~Bp <=> B~p (disbelief in something is the belief of the absence of that thing).
So we can all agree that if someone doesn't believe in p then he believes that p is false, and vice versa.
So we actually have ~Bp <=> B~p.
That's interesting are there any more modal operators that satisfy such equivalence in our informal languages?
So we actually have ~Bp <=> B~p.
That's interesting are there any more modal operators that satisfy such equivalence in our informal languages?
Comments (36)
No.
I don't believe that you had Weetabix for breakfast and I don't believe that you didn't have Weetabix for breakfast.
Poor comedian...
I'm not trying to be funny. I'm pointing out that not believing that something is true is not the same as believing that it's false.
I have a brother and his name is Andrew.
Do you believe that the above is true? Do you believe that the above is false? It's appropriate to not believe either, especially if you have no evidence either way.
Is a bit counter intuitive.
Bob: 'I don't believe in anything you said'
Alice: 'Why?'
Bob: 'Because the first thing you said was wrong'
I know. I'm pointing out that ¬Bp doesn't entail B¬p. It's possible for ¬Bp ? ¬B¬p.
You can just substitute ¬p=q.
And that's wrong, because they're not equivalent, as I tried to explain. I don't believe that you had Weetabix and I don't believe that you didn't have Weetabix. 1) is true but 2) is false.
No... I wanted to show that not believing in a disjunction implies believing in the negation of each disjunct. This is very strange behaviour, and is implied by the OP.
1. There are <10000 asteroids around Saturn.
2. There are 10000 asteroids around Saturn.
3. There are >10000 asteroids around Saturn.
I believe in (1 v 2 v 3). I don't believe in (1 v 2), that doesn't mean I believe that ~1 &~2, since that commits me to 3. I suspend belief in each disjunct while believing the disjunction as a whole is true (it's a tautology).
I have a brother.
Do you believe that the above is true or false? Do you understand that it is possible (even appropriate) to believe neither, i.e. to withhold judgement?
Not if we're being strict about the logic. It is possible to not believe that p is true and to not believe that p is false. This is the position we (should) have when we have no evidence either way.
Do you believe that I have a brother? Do you believe that I don't have a brother?
Sure, but I was addressing the OP which uses formal logical syntax. My ordinary language response was just an attempt to show why it's wrong.
~B(Ex~P(x)) <-> B(~Ex~P(x)) <-> (B(AxP(x))
"The agent withholds belief about whether there exists an entity without property P"
iff
"The agent believes every entity has property P"
I lack belief in some things which are not gazoompas. Therefore I believe everything is a gazoompa.
I think we can see that lack of belief and disbelief are different now.
Wouldn't that mean you lack the belief that there is some thing that is not a gazoompa? If you want to say you lack belief in some thing, shouldn't you put ? before the B?
They're different.
~B(Ex~P(x))
~ExB(~P(x))
Agent withholds belief that there is an X which is not P.
There is no X which A does not believe is P.
Do you agree with that holding when B~ <=> ~B (abusing notation)?
How would you translate it?
not believing there is any x without the property p?
Yeah. I agree that it's a more precise translation. My motivation for translating the ~B as withholding belief or suspending belief was to highlight the contrast between it and B~; both can be translated as 'doesn't believe' and it depends on the context which means which. Equating the two is an error made tempting by 'does not believe' meaning both in ordinary language. There are good reasons to maintain that they are distinct despite this, however.
Quoting fdrake
(Should be "I lack the belief that there are things that are not gazoompas")
Going through my comments there's a load of mistakes so they were probably terrible to read through (sorry :p).
Yeah this is fair enough. I've been imprecise a few times in the thread, which is certainly a shame since it's a thread about logic; and no one is less sympathetic to imprecision (rightly, probably) than logicians.
The first two contradict each other, as do the last two.
The first and the third contradict each other; but not the second and the last. One can consistently hold that
~B(p) & ~B(~p)
if one chooses not to have an opinion about (p). That's the point of view of an agnostic when asked if they believe in god, for example.
So while ~B(p) implies ~B(~p), it does not imply B(~p).
Hence the OP is incorrect in saying ~B(p) <=> B(~p).
You don't believe I ate W and you don't believe I didn't eat W, then what do you believe: did I eat or didn't I?
We have ~Bp & ~B~p, the question is do we have BpvB~p?
How can someone believe something and also believe its negation at the same time?
What is the semantics of 'belief'?
Can someone both believe in God and believe there's no God?
Besides me of course.... :=)
What colour do you believe the shirt I am wearing right now is?
Quoting MathematicalPhysicist
No.
Quoting MathematicalPhysicist
They can't.
It's not relevant.
To your question the answer is "I don't know", while the answer to do you believe I ate or do you believe I didn't eat has a definite answer; while to your question there are more than two options, it's not binary, either 0 or 1.
There's no room for comparison between the two questions.
The question can be rephrased as "for any given colour, do you believe it is that colour or not?".
Quoting MathematicalPhysicist
Whether the options are binary is not relevant, you believe that one must believe in some value.
If only I could upvote you! Thank you, my friend.
It has no answer because it is a false dichotomy. The question presupposes that one either believes W or one believes ~W. It erroneously omits the possibility that one could believe neither. It also omits the somewhat more controversial possibility that one could believe both. One can debate whether that last is possible. But it nevertheless needs to be included in the quartet, to complete the analysis.
For the vast majority of well-formed propositions that can be made, I believe neither the proposition nor its negation, because I am in no position to form an opinion either way.
It would be very sad if we were to ban the state of suspended judgement. It seems to me that that is one of the most beautiful states there is.