Applied philosophy refresher

This is a discussion on Applied philosophy refresher within the A Brief History of Cprogramming.com forums, part of the Community Boards category; Originally Posted by CornedBee Isn't Curry's paradoxon essentially the mathematical formulation of circular reasoning? That's a good question. I've just ...

  1. #16
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    Quote Originally Posted by CornedBee View Post
    Isn't Curry's paradoxon essentially the mathematical formulation of circular reasoning?
    That's a good question. I've just spent five minutes thinking about it and came up with "no", but I'm not really sure. The difference is that the proof in Curry's paradoxon uses logical equivalence (denoted by the "<=>" sign, or "iff") which is inherently circular all by itself: in step n of the proof, I can put in any statement from some step m with 0 < m < n and start again, making it circular. In mathematics, a circular proof usually consists only of implications, i.e. I have a number of theorems T(1), T(2), ..., T(n) and in order to prove all of them, I show that T(1) implies T(2), T(2)->T(3), ..., T(n-1)->T(n) and finally show that a single T(i) for 0<i<n+1 holds true, which implies that all the other theorems are also true.

    Another argument in favor of a negative answer is the fact that a proof in propositional logic usually consists of expressions implying other expressions, while circular reasoning consists of theorems implying other theorems, so there's sort of a type mismatch.

    Additionally, circular reasoning isn't bad at all. It's a common technique often used in practive, and if we disallow it in propositional logic, we'd also have to disallow it in mathematics.

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  2. #17
    Cat without Hat CornedBee's Avatar
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    Additionally, circular reasoning isn't bad at all. It's a common technique often used in practive, and if we disallow it in propositional logic, we'd also have to disallow it in mathematics.
    Huh? Circular reasoning is when not all your premises are proven or axioms, and you later prove a premise by a reasoning chain that depends on the premise.
    How can that ever not be bad?


    There is something intuitively very wrong about Curry's paradoxon, but it's hard to show what mathematically. Which is pretty much the point of it. In the English formulation:
    "If this sentence is true, then Santa exists."
    This is what I would call a baseless assumption. It's pretty much the logical equivalent of the Chewbacca defense.
    You can prove that the sentence cannot be false, because that leads to a logical contradiction. Therefore it must be true, which means that Santa must exist.

    I think one problem here is that the natural language construct "if ... then" is mapped to logical implication. That's however not true. In propositional logic, if the condition is false, then the entire statement is true regardless of the condition. This is what leads to the contradiction. In natural language semantics, however, if the condition is false, the entire statement doesn't have a truth value. (Not always, at least. Natural language is fuzzy like that.) So perhaps Curry's paradox is a limitation of two-state logic. If we used tri-state logic, we might be able to avoid the paradoxon by creating the following truth table for implication:

    Code:
    true -> true = true
    true -> false = false
    false -> * = none
    none -> * = none
    In this situation, there is no contradiction, for the false case, and the whole statement is endlessly recursive as it should be.

    However, I don't know the full implications of tri-state logic.
    Last edited by CornedBee; 04-28-2009 at 06:07 AM.
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  3. #18
    Registered User hk_mp5kpdw's Avatar
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    From a recent NY Times article by Jimmy Carter:
    I have used weapons since I was big enough to carry one, and now own two handguns, four shotguns and three rifles, two with scopes. I use them carefully, for hunting game from our family woods and fields, and occasionally for hunting with my family and friends in other places. We cherish the right to own a gun and some of my hunting companions like to collect rare weapons. One of them is a superb craftsman who makes muzzle-loading rifles, one of which I displayed for four years in my private White House office.

    But none of us wants to own an assault weapon, because we have no desire to kill policemen or go to a school or workplace to see how many victims we can accumulate before we are finally shot or take our own lives. That’s why the White House and Congress must not give up on trying to reinstate a ban on assault weapons, even if it may be politically difficult.
    Does this qualify as one of these? "I own guns but I don't want to own an assault weapon because I have no desire to kill policemen." If one owns an assault weapon then they must therefore be a killer?
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  4. #19
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    Quote Originally Posted by hk_mp5kpdw View Post
    From a recent NY Times article by Jimmy Carter:


    Does this qualify as one of these? "I own guns but I don't want to own an assault weapon because I have no desire to kill policemen." If one owns an assault weapon then they must therefore be a killer?
    sounds like a slippery slope argument
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    Quote Originally Posted by CornedBee View Post
    Huh? Circular reasoning is when not all your premises are proven or axioms, and you later prove a premise by a reasoning chain that depends on the premise.
    How can that ever not be bad?
    Well, exactly as I said. Suppose you want to show that A <=> B <=> C <=> true. The usual way to prove this is to do it in the following way:

    First, you show the following implications:
    A=>B
    B=>C
    C=>A

    Clearly, this is a circular reasoning chain, as for instance A depends on A and none of the premises are proven to be true. Hence, you only know that A, B and C are either all true or all false. So what you have to do now is to prove one of A, B, C to be true and thus determine the other two statements to be true also.

