Hi Bruno, I'm back :-)
The axiom B(Bp->p)->Bp seems very strange to me.
Is it applicable only to machines or also to humans ?
Intuitively, let us consider for example p = "Santa Claus exists".
I don't believe that Santa Claus exists (~Bp).
If I consider the proposition "Bp->p" which means "If I believe that Santa Claus exists, then Santa Claus exists", this proposition seems true to me, because of le propositional logic rule "ex falso quodlibet sequitur" or false->p : the false proposition Bp implies any proposition, for example p.
So I can say thay I believe in the proposition Bp->p : B(Bp->p). According to the lobian formula B(Bp->p)->Bp, this implies Bp (I believe that Santa Claus exist) !
More formally :
The axiom ~Bp->B(~Bp) seems correct to me, isn't it ?
It seems also correct to me to say that the logical rules are valid inside believes, for example : (B(p->q) and B(q->r)) -> B(p->r), or B(F->p) where F is the false proposition.
Let us consider a p such that ~Bp, which is equivalent to (Bp)->F.
Then we have B(~Bp), which is equivalent to B(Bp->F).
From this and B(F->p) we can infer B(Bp->p).
Finally with the lobian formula B(Bp->p)->Bp we get Bp.
Is there an error anywhere ?
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