It is very common for people who are putting forward a claim to say something like, "you must consider this with an open mind" or, if their claim is rejected, they will say something like, "well of course you don't believe it, you're closed-minded". It's often used by those who believe in unlikely, or even disproved, ideas.
There are many ways that this 'appeal to open mindedness' manifests itself, so let's have a look at why it is not usually a good claim.
What is an open-minded person?
An open-minded person is someone who is willing to consider ideas, opinions and arguments purely on their merit. If a claim can be shown to be correct then an open-minded person will alter, or add to, their worldview with this new-found knowledge.
Although being open-minded means a willingness to consider that a claim may be true, it also means considering the possibility that it may be false. A truly open-minded person's mind is open to both possibilities. An open-minded approach to assessing claims does not preclude rejecting claims if they're found wanting.
Misuse of the term "open-minded"
Exchanges that include constant references to how open people's minds are can be quite common in debates and can be very frustrating to those on the receiving end. This is because we generally regard open-mindedness to be a virtue, so any suggestion we're not open-minded is taken as a personal slur or attack – which it is. An attempt to win a debate by attacking the person rather than their argument is, of course, simply a fallacious ad Hominem.
A deeper problem with this issue, however, is that the term “open minded” is wrongly characterised as simply meaning “accepting claims”, or worse, “accepting claims without good reason or evidence”. This is not open-mindedness. The actual word for a person who is too willing to believe things is: credulous.
The appeal to open-mindedness, when used in this context, is really an appeal to relinquish one's rational integrity. It's an appeal to accept something without good reason under the guise that it is virtuous to do so. Of course, once it's realised that the virtue of being open-minded has been substituted with the folly of being credulous, the absurdity of doing so is clear.
Open-mindedness is considered a virtue, and true open-mindedness is.
The appeal to open-mindedness is frequently used by people who wish to sound virtuous, and simultaneously make their opponent sound intolerant, while defending or promoting their ideas and beliefs. However, this argumentative tactic relies on using the term “open-minded” to describe uncritical belief acquisition or defend cherished beliefs when “credulous” would be a more appropriate epithet.