I titled this post Paradox 1 with the optimistic expectation there will be a 2. We shall see. There is a very nice Blog titled THINKING ON THINKING at http://pmulder.blogspot.com/ It has a posting on the Russel Paradox. (By the by, we are talking Bertrand Russell here, not Jack Russell.)
Tuesday, February 20, 2007 Russell's Paradox "Nothing contains everything", applied: A man of Seville is shaved by the Barber of Seville if and only if the man does not shave himself. Does the barber shave himself ? * If the barber does not shave himself, he must abide by the rule and shave himself. * If he does shave himself, according to the rule he will not shave himself. And, another one:"One of themselves, even a prophet of their own, said, the Cretians (sic) are alway liars, evil beasts, slow bellies. This testimony is true."Titus 1:12-14 (King James Version)
Much has been written about the paradoxes Barbieri di Seviglia and Cretan Liar.
We may describe the Cretan Liar as follows: Epimenides says "All Cretans are liars." Epimenides is a Cretan. If Epimenides' statement is true, then this implies that Epimenides always tells falsehoods. Hence, Epimenides' statement is false. Or, the truth of Epimenides statement implies the falsity of Epimenides' statement. Both paradoxes have been seen as problems in statements that are self-referential.
I do not believe that the Cretan Liar is a "nothing contains everything" type of paradox. Observe that the Cretan Liar scenario is a paradox when there is exactly 1 Cretan liar. It is also a paradox when the scenario allows for exactly 2 Cretans who lie. And so on, up until all Cretans surveyed are liars. The scenario where all Cretans are liars still leads to the paradox: the truth of the statements of 1,000 Cretans that all Cretans are liars implies the falsity of the 1,000 statements. I conclude that such a paradox as the Cretan Liar indicates that the original statement was not the type which may be assigned a truth value.
I do not believe that all propositions may be assigned such truth values. When a proposition, or statement such as "The Cure concert rocked!" is treated as if it may be assigned a truth value, it always leads to paradoxes. Hence, it was not the type of things to assign a truth value to. Epimenides statement is an appraisal, not a proposition of the form (x) {if x is a Cretan, then x is a liar}. It has only the appearance of being so. This is not the first instance of mimicry, mummery, or disguise in Nature.
Now, the Barber. Observe that the Barber paradox disappears if we allow that there are 2 barbers. Then barber 2 may shave barber 1, and barber 1 may shave barber 2. It disappears if there are 3 barbers. Barber 1 shaves barber 2, barber 2 shaves barber 3, barber 3 shaves barber 1. { Of course, we have to transform the original statement of the paradox to: A man of Seville is shaved by a Barber of Seville. ( not "the" barber, not just one and no more.) } If all men in Seville are barbers, they all shave each other in some sort of ghastly daisy-chain rig-a-marole.
In fact, they form sort of a circular group of order N, where N equals the number of men in Seville.
From this I conclude that the Barber Paradox is a Singularity Paradox. It is not dependent upon Set Theory and whether there is a Set that contains all items of a population. It is dependent on the fact that Singularities are inherently paradoxical.
For example, look at the work done by Linde on Inflation Theory cosmology. This was inspired, in part, by the cosmological problems created by the singularity of the original Big Bang theory. In essence, where did the singularity come from?
Similarly, we see the problem in arguments for the existence of God from a chain of causes, wherein God becomes the First Cause. As observed by Schopenhauer, the Principle of Causality is not a hansom cab that we take only so far, and then send away when we judged that we have arrived at our destination. Thus, the Principle of Causality cannot be used to establish a Terminus at the beginning of a chain of caused events. I have often wondered "I wonder what it was like for the first men or women who uttered speech? Was there anyone else around to understand them?" Scenarios like this, dealing with the first occurence, are very, very foggy. Was there a "first" person to speak? We often talk as if there were. So, prove it. Singularities are funny creatures. Ask Professor Hawking if you do not believe me. Two paradoxes down.
Tuesday, February 20, 2007 Russell's Paradox "Nothing contains everything", applied: A man of Seville is shaved by the Barber of Seville if and only if the man does not shave himself. Does the barber shave himself ? * If the barber does not shave himself, he must abide by the rule and shave himself. * If he does shave himself, according to the rule he will not shave himself. And, another one:"One of themselves, even a prophet of their own, said, the Cretians (sic) are alway liars, evil beasts, slow bellies. This testimony is true."Titus 1:12-14 (King James Version)
Much has been written about the paradoxes Barbieri di Seviglia and Cretan Liar.
We may describe the Cretan Liar as follows: Epimenides says "All Cretans are liars." Epimenides is a Cretan. If Epimenides' statement is true, then this implies that Epimenides always tells falsehoods. Hence, Epimenides' statement is false. Or, the truth of Epimenides statement implies the falsity of Epimenides' statement. Both paradoxes have been seen as problems in statements that are self-referential.
I do not believe that the Cretan Liar is a "nothing contains everything" type of paradox. Observe that the Cretan Liar scenario is a paradox when there is exactly 1 Cretan liar. It is also a paradox when the scenario allows for exactly 2 Cretans who lie. And so on, up until all Cretans surveyed are liars. The scenario where all Cretans are liars still leads to the paradox: the truth of the statements of 1,000 Cretans that all Cretans are liars implies the falsity of the 1,000 statements. I conclude that such a paradox as the Cretan Liar indicates that the original statement was not the type which may be assigned a truth value.
I do not believe that all propositions may be assigned such truth values. When a proposition, or statement such as "The Cure concert rocked!" is treated as if it may be assigned a truth value, it always leads to paradoxes. Hence, it was not the type of things to assign a truth value to. Epimenides statement is an appraisal, not a proposition of the form (x) {if x is a Cretan, then x is a liar}. It has only the appearance of being so. This is not the first instance of mimicry, mummery, or disguise in Nature.
Now, the Barber. Observe that the Barber paradox disappears if we allow that there are 2 barbers. Then barber 2 may shave barber 1, and barber 1 may shave barber 2. It disappears if there are 3 barbers. Barber 1 shaves barber 2, barber 2 shaves barber 3, barber 3 shaves barber 1. { Of course, we have to transform the original statement of the paradox to: A man of Seville is shaved by a Barber of Seville. ( not "the" barber, not just one and no more.) } If all men in Seville are barbers, they all shave each other in some sort of ghastly daisy-chain rig-a-marole.
In fact, they form sort of a circular group of order N, where N equals the number of men in Seville.
From this I conclude that the Barber Paradox is a Singularity Paradox. It is not dependent upon Set Theory and whether there is a Set that contains all items of a population. It is dependent on the fact that Singularities are inherently paradoxical.
For example, look at the work done by Linde on Inflation Theory cosmology. This was inspired, in part, by the cosmological problems created by the singularity of the original Big Bang theory. In essence, where did the singularity come from?
Similarly, we see the problem in arguments for the existence of God from a chain of causes, wherein God becomes the First Cause. As observed by Schopenhauer, the Principle of Causality is not a hansom cab that we take only so far, and then send away when we judged that we have arrived at our destination. Thus, the Principle of Causality cannot be used to establish a Terminus at the beginning of a chain of caused events. I have often wondered "I wonder what it was like for the first men or women who uttered speech? Was there anyone else around to understand them?" Scenarios like this, dealing with the first occurence, are very, very foggy. Was there a "first" person to speak? We often talk as if there were. So, prove it. Singularities are funny creatures. Ask Professor Hawking if you do not believe me. Two paradoxes down.
Posts
No comments:
Post a Comment