Is Logic Necessarily Infallible?

[thegnome54]

We, as humans, tend to take logic as an 'absolute truth'. Our logic, at its most basic levels, appears to be based on the concept of 'objects', or 'things'. Syllogisms, for example - Thing A is equal to Thing B Thing B is equal to Thing C Therefore: Thing A is equal to thing C In order for the concept of 'equals' to work properly, we need to consider 'things'. Two apples, for example, can be considered 'equal things', even though the atoms which actually make them up are obviously not the same atoms. The ability to group matter into functional Things is clearly very useful when it comes to interpreting the world around us - it allows us to recognize Thing A as an apple, which must be good to eat (like other apples) and Thing B as a rock, which is of little interest when it comes to food (like other rocks). If you step outside of our frame of reference, though, there's much less reason to treat the world as Things than we think. The apple is constantly losing and gaining atoms, processing chemicals and growing, to eventually die - the apple itself is made up of atoms borrowed from previous Things, and when it dies, those atoms will form into new ones. The point is, the concept of a Thing, though useful for human purposes, is somewhat meaningless in the long run. A Thing is really just a momentary conglomeration of atoms with emergent properties which are of note to us homo Sapiens. If our entire system of logic is based on this conveniently evolved method of perception, why should it be trusted beyond what we know it applies to? Do you think it's possible that mankind will reach a point where we are unable to use 'traditional' science to understand the world (or maybe even understand it at all), because we find that we have reached a level where our logic system no longer applies?

[R.D.735]

It was once thought that rational thought alone could unravel every truth, and this belief was called rationalism. However, human logic is impure and inexact because it is influenced most by past experience and emotion and least by fact. Mathematics is the only pure logic system available to us, but Alan Turing already proved that there are some mathematical problems that cannot be definitively solved, no matter how much information is available or how powerful one's logic is. I'm quite certain that mankind has already discovered some mysteries that are beyond our ability to solve, either because our own minds are too limited or because no solution can be reached. No one has yet figured out a method that can test whether a problem is impossible to solve or not, though, so it may be that the cause of the Big Bang can be discovered, but how the brain thinks cannot be revealed, or, perhaps, all of the problems that cannot be solved are far more trivial.

[thegnome54]

However, human logic is impure and inexact because it is influenced most by past experience and emotion and least by fact. Mathematics is the only pure logic system available to us, but Alan Turing already proved that there are some mathematical problems that cannot be definitively solved, no matter how much information is available or how powerful one's logic is. I think you're missing the point. I'm questioning logic itself. Not the imperfect application of logic by fallible human beings.

[R.D.735]

I pointed out that there are mathematically-proven limits to mathematical logic. Isn't that relevant? If we exclude human fallibility and mathematical impossibility, are there any ways for logic to fail at all? By what other means could logic fail?

[thegnome54]

I pointed out that there are mathematically-proven limits to mathematical logic. Isn't that relevant? Not really, because math itself is based on mathematical logic. Since I'm calling all logic into question, the conclusions are... well, questionable. If we exclude human fallibility and mathematical impossibility, are there any ways for logic to fail at all? By what other means could logic fail? By simply not applying - being wrong, although it seems correct to our limited brains.

[Xaqaria]

I think you're missing the point. I'm questioning logic itself. Not the imperfect application of logic by fallible human beings. Its not that the application is done by fallible human beings, it is that the concept of logic that is being applied was created by fallible human beings

[R.D.735]

Ah. I see now, though I can't imagine a case in which logic would be inapplicable. I think logic is not something that humans invented, but that it is something that humans simply use, not unlike light or fire. I think it's universally applicable.

[thegnome54]

Ah. I see now, though I can't imagine a case in which logic would be inapplicable. I think logic is not something that humans invented, but that it is something that humans simply use, not unlike light or fire. I think it's universally applicable. Yes, that's my point. I don't think it is. Do you have a counterargument to my statements that supports logic as universally applicable?

[skysaw]

Yes, that's my point. I don't think it is. Do you have a counterargument to my statements that supports logic as universally applicable? Gnome, You may be interested in Gödel's Incompleteness Theorem. There is some heady stuff there, but the crux of it is that a complete and consistent set of axioms for all of mathematics is impossible. It's quite a mind-bender, and touches on mathematics, logic, and philosophy. I really think this is right up your alley. For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

[R.D.735]

From thegnome54 Yes, that's my point. I don't think it is. Do you have a counterargument to my statements that supports logic as universally applicable? Using logic to determine the limits of logic is the very problem you mentioned. I can't put forth any argument based on logic to establish the limits of logic either in its completeness or its applicability, since logic may be inherently inadequate. However, since illogical arguments are also inadequate, there can be no counter-argument to the statement that logic is inadequate. One could argue that, since there can be no counter-argument, the statement must be true, though that is a deduction based on faulty logic.

