logical and causal possibilities and proofs

Originally posted by twhitehead
...

Even more fun would be to let all irrational numbers be twice as likely as rationals. I wonder if a probability density function can be defined in that case.
....

Oh hell, please lets no go there. That looks to me like a conceptual nightmare guaranteed to cause complex confusions.
The concept of just the 'ordinary' probability density function cause people enough confusion as it is.
I think this thread has proved that.

One of the big sources of confusion is the name "probability density" which people naturally but erroneously assume implies a density is a probability! It isn't! But you can readily get probabilities directly from its integration.
The problem is, even if you know that, like I do, if you keep calling it "probability density", your constantly telling your subconscious that it is a probability, and then, as I know from experience, you unconsciously treat it as a probability and then you wonder why the equations you make for it are spewing out complete nonsense!

And just calling it "density" is often a bad idea also because the word "density" has a wider generic meaning, asp in physics, and in some contexts you want to avoid confusing the two meanings of the word.

For these reasons, I have invented a new name for "probability density"; of just "densi". I intend to put that in my book and explain the above reasons for it, and then recommend and hope that this new word will become the new standard word to use for probability density.

Certainly since I have been using the word densi for probability density, I have stopped making the unconscious error of thinking of it as a probability and thus stopped occasionally making mathematical mistakes from that. So it works for me and I would guess it would work for most people.


Another huge potential source of total confusion:

The word 'range' can mean range of the output of the function:

http://www.purplemath.com/modules/fcns2.htm

or it can mean the range of the input of your function;

http://www.mathgoodies.com/lessons/vol8/range.html

So lets get this straight;
If a statistician speaks of the 'range' in the context of a distribution, he can either be referring to:

EITHER

The input of the function, NOT to ever be confused with its output!

OR

The output of the function, NOT to ever be confused with its input!

GREAT! Well I am glad we have that all straightened out.😕

This is why I refuse to ever use the word 'range' for the output of the function and always insist, like I will do so in my book, to use the word codomain for that output.

Originally posted by humy
Oh hell, please lets no go there.

But ... but ....
Think about it for a moment. There are infinitely more irrationals than rationals. So did I mean that picking a number at random would be twice as likely to result in an irrational, or did I mean that any given irrational is twice as likely as any given rational? In the latter case, there would be virtually no difference from the uniform distribution as rational's have only an infinitesimal chance of being picked anyway. The former case is just a recipe for madness.

two new developments;

firstly:

The brother of mine that is an extremely well qualified professor in philosophy has e-mailed me in response to my questions and basically said things in flavor of what I intuitively thought about probability, definition and truth values.

secondly, I can take the my definition of probability to be an 'axiom' and I found this:

https://en.wikipedia.org/wiki/Axiom
"...As used in modern logic, an axiom is simply a premise or starting point for reasoning. Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is a central question in the philosophy of mathematics, with modern mathematicians holding a multitude of different opinions. ..."

In other words, and not just from this source but others, just as I have recently come to strongly suspect, the is NO 'standard' or formal agreement in maths or philosophy to whether definitions have truth values.

To sum up: I take all that (not just from the above sources alone ) to mean I am completely free to decide for myself whether definitions have truth values and whether truth values can be said to have binary probability values. Since it both simplifies my definition of probability to have both those two things and I now recently found actual examples of usefulness in certain situations of having both those two things, and since having both better conforms to what I and my brothers and many other people I have spoken to mean by probability, and it leads to no paradox or contradiction in having both, this is a total none brainier to me: I will now accept both.

So, at last this issue is settled once and for all for me:
In my book, I will say:

probability can take on any value in the [0,1] interval, depending on what sort of probability it is;

truth values have probabilities (only 0 or 1);

definitions (and analytic propositions ) have truth values.

There will sometimes be ambiguity in regard to what is the 'agreed' definition and therefore the 'true' one. But, since my work isn't about that, that is irrelevant to my work thus I don't need to discuss and will not discuss that issue in my book.

Settled.

