Doxastic logic is concerned with beliefs. The idea "it is believed that" is taken as a modal operator.

So for a proposition p, we could write

**B**p to mean "it is believed that p".

Presumably, if we have more than one person in view (say, X and Y), we could mark the belief operator to say which believer we're talking about. So

**B**_{X}p would mean "X believes that p".

I've taken a look at the

Wikipedia article on doxastic logic, and parts of it strike me as awfully weird.

A big chunk of the article is concerned with categorizations of different "reasoners" ("believers", I suppose), as defined by Raymond Smullyan.

For example, a reasoner is described as "accurate" if he/she never believes anything that is false:

∀p:

**B**p→p

and a reasoner is described as "consistent" if he/she never believes both a proposition and its negation:

∀p:

**B**p→¬

**B**¬p

OK, those makes sense. Similarly, there are (rather peculiar) definitions for "conceited", "consistent", "normal", "peculiar", and "regular" reasoners that also make sense and are (I think) adequately non-weird.

But I'm reduced to "what the actual fuck" when I read about his definition of a "reflexive" reasoner:

∀p:∃q

**B**(q≡(

**B**q→p))

I mean, I think I can make some sense of this, but why is this case interesting? When would it ever arise?

If we take p = "Hillary Clinton is POTUS", for example, and imagine that I'm a "reflexive" reasoner for the sake of argument, what the hell might the corresponding q look like?

I'm nearly as befuddled by his definition of a "modest" reasoner:

∀p:

**B**(

**B**p→p)→

**B**p

What is "modest" about this? And how is it anything but weird? Take any proposition having the property that I believe that I don't believe it:

**B**(¬

**B**p)

Then, assuming I can handle basic logic, we'd have

**B**(¬

**B**p∨p)

So (under the same assumption) we'd have

**B**(

**B**p→p)

So "modesty" (by Smullyan's definition) would entail

**B**p.

IOW, if you're "modest", and you can handle basic logic, then for any p, if you believe that you don't believe p, then you actually believe p. Utterly bizarre.