To play devil's advocate, we all receive information (and probably some amount of disinformation) about the world via media channels, whether traditional ones like newspapers and TV news, stuff on the internet or whatever. Even if it's just people's opinions on Facebook or right here on Dissensus. So what do we mean by 'core convictions'?
In other words, it's easy to dismiss someone who thinks "Jeremy Corbyn was in the IRA" because they've got a distorted memory of something they read in the Sun that was already pretty biased and perhaps inaccurate. But each of us has a store of opinions and facts, or things we think are facts, that we've read or heard somewhere else. Perhaps these ideas are more accurate and complete for having been picked up from the Guardian or the New Statesman or New York Times or whatever than those garnered from the Sun - I'd certainly hope so, at least - but the basic principle is the same, isn't it?