SANITY AND INSANITY Sanity means: More clear: Original: IS means IS. IS is IS IS does not mean IS NOT. IS is not IS NOT IS NOT means IS NOT. IS NOT is IS NOT IS NOT does not mean IS. IS NOT is not IS Insanity means: More clear: Original: IS means IS NOT. IS is IS NOT IS does not mean IS. IS is not IS IS NOT means IS. IS NOT is IS IS NOT does not mean IS NOT. IS NOT is not IS NOT Some people might say, perhaps there IS a universe where IS means IS NOT, where IS and IS NOT are equal in meaning. Well if there IS such a universe, then also there IS NOT such a universe, so what have they said? Nothing. THE FOLLOWING ARE 3 FAUX PROOFS THAT LOGIC IS ILLOGICAL. THE UNEXPECTED TEST PARADOX The unexpected test paradox is touted by confusion apologists as proof that logic is invalid or incomplete. Its no surprise that the paradox is not a paradox at all except to the confused mind. The paradox goes as follows. A teacher of Illogic 101 tells his students on friday that they will be given an unexpected test some time during the next week but won't be told before what day it will happen. He tells them they DEFINITELY will get a test, and it will DEFINITELY be within next week, but they DEFINITELY will not know what day the test will happen before it happens. His intent of course is to get them to study every night as if the test will be the next day as they will 'never know what day it will be on'. A logic student brighter than most argues that the teacher's proposition is impossible. He argues as follows. Say the teacher waits until the end of Thursday, and still hasn't given the test. Then the students KNOW it will be given the next day on friday so the test won't be unexpected when they come to class in the morning. They will be very well prepared by studying the night before. So the teacher can't give it Friday. So on Wednesday night if the test has not yet been given, the students know that the test can't be given Friday, so must be given Thursday, and hence again will not be unexpected. So the teacher can't give it Thursday either. So on Tuesday night if the test has not yet been given, since it can't be on Friday or Thursday, it must be given on Wednesday and again will not be unexpected. Repeating the above it is easy to prove the same for Tuesday and Monday, so the student concludes that an unexpected test can not be given. Although perhaps counter intuitive, the argument is in fact valid, it is not possible to give an unexpected test in a finite amount of time IF the test is guaranteed to be given. The problem arises from a misuse of the word 'unexpected'. By unexpected the teacher means expected with a probability less than 100 percent. Truly unexpected means utterly without clue, no idea if they will have a test or not, not even an ability to give a probability of whether a test will happen or not. No clue means they are not able to make an intelligent bet on whether the test will happen on any given day. For example say the teacher said instead, "Tomorrow I will toss a coin and if it comes up heads you will get a test and if it comes up tails you won't." Is that an unexpected test? No of course not, the 'expectation' of the test would be a 50 percent chance of having the test. That's not 'unexpected'. So say the teacher has a 6 headed dice, and every day he tosses the dice, and if it comes up on side 1 he gives the test, but if it comes up on on any other side he doesn't give a test. Is that an unexpected test? Again, no, on each and every day the students have a reasonable 'expectation' of 16.66 percent or so of having that test. Hardly unexpected. Notice however they may never get a test! So say the teacher instead says "I will place 5 pieces of paper in a bowl, each piece has a day of the week on it, Monday through Friday. On Sunday night I will mix up the papers and randomly pick one out. Whatever day of the week is on that piece of paper is the day I WILL give the test, which I will know on Sunday night, but you will not know until the day of the test. Is that an unexpected test? Well the students know that on Monday there is a 1 in 5 chance the test will fall on that day, so no, they have a very clear expectation of a 20 percent probability that the test will happen on that day. On tuesday however, if the test did not happen on Monday, they know they have a 1 in 4 chance of getting the test on Tuesday. Far from being unexpected, they actually have a GREATER expectation of getting the test on Tuesday at 25 percent probability, than they had on Monday at only 20 percent probability. On Wednesday, if the test did not happen on Tuesday they have a 33 percent chance of getting the test, and if not, then on Thursday a 50 percent chance of getting the test, and if not, then on Friday a 100 percent chance of getting the test. Is it clear then that it is in fact impossible to give a test in a finite amount of time, that the students WILL NOT KNOW FOR SURE before the day of the test that it will be given. But WILL NOT KNOW needs to be redefined as WILL IN FACT KNOW ALTHOUGH WITH ONLY A NON ZERO PROBABILITY. If the Teacher said "You will not know what day the test will be given UNLESS it is given on the last day" then he would have stated a consistent position. On every day before the last day, they won't be sure they will get the test, but they will have a reasonable finite probability that they will. On the last day, they have a certainty. If you guarantee that the test WILL be given, and guarantee that it will be given during a finite period of time, then the probability of the test being given on any day increases as the days go by, until on the last day the probability is 100 percent. Thus the test is 100 percent expected on the last day, but only 80 percent expected on the day before, and 60 percent on the day before that etc. 80 percent expected is not 'unexpected', you see? Even 5 percent expected is not 'unexpected'. In truth only 0 percent probability is truly unexpected. The statement that a test WILL be given in a finite amount of time implies that the students will expect the test sometime during that finite amount of time. Clearly if the time limit has run out, it must be given on the last day, if the teacher is to be true to his word. So what day the test is given will remain unknown with certainty but expected more and more until the last day at which point it will be known with certainty. That is the best the teacher can promise, you won't know when the test will occur unless he lets it go to the last day, in which case you will know very well to study Thursday night for it. So the student is right, it is impossible to give a totally unexpected test in a finite period of time. The only way a test can be totally unexpected is to threaten it in an unlimited amount of time, or indicate that maybe the test won't happen at all. For example if the teacher has an infinite sided coin, and only one side is marked 'yes you get a test', then every day he flips the coin there is a 1/infinity of a chance of getting the test, and since that is as close to zero as possible, then you can