Chapter Two in particular makes it clear the kind of world that David and Sophie are living in. We are presented with a world in which abominations or mutants are deplored and to be avoided. If they are discovered, there are serious ramifications. Note what happens when "Offences" are discovered in David's household:
We would all kneel while he proclaimed our repentance and led prayers for forgiveness. The next morning we would all be up before daylight and gather in the yard. As the sun rose we would sing a hymn while my father ceremonially slaughtered the two-headed calf, four-legged chicken, or whatever other kind of Offence it happened to be. Sometimes it would be a much queerer thing that those...
Thus we can understand the atmosphere of profound secrecy and concern that David and Sophie are brought up in. Both, of course, have reason to fear discovery of the way in which they are "Offences," just as the two-headed calf is an offence. The consequences of discovery are terrifying.
This is why the characters of David and Sophie are characterised by the necessity of keeping a dreadful secret that could endanger their lives and the lives of those around them. Living in such a society that is so vigilant for "Offences" does not bode well for either of them, and makes the threat of discovery ever-present.