    You save time compared to the naive approach, i.e. show that A<=>B, B<=>C, C<=>1, which would yield the following tasks:

    A=>B
    B=>A
    B=>C
    C=>B
    C<=>1


    There is something intuitively very wrong about Curry's paradoxon, but it's hard to show what mathematically. Which is pretty much the point of it.
    That's the intersting thing about the paradoxon: every step is perfectly legal, yet the overall result is contradicting to common sense.

    In the English formulation:
    "If this sentence is true, then Santa exists."
    I think X->Y is actually a little bit more complicated. In English, it would be "'If X, then Santa exists' is true (if X is true and Santa exists) or (if X is false)".

    I think one problem here is that the natural language construct "if ... then" is mapped to logical implication. That's however not true. In propositional logic, if the condition is false, then the entire statement is true regardless of the condition. This is what leads to the contradiction. In natural language semantics, however, if the condition is false, the entire statement doesn't have a truth value.
    I agree, the natural language construct "if...then" is only defined for conditions which are true. But you may as well define Y to be "Z and not Z", which is clearly false, regardless of the truth value Z. However, the technique in Curry's paradoxon still allows you to show that it's true.

    Code:
    true -> true = true
    true -> false = false
    false -> * = none
    none -> * = none
    In this situation, there is no contradiction, for the false case, and the whole statement is endlessly recursive as it should be.

    However, I don't know the full implications of tri-state logic.
    And furthermore, there exists more than one tri-state logic. But I know what you mean, and you're probably right.

    Greets,
    Philip

    PS: Please excuse my sloppy answers. ATM I'm at work and there's constantly someone dropping into my office bugging me with silly requests.
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  6. #21
    spurious conceit MK27's Avatar
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    Quote Originally Posted by hk_mp5kpdw View Post
    Does this qualify as one of these? "I own guns but I don't want to own an assault weapon because I have no desire to kill policemen." If one owns an assault weapon then they must therefore be a killer?
    This is why Noam Chomsky's "generative grammar" (which I believe was built up from propositional logic) is considered contentious (eg, interesting, but reductive) in linguistics and near-useless in literary theory (to which the study of a Jimmy Carter Editorial more properly belongs) but ended up, in the course of the late 20th century, being most useful in computer science (eg, the Chomsky heirarchy in automata theory, about which I know zip).

    My point is that hk_mp5kpdw demonstrates well how ridiculous applying a restricted set of tools can get -- syllogistically, you might say that just because your "logic" works in one situation does not mean in will work in all situations. Maybe Snafuist wants to take this all the way into town?
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  7. #22
    Super Moderator VirtualAce's Avatar
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    I think the biggest one that is misused is:

    If A then B
    If B then C

    Therefore if B, then A and C
    If C then A and B.

    If not A then not B and not C since if not B then not C
    If not B then not C
    If not C then not B

    This is what most of the recent fear-mongering is driven by. Just because event A has happened before which led to event B and then C does not mean that every current event A will lead to event B. It could also mean that event A may happen, B might not, and C may happen. Or it could mean that A, B or C happens by itself and/or A may happen without a B and without a C. For sure just because A happens does not mean that B and C will just because they have before. The axiom does hold true in all cases and history and world events are never this predictable. There are thousands of variables that are involved in any world event and to summarize it into a simple 3 line axiom is ludicrous.

