August 25, 2011
The logic of science-15: Truth by logical contradiction
(For previous posts in this series, see here.)
Theologians often try to claim that they can arrive at eternal truths about god by using pure logic. In some sense, they are forced to make this claim because they have no evidence on their side but it is worthwhile to examine if it is possible to arrive at any truth purely logically. If so, we can see if that method can be co-opted to science, thus bypassing the need for evidence.
In mathematics, there is one way to prove that something is true using just logic alone and this is the method known as reductio ad absurdum or reduction to absurdity. The way it works is like this. Suppose you think that some proposition is true and want to prove it. You start by assuming that the negation of that proposition is true, and then show that this leads to a logical contradiction or a result that is manifestly false. This would convincingly prove that the starting assumption (the negation of the proposition under consideration) was false and hence that the original proposition was true.
The most famous example of this kind of proof is the simple, short, and elegant proof of the proposition that √2 (the square root of 2) is NOT a rational number. I believe that everyone should know this beautiful proof and so I will give it here.
This proof starts by assuming that the negation of that proposition is true, i.e., that the square root of two IS a rational number. You can then show that this assumption leads to a logical contradiction, as follows.
A rational number is one that can be written as the ratio of two integers. For example, the number 1.5 is rational because it can be written as 6/4, 12/8, 3/2, and so on. Similarly 146.98 is a rational number because it can be written as 14698/100. Conversely, the famous number π=3.1415927… is not a rational number. It cannot be written as the ratio of two integers since the number does not terminate AND there is no repeating pattern of digits.
(As a slight digression, to see why an infinite but repeating pattern is a rational number, take the number 4.3151515… where the sequence 15 is repeated indefinitely. Call this number y. If we multiply y by 10, we get 10y=43.151515… If we multiply y by 1000, we get 1000y=4315.151515… Subtracting 10y from 1000y, we get 990y=4272 exactly, since the repeating numbers after the decimal points are equal in both cases. Hence y=4.3151515… =4272/990 exactly and is thus a rational number. Similar reasoning can be applied with any repeating sequence.)
So IF √2 is a rational number, then it can be written as the ratio a/b, where a and b are integers. We then make sure that the ratio has been 'simplified' as much as possible by getting rid of all common factors. For example in the case of 146.98 discussed above, the ratio 14698/100 can be simplified to 7349/50 by cancelling the only common factor that the numerator and denominator share, which is the number 2. In the case of 1.5, the ratio we would use is 3/2, since the others have common factors.
So our starting assumption becomes that √2=a/b where a and b are integers that do not have any common factors. We can now multiply each side by itself to get 2=a2/b2. Hence a2=2b2. This implies that a2 is an even number (because it has a factor of 2). But if the square of a number is even, that means the number itself must be even. Hence a=2c, where c is also an integer. This leads to (2c)2=2b2 and thus b2=2c2. This implies that b2 is an even number and hence b is also an even number. Thus b also has a factor 2 and we have arrived at the conclusion that a and b both have the common factor 2. But if a and b have a common factor, this contradicts what we did at the start of the proof where we got rid of all their common factors. We have thus arrived at a logical contradiction. Hence our starting assumption that the square root of 2 is rational must be wrong. Since there are only two possible alternatives (the square root of 2 is either rational or not rational), we can conclude that it is not rational.
Note that we have proven a result to be true without appealing to any experimental data or the 'real' world. As far as I am aware, the only way to prove that a proposition is true using pure logic alone is of this nature, to show that the negation of the proposition leads to a logical contradiction of this sort.
Philosophers and theologians down the ages have tried to apply the reductio ad absurdum argument to prove the existence of god using logic alone. But the problem is that assuming that there is no god does not lead to a logical contradiction. So instead they appealed to what they felt was manifestly true, that the assumption that god did not exist meant that the existence and properties of the universe were wholly inexplicable. Almost all arguments for the existence of god are at some level appeals to this kind of incredulity.
But this is not a logical contradiction, since they are after all appealing to the empirical properties of the universe. In days gone by when much of how the world works must have seemed deeply mysterious, this subtle equating of empirical incredulity with logical contradiction may have passed without much notice. Even if what was shown was not strictly a logical contradiction, if the negation of a proposition 'god exists' seemed to lead to an obvious disagreement with data in that the properties of the world could not be explained, the negation of the proposition could be rejected, thus proving the original proposition to be true and that god exists.
But those arguments no longer hold since science has explained much of how the would works. Assuming that god does not exist no longer leads to either a logical or empirical contradiction.
Next: Some concluding thoughts