[thegnome54]

Using logic to determine the limits of logic is the very problem you mentioned. I can't put forth any argument based on logic to establish the limits of logic either in its completeness or its applicability, since logic may be inherently inadequate. However, since illogical arguments are also inadequate, there can be no counter-argument to the statement that logic is inadequate. One could argue that, since there can be no counter-argument, the statement must be true, though that is a deduction based on faulty logic. Very well put. The only practical conclusion I can think of from this dilemma is this: If anyone ever develops a model of the universe which makes absolutely no logical sense, and yet provides accurate results, it should be accepted by the scientific community. Of course, it would be much better to have an equally successful model that actually makes sense, but if that is not available... Of course, the very concept of accepting illogical means which provide useful ends seems to be based on logic as well. I don't believe any human being will ever be able to really address this issue, since our thoughts are so helplessly entwined in logic. I don't know, I feel like I'm going in circles, but at the same time there should be some useful conclusion to be taken from this realization that logic is not necessarily infallible.

[O'nus]

Very well put. The only practical conclusion I can think of from this dilemma is this: If anyone ever develops a model of the universe which makes absolutely no logical sense, and yet provides accurate results, it should be accepted by the scientific community. Of course, it would be much better to have an equally successful model that actually makes sense, but if that is not available... Of course, the very concept of accepting illogical means which provide useful ends seems to be based on logic as well. I don't believe any human being will ever be able to really address this issue, since our thoughts are so helplessly entwined in logic. I don't know, I feel like I'm going in circles, but at the same time there should be some useful conclusion to be taken from this realization that logic is not necessarily infallible. I think you should consider reading Wittgenstein's Tractatus. He discusses this very thing in regards to logic, nature, and thought. Here's the synopsis of the Wittgenstiens observations: - Thought: You can think to yourself, but when you say "I am hungry" or "I do not like you" - who is the "I" that you are referring to you? Where can we find that "I"? The "I" is what Wittgenstein would refer to as the "facts" of the world and are things which we cannot truly see/define/exeperience. However, we can make propositions about them. Consider the analogy he gives us: - Imagine your eyes are covered and you have to walk forward. There is a hand guiding you every now and then. You may begin to expect it or look for it. This is the analogy for logic, thought, math, everything us humans do. - Logic/Math: While keeping the abovementioned analogy in mind, we can only predict, describe, and purport propositions about the world. It may be absolutely true that 2+2=4 but we are the authors of these variables and logic. There is no body or entity of "2" or "addition" it is a concept developed by humans in order to functionally understand and interpret the world. I think this is why Wittgenstein eventually gave up on math and aeronautics. - Langauge: Language is limited in its expression. By now, I hope you see how this has been shown. We can only express our thoughts and feelings about the interpretations we make of observations in our environment. The mistake we often make is attributing these interpretations as manifest objective entities that exist independent of us. What do you think...? ~

[thegnome54]

"While keeping the abovementioned analogy in mind, we can only predict, describe, and purport propositions about the world. It may be absolutely true that 2+2=4 but we are the authors of these variables and logic. There is no body or entity of "2" or "addition" it is a concept developed by humans in order to functionally understand and interpret the world. I think this is why Wittgenstein eventually gave up on math and aeronautics." Wonderful! I've reached the same conclusion myself, to the point that I get irritated when people insist that mathematics are an absolute truth. I've just gone a bit farther than math and numbers, and realized that the very concept of an 'entity' is a human construct - all matter is presumably made up of the same basic elements, and the way we group matter into entities is not objectively relevant. I will definitely try to find that, thanks for the reference. Wittgenstein sounds like a guy with a head on his shoulders. *EDIT* I found a copy of Tractatus on amazon, and was able to preview the first two or three pages. Unfortunately, most of the meaning within it washed over me unabsorbed. I am forced to conclude that I am simply too young and inexperienced to grapple with it yet. The terminology he uses seems vague and counter-intuitive, and I can't really see the flow of ideas. Hopefully someday when I'm older I will be able to properly read through it, but until then I'll have to go back to my own personal musings.

[O'nus]

I will definitely try to find that, thanks for the reference. Wittgenstein sounds like a guy with a head on his shoulders. Maybe, but I do not think he was a very happy guy. Be careful, the road you are on can be very lonely and suicidal if you start doubting the very existance you are in. The truth is, you are still existing and interpreting an environment. Even Wittgenstein needed some scaffolding before he could come to his conclusion. In fact, he described math, logic, etc. as a ladder that you climb up but, once you get to the top, you get rid of the ladder and do not use one again. ~

[thegnome54]

Maybe, but I do not think he was a very happy guy. Be careful, the road you are on can be very lonely and suicidal if you start doubting the very existance you are in. The truth is, you are still existing and interpreting an environment. Even Wittgenstein needed some scaffolding before he could come to his conclusion. In fact, he described math, logic, etc. as a ladder that you climb up but, once you get to the top, you get rid of the ladder and do not use one again. ~ I'm not too worried - illusory happiness is still happiness, and as yet I still believe in an objective reality anyways. A friend of mine has referred me to Bertrand Russel's The Problems With Philosophy, or something along those lines. I'll check it out and see if I can get anything out of it. I'm terribly unexposed to this sort of stuff, somehow. Public education can really get in the way of learning sometimes.

