any examples of 'modeless' continuous probability distributions?

Originally posted by twhitehead
...
I can imagine a world in which coins have snakes on them.

but can you imagine a world which coins have no snakes but where you can select snake on a coin?

More generically:

Can you imagine a logical contradiction in the external world? (external world means external to your mind, concepts, feelings, etc )

Originally posted by humy
but can you imagine a world which coins have no snakes but you can select snake on a coin?

But that is not what the definitions for causal vs logical state, although the definitions are not very clear and different sources have different definitions.

Suppose we knew absolutely that faster than light travel was impossible in our world, but possible in a different universe. Would fast than light travel in our world be logically impossible or causally impossible?

This reference would seem to suggest it would be logically possible but not causally possible:
http://storiesandsoliloquies.com/2015/05/01/the-philosophers-lexicon-logical-and-causal-possibility/

If you can find a clear definition I would be interested. I am mostly finding vague descriptions followed by vague examples.

Originally posted by twhitehead
We should not strictly call them 'facts'. Nor should we reject them, but accept that we do not have absolute certainty of their validity.
So earlier in the thread when you talked of something having a probability of absolute zero, were you allowing for the kind of uncertainty here or not?
In your OP, does the zero refer to the kind of uncertainty here? ...[text shortened]... locity of zero. I still say that assigning a probability of zero to such an event is incoherent.

We should not strictly call them 'facts'. Nor should we reject them, but accept that we do not have absolute certainty of their validity.
So earlier in the thread when you talked of something having a probability of absolute zero, were you allowing for the kind of uncertainty here or not?

For logical possibilities, yes. For causal possibilities, no. That is because causal possibilities always ASSUME we know correctly the natural laws we think we know correctly.
If I drop an apple, it is 'causally' a fact that it will fall down and not up and causally impossible for it to fall up. That is because if the law of gravity is what I assume it to be, it is impossible for the apple to fall up. But it is not 'logically' a fact because there is no logical contradiction in me assuming incorrectly natural law, including the law of gravity, thus it being logically possible for the apple to fall up.

Note " 'causally' a fact " and " 'logically' a fact" isn't conventional terminology; but I don't know how else to say it.

Let me put it this way:

A 'fact' (esp a scientific 'fact' ) about the world is 'causally' a fact, because it is causally impossible for it to be wrong, but never 'logically' a fact because it is logically possible for it to be wrong.

Originally posted by twhitehead
We should not strictly call them 'facts'. Nor should we reject them, but accept that we do not have absolute certainty of their validity.
So earlier in the thread when you talked of something having a probability of absolute zero, were you allowing for the kind of uncertainty here or not?
In your OP, does the zero refer to the kind of uncertainty here? ...[text shortened]... locity of zero. I still say that assigning a probability of zero to such an event is incoherent.

One the basis that there is epistemic uncertainty about all statements of fact, insisting that they are not facts makes discourse impossible. So we may as well call them facts.

Originally posted by humy
According to you, would the probability of an earthworm burrowing (or perhaps 'swim', rather than 'burrow', for at least part of the way ) to the exact center of the Earth be 'zero' probability?
Or do you say that even that probability is 'undefined' because there is no such thing as 'truly' zero probability?

In light of recent discussions I must say that the probability of an earthworm burrowing to the centre of the earth is in fact positive and non-zero.
Picking a random real number and getting a rational is less likely than a worm burrowing to the centre of the earth.
Since all scientific knowledge is based on experiment and all experiments are finite in nature, all prior/posterior probability calculations must necessarily result in a non-infinitesimal number. (is there a word for non-infinitesimal? )

Originally posted by humy
I am using the words 'causally' and 'logically' rather loosely and informally here and this isn't quite conventional terminology.

(I quoted you before you edited this out 🙂 )

Loose definitions may only lead to more confusion. Anyway, I believe we have largely exhausted the causally vs logically distinction and I don't think we have resolved anything by it. From your usage it would appear that 'causally' rules out absolute zero probabilities. Thus we could assign a probability of zero (but not absolute zero) to a causally 'impossible' event (where 'impossible' simply means highly improbable to the point of virtual certainty). I still hold that assigning such probabilities is not useful as we typically do not actually talk about the real world but rather hypothetical worlds in which the laws of physics are assumed or rather defined to be true and thus any violations become logical impossibilities.
Further I hold that assigning a probability to a logically impossible event not only is not useful but is incoherent and violates the definitions.

Originally posted by DeepThought
One the basis that there is epistemic uncertainty about all statements of fact, insisting that they are not facts makes discourse impossible. So we may as well call them facts.

Agreed. And in such discourse, because we assume said facts, any violation of said facts is a logical impossibility.

Originally posted by twhitehead
Agreed. And in such discourse, because we assume said facts, any violation of said facts is a logical impossibility.

No it just means that the fact wasn't a fact.

Originally posted by twhitehead
..., any violation of said facts is a logical impossibility.

no, causally impossible. That is assuming the type of fact you refer to is a fact about the external world (which we can only know by direct/indirect observation ) and not, say, a purely mathematical fact; else that said fact can never have absolute logical impossibility (i.e. with absolute zero probability), only possibly causal impossibility.

Originally posted by DeepThought
No it just means that the fact wasn't a fact.

You are misunderstanding.
If I were to say "I toss a coin, what is the probability of it landing 'heads'". You will not include in your sample space the possibility that the coin will fly off into space and never come down.
If I were to ask you what is the probability curve for cars going at different speeds, you will not allow for some cars going faster than the speed of light.
Within our conversation it is assumed that the laws of nature are fact and anything that violates them is not merely highly improbable but logically impossible.
We do not say 'well in some possible world a car may go faster than light, so that is only causally impossible so we must assign an infinitesimal probability to that' - unless you are doing a course specifically on the difference between causal and logical impossibility.

Originally posted by humy
no, causally impossible. That is assuming the type of fact you refer to is a fact about the external world (which we can only know by direct/indirect observation ) and not, say, a purely mathematical fact; else that said fact can never have absolute logical impossibility (i.e. with absolute zero probability), only possibly causal impossibility.

What I am saying is that the vast majority of probability scenarios are not about the true 'external world' but about a mathematical model we have made of what we think the ideal external world is. As such we are really talking about our mathematical model and not the real external world.
Coins may exist in the real external world by you readily accepted that whenever I gave a coin flip example I was talking about a hypothetical imaginary world in which coins had certain properties quite separate from natural law.

Originally posted by DeepThought
No it just means that the fact wasn't a fact.

I suppose one could always define a 'fact' (about the external world) as a theory that has some totally arbitrarily defined infinitesimally small totally rationally assigned non-zero probability of being false? ( say, less than one chance in a trillion )
In which case there is such thing as 'facts' about the external world including 'scientific facts' else no such thing!

In either case, there is no such thing as 'absolute facts' (meaning with absolute rational certainty with no logical possibility of being false, even if it is causally impossible for it to be false) about the external world.