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Contrary to the assertion of the scandalous propaganda on the left, Santa Claus does indeed exist. And I can prove it. Start with proposition A: A . If A is true, then Santa Claus exists. Now, suppose A were true. Then it would follow that if A is true, then Santa Claus exists , and again since we're supposing A is true, it would follow that Santa Claus exists. So we've shown that if A is true, then Santa Claus exists . But that is proposition A, so we've proven that proposition A is true. So that means that Santa Claus exists! (A remarkable conclusion given my recent post on things that probably don't exist .) The only trouble is that the reasoning above lets you prove anything (e.g. that penguins rule the universe ). It's an example of Curry's paradox , which can't be easily explained away, and is the subject of ongoing research by logicians. Bah humbug! Still not convinced? Thomas Aquinas to the rescue! Well, actually, his modern admirers. Aquinas came u...

It turns out that the number 1 reason people visit this blog is to calculate log base 2 of an integer. So here is log 2 of 1 through 10, to 16 digits precision: log 2 (1) = 0 log 2 (2) = 1 log 2 (3) = 1.584962500721156 log 2 (4) = 2 log 2 (5) = 2.321928094887362 log 2 (6) = 2.584962500721156 log 2 (7) = 2.807354922057604 log 2 (8) = 3 log 2 (9) = 3.169925001442312 log 2 (10) = 3.321928094887362 Note that log 2 ( x ) is defined for any x greater than zero. If you have a calculator than computes the natural logarithm (often denoted ln), then you can calculate log 2 ( x ) = ln( x )/ln( 2 ). The same thing works with log base 10, i.e. log 2 ( x ) = log 10 ( x )/log 10 ( 2 ). But what does it mean ? log 2 ( x ) means the power you have to raise 2 in order to get x . For example, 2 2 = 4, so log 2 (4) is 2. Similarly, 2 3 = 8, so log 2 (8) = 3. It turns out that 2 1.58496 is very nearly 3, so log 2 (3) is roughly 1.58496. Some cases deserve special mention. log 2 (2) = 1 because 2 1 is...

A recent article by philosopher Steven Hales is titled " You Can Prove a Negative " (a slightly different version of the article is available as a pdf file ). Hales argues that the "principle of folk logic" saying you can't prove a negative is just plain wrong . He points out that " any claim can be expressed as a negative, thanks to the rule of double negation." So it's easy to come up with examples of proving a negative. Hales goes on to say that "Some people seem to think that you can’t prove a specific sort of negative claim, namely that a thing does not exist." He counters this with an example of a valid proof that something doesn't exist: 1. If unicorns had existed, then there is evidence in the fossil record. 2. There is no evidence of unicorns in the fossil record. 3. Therefore, unicorns never existed. Of course, the difficulty here is with the truth of the premises (1 and 2). In particular, it could be that we just haven...