The first claim, while difficult to prove conclusively, has a decent amount of evidence in its favour. Vote Leave, the official leave campaign, consistently insisted that Brexit would be a relatively smooth process, and that the EU almost certainly agree to arrangements which did not disrupt the flow of trade, having an enormous incentive to do so. While it is difficult to prove the general thrust of a campaign, particularly when many contradictory promises are made, the campaign leaflets are illustrative. They emphasized continuity in trade and cooperation, achieved by an alternative agreement.
Of course, the messages of the campaign itself does not necessarily imply beliefs on the part of the target audience. But polling taken in July 2016 about expectations of Brexit suggest that beliefs were indeed congruent with the campaign message. In this comres poll, 70% of respondents expected some sort of continued single market membership, either unmodified or with some changes on Freedom of Movement. More surprisingly, these expectations were not hugely divided by how people had voted. 61% of leave voters expected the same, with the bulk (54%) expecting single market membership with new rules on Freedom of Movement. \
But this does not in and of itself mean that leavers are being disingenuous if they say that 'no deal' Brexit was in fact what they had in mind when they voted to leave. Research on the psychology of attitude formation suggests that people's memory of past behavior and beliefs is affected by current attitudes. One example of this is an experiment carried out by Michael Ross in 1981. Half the participants were presented with information on dental hygeine, the other were not. They were then then asked how often they had brushed their teeth in the past two weeks. Those who had been given information on dental hygiene were significantly more likely to say they had brushed their teeth more often. While there are problems with this specific experimental set up, similar results have been found in numerous other experiments, particularly when memories relate to attitudes that are value laden.
Cognitive dissonance theory suggests that this extends to memory of attitudes and attitudes about the past, when these are in tension with current, more salient attitudes. Think about going for a meal and discovering that your favorite dessert had run out. It wouldn't be too strange to decide that you had never really wanted it in the first place. It's particularly easy to imagine this occurring when what is at stake is your own sense of self, and whether you were duped into false expectations. In the case of Brexit, current attitudes can be a strong part of people’s sense of self.
Finally, the contradictory and impossible nature of the Leave campaign’s promises, and absence of any specific conditional commitments as to the relative importance of leaving or what would be done in what circumstances allows for the selection of certain real past memories which are more consistent with the present reality. You could remember the campaign promise that the mechanisms for maintaining the regulatory framework of the single market would not apply after Brexit, but fail to recall how important you thought this was, relative to other, contradictory promises (continued market access).
This is important when thinking about how to argue the case against no deal. Arguments which start from the premise that leavers are being disingenuous if they say they voted for no deal are not necessarily true (though some prominent politicians and journalists almost certainly are being disingenuous). In any case, that is unlikely to work well as a persuasion strategy. Prominent political actors who are dishonest should be called out for it. But when addressing the broader base of leave voters, assuming sincerity is the better strategy.
