A coin is tossed in the air five times and each time it lands "heads"
- what will happen when it is tossed the sixth time? The answer is logical
isn't it? But which kind of logic? Here are some possible "logical"
approaches, but they all give different answers!
Inductive Logic
This is based on recognising patterns and then assuming that the pattern is going to continue. In this case the pattern is obvious, the coin when tossed comes down "heads" so therefore it will probably come down "heads,' the next time.
Inductive logic is the type of reasoning that, in spite of what Popper said, most of our scientific progress is based upon. Not only this, we tend to rely on it throughout our daily lives - when our foot hits the brake pedal we expect the car to stop, not because we understand about the levers, hydraulics and friction involved, but because it has always caused the car to stop in the past (except of course for the time when there was a truck coming at the intersection and the cylinder was empty of fluid!! see "Murphy's Law" below). Still inductive logic seems to work most of the time so it will obviously be "heads".
Deductive Logic
Now deductive logic is the one that gives us certain answers based on forms of argument like the syllogism:
When an evenly weighted coin is randomly spun in the air the probability of it landing "heads" and "tails" are equal. This coin is evenly weighted and is being randomly spun in the air. Therefore the chances of its landing "heads" and "tails" are equal.
The result of tossing a coin does not depend in any way on the previous results - an unlikely occurrence is just made up of a chance encounter of such events (in the same way that I am a random encounter of my various cells?), so that settles that - or does it?
A correctly reasoned syllogism only guarantees a valid answer - whether it is true or not is a very different matter. It depends on the truth of the premises (if we started with" All things with hair are monkeys," what does that make you and me'). So how are these premises arrived at? Usually they are assumptions (i.e. guesses!) or they result from inductive logic, which, as we know, is never certain, so…say, what proof do we have that it is evenly weighted, maybe the guy tossing it is a conjurer of some sort? (Now would we be more or less skeptical if a machine were tossing it?). How certain are we really that past events don't affect probabilities (it would fit in with holistic philosophies.)? etc. etc.
This is superstitious, non-scientific claptrap! The chances are obviously 50:50!
Folk Logic
Or plain old "common sense". The gut reaction of the average, uneducated (by which we presumably mean not brainwashed into accepting our current obsession with scientific methodology?) Joe in the street. Well you just go and ask a sample of such specimens what the answer is and they will no doubt tell you that by "the law of averages" it's about time it came down "tails", so there is a greater probability of this.
Gut reaction is the same as intuition, what we have acquired as a result of generations of evolution and natural selection. How many times have our instincts proved to be right in the past (Oops - inductivism again!) and what would be the use of having such instincts if we can't rely on them? The more mathematically minded might also point out that the chances of five successive "heads" are 1 in 32 and this would increase to 1 in 64 if it landed "heads" the sixth time, therefore 31 times out of 32 it will come down "tails" on the sixth toss! There seems little doubt it will land "tails".
So what is the right answer? Logic can't help at all unless we can decide which logic to use and it is fairly certain that each and every one of us has applied all of these types of logic to some situation in the past. Maybe the only sensible way to decide between them is to “toss a coin”?
P.S. If you want to use this as a teaching situation, then why not actually toss a coin until it comes down "heads" five consecutive times it shouldn't take too long as we know that on average it will occur once in every 32 starts. There again we've all been in teaching long enough to take into account "Murphy's Law" ("Probability" = 1/"Desirability" the truth of which we have noted so often, the ultimate in folk logic!)... that we may still be tossing the coin at the end of the lesson!
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