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What can we prove?

69dodge said:
I have a theory that all dogs bark loudly. I see a cat. It doesn't bark loudly and it isn't a dog. So it's evidence in favor of my theory.

You have a theory that all dogs bark softly. You see the same cat. It doesn't bark softly and it isn't a dog. So it's evidence in favor of your theory too.

Oops. :p

How can the same observation be evidence in favor of two contradictory theories?
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That's interesting. Our two theories are now:

Your's: All non-loudly-barking objects are not dogs.

Mine: All non-softly-barking objects are not dogs.

The cat that doesn't bark is indeed evidence in favour of both theories.

Imagine we were mad enough to actually try to prove our theories by this long-winded method. After billions of years of effort we have examined all the non-barking objects we can find - grains of sand, coffee machines, everything. None of them turned out to be dogs. Also, of course, none of them turned out to be seals. At this point we have proved at least two theories: 'All dogs bark', and 'All seals bark'.

Now we are in a position to test our competing theories - there are no more non-barking objects left to examine, but there will be a mixture of loudly-barking and softly-barking things out there. To gather evidence for your theory, you will be looking for non-loudly-barking objects. I will be searching for non-softly-barking objects. We might both find seals, audio playback devices, mechanical models of dogs, etc. but eventually, either you will find a non-loudly-barking object that is a dog, and/or I will find a non-softly-barking object that is a dog. At this point, one or both of our theories will have been disproved.

For those that missed it above, I have already admitted that the method I propose is a ridiculous way of going about the research. However, from a logical standpoint every non-barking non-dog observed does add an infinitesimal amount of evidence in support of the theory.
 
ceptimus said:
You don't even have to observe the things that your theory is about to gather supporting evidence.

Suppose my theory is, 'All dogs bark'.

Then by pure logic, I rearrange this to say, 'Any non-barking object is not a dog'.

If the second version of the theory is correct then so is the first.

So I can go about my daily business, observing as I go any non-barking objects, say my car, the curtains, a paving stone, a lump of coal, etc. And so I gather supporting evidence for my theory.
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My gran had a box to warn would be burglars off that barked. 'twas't a dog though
 
I see what ceptimus is getting at, albeit rather clumsily.

What he is doing is trying to disprove his hypothesis. The scientific method can be applied in a number of ways, however always amounts to taking a hypothesis and trying to disprove it.

Why? Because as has already been said, you can 'prove' things in maths and law, but in science we explore the weight of evidence.

This can be messy, as there is no quantitative amount of evidence that amounts to proof. A hypothesis becomes a theory if it can be used to correctly anticipate an event. Using this 'dog' example, if you use the hypothesis 'all dogs bark', and you look at an animal and say 'it's a dog, it should bark', and it does, the hypothesis becomes a theory (this barking dog thing is a little barking mad, but I'll play along).

A theory becomes 'robust' if it can be repeated.

Now, a hypothesis cannot truly be proven. There could always be a 'negative' lurking unseen somewhere you haven't looked. It can, however, be disproven. 'All four legged animals are pink' is a hypothesis that can be supported (every four legged animal seen to be pink adds weight) or, in one foul swoop, disproven (oh look! a purple four legged animal!). One way, you could still be wrong. The other, you will have your answer immediately.

Hence 'all dogs bark' can be given the weight of evidence, or disproven. Just find a dog that doesn't bark and your hypothesis has been disproven. The scientific method works best if you can disprove things rather than 'prove' them.

Athon
 
athon: I see what ceptimus is getting at, albeit rather clumsily.
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What ceptimus is getting at with the barking dog is also known as Hempel's paradox
http://www.google.com/search?q="Hempel's+paradox"

In Hempel's 1937 version, it was black crows, summarized by some un-named smart-ass:

<blockquote>I never saw a purple cow
But if I were to see one
Would the probability ravens are black
Have a better chance to be one?</blockquote>With apologies to Frank Gelett Burgess, author of the original Purple Cow poem.
 
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