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Topic: Help me understand doxastic logic (Read 633 times) previous topic - next topic

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  • Brother Daniel
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Help me understand doxastic logic
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 Bp 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 BXp 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:Bp→p
and a reasoner is described as "consistent" if he/she never believes both a proposition and its negation:
∀p:Bp→¬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:∃qB(q≡(Bq→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(Bp→p)→Bp
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:
BBp)
Then, assuming I can handle basic logic, we'd have
BBp∨p)
So (under the same assumption) we'd have
B(Bp→p)
So "modesty" (by Smullyan's definition) would entail Bp.
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.

Re: Help me understand doxastic logic
Reply #1
IOW, if you're "modest", and you can handle basic logic

If reasoners were always able to handle basic logic, there's be no need for the 'consistent' category. So presumably illogical reasoning is permitted.

Re: Help me understand doxastic logic
Reply #2
Yeah. I think that's right. Where I see this going is into horrible looking lines that describe a propaganda effect.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
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Re: Help me understand doxastic logic
Reply #3
Fair enough.  So perhaps "modest" is intended to be one of the undesirable categories, like "inconsistent" or "peculiar" or "conceited".

But in the latter cases, I can match the category names to some easily understandable (but undesirable) doxastic phenomena.  Not so much with "modest".  I don't have an intuitive grasp of what the label "modest" is trying to convey here.
  • Last Edit: March 08, 2018, 11:15:41 AM by Brother Daniel

Re: Help me understand doxastic logic
Reply #4
Folk psychology and intentional stance logic in general is notoriously hard to read as cybernetics.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
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Re: Help me understand doxastic logic
Reply #5
To make things weirder:  One of the headings within the article is "increasing levels of rationality".  It goes on to define "type 1", "type 1*", "type 2", "type 3", "type 4", and "type G".

Type 1 (I think) corresponds to "can handle basic logic".

Type 2 (I think) means being type 1 and believing that you're type 1.

Type 3 means being type 2 and also being "normal", where "normal" means
∀p:Bp→BBp
(i.e. if you believe something, then you believe that you believe it).

Type 4 means being type 3 and believing that you're "normal".

Type G means being type 4 and believing that you're "modest".  Which makes me suspect that being "modest" is intended to be a desirable thing.

But I've shown above (I think) that for anyone who is both "modest" and "type 1", we have
∀p:BBp)→Bp

And that means that for anyone who is "modest" and "type 1" and "normal", we have
∀p:BBp)→BBp

So such a reasoner cannot believe that he/she doesn't believe something, without then believing a contradiction.  By type 1, then,
∀p:¬BBp)
There is no proposition that such a person knowingly fails to believe!


Maybe the Wikipedia writers garbled something, and I should look to the sources.

  • Pingu
Re: Help me understand doxastic logic
Reply #6
You lost me at "it is believed".  Why no subject for the verb?
I have a Darwin-debased mind.

  • Brother Daniel
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Re: Help me understand doxastic logic
Reply #7
I assume that the subject is intended to be clear from the context.

But yeah.  That's why I said
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 BXp would mean "X believes that p".

  • uncool
Re: Help me understand doxastic logic
Reply #8
The wiki links to https://en.m.wikipedia.org/wiki/L%C3%B6b%27s_theorem to explain why it should be true, and the theorem seems to fall into the same problem.

Also, it says that type 4 reasoners are modest, and implies that type 4 is pretty good ("increasing levels of rationality"), so
"modest" seems like a "good" descriptor.

  • Brother Daniel
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Re: Help me understand doxastic logic
Reply #9
The wiki links to https://en.m.wikipedia.org/wiki/L%C3%B6b%27s_theorem to explain why it should be true, and the theorem seems to fall into the same problem.
Specifically, to explain why it should be true that "any reflexive reasoner of type 4 is modest".  As I hinted earlier, I have even more trouble trying to grasp what doxastic phenomenon the descriptor "reflexive" is supposed to capture.
Quote
Also, it says that type 4 reasoners are modest, and implies that type 4 is pretty good ("increasing levels of rationality"), so
"modest" seems like a "good" descriptor.
Yes, I noticed that.

So (in decreasing order of probability) it seems that

(1) I'm being stupid

or

(2) Wikipedia screwed up

or

(3) The original ideas from Smullyan were put together sloppily.

  • Brother Daniel
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Re: Help me understand doxastic logic
Reply #10
I can't help wondering whether propositions are even the right category for things to be believed.

Normally, when the propositions p and q are equivalent, you can substitute one for the other in any complex proposition that refers to them.  Clearly you can't do that when talking about belief.  Someone can believe p while failing to believe q even if p and q are equivalent:  (p↔q)∧(Bp)∧(¬Bq) is a possible state of affairs.

Moreover, the object of a person's belief may be so nonsensical that there is no corresponding proposition.  (Trinitarian theology may be an example here.)

I guess what I'm suggesting is that "belief" doesn't work on the space of propositions, but rather on the space of ... something else.  Let's call them "notions".

A notion may (or may not) correspond to a particular expression of a proposition, in which case belief in the former notion can loosely be described as belief in the latter proposition.

Now I'm not at all sure that this suggestion will really shed light on anything.  Just "thinking out loud" here.

Re: Help me understand doxastic logic
Reply #11
This is why you dont get work done.

  • uncool
Re: Help me understand doxastic logic
Reply #12
Might be a good place to ask @Quizalufagus to comment on Löb's theorem.

  • Brother Daniel
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Re: Help me understand doxastic logic
Reply #13
This is why you dont get work done.
shut up Perseus

fake eta:  This isn't the reason; this is just a symptom

Re: Help me understand doxastic logic
Reply #14
I've been working on a screenplay for several years about a guy who's super add but also a super genius. His boss at a defense contractor figures out that the hero can't finish a task because he gets distracted so he starts giving the hero immensely tedious jobs but adds in "other projects" the hero can work on in his spare time. The hero never finishes his "jobs" but always comes up with insanely brilliant solutions to the "side projects" because they are what he does when he's distracted. The fact that it's a defense contractor is the plot device for the story. You can probably guess at some of the story arc.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
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  • predisposed to antagonism
Re: Help me understand doxastic logic
Reply #15
I like it.  I can half relate to that character*.  Lots of times I've gotten good work done that wasn't on my task list.  If you want to stop me from making progress on anything, just add it to the list of things I'm officially supposed to be working on.


[* I have the super ADD part, not the genius part obviously ]

Re: Help me understand doxastic logic
Reply #16
Me too. I'm always busy but not always with tasks from my list.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor