Withholding belief v. denying knowledge - Page 10

pseudonous · 05-04-2008, 03:53 PM

Quote:

Taking your two (three) axioms A B (C): should we not also include the probabilities of there being no other axioms say O(AB) and O(ABC)

The two probabilities become:

1/xyO(AB) and 1/xyzO(ABC)

Romansh,

I am not sure I understand your symbolism can you please elaborate.

romansh · 05-04-2008, 03:58 PM

O(AB) is the probability that there are no other axioms other than A and B in play

O(ABC) is the probabilty that A B and C are the only axioms

pseudonous · 05-04-2008, 04:06 PM

Romansh,

OK I see what you are saying but I made the assumption that only A, B and C are relevant since I was only attempting to show that there is at least one case where an argument with fewer axioms is more likely to be true.

Do you think I am over simplifying or misconstruing the question I was asked?

pseudonous · 05-04-2008, 05:18 PM

Let's try this for an example.

We have one argument,
1. If P then Q (Axiom A)
2. P (Axiom B)
therefore we conclude
3. Q

And a second argument
1. If P then Q (Axiom A)
2. P (Axiom B)
3. T (Axiom C)
therefore we conclude
4. Q&T

As before we have the probablity of AB is 1/xy and the probablity of ABC is 1/xyz. In this case, there is also the possibility of other axioms that will justify the conclusions i.e.
1. P or Q (Axiom D)
2. not P (Axiom E)
therefore we conclude
3. Q
There may be an infinite amount of other axioms such as these that will justify the conclusion of either argument. However, any O(ABC) will justify the conclusion of both conclusions where as any O(AB) will only justify the first conclusion. This follows since the conclusion of the first argument is contained in the conclusion of the second argument.

So, the probability O(AB) > O(ABC) and hence, [1/xy + O(AB)] > [1/xyz + O(ABC)]

Given this, the probability that the conclusion of the first argument is true, is greater than the probability that the second more complex argument is true.

romansh · 05-04-2008, 05:36 PM

Nous

I think that when we say A and B are relevant we are making the implicit assumption that C, E, F, G, H, I ..... are not relevant. So I suspect we need a term when calculating the probability for the irrelevance of all the axioms.

we have no way of telling whether:
1/O(AB) gives us better odds than 1/zO(ABC)

I suspect adding and tweaking how the theories are joined doesn't necessarily weaken the overall theory.

Just the same way the "standard model" is fusion/unification a range of theories and explanations. Each in itself is incomplete but the whole is greater than the sum of the parts ... so to speak. Nevertheless the standard model remains incomplete, as it does not explain gravity.

pseudonous · 05-04-2008, 06:01 PM

Romansh

I agree largely with what you are saying but I would add that in some situations we do have a way of telling that 1/O(AB) gives us better odds than 1/O(ABC). This is what I was trying to illustrate in my previous post perhaps I need to rethink this. Did you have any specific objection to my previous post or was it just in general unclear? I am thinking it may be that I am not using O(AB) in quite the same way as you.

Remster · 05-05-2008, 09:09 AM

Pseudonous

Can you please explain how the mathematics relates to the sort of example we've been discussing? To recap, Romansh tells me that Newton's theory seems to work well for medium-size objects but not for very large or very small objects. So we have (among others) two options:

Theory 1. Such-and-such anti-Newtonian theory.
Theory 2. Newton's theory combined with the theory that some powerful though otherwise undetectable being makes it seem to us that very large and very small objects behave in anti-Newtonian ways.

I'd agree that Theory 1 is the simpler theory, but I'm interested to know why it is more likely to be true than Theory 2 – in particular how the mathematics shows this. Here are a couple of points to note:

The actual and possible evidence for the two theories is exactly the same.
No part of Theory 2 is dispensable: you can't explain the evidence with any part taken in isolation.

Quote:

It sounds to me that you are denying the existence … skeptical people that merely require the appropriate justification to hold a belief.

That's right, because those people hold 'structural' beliefs (such as that simpler theories are more likely to be true) that are in principle not amenable to evidential support. I realise you haven't granted this yet though.

Remster

Remster · 05-05-2008, 09:11 AM

Rom

Quote:

For example I have mild tinnitus .... It's only there when I concentrate on it.

That's a different issue (if an interesting one). What I'm concerned about here is whether you can believe that the symptoms of your tinnitus – ringing in the ears, right? – are there when they're not.

Quote:

regarding self evident truth ... I agree ...

Are you agreeing with me that the self-evidence to us that x = x is no more evidence that x = x than the self-evidence to some theists that God exists is evidence that God exists?

(Obviously I'm not using 'self-evidence' as a success term here. If you prefer to use it that way, you may replace 'self-evidence' above with 'apparent self-evidence' or 'seeming self-evidence'.)

Rem

shadowind · 05-05-2008, 04:51 PM

Quote:

Originally Posted by pseudonous

Remster

Since 1/xy > 1/xyz, it follows that the arguement based on less assumptions is more likely to be true.

do you mean that if each axiom had a equal probability of being correct
that the probabilitys of just 2 axioms being correct to make the conculsion correct is greater then having three axioms?

if so how is one to tell the probabilitys of the axioms with out assuming?

also arguments are not numbers.

pseudonous · 05-05-2008, 08:37 PM

Quote:

Originally Posted by Remster

Pseudonous

Can you please explain how the mathematics relates to the sort of example we've been discussing? To recap, Romansh tells me that Newton's theory seems to work well for medium-size objects but not for very large or very small objects. So we have (among others) two options:

Theory 1. Such-and-such anti-Newtonian theory.
Theory 2. Newton's theory combined with the theory that some powerful though otherwise undetectable being makes it seem to us that very large and very small objects behave in anti-Newtonian ways.

I'd agree that Theory 1 is the simpler theory, but I'm interested to know why it is more likely to be true than Theory 2 – in particular how the mathematics shows this. Here are a couple of points to note:

The actual and possible evidence for the two theories is exactly the same.
No part of Theory 2 is dispensable: you can't explain the evidence with any part taken in isolation.

Remster

Remster,

The mathematics was intended to answer your request for a justification of my answer to your original question:

Quote:

And do you hold anything like the view that of two beliefs that fit the same evidence, the simpler of the two is more likely to be true?

the answer I gave being, "yes, it would be true of some axiomatic arguments."

The justification I gave would not apply to the example you have been discussing with Romansh. I must have been a bit confused about the intentions behind your question. Sorry, about getting off track. Give me a bit of time to think about these two hypothetical theories and I'll see what I can come up with. It seems to me there might be a metaphysical aspect to theory 2.

Pseudonous