Suppose you think reasons are objective matters of fact rather than psychological states of an agent. I'm thinking of guiding or justificatory reasons here. If what I say turns out to be true of motivating or explanatory reasons, I'd be surprised. I'm not not open to surprises, however. If you think of reasons as considerations that favor an action you will probably be sympathetic to this sort of view. Lots of people think that that's what reasons are, so I suspect lots of people will be sympathetic.
I'm of the view that belief aims at the truth, but I'm not sure that this is something we ought to take as basic. If an explanation can be given, perhaps this will suffice. The aim, purpose, or function of belief is to provide the agent with reasons for the purpose of practical and theoretical deliberation. Of course, we're talking about guiding or justificatory reasons. Any false belief will pass off a non-reason as a reason. Any false belief will fail in its aim, function, purpose, etc... So, any false belief will fail to fulfill its aim. Extensionally, as it were, the claim that all beliefs aim at the truth is correct. The fact that beliefs all aim at the truth, however, is not a brute fact.
Think about the belief's justification. There's justification for doing things or believing things that we have reason not to do or believe--sometimes. There is when there is a stronger reason to the contrary that defeats the reason(s) against. As there's no aim, function, or purpose of belief that gives us reasons that override the reasons to refrain from holding those beliefs that will pass off non-reasons as reasons, it seems there cannot be justification for adopting, holding, or deliberating from the false beliefs. If a belief cannot be justified when there is a reason not to hold it and no accompanying overriding justification that defeats this reason, there are no false beliefs that are also justified. The justification would have to involve an overriding normative or guiding reason that defeats treating a non-reason as a reason for the purposes of theoretical or practical deliberation. I don't think anyone actually believes in such things. I've never met someone who has.
In short, here's the argument. Every false belief passes off a non-reason as a reason. Beliefs are supposed to provide us with reasons, not non-reasons, for the purpose of deliberation. There could be a justification for this only if there were overriding reasons for treating non-reasons as reasons. There aren't. There is not sufficient justification for adopting, holding, or deliberating from a false belief or a belief that passes non-reasons off as reasons. Doxastic justification is factive.