logical and causal possibilities and proofs

(edit error;
the title of this thread was meant to be:
" causal and logical probability/proof/fact " )

I have noticed that people and science in general implicitly deal with two different categories of probability with neither category currently having been formally identified nor have any name whatsoever, neither formal or informal. Thus I see usefulness in explicitly defining and naming these two categories which I will do here:

When most people, including scientists, calculate probabilities for different outcomes, they often implicitly take no account of the logical possibilities and therefore the logical probability that the the natural laws, as they believe them to be so, are entirely as they believe them to be so. In other words, when they calculate probabilities, they typically treat that probability of them be wrong about what they believe to be the actual natural laws, to be exactly zero probability i.e. they assume they assumed natural law correctly. They implicitly have to do this for very good practical reasons! To take into full account the logical possibility that they could be wrong about the relevant natural law by calculating the probabilities for that before calculating the probability of a causal possibility would generally be extremely arduous and mathematically complex and yet would typically make very little difference in practical terms to the numerical value of that calculated probability outputted for that causal possibility.

I call the typically more convenient type of probability of causal possibilities/impossibilities that takes no account of probabilities of what we think we know about natural law to be entirely correct

'causal probability'.

And I call the often less convenient type of probability but arguably more valid type of probability of either causal possibilities/impossibilities or logical possibilities/impossibilities that takes full account of probabilities of what we think we know about natural law to be entirely correct

'logical probability'.

Thus logical probabilities are arguably the only really 'true' probabilities as they don't make any unqualified assumption that we know natural law entirely correctly while causal probabilities, although generally better for more practical purposes, are just a convenient approximation of logical probabilities.

Also, when we speak of 'proof', there are two kinds of proof which I will give new names to here;

A proof that shows the causal probability, but not necessarily also the logical probability, of something, to be exactly 0 or 1, it is a

'causal proof'.

A proof that shows the logical probability, and therefore necessarily also the causal probability, of something, to be exactly 0 or 1, it is a

'logical proof'.

In addition;
Anything that has been causally proven is a

'causally fact'.

Anything that has been logically proven is a

'logical fact'.


OK;

So, when a scientists, say, using is assumed knowledge of the laws of gravity, calculates the probability of a planet having a certain orbit and finds that probability to be exactly 0 and then says he has 'proved' that it is 'impossible' for a planet to have that certain orbit, what he means is that he has 'causally proved' it causally impossible by proving it has a 'causally probability' of exactly 0. And now it may be a 'causal fact' that a planet cannot have that orbit.
But he has not 'logically proved' anything and the 'logical probability' is not an absolute probability of zero but rather a very close non-zero approximation to zero.
And it isn't a 'logical fact' that a planet cannot have that orbit.

Any criticisms; thoughts; comments?

Originally posted by humy
Any criticisms; thoughts; comments?

Your usage of the term 'exactly zero' does not make sense. Experiments are necessarily finite and thus causal probability is necessarily non-zero. To call it 'exactly zero' rather than 'as good as zero' seems to me to be unfounded. Or maybe that is what your post is saying, its not very clear.

Originally posted by twhitehead
Your usage of the term 'exactly zero' does not make sense. Experiments are necessarily finite and thus causal probability is necessarily non-zero. To call it 'exactly zero' rather than 'as good as zero' seems to me to be unfounded.

You haven't understood what I am saying at all:
I am saying a 'causal probability' can NEVER have 'exactly zero' 'logical probability' AND only that 'logical probability' is the 'true' probability, NOT the 'causal probability' which merely is the less valid BUT more convenient (often wildly by far more convenient to avoid massively arduous calculation ) estimate of the 'true' probability.

I also think that most scientists, when they make a claim include two things:
1. an implicit assumption about certain scientific causal facts. ie they would essentially be saying "Assuming Einstein's relativity is correct, a planet cannot be in that orbit". I would claim that this makes any violations logical violations as the assumptions become axioms.
2. error bars. For anything that they know is not completely assumed to be a law but is based on measurements, the estimate the error. This may come in the form of a direct cut-off figure (that really only means 'its probably in this range' ) or a measure of certainty (standard deviation) such as we see in CERN reports.

