Worldviews and mistaken beliefs, a paradox
Here's a little paradox that I've come across while thinking. It's about worldviews and knowing that at least one thing I currently believe to be true is actually false.
1. For all x, if x is a belief in my worldview, then I hold that belief. [Premise]
2. If for all x, if x is a belief in my worldview, then I hold that belief, then I don't believe that my worldview contains a mistaken belief. [Premise]
3. Thus, I don't believe that my worldview contains a mistaken belief. [from 1, 2, MP]
4. There is at least one proposition such that I believe it and it is false. [Premise]
5. If there is at least one proposition such that I believe it and it is false, then my worldview contains a mistaken belief. [Premise]
6. Thus, my worldview contains a false belief. [From 4, 5, MP]
7. Thus, my worldview contains a mistaken belief and I don't believe that my worldview contains a mistaken belief. [3, 6, Conj.]
This conclusion seems paradoxical to me. It's not a contradiction at it stands, it's just fishy. If I form the belief as a result of this argument, that my worldview contains a mistaken belief, then I hold two contradictory beliefs. It seems that I have to reject a premise, but I don't find any of them weak. In fact, three of them are logical tautologies, and the last is proven by induction. I'll discuss the premises below.
This one may seem a little unnecessary. I had no luck formulating the argument without this premise. I think it is a logical tautology, that is, true per definition. Let me first define worldview as I use it here.
Worldview =df The (complete) set of beliefs a person has.
Then, given the above definition, it is clear that if a belief is part of a person's worldview, then that person holds that belief.
When I wrote this article and when I was thinking of the paradox, I noticed that it is easy to make a category error and call a worldview false. But that doesn't make sense. ‘True' and ‘false' are meaningful in relation to propositions and not to beliefs. We may instead say that a worldview is mistaken which just means that the worldview contains at least one mistaken belief. A mistaken belief is a belief in a false proposition.
Common language may be broader in the use of ‘true' and ‘false' but here I will restrict myself for the sake of clarity of thought.
This one is similar to the first, as it seems unnecessary and is a logical tautology.
One could argue it with a reductio. Assume that I believe that my worldview contains a mistaken belief. If I believe that, then I don't hold the belief that is mistaken. (Since if I did, I would have two contrary beliefs.) If I don't hold that belief, then it isn't part of my worldview. But from the assumed we can deduce that it is part of my worldview. Thus, contradiction, and the assumption is, thus, false.
This one is not a tautology for a change, but I think it is uncontroversial. I have a large number of beliefs, all grown-ups do, and in the past it has always been the case that a belief I had turned out to be false. Similar behavior has been observed in other humans. By induction we have good reason to believe that some of my current beliefs are false. The trouble is that I don't know which of them it is!
This is another tautology. By definition my worldview is the set of beliefs that I have, and if I hold a mistaken belief then it follows that my worldview contains at least one mistaken belief.
I don't know. I haven't found one, if there is one.
 The argument is valid in propositional logic but some of the propositions are formulated in predicate logic for extra clarity.