A question on the belief operator in Doxastic Logic

Question

Let Bp be the statement "it is believed that p".

Why is ~Bp not equivalent to B~p? in words it amounts of saying that: "it's not believed that p" equivalent to "it's believed that not p".

If we are dealing in a classical two truth valued doxastic logic the above equivalence should be apparent. But you can also say that you don't believe that p and that you don't believe that ~p, in which case in what do you do believe? something all else, q which is not p and not ~p, and if it's intutionistic logic then ~~p isn't equivalent to p, which is quite hard to fathom. (I like this word "fathom", reminds me of phantom of menace...).

You can replace p with anything, God, Jesus, Moses, whatever... :-D

Answer 1

Why is ~Bp not equivalent to B~p? in words it amounts of saying that: "it's not believed that p" equivalent to "it's believed that not p".

To believe and not believe are propositional attitudes. Propositional attitudes (a species of proposition) assert claims about the mental states of agents about other propositions. Thus, one can negate both claims about propositional attitudes (perhaps better named attitudinal propositions) and claims about the propositions themselves. An example will clarify.

Speaker 1 I do not believe that grue exists.
Speaker 2 Ah ha, so, you believe grue DOESN'T exist.
Speaker 1 I can understand why you might think that, but I don't know what 'grue' is, have never heard of 'grue', and have no beliefs about 'grue' one way or the other. How am I to believe or disbelieve in something that is meaningless to me?
Speaker 1 Well, 'grue' is a philosopher's color that is green and then at some points becomes blue in order to drive home a philosophical point.
Speaker 2 Well, that's absurd. Now, I certainly don't believe grue exists.

And here then is the difference between the natural language interpretation of ~Bp and B~p. In the first case, the belief doesn't exist (what ontologists call negative existential quantification) and in the second case, the belief does exist, but the belief is of a proposition that contains negative existential quantification. The first statement's assertoric force is directed towards imputed belief, and the latter towards the object of belief. It's not any more complicated than that.

Answer 2

Consider a proposition you've never thought of before, perhaps, "The Yodeling Lords of Darkness reside in the Castle of Pancakes atop Kangaroo Mountain." Granted, you've now heard of this proposition (as of reading this post), but imagine a scenario where you haven't read this post. Now, if you've never entertained such a proposition in your thoughts, then "intuitively," you aren't in a position to believe it. So ~Bp.

However, since you've never entertained this thought, how should you think that B~p? This would be, "The Yodeling Lords of Darkness don't reside in the Castle of Pancakes atop Kangaroo Mountain." But you don't have any opinions about the evil yodelers since you have no thought of them; you perhaps don't even know about the Castle of Pancakes or the Kangaroo Mountain. Now, if there is doxastic closure, you might be thought to implicitly believe all the things entailed by what you explicitly believe, and perhaps there is some obscure hypothetical awareness, within you, of the general possibility of evil yodelers, breakfast-themed fortresses, and mountains named after Australian animals. So perhaps there is some "virtual," subconscious sense in which you have thoughts about these things after all.

But epistemic closure is suspect, and the belief operator is not confined to doxastic logic alone but is of a piece with many a normal epistemic logic. More precisely, the problem of logical omniscience stands atop its own weird mountain in this realm, so to speak, and expresses the problem of doxastic/epistemic closure. So again "intuitively," one need not believe all the formal implications of one's other, noninferential beliefs. And so again, if you never consciously entertain a given proposition, perhaps because said proposition is so random as to never have been filtered out of your inner doxastic aether, it is hard to say that you actively believe or disbelieve in such propositions. But so ~Bp holds even though B~p doesn't, here.

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Addendum

There is a different context in which ~Bp does often go with B~p, however. In terms of linguistic pragmatics rather than logical semantics, if someone says, "I don't believe that [insert some proposition p]," this often carries the pragmatic implication that this person also does actively believe that ~p. For example, if a religious fanatic says, "I don't believe in the Zero-gods," they probably also mean to say, "I believe that religions that worship the Zero-gods are false." So your familiarity with this context might be what is prompting you to question the counterpart moment in doxastic logic. You might be interested in dialogical logic in this connection, then.

Answer 3

The point is precisely that you do not believe anything, which is fine. ~Bp is not considered as B~p because belief is not necessary. That is, we cannot form a disjunction of the form "Bp ∨ B¬p", there are three positions which one can have towards a proposition:

Belief (which is then sub-divided into Bp and B~p)

Non-belief (suspension of belief, or the classic state of being adoxastos, you simply lack belief or judgement towards the proposition. This is expressed by ~Bp ∧ ~B~p and it is totally acceptable.)

And this does not deny bivalence. The principle of bivalence is just that Bp ∨ ¬Bp, which is satisfied. if someone presents to u a proposition which seems very confusing and hard to make a judgement, you can simply say "i dont believe that this is true nor that this is false, i just dont believe", this is not rare. If ~Bp → B~p, then belief would end up being necessary (through elimination of disjunction, if Bp v ~Bp, and Bp → (Bp v B~p) and ~Bp → (B~p v Bp), then B~p v Bp). You would have to necessarily make a judgement and ascribe a truth value to a proposition rather than possibly suspend judgement about which truth value it has, and this necessity does not seem intuitive at all. The principle of bivalence is not revoked by granting the lack of this equivalence and of this necessity. Everything ends up being fine.