Originally posted by humy
You haven't understood what I am saying at all:
I am saying a 'causal probability' can NEVER have 'exactly zero' 'logical probability' AND only that 'logical probability' is the 'true' probability, NOT the 'causal probability' which merely is the less valid BUT more convenient estimate of the 'true' probability.

Well you confused me with this definition:

A proof that shows the causal probability, but not necessarily also the logical probability, of something, to be exactly 0 or 1, it is a
'causal proof'.

That directly contradicts what you have now said.

Originally posted by twhitehead
Well you confused me with this definition:

A proof that shows the causal probability, but not necessarily also the logical probability, of something, to be exactly 0 or 1, it is a
'causal proof'.

That directly contradicts what you have now said.

how do the two statements contradict?

Originally posted by humy
how do the two statements contradict?

Statement A:

A proof that shows the causal probability ... to be exactly 0

Statement B:

a 'causal probability' can NEVER have 'exactly zero'

Originally posted by twhitehead
Statement A:

A proof that shows the causal probability ... to be exactly 0

Statement B:

a 'causal probability' can NEVER have 'exactly zero'

statement b actually reads:

"a 'causal probability' can NEVER have 'exactly zero' 'logical probability' "

thus there is no contradiction as a 'causal probability' is not a 'logical probability' thus the two kinds of probability can have different numerical values for their probabilities without contradiction.

Actually, I used a very poor choice of words there because it can be misread as a statement about a probability of a probability as opposed to a probability of something that is either possible/impossible but isn't itself a probability.
To avoid that confusion, I should have said that as something more like:

"a 'causal probability' of a causal possibility can be 'exactly zero' but the 'logical probability' of that same causal possibility (or any other causal possibility or any logical possibility ) can NEVER be 'exactly zero' "

-to make it clear what the probability is of.

Originally posted by humy
statement b actually reads:

"a 'causal probability' can NEVER have 'exactly zero' 'logical probability'

I still find it confusing and unnecessarily complicated. For a start logic does not have probabilities. Something is either logically possible or not logically possible (impossible). To assign a probability is unnecessary and confusing as it suggests the availability of alternatives of given frequencies etc which is not the case at all. Saying that square circles have a 'logical probability' of zero of existing seems unnecessary.
2+2 has a logical probability of zero of being equal to 7. I don't like it. Just say its not 7 or that 2+2=7 is illogical.

I have I believe challenged you in the other thread (and if I haven't then consider it the challenge here) to provide a reference to a definition of the term 'probability' that allows for events that are not possible or more accurately allows for things that are not events, as events are defined as being possible.

Originally posted by twhitehead
I still find it confusing and unnecessarily complicated. For a start logic does not have probabilities. Something is either logically possible or not logically possible (impossible). To assign a probability is unnecessary and confusing as it suggests the availability of alternatives of given frequencies etc which is not the case at all. Saying that square ...[text shortened]... more accurately allows for things that are not events, as events are defined as being possible.

logic does not have probabilities. Something is either logically possible or not logically possible (impossible).

Doesn't that depend on exactly how you mean and therefore how you should exactly define what is a probability?
I have been working on that very problem for years and I am still very far from satisfied that I have decided what meaning and definition of the word 'probability' should be.

To assign a probability is unnecessary and confusing as it suggests the availability of alternatives of given frequencies etc which is not the case at all.

Not if you assign exactly either 1 or 0 logical probability, as this tells you there is zero probability of an alternative therefore there is no alternative.

Saying that square circles have a 'logical probability' of zero of existing seems unnecessary.

only because in this case it is far too obvious that square circles are a contradiction; it wouldn't be even worth the minuscule effort of just saying it.

But I am aware many people would say that something that is true/false by definition (or by pure deductive logic ) "has no probability"; But there are also many that say it does. This brings us back to the thorny problem of what do you mean and therefore how should you define probability? It is 'thorny' because different people mean slightly different things by it so what none arbitrary criteria one should use to narrow down our valid choices of meaning? -I currently have no answer for that question. Maybe there really is no answer so its a stupid question? If you (or anyone else here) do have an answer, I love to hear it.

Originally posted by humy
Doesn't that depend on how you define what is a probability?

Which is why I have challenged you to find a formal definition that doesn't agree with me but agrees with you.

