory describes as 'time-dilation'. It endorses, in commonsense terms, the relativistic revolution which puts time in bodies, not the bodies in time. Q.E.D