[O'nus]

I'm not too worried - illusory happiness is still happiness, and as yet I still believe in an objective reality anyways. A friend of mine has referred me to Bertrand Russel's The Problems With Philosophy, or something along those lines. I'll check it out and see if I can get anything out of it. I'm terribly unexposed to this sort of stuff, somehow. Public education can really get in the way of learning sometimes. Good - I am glad to hear this. I should note to you that Bertrand Russell was Wittgensteins teacher. However, when assessing Wittgensteins thesis for his Ph.D - even Russell admitted to Wittgenstein's genius. ~

[skysaw]

I found a copy of Tractatus on amazon, and was able to preview the first two or three pages. Unfortunately, most of the meaning within it washed over me unabsorbed. I am forced to conclude that I am simply too young and inexperienced to grapple with it yet. The terminology he uses seems vague and counter-intuitive, and I can't really see the flow of ideas. Hopefully someday when I'm older I will be able to properly read through it, but until then I'll have to go back to my own personal musings. Try Godel, Escher, Bach: An Eternal Golden Braid by Douglass Hofstadter. It's an easy (and actually extremely fun) read, and covers much of this territory, as well as many other fascinating topics such as artificial intelligence and philosophy. It didn't win the Pulitzer for nothin' you know!

[thegnome54]

Try Godel, Escher, Bach: An Eternal Golden Braid by Douglass Hofstadter. It's an easy (and actually extremely fun) read, and covers much of this territory, as well as many other fascinating topics such as artificial intelligence and philosophy. It didn't win the Pulitzer for nothin' you know! I'll be sure to look into that as well, thanks. Maybe this weekend if the homework mire isn't too deep I'll go over to the library and dig some of these out.

[tkdyo]

I have had many thoughts about this complex problem as well. It only makes sense to me that our current logic cannot explain everything that happens in our lives. I really like the eyes closed analogy and completely agree with the problem of math being absolute. I am in level 4 calculus and thus far I have seen a lot of things that make me think mathematicians just dont address the things that math cant answer..such as the whole concepts of an integral. Sure it may be able to estimate the sum of shapes under a curve, but at the same time, how do we know that infinitely small things will still completely fill this curve? We cant test this, we can only use mathematical logic to say it works. or, even simpler, why not divide by 0? I has to stand for something or we would not get it in complex problems, I believe we still just dont know what it would actually do.

[O'nus]

I have had many thoughts about this complex problem as well. It only makes sense to me that our current logic cannot explain everything that happens in our lives. I really like the eyes closed analogy and completely agree with the problem of math being absolute. I am in level 4 calculus and thus far I have seen a lot of things that make me think mathematicians just dont address the things that math cant answer..such as the whole concepts of an integral. Sure it may be able to estimate the sum of shapes under a curve, but at the same time, how do we know that infinitely small things will still completely fill this curve? We cant test this, we can only use mathematical logic to say it works. or, even simpler, why not divide by 0? I has to stand for something or we would not get it in complex problems, I believe we still just dont know what it would actually do. You should go see the movie pi. The mysteries of the number are actually quite astounding. The movie takes it a little far in some contexts but.. it is still interesting. ~

[skysaw]

You should go see the movie pi. The mysteries of the number are actually quite astounding. The movie takes it a little far in some contexts but.. it is still interesting. ~ If you're talking about the Aronsky film, yes, it's a great flick. Disturbing and provokative. It's not actually about the number pi, rather a ficticious "God" number that embodies the mysteries of the universe.

[grasshoppa]

the spiral number? 1.168?

[skysaw]

the spiral number? 1.168? You're talking about phi, or the Golden Ratio. They do talk about a little bit in the movie, but it's not the "God" number in the story.

[O'nus]

If you're talking about the Aronsky film, yes, it's a great flick. Disturbing and provokative. It's not actually about the number pi, rather a ficticious "God" number that embodies the mysteries of the universe. Was not the "God" number intended to be the 268 digit-length of pi? Did we have that many numbers of pi at the time of that movie? Either way, the movie was inspired by the number pi. ~

[skysaw]

Was not the "God" number intended to be the 268 digit-length of pi? Did we have that many numbers of pi at the time of that movie? Either way, the movie was inspired by the number pi. ~ No, it was not pi. The movie is not that old. Note that the choice of black and white film was an artistic one; the film came out in 1998. Pi had already been calculated to millions of digits. Here's the plot summary.