Originally posted by twhitehead
But ... but ....
Think about it for a moment. There are infinitely more irrationals than rationals. So did I mean that picking a number at random would be twice as likely to result in an irrational, or did I mean that any given irrational is twice as likely as any given rational? In the latter case, there would be virtually no difference from the uniform ...[text shortened]... ly an infinitesimal chance of being picked anyway. The former case is just a recipe for madness.

This won't work. The rationals are a zero measure set. This means they do not contribute to any integral over the reals. The density function gives you the probability of the result being in some range, rather than at a particular point. Suppose the probability density is uniform in the range. We can go through setting the probability density function to zero on the rationals without altering the probability of the random variable lieing in any given interval. So increasing the probability of the rationals won't alter the probability of the result being in a given interval either. See the Wikipedia on the Dirichlet function (they call the indicator function).

https://en.wikipedia.org/wiki/Indicator_function

Originally posted by DeepThought
This won't work.

What do you mean by 'won't work'?

So increasing the probability of the rationals won't alter the probability of the result being in a given interval either.
Yes, I said that in the post you replied to.

What you haven't addressed though is my other option ie suppose I tell you that on picking a number in a given interval you will be twice as likely to pick an irrational than a rational. How would you express that in probability densities?
My guess is one would need to simply separate the two sets and have two separate functions.

Originally posted by humy
To sum up: I take all that (not just from the above sources alone ) to mean I am completely free to decide for myself whether definitions have truth values and whether truth values can be said to have binary probability values.

I disagree. I don't think you have understood the difference between an axiom and a definition. They are not the same thing.
An axiom is assumed to be true, therefore it seems reasonable but trivial to say it has a truth value. An axiom does not introduce new terminology.
A definition on the other hand does not have a truth value as no assumptions are necessary as a definition isn't a statement about reality but a description of the meaning of a word.
If I say a squark is a red cow. That is a definition. It holds no truth value. Red cows may or may not exist. It makes no assertions about that.
If I say 'there are no squarks'. Then that may be an axiom, and whether or not it is a valid axiom depends on whether or not red cows actually exist.
The standard definition of probability is a definition not an axiom. I am less sure about your definition. You may have both axioms and definitions in it.

Originally posted by twhitehead
I disagree. I don't think you have understood the difference between an axiom and a definition. They are not the same thing.
An axiom is assumed to be true, therefore it seems reasonable but trivial to say it has a truth value. An axiom does not introduce new terminology.
A definition on the other hand does not have a truth value as no assumptions are n ...[text shortened]... an axiom. I am less sure about your definition. You may have both axioms and definitions in it.

https://en.wikipedia.org/wiki/Theorem
" A theorem is a logical consequence of the axioms. "

I have deduced theorems from my definition of probability (which includes the tie axiom but which I didn't explicitly show here in my definition due to risk of pre-publication plagiarism).

Therefore I see no reason I am aware of to not call my definition a group of axioms (I previously said 'axiom' as in singular which I now presume to be incorrect )

So I take it a 'definition' can also (but not necessarily) be and consist of a group of 'axioms'?
ANYONE; can anyone with expertise in this confirm/refute that?

However:
Even if there is such a reason that I am unaware of to say there are no 'axioms' in my definition, or if, perhaps, my said 'definition' is not a 'definition' but just a 'group of axioms' (which if either would you say? ), as long as I am careful to make sure that my whole system of logic is entirely logically self-consistent i.e. free of any internal contradiction, it makes no difference to my logically valid freedom to decide for myself whether definitions have truth values and whether truth values can be said to have binary probability values esp I thing as I have spoken to several people, including a top professor in philosophy, that say yes to both those things.

I have just thought of another use for giving a definition a truth value (of 'true' ) ;
from my university studies in artificial intelligence, in an 'expert system';

https://en.wikipedia.org/wiki/Expert_system

when the user of it presents an 'query' to its 'inference machine', that inference machine must make no distinction between a 'fact', which may or may not be a definition of something, in its 'knowledge base' and a fact that can be deduced from 'facts' (and, in addition, 'rules' ) in its 'knowledge base' when deciding whether to return the value 'true' or 'false' (or 'yes' or 'no' depending on the setup) to the user. (see https://en.wikipedia.org/wiki/Prolog )
Thus it must tell you something is true if it is true by definition.
It would be extremely unreasonable for it to return the message "syntax error, definitions don't have truth values!" as this very unhelpfully wouldn't give the user a straight answer to his question.

