Section 4.1 is fairly unpolished. I’m still looking for better ways of handling the problems it brings up; solutions 4.1 and 4.2 are very preliminary stabs in that direction.
The action condition you mention might work. I don’t think it would re-introduce Löbian or similar difficulties, as it merely requires that ¯¯¯a implies that G is only true, which is a truth value found in LP. Furthermore, we still have our internally provable T-schema, which does not depend on the action condition, from which we can derive that if the child can prove (¯¯¯a→G)∧¬(¯¯¯a→¬G), then so can the parent. It is important to note that “most” (almost everything we are interested in) of PA⋆ is consistent without problem.
Now that I think about it, your action condition should be a requirement for paraconsistent agents, as otherwise they will be willing to do things that they can prove will not accomplish G. There may yet be a situation which breaks this, but I have not come across it.
Section 4.1 is fairly unpolished. I’m still looking for better ways of handling the problems it brings up; solutions 4.1 and 4.2 are very preliminary stabs in that direction.
The action condition you mention might work. I don’t think it would re-introduce Löbian or similar difficulties, as it merely requires that ¯¯¯a implies that G is only true, which is a truth value found in LP. Furthermore, we still have our internally provable T-schema, which does not depend on the action condition, from which we can derive that if the child can prove (¯¯¯a→G)∧¬(¯¯¯a→¬G), then so can the parent. It is important to note that “most” (almost everything we are interested in) of PA⋆ is consistent without problem.
Now that I think about it, your action condition should be a requirement for paraconsistent agents, as otherwise they will be willing to do things that they can prove will not accomplish G. There may yet be a situation which breaks this, but I have not come across it.