Logic: Bayes and Popper

Date: 06/24/2003 at 03:26:28
From: Ham
Subject: Logic: is p->q totally equivalent to ~q->~p

Is p->q totally equivalent to ~q->~p in practice?

In science, we build up theories where, based on the logic p->q, 
something implies something. For example, 

   p: X is a man
   q: X has red blood.
   if X is a man then he has red blood. 

And we confirm this by finding examples of man. 

However, can we find examples of ~q-->~p to support my theory? If  
p->q totally equivalent to ~q->~p, then using the previous example, 
the theory will be supported by almost everything, like a chair, a 
computer, and pen... etc. In this way I can also construct theories 
like God has 5 hands, Alien is red in colour, etc., because there is 
an infinite number of ~q-->~p examples to support my theory, (Not 5 
hand --> not a God, Not red in color --> not an alien). 
So, is p->q really equivalent to ~q->~p in practice?

Date: 06/24/2003 at 22:37:37
From: Doctor Achilles
Subject: Re: Logic: is p->q totally equivalent to ~q->~p

Hi Ham,

Thanks for writing to Dr. Math.

That's a very insightful question.  It is actually a fairly well-
known issue among philosophers of science. It is traditionally 
referred to as the "Raven's paradox" because it was originally 
formulated by a man named Hempel as follows:

In science, we can make a theory, e.g. "All ravens are black."  This 
theory is confirmed by a black raven, but the logically equivalent 
theory "all non-black things are not ravens" is confirmed by any non-
black non-raven.

Another name for this type of problem is "induction with negative 
inference."

First of all, you are right: because the two statements are logically 
equivalent, any evidence that confirms one also confirms the other.

In my opinion, the most reasonable response to the Raven's paradox 
was formulated by a man named Bayes. The basics of it are as follows:

The general idea behind the scientific method is to make predictions 
and then test them. Predictions come from theories. So, if I have a 
theory that says "all ravens are black," I can make two predictions:

   1) The next raven I see will be black.

   2) The next non-black thing I see will be a non-raven.

Which of these two should I test? To get at that, let me take a 
simpler example.

Let's say I'm in a room with two other people, and I have a theory 
"All people in this room are older than 20." There are two predictions 
that this theory makes:

   1) The next person I ask in this room will be over 20.

   2) The next person I find who is under 20 will not be in this 
      room.

So I can do one of two things: I can either ask one of the two people 
in the room with me what his/her age is, or I can go find a person 
who is under 20 and then see if s/he is in the room I was in.

Let's say I decide to do the latter (go find a person under 20 and 
then see if s/he is in the room I was in). The way I do that is to go 
ask everyone I see his/her age until I find someone under 20, then 
check whether that person is in the room in question. Let's say that 
the first under-20-year-old I find turns out to be outside the room in 
question: how confident am I of my theory based on that one piece of 
data? Not very confident, right? There are probably well over 1 
billion people who are under 20, and I've discovered that one of them 
is not in the room, but I haven't determined the whereabouts of the 
remaining billion or so.

Now let's say that I decide instead to do the first test (ask someone 
in this room his/her age). I turn to the person next to me and ask her 
how old she is; "25," she replies: how confident am I now of my theory 
based on that one piece of data? Pretty confident, right? I still 
don't know the age of everyone in the room, but (given that I know I'm 
over 20), I now know that 2/3 people in the room are over 20, and 
there's only one person left whom I don't know.

In other words, using the method of finding a person under 20 and then 
checking to see if that person is in the room in question confirms my 
theory, but I am only able to survey about one billionth of the under-
20 population at a time; whereas using the method of asking a person 
in the room what his/her age is allows me to survey about one third of 
the population at a time. So each test of the "find an under-20-year-
old and see where s/he is" type gives me virtually zero confirmation, 
whereas each test of the "ask someone in the room his/her age" type 
gives me a lot of confirmation.

What does this have to do with ravens? Well, there are maybe 300,000 
ravens in the world (I just made that number up; it may actually be 10 
times greater or 10 times smaller, I don't know for sure). How many 
non-black things are there in the world? Hundreds of trillions at 
least, which means that there are at least one billion non-black 
things for every raven in the world.

Remember, I can test the theory that "all ravens are black" by either 
finding a raven and then checking its color or finding a non-black 
thing and then checking if it's a raven. So when I find a non-black 
thing and discover that it's not a raven, I confirm my theory, but 
just a very little bit. Whereas, when I find a raven and discover it 
is black, I confirm my theory much more (not completely, but certainly 
much more than using the other method).

Think of it this way: Let's assume that hiding out in the world 
somewhere is a non-black raven. Am I more likely to find it first by 
systematically searching through all the ravens in the world or by 
systematically searching through all the non-black things in the 
world?

This is just a short introduction to the philosophy of science that 
has gone into investigating this problem. An Internet search on the 
phrase "ravens paradox" turns up over a hundred sites, if you also 
search for "bayes" that number goes down around 50, so you can read 
up on it online. I'd also be happy to discuss this further with you 
if you want.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/26/2003 at 10:56:00
From: Ham
Subject: Logic: is p->q totally equivalent to ~q->~p

Thank you very much!

Another thing:

Using p-->q, a black raven gives stronger support to our theory: 
all ravens are black.

Using ~q-->~p, a white pen also gives support, but much much weaker 
support, to our theory. 

However, a white pen also gives support to "All ravens are green/red/ 
yellow/purple, etc." Isn't it contradictory? This gives support to 
completely opposite theories.

Using p-->q does not have this problem at all. So, besides a matter of 
degree in supporting the theory, their function seems to be different 
too. Are they equivalent?

Thank you again, Dr. Math, for your quick response and very detailed 
explanation. I am inspired by your response!

Date: 06/26/2003 at 16:53:45
From: Doctor Achilles
Subject: Re: Logic: is p->q totally equivalent to ~q->~p

Hi again Ham,

Thanks for writing back to Dr. Math.

You are correct that a white pen gives a VERY tiny bit of support to 
the theory "All ravens are red." However, this theory makes an 
absolute claim. It is equivalent to the statement "There is nothing 
that is both a raven and not-red." The theory is false if there is 
even one raven out there which is not-red. So while the white pen 
gives a tiny bit of confirmation to our theory, the first black raven 
we find disproves it.

This line of reasoning follows the theory of a man named Karl Popper.  
Popper argued that scientific theories can never be proven true, but 
that a counter-example can prove them false. Whether or not Popper's 
argument works for all scientific theories, it certainly works for 
theories that make absolute claims, e.g. "All X are Y" or "No A is B."

Best,

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/