Laure

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  1. Thom, that's a lot of words, there. (So here are a lot of words back at ya!) I still don't see your point with the M statements. If I paraphrase them to be more exact, they all say "if H was true, I just added (or will add) a dollar". That statement is always true if we accept that the person will do what he says. So, the 2nd line in the truth table, "H was true but I didn't add a dollar", does not occur. So, I don't see a problem. Unless I'm still missing your point. I re-read what you wrote about the "paradox of entailment". I don't think that modern logicians would conclude "the U.S. economy negotiates one trillion decisions per microsecond" from a false premise (P & not P). Modern logicians would say that "(P & not P) --> Q" is always true, because the antecedent is always false, but that doesn't tell you whether Q is true or false. It could be either. When we say that the argument is "valid," all we are saying is that Q is derivABLE from "P and not P." Yeah, sure, in a world where P and not P are both true, it would follow that Q was true. Now I dare you to prove to me that "P and not P" (i.e. Check Your Premises!), and if you can do it, I'll grant you that Q is true! I think it just means that "if contradictions exist, then everything's true!" It has been a while since I read The Analytic Synthetic Dichotomy, and I guess I have a little trouble with understanding the word "dichotomy." So I went and looked it up again. :-) "A division into two especially mutually exclusive or contradictory groups or entities." They give the example of "theory and practice" (which of course I don't think of as a dichotomy). I've also heard of the "mind-body dichotomy", I guess meaning that the mind can survive independent of the body (which of course I also reject). When Peikoff discusses the Analytic-Synthetic Dichotomy, if I recall, his complaint was that philosophers who buy into this dichotomy believe that if something is provable, it has nothing to do with reality, and if it has to do with reality, that is, with something we actually see with our own eyes, that it can't be proven. I reject that, too. However, with symbolic logic (as with all of mathematics), what we are doing is abstracting away the particulars, ignoring the semantics, and dealing with pure symbol manipulation or syntax. I don't think there is anything wrong with doing that. If we adopt the language from that Wikipedia article, we can determine if an argument is "valid" just by looking at the syntax, the symbol manipulation, without worrying about the semantics. We also have to look at the semantics to determine if an argument is "sound." But, I think it makes the job easier if we can ignore semantics in the checking of "validity," and I don't see anything wrong with ignoring semantics at that stage. Just like with an algebra problem -- once I encode it in a formula, I don't have to think about the meaning until I get my final answer and then check it for reasonableness. Say I'm trying to calculate the average speed of a car, and I get 1000 mph -- then I need to go back and check my work! But while doing the calculation, I don't need to keep anything in mind about the meaning of the problem I'm solving. I start out with my "story problem," encode it into a string of symbols, pop down into the realm of pure syntax, manipulate the symbols, then pop back up to the realm of semantics to check the result, keeping in mind the meaning of what I was trying to calculate. So, while it's not a dichotomy, I think it is very useful to be able to examine the analytic or syntactic aspect of a problem, independent of the meaning.
  2. Thom, I still don't see what you're getting at with the "M" statements, but I think I understand your problem with statements such as HP4. Please see the thread I started, "Question on Conditionalizing", and visit the Wikipedia links mentioned there. I think you're a fan of "strict implication", which means that we look at the meaning of the P's and Q's and ask ourself if the P being true would in any way cause the Q to be true, and if not, we consider "P --> Q" to be false, even though the normal symbolic logic we learn in school says that it evaluates to true if P is false. I don't see any problem with conventional symbolic logic. We just need to define our terms so everyone is on the same page. In the link in the other thread on "paradoxes", the author mentions the idea that an argument with false premises can be logically "valid" although it is not logically "sound." I can go along with that, and suspect that you can, too. It's just a matter of defining our terms.
  3. A basic question, Roger. Do you agree that for all x, x+0=x? If not, how do you do the algebraic simplification step that gets us from: x + 0 = y + 1 to x = y + 1? If you agree that we can "drop the '+ 0'", how is that in any way different from "substituting 'x' in place of 'x + 0'"? Like my hero Spock says, "A difference that makes no difference is no difference." (He probably got it from some philosopher but I dunno... )
  4. Come on, Merlin, pay attention, fer chrissake! Laure (whom you were defending) was trying to show how ~MY PREMISES~ lead to a contradiction, and I was showing how they do not, that any attempt to use the distribute law gambit ended up with OTHER expressions that were undefined BY MY PREMISES. In other words, she AND YOU have failed to show a contradiction in my argument so far. "Fatally flawed" my pasty, white ass. You people are better than this, aren't you? REB I was trying to show the contradiction in your saying that 0*1=0 but 1*0 is "undefined". I think what I showed was that for your system to be consistent, 0*1 must also be "undefined". I think if we go through all the symbol manipulations, we will end up with everything in Roger's system being "undefined." Go through my postulates carefully and indicate all the modifications you would make, then we will see how things shake out.