    To put this into context using a positive example:

    Just because we have witnessed 1 airliner that was able to land in the water under no power and survive does not mean it is possible in all situations nor does it mean it is the best alternative in every engine loss situation. Each event must be analyzed for what it is instead of completely based on a prior event where other things happened as a result of the event.
    The plane made it safely to the Hudson but one misstep could have sheared off the wing (since as the plane stops 'flying' one wing will drop as it stops generating lift) which would have then exposed the fuel to the elements. Now the fuel may not have ignited or it could have sparked if the torn wing and fuesalage structures sparked somehow. Also if you touch down too hard the fuesalage will break apart. Even though it's water it acts like concrete at high speeds and has a very high resistance to motion which means the aircraft can violently decelerate. Most large planes that land in the water will cartwheel as one wing dips into the water and the force shears the wing roots right out of the fuesalage. This will open up the passenger cabin like a can of sardines. Shortly after this the tail section will be forced upward as the plane digs into the water and it most likely will shear off as well since the plane is not designed to handle the g-forces involved in the rapid deceleration.

    I illustrate this because immediately after this incident we had several stories about similar event where planes landed in the water. You also had people who honestly believe because of the way in which the news reported the story that if a plane gets into trouble all they need to do is find some water and ditch it. It is not unheard of but any good captain/pilot of an aircraft knows that just b/c the Hudson incident just happened to turn out good - landing in the water is definitely not the best option in an engine loss scenario. Best bet is to try to get back to the airport and land it dead stick/yoke and/or pick a cornfield or a highway to land on. Again every scenario must be analyzed.

    It is simple to see that in this case the clear answer is that of course ditching in the water is not the best option in all cases just because it worked one or two or three times. Yet with other incidents that occur this insane logic is applied and we say that since A happened we know for sure that B will follow and then C. This is just not true.

    But it is used time and time again to say that oh man since A happened and in the past B followed suit we must prepare for B b/c it is definitely going to occur. And likewise for C. While preparedness is a great thing and axioms like this help us prepare they do not for certain tell us the future and therefore scaring people based on these types of axioms is completely illogical.
    Last edited by VirtualAce; 04-28-2009 at 05:55 PM.

  8. #23
    int x = *((int *) NULL); Cactus_Hugger's Avatar
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    Quote Originally Posted by CornedBee View Post
    Unicode, and Firefox is not handling it well. It overflows the sig box and the quote box.
    Firefox is handling about as well as I think could be asked of it. Most UI's I've seen degrade in that fashion when confronted with excessive combining marks... (I take advantage of it sometimes - fun & harmless practical joke in an IM, if the other side has unicode support.)
    It's a bunch of Unicode - the spaces (and some of the characters) have a ton of combining marks on them. Quite a few, including some spaces that have multiple occurrences of the same combining mark (I have no idea if Unicode allows such things...). One of those "waterfalls" has 17 combining marks...

    You also need the font to display this, otherwise you'll probably miss the ​ ​ ​ ҉ (combining cyrillic millions sign - should look like a shuriken...)
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  9. #24
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    I'm not going to hijack your thread here, but I really think you're confusing logical fallacies with hypothetical correlations. I basically said "if A (an event) is true, and B (a cycle) is true, and C (the coincidence of A and B) is true, then D (some outcome) could possibly be true". It's simply an unconfirmed correlation.

    >> While preparedness is a great thing and axioms like this help us prepare they do not for certain tell us the future and therefore scaring people based on these types of axioms is completely illogical.

    I'll address this in more detail in the other thread, but I apologize for any hysteria I may have created; please just keep in mind that anything I say should be taken with a grain salt, OK?
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    Registered User whiteflags's Avatar
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    Quote Originally Posted by Bubba View Post
    I illustrate this because immediately after this incident we had several stories about similar event where planes landed in the water. You also had people who honestly believe because of the way in which the news reported the story that if a plane gets into trouble all they need to do is find some water and ditch it.
    Isn't that just like the news to latch onto a type of story because it's something apart from the routine? It's happening now with the swine flu, but the only reason everyone and their mom is reporting it is because it's something other than the war and President Obama's approval rating that might be a major headline.

    I also wonder if people who aren't experts are simply reading things that aren't being said. The swine flu is dangerous in that a pig is a mammal, no vaccines are available, and it kills young people, but the mortality rate is way too low for it to be the next pandemic - we just don't know yet. The news I pay attention too hasn't reported it like this, hasn't issued particular warnings beyond "wash your hands." I don't think misinterpretation is all that hard among readers, and I'm not sure the media is entirely to blame.
    Last edited by whiteflags; 04-28-2009 at 08:02 PM.

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    Quote Originally Posted by Cactus_Hugger View Post
    Firefox is handling about as well as I think could be asked of it.
    Firefox does a pretty good job imo
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