Originally posted by humy
I have deduced theorems from my definition of probability

Then it isn't purely a definition. It may be a combination of definitions and axioms.

Therefore I see no reason I am aware of to not call my definition a group of axioms
Because there is a difference between a definition and an axiom. They are not the same thing. It would be wise to try and separate them and label them appropriately.

So I take it a 'definition' can also (but not necessarily) be and consist of a group of 'axioms'?
I think axioms may be implicit in a definition, but generally, no a definition is not a group of axioms. A good definition tells you nothing new but only establishes language. An axiom on the other hand makes an assertion.
It should be impossible to draw a conclusion from definitions alone.

it makes no difference to my logically valid freedom to decide for myself whether definitions have truth values and whether truth values can be said to have binary probability values esp I thing as I have spoken to several people, including a top professor in philosophy, that say yes to both those things.
If you want to cause immense confusion, then go right ahead.
I find it interesting that earlier you mentioned that you took the trouble to rename probability density due to the potential for confusion, but seem to have no problem creating confusion for others by creating new meanings for words that already have well established meanings.

Originally posted by twhitehead
... seem to have no problem creating confusion for others by creating new meanings for words that already have well established meanings.

I haven't created any new meanings whatsoever but now I choose (for a short while there, you had persuaded me to make a pretty bad choice which would have been a disaster for my book ) to stick rigidly to the well established meanings the majority (but far from all) of people I have spoken to, including a top professor in philosophy I may add, agree with, which is the same meaning as mine but not yours. I may cause considerable confusion (to the majority) if I changed that meaning. If you have some evidence, or at least some sort of vague indicator, that the majority of people agree with your definitions here as apposed to mine, I would be very grateful if you would show it to me here.

Originally posted by humy
I haven't created any new meanings whatsoever but now I choose (for a short while there, you had persuaded me to make a pretty bad choice which would have been a disaster for my book ) to stick rigidly to the well established meanings the majority (but far from all) of people I have spoken to, including a top professor in philosophy I may add, agree with, which ...[text shortened]... r definitions here as apposed to mine, I would be very grateful if you would show it to me here.

I need to go back through the thread at some point as I have forgotten exactly what your definition is or where you defined it.

There can be no doubt however that a proper definition does not have a truth value.
An axiom is assumed to be true, so its truth value is not important in the sense that it could be true or false. It is always true.

at this philosophy forum;
http://forum.philosophynow.org/viewforum.php?f=21

I made the thread:

Can 'definition' be 'axioms'?

where I asked:

"...In formal logic, can a 'definition' be ( but not necessarily I presume?) or consist of one or more 'axioms'?

And can an 'axiom' itself have a 'truth value'? ( albeit in a trivial way I presume)
..."

and I got a reply of:

"...
A definition is an explanation of the meaning of a word. It can be an incorrect definition, the wrong explanation, if the word is taken as belonging to a standard in the language. If the word is taken as a temporary usage for the duration of a document or discussion ( "what this word means in this document" ) and is given by the author of the document, it cannot be incorrect, although perhaps a poor, possible ambiguous definition.

Every axiom, of any kind, is really saying:
IF this is true...

Axioms usually leave of the "IF" because they presume that there would be no argument. All axioms are presumed to be true before the rest of the argument has any meaning. If the axioms cannot be accepted as true, the argument is void, not necessarily false.

So
Yes a definition can be an axiom, and
Yes axioms have truth value.

And declarative definitions are always true ( eg. "This is what I mean when I say this word" ).
..."

At least just personally, I find his reply concise and informative.

Originally posted by humy
So
Yes a definition can be an axiom, and
Yes axioms have truth value.

That is an odd conclusion. The answers match precisely what I have been saying which is basically the opposite of that conclusion.

Originally posted by twhitehead
That is an odd conclusion. The answers match precisely what I have been saying which is basically the opposite of that conclusion.

I don't know how you are reading that.