  5. Roger, of course you don't use "0" to "do" things, any more than you use "5" to "do" things. "0" and "5" are symbols; they are nouns. The operators (+,-,x,/) are the verbs; they are what "do" things. I say again, show me your equality principles and postulates.
  6. I showed you my equality principles and postulates. Show me yours, Roger. Let's nail down the specifics of your system, and then we will see whether it is logically inconsistent or just consistent and useless!
  7. Well,... yeah! The whole idea of mathematics is to abstract away the units, or the referents in reality, so that we can just manipulate the symbols. That's why we can say 2+2=4 and it doesn't matter 2 of what. We don't have to stop and think, "OK, if I take 2 apples and add 2 apples, I have 4 apples. But, what if I had 2 oranges and add 2 oranges?? Gee, what could the answer be?"
  8. Roger, "for all x, y, z, (x+y)+z = (x+z)+y" is a postulate in my math. "For all x, y, x+y = y+x" is another postulate. So, (1-1) + (1-1) = (1+(1-1)) -1 = ((1-1)+1)-1 = 0+1-1 = 1-1 = 0. Let's keep it real simple. Roger says x+0 is incalculable. Roger, would you agree that if you see "x+0" in an equation, that you can simply substitute "x", since the "+0" does nothing? Well, when we say that "x+0=x", we are saying precisely that "we can substitute 'x' for 'x+0'". That is what it MEANS. "x+0" is synonymous with "x". Another example, this idea that 0*1 is 0 but 1*0 is undefined. Let's go back to Montessori Preschool for a moment. If we want to show what 2*3 is, we can lay out some pennies in 2 rows and 3 columns. If we want to show what 3*2 is, we can lay out the pennies in 3 rows and 2 columns. Now let's try 0*1. Roger says that's OK. 0 rows and 1 column. 0 pennies. How about 1*0? Oh, we could have 0 rows of pennies, but 0 columns is verboten! What if we go round to the side? So Roger, if 1 row of 0 columns is verboten, so is 0 rows of 1 column. So you can't have it both ways. If you object to 1*0, you can't "do" 0*1 either. The empty set means a set with no elements. Saying that there is no such thing as the empty set is as silly as objecting to the use of the word "nothing". If we have 5 chairs and add none, we still have 5 chairs. Why is it "forbidden" to say that we have added none? Just because Rand made a comment disparaging the "reification of the zero"? She was just complaining about philosophers who glommed onto the concept "zero" and tried to give it some mystical interpretation. You say "there isn't any kinetic energy", and we say "the kinetic energy is zero". It's the same thing! There is nothing to be gained by avoiding the word "zero" like a taboo! Here are the equality principles and postulates of my math: Show me yours. I think what you'll find (best-case scenario) is that if you create a new symbol for your special "I can't say 'zero' so let's say 'undefined'", (0*0=#) you will find that that symbol "#" is redundant, and is equivalent to "0". Worst-case, you end up with a system that contains a contradiction.
  9. Interesting. The "strict implication" article goes to the heart of Thom's "Iffy" thread. I think what Thom is getting at is that "strict implication" is the way we intuitively interpret "if" in natural language. Taking an example from the article, "If Bill Gates went to medical school, then Elvis is still alive". A true statement, but intuitively false, since there's nothing about Bill Gates going to medical school that would cause Elvis to still be alive. Clicking around in Wikipedia, I found this article: Paradoxes of Material Implication. (It has a few "broken" formulas, unfortunately, but I think it's an interesting article anyway.) It says, "The paradoxes of material implication are a group of formulas which are truths of classical logic, but which are intuitively problematic." which I'm not sure is a good definition of paradox. Just because something is intuitively problematic (like my example in the first post) doesn't mean it's a real paradox. Anyway, I thought this article was relevant to some of the other discussions going on here. It points out that from a contradictory set of premises, anything follows. It says that such arguments are "valid" but not "sound". "Valid" = there's nothing wrong with the logic itself. But not "sound" because it contains faulty premises. That's why the lady said "check your premises" after all. Your argument can be perfectly valid, but if you have a false premise, you can "prove" anything!
  10. Thanks for the quick reply! So, given that the moon is a satellite, "if the moon is not a satellite, then the moon is a satellite" is just a funny little tautology and not a paradox.
