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/
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