# Archive for March, 2013

## When You Have Eliminated the Impossible…

Posted by allzermalmer on March 20, 2013

Sherlock Holmes is written to have said “when you have eliminated the impossible, whatever remains, however improbable, must be the truth.”

The full quote is found in the book The Sign of Four: “You will not apply my precept,” he said, shaking his head. “How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth? We know that he did not come through the door, the window, or the chimney. We also know that he could not have been concealed in the room, as there is no concealment possible. When, then, did he come?”

The method that Holmes is using is an Eliminative Method, or a form of Eliminative Inference. This form of reasoning has at least two forms of inference. They are the Disjunctive Syllogism and Modus Tollens.

Disjunctive Syllogism: Because at least one disjunct must be true, by knowing one is false we can infer that the other is true.
Premise 1: Either A or B
Premise 2: ~A
Conclusion: Therefore, B

Premise 1: Either they Came through the Door, or Came through the Window, or Came through the Chimney, or Came through the hole in the roof.
Premise 2: Didn’t come through the door and Didn’t come through the window and didn’t come through the chimney.
Conclusion:Therefore, they Came through the hole in the roof.

This is an eliminative inference. We have some possible solutions to the mystery that Holmes is investigating. Holmes has also made some observations that are inconsistent with some of those possible solutions. These observations contradict the possible solutions, and this makes them impossible. Holmes rejects some of the conjectured solutions and has now limited possible solutions that are still consistent with his observations.

Suppose Holmes were to list alternative hypothesis. Suppose that Holmes ordered them in preference, and those that he preferred the most were considered to be the most probable. For example: Come through the Window (65%) > Come through the Door (20%) > come through the Chimney (10%) > Came through the hole in the roof (5%).

Here are four alternative hypothesis, and each one of them is graded on their probability of being true and his preference going from most probable to improbable. Holmes collects some observations. Some of the hypothesis are inconsistent with the observations. The more probable hypothesis are eliminated by the observations collected. To hold to the observations collected and hypothesis that are inconsistent with those observations, would be equivalent to holding to what is impossible. The hypothesis are contradicted by the observations. No matter how probable the impossible is, it still isn’t possibly true. And once we have eliminated the impossible, whatever remains, however improbable, must be the truth.

## Superbowl and Falsifiability

Posted by allzermalmer on March 20, 2013

There was a Superbowl a couple of months ago. Now keep in mind that with the rules of NFL and Superbowl, no Superbowl can end in a tie. One of the two teams must win the game. But let us take a look at falsifiability and unfalsifiability.

Suppose that the Superbowl is only between the Baltimore Ravens and the San Fransisco 49ers. You must predict who will win the game.

We can make the claim “The team with the most points will win”. Now this statement is true and cannot be false. This statement is necessarily true. “The team with the fewest points will lose”. This statement is also true and cannot be false, so it is necessarily true. We cannot show that this statement is false, since it is true, so it is unfalsifiabile.

We may now take a further step and predict that either the Ravens will win. This is falsifiable, since we can watch the game and see if the Ravens won or lost. Further, we make take the opposite position and predict that the 49ers will win. We know that either the Ravens win or the 49ers win. Only one of these options can be right.

But we may take a further step and predict who will win the game and what the score will be. We can predict that the Ravens will win and that they will win 14 to 7. We may also do the same with the 49ers. But now we are getting more specific with our predictions. It makes it easier and easier to show that our prediction is wrong. The score could be that the 49er won 14-7, which shows the prediction of Ravens winning 14-7.

Let us further assume that the total amount of points that can be scored in the game is 40 points, and there are a total of 2 teams. We may use the formula of computational probability, nCr = ( n! ) / (r! (n-r)! ). n= total number of subjects, r= number of objects in arrangement, != factorial. Through computation of (40!)/(2!(40-2)!)=780 different possible combination of the =score. This means we can make 780 different predictions. For example, Ravens win 14-7 or 49ers win 14-7, and etc. Maybe the Ravens win 21-14 or the 49ers win 21-14. There are 780 different specific predictions we can make which contain (1) the team that wins of the two, and (2) the score.

Now if we pay attention, we start from the most general and further move down from less general to more and more precise. We can start with a general prediction of “The team with the most points will win”. It is true but it is not falsifiable. We may further move down with a more precise prediction of “The Ravens will win”, which is falsifiable. We may further move down with a more precise prediction of “The Ravens will win 14-7”. Each stage down we go, the claim is more and more falsifiable. It eliminates other possible outcomes, and can be shown if one of these other possible outcome were to be obtained.

We have two general claims that we know are necessarily true and not informative about who will win the game between the Ravens and 49ers, which is 100% you will get the answer correct. We have two further general claims which are not necessarily true and are informative. This claim is general and can be shown to be right or wrong, but it 50% you will get the answer correct. When we further move down to what the score will be, then we have 0.00128% of getting the answer correct. We have 780 possible correct predictions, and only one of them can be correct, so we only have a 0.00128% of getting the correct prediction.

We may make a further specification that you must pick the winner, the points, and how they obtained those points. For example, the only ways to obtain points would be (1) 6 for a touchdown, (2) 3 for field goal, (3) 2 for safety, (4) 1 for extra point. This now means one may say, “The Ravens will win 28-14, with Ravens getting 3 touchdowns, 3 field goals, 1 extra point.” Things are becoming more and more specific, which makes it easier and easier to show it is false.

Suppose we have these claims: (1) All Ravens on Earth are Black, (2) All Ravens on the Northern Hemisphere are Black, (3) All Ravens in North America are Black, (4) All Ravens in the United States are black, (5) All Ravens in California are Black, (6) All Ravens in Los Angeles are black.

These claims are based on the most general of them all and is moving down to more and more precise general claims. We know that If all ravens on earth are Black and all Ravens in Los Angeles are not black, then we know that all ravens on earth are not black. But if all ravens on earth are not black, that does not mean that all ravens in Los Angeles are not black. All the Ravens in Los Angeles can still be black.