But I am aware many people would say that something that is true/false by definition (or by pure deductive logic ) "has no probability"; But there are also many that say it does.
The average person simply has not thought it through. In addition there is the issue of causal probability that most people are intuitively aware of even if they have not thought it through ie when they say there is 'zero probability' they really mean 'very very unlikely' rather than 'logically impossible', or they just equate the two situations for simplicities sake.

It is 'thorny' because different people mean slightly different things by it.
It should only be thorny if you are a dictionary writer and you are interested in what public usage is. And even then you can get away with listing both uses and not taking sides.
In mathematics however you simply write a definition.
It remains the case that the whole point of probability is to deal with situations that occur at certain frequencies and involve an assumed randomness. To include in this situations that cannot occur and do not involve randomness just seems unnecessary. And it would appear that every text book I have looked at so far agrees with me.

Originally posted by twhitehead
Which is why I have challenged you to find a formal definition that doesn't agree with me but agrees with you.
...

My current definition of probability (which may keep on evolving just as it has been slowly evolving over time and I hadn't bothered to try and write it down before so me exactly choice of words below are improvised and likely very far from perfect )
is:

1, Given all what you know, if there there exists a most rational degree of certainty for something with all that knowledge you know and no other knowledge i.e. not including knowledge you don't have, then, for you, the probability for that something is that most rational degree of certainty.

2, if there doesn't exist a most rational degree of certainty for something given that limit knowledge then the probability is undefined i.e. it doesn't exist else it does exist.

3, if it is true by definition or by pure deductive reasoning, its not only that probability for 'you' in particular but that probability for everybody i.e. it is objective as in independent of personal knowledge.

( I can just imagine some (not all) qualified philosophers reading all the above and shouting "NO! NO! NO! ..." but, what the hell )

OK

using criterion 2
There does exist a most rational rational degree of certainty that 1+1=2 therefore there exists a probability for that.

using criterion 1:
The most rational degree of certainty I have that 1+1=2 is true is total certainty therefore the probability, for me, is 1.

using criterion 3:
1+1=2 is purely deductive thus it's probability of 1 isn't just true for me in particular but everybody i.e. it is objective.


Note that I have deliberately left out an axiom in my above criteria, which I call the 'tie axiom' of logic, because I cannot afford the risk of plagiarism before publication. As a result, in its said form above it is subtly incomplete and completely fails to deal with certain paradoxes and problems in philosophy and probability theory -especially the 'asymmetrical reasoning problem'. With the tie axiom placed there, none of those problems ever arise.

What is your definition? I am very eager to compare.

Originally posted by humy
What is your definition? I am very eager to compare.

I stick with the textbook definition:

Given a set of possible outcomes that occur at specific frequencies, the probability is a measure of those frequencies.

Or in the words of Wikipedia:

Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty).

Note that it is not clear in the above sentence but I would add that 1 and 0 are exclusive. ie a probability lies in the open set (0,1)


I note that your definition is 'applied probability'. Your probability is to mathematical probability as engineering is to physics or physics is to maths.
You are talking about probability specifically as it may be applied to reality and our knowledge of reality.

In addition your definition appears to focus on the 'zero' and 'one' ends of the scale whereas I see probability as being confined between the two. You do not even mention frequency which I see as the very heart of probability.

When I see something happens at a frequency of 90% you will only ask 'how sure are we that it will happen'? So whereas a 50/50 result is fantastic under my definition it is unsatisfactory failure under yours.

Originally posted by humy
(edit error;
the title of this thread was meant to be:
" causal and logical probability/proof/fact " )

I have noticed that people and science in general implicitly deal with two different categories of probability with neither category currently having been formally identified nor have any name whatsoever, neither formal or informal. Thus I see use ...[text shortened]... 't a 'logical fact' that a planet cannot have that orbit.

Any criticisms; thoughts; comments?

There's a concept called likelihood, which roughly speaking is the probability of a probability. I'm not sure if it fills the gap that you are talking about, it's a while since I read about it.

Originally posted by DeepThought
There's a concept called likelihood, which roughly speaking is the probability of a probability. I'm not sure if it fills the gap that you are talking about, it's a while since I read about it.

Not to be confused with standard English usage:
https://en.wikipedia.org/wiki/Likelihood_function

It does appear that Humy would be better served by that definition.