claim the test is surely unexpected or at least unpredicatable. As long as the test is guaranteed to happen in a finite amount of time, the test by definition becomes an EXPECTED test in that finite amount of time, and the closer you get to the end of that time, the greater the probability of the test occuring becomes until the last day arrives at which point the probability is 100 percent. As time goes on through the week the test becomes increasingly more expected, but it was never UNEXEPECTED from the word go. Unexpected means they didn't see it coming at all. It thus becomes impossible to give an evenly unexpected test during a finite amount of time if the test is guaranteed to be given. The teacher's use of the word 'unexpected' is a loose usage of the word, not literally correct and is in fact impossible in the scenario the teacher has laid out, as correctly indicated by the student who got an A+ for the course. Notice that the verbal shenanigans of the teacher in no way invalidates the usefulness or correctness of logic, but only indicates that sloppy use of words can be abused in subtle ways that lead to illogical conclusions. The problem is not with logic, nor with words, but with the sloppy mind. IS remains IS, and IS NOT remains IS NOT. RUSSELL's PARADOX. Let's define an object as anything that has qualities. Let's define a class of objects as an object grouping other objects or classes together according to qualities that are common to all of them. A member of a class is simply an object in the class. An object in a class may either be another simple object or class of objects. Not all objects are classes, but all classes are objects. Let's define a small class as a class with 5 or less members in it. A member can be a number or a thing or anything at all. Thus {0,1,2,3,4}, {1,2,3}, {2,3,4,5,6} etc are all small classes. A large class would be any class with more than 5 members in it, {0,1,2,3,5}, {0,1,2,3,4,5,6} etc. So the first question is, is the class of small classes, a small or large class? The answer is the class of small classes is a large class as there are an infinite number of small classes. The second question is, is the class of large classes a small or large class? It too is a large class for the same reason. Thus the class of small classes does not belong to itself. And the class of large classes does belong it itself. So let's simplify this with something more real. Classes are groups of objects with common qualities, such as the class of dogs. Classes themselves are also objects and thus can themselves be the members of another class. Classes divide into two groups, those that are members of themselves and those that are not members of themselves. So we consider the class of dogs which is itself a class but not a dog. Thus the class of dogs IS a member of the class of classes. However the class of dogs is not a dog, its a class. In other words the class of dogs is not a member of the class of dogs, and therefore the class of dogs is not a member of ITSELF. Get this, the class of dogs is a member of the class of classes, but is not a member of the class of dogs and so it not a member of itself. Thus the class of dogs IS a member of the class of classes that are not members of themselves. Russell spent an entire summer looking at a blank peice of paper with the following question in mind. Is the classs of classes that are not members of themselves, a member of itself? In an absence of being able to directly determine an answer to the question we revert to reductio ad absurdum. That means we assume an answer of true and see if it leads to a contradiction. Then we assume an answer of false and see if it leads to a contradiction. So if we assume yes, the class of classes that are not members of themselves IS a member of itself, then it does not belong in itself as only classes which are NOT members of themselves belong in it. That's a contradiction. If we assume no, the class of classes that are not members of themselves is NOT a member of itself, then clearly it does belong in itself as just another class that is not a member of itself. That is a contradiction. Either way we get a contradiction. If we say yes, we get no. If we say no, we get yes. If we assume it is a member of itself, we get it shouldn't be a member of itself. If we assume it isn't a member of itself, we get it should be a member of itself. Now, let's look at classes that ARE members of themselves. Take for example the class of existing objects. Well clearly the class of existing objects is itself an existing object, hence belongs in its own class of existing objects. Another example is the class of classes in general, clearly the classes of classes is a class and thus is a member of itself. Another example is the class of human inventions, clearly the class of human inventions is a human invention, so belongs to itself. So what about the class of all classes that are members of themselves? Is it a member of itself? Again we apply reductio ad absurdum. If we assume that the class of all classes that are members of themselves, IS a member of itself, them it should be a member of itself, which agrees with our assertion. If it is a member of itself, then it belongs in itself, which is fine. Thus there is no contradiction. If we assume that the class of all classes that are members of themselves IS NOT a member of itself, then it shouldn't be a member of itself, which also agrees with our assumption. Again there is no contradiction. If it is not a member of itself, then it doesn't belong it itself and again this is fine. Thus in this case we do not have a contradiction, and the statement can be considered true or false at will, or undecidable. The contradictoryness or undecidability of a mal formed self referencing statement does not in any way invalidate the validity of logic or logical arguements that are not made of mal formed statements. The above two examples of classes belonging to themselves or not, correspond to much simpler statements of the form: This statement is false. Contradiction, carries no meaning. This statement is true. Can be true or false at will. This statement is uncertain. Left as a homework problem. This statement is undecidable. Left to Godel and Russell to fight it out. THE OMNIPOTENCY OF GOD If God can do anything, can he create a stone so big he can't lift it? The answer is God can do anything that is not illogical, and does not violate the fundamental nature of God. Remember even in the Bible it says 'From the word came everything that came.' That implies there are pre existing things that were not created, and thus God, which is part of those things, can not touch or change them. Anyhow the answer to the question is yes. God can create a stone of any size, and then limit himself down to a point where he can not lift it. Any creature or human who is God in Carnation has done that to himself in transiting from Creator to Creature. A higher question would be can the unlimited limit itself? Again the answer is yes. If it couldn't, it wouldn't be unlimited now would it? Homer