  11. I have a question for the logic experts on the forum. (I'm not an expert, but I had a symbolic logic course in college and was the top student, and I had the benefit of a high school math curriculum that emphasized proofs. As a programmer, I use boolean logic just about every day.) In looking back at my high school math book, it describes Modus Ponens, Modus Tollens, and all that good stuff. It describes "Conditionalization": a conditional sentence follows from its consequent. So, if Q is true, P-->Q is true for any old P. 1) Is my interpretation correct, or does P have to be something that was previously used in order to derive Q? 2) My textbook gives an example that has Q as "The moon is a satellite", and two possible P's: "The moon is made of green cheese" and "The moon is not made of green cheese". It says either of the green-cheese statements can be used as P, and P-->Q is true in either case. This kind of makes sense, since we KNOW that the moon is a satellite, no matter what, so if the moon is made of green cheese, it's a satellite, and if it's not made of green cheese, it's a satellite. My question is, what about "If the moon is not a satellite, the moon is a satellite"? True statement? Is this considered a paradox, or is it just a perfectly valid, if strange, application of conditionalizing? (This post is not trying to start an argument; I just want to learn something from the experts. I defer to Ba'al & Merlin (and maybe even Dragonfly!).)
  12. Question: what is the square of 0? It is 0*0. And what is 0*0 ? It is 0. In fact 0^n, for n > 0 is 0. Proof: 0^1 = 0 so the thing is true for n = 1. Suppose 0^n = 0. Then 0^(n+1) = (0^n)*(0^1) = 0*0 = 0 The induction completes the proof. For all n >= 1 0^n = 0 Ba'al Chatzaf 0 does not have a square. 0*0 is undefined. 0^n is NOT 0*0 n times. It is 1 * n factors of zero. There is no number that corresponds to n factors of zero. 0*0 n times is just as undefined as 0*0 is. So 0^n = 1. No induction necessary here. REB *sigh* Roger, now you're saying that 0*0 is undefined, but 0^n = 1. But 0^2 is just another way of writing 0*0. So, if 0*0 is undefined, then 0^2 must be undefined; it's the SAME THING. Are you saying that 0^n is undefined for n=2, but 1 for n not equal to 2?? Here's another grade-school example of why that just doesn't work. What's (1-1) * (1-1) ? If we simplify what's inside the parentheses first, we get 0 * 0, which in RogerLand is Undefined. But, if we use the good old "FOIL" method to distribute it out, we get 1 - 1 - 1 + 1. Compute it left to right, we get 0 - 1 + 1, or -1 + 1, or 0. So (1 - 1) * (1 - 1) both equals zero and is undefined. In your math, can something equal zero AND be undefined? Or don't you believe that (x + y)*(z + w) = x*z + x*w + y*z + y*w? Your system has a contradiction. Face up to it.
  13. Thom, I'm sorry, but I think there's something I don't understand. I don't know the point you are trying to make about these "M" statements. You say, "MI1. Add dollar now if HP1." I interpret that as "if HP1 is true, then add a dollar". Is that correct? That is an imperative statement, not a material implication. If you want to phrase it as a material implication, you could say, "if HP1 was true then a dollar was just added", and you assume that your person with the money is "obedient", then the "M" statements are always true, because it's always the case that either a dollar was added or the HPn was false. So, what is it that I'm not understanding? I really think we may be able to get to the point where one or the other of us says, "Oh, OK, I get it."
  14. Oh, that's interesting... so 0^1 = 0^2 = 1 or in other words, 0 = 1 and 0 * 0 = 1 ? That is Objectivist mathematics? Bingo! Roger, you have just stated that 0 = 1. You can certainly come up with your own mathematical system where 0 = 1, but I thought you liked math to be useful in practical applications!
  15. Roger, the simplest way to explain why 0^0 should be thought of as undefined is to note that for nonzero n, 0^n = 0, and n^0 = 1. You'll accept that, right? So, what happens at zero? Is 0^0 = 0 or is 0^0 = 1? It depends which formula you use. Since it can't be both zero and one, it must be undefined. Also, you've got a contradiction in your post 106. First you say, Then you say, Here, you are taking (1 * (0^n)) / (1 * (0^n)) and you say 0^n is "nothing", i.e. 0. So, you are here assuming that 1 * 0 = 1, when above, you said that x * 0 = 0. *edit* Also, defining 0/0 as 1, as you want to do, can allow you to "prove" contradictions, and that's not a good thing! Here's a link to a good example of this: *Algebra Quandry*. I think you're trying to do math with the right side of your brain. Question for you: explain to me what a number like, say, 8 to the 0.4325 power corresponds to in reality?