The Validity of Logic Itself

[Infidel7]

I have recently been arguing with a classmate about the differences between science and religion. His main argument focuses on the 'fact' that science requires as profound a leap of faith as any religion. Easy to refute, right? Well, then he proceeds to attack the very foundation of science itself.

He claims that logic is merely a human construct and is therefore fundementally flawed and cannot be trusted. I was under the impression that logic was independant of human input (minus the perspective issue).

He goes on to claim that science is based on logic which makes only assumptions. We can only assume that a stone will fall to the ground once dropped, but what's to say it really will. I know I cannot say that it will because it has every other time I dropped a stone it fell. That's a logical fallacy.

What are your ideas in this area? Is logic subjective/flawed? Is it logical to asssume the sone will fall?

[NoXion]

Go tell him to jump off a cliff. By his own reasoning, there's no 100% garauntee that he will die when he reaches the bottom or even fall in the first place.

[Spacebeard]

This solipsist bullshit was also dealt with in this thread a few days ago.

I second NoXion's suggestion.

[Kuroneko]

He claims that logic is merely a human construct and is therefore fundementally flawed and cannot be trusted. I was under the impression that logic was independant of human input (minus the perspective issue).

Well, logic is a human construct, yes. As for its status, one can view logic as theory of correct thinking. In this way, it is really not much different than any other scientific theory--this view is even bolstered by the fact that there are many types of logic, some of which are better suited for different applications. A hallmark of a superior scientific theory is giving good results where competing theories fail. It is analogous with logic.

He goes on to claim that science is based on logic which makes only assumptions. We can only assume that a stone will fall to the ground once dropped, but what's to say it really will. I know I cannot say that it will because it has every other time I dropped a stone it fell. That's a logical fallacy.

Tell him that the old Humean trick of criticising the principle of induction is about a century out of date. There are other models in scientific philosophy that do not share the same problems, the most famous one of which is falsificationism.

What are your ideas in this area? Is logic subjective/flawed? Is it logical to asssume the sone will fall?

In cases such as this where the discussion forces one to split hairs, I prefer to avoid the somewhat loaded term "logical" in favor of "rational". And that's really a more pertinent issue--it is rational to assume that the stone will fall, and irrational to assume that it will not.

[Infidel7]

Excellent input.

I confronted him about this Solipsist bullshit he was spouting, but he fell back onto a somewhat "agnostic" approach to everthing, not so much solipsist. He claimed that everything was a leap faith and that therefore there was no need in worrying oneself about anything. There could not be a worse worldview IMHO.

His argument is obveiously flawed, however I am having a hard time poking holes in it without reverting to the "logic" he is criticizing me about. By supporting a scientific worldview using logic, he claims my beliefs are incestuous. Is this circular reasoning on my part or his?

[NoXion]

His argument is obveiously flawed, however I am having a hard time poking holes in it without reverting to the "logic" he is criticizing me about. By supporting a scientific worldview using logic, he claims my beliefs are incestuous. Is this circular reasoning on my part or his?

Correct me if I'm wrong, isn't this an example of special pleading?

[Kuroneko]

I confronted him about this Solipsist bullshit he was spouting, but he fell back onto a somewhat "agnostic" approach to everthing, not so much solipsist. He claimed that everything was a leap faith and that therefore there was no need in worrying oneself about anything. There could not be a worse worldview IMHO.

The problem with the "everything is uncertain" type of argument is, of course, that not everything is uncertain to the same degree. I suppose he could then claim that the relative amount of uncertainty is uncertain, and so on ad nauseum, but the real issue is that both science and logic provide verifiable and repeatable results. Don't let him treat the issue of not being "fully certain" with more importance than it deserves--it's not a question of what's absolutely perfect, but what's the best we have available. If he has an alternative method that verifiably performs better, ask him to present it.

His argument is obveiously flawed, however I am having a hard time poking holes in it without reverting to the "logic" he is criticizing me about. By supporting a scientific worldview using logic, he claims my beliefs are incestuous. Is this circular reasoning on my part or his?

If science is based on logic as he claimed, then this cannot be circular reasoning. If logic is a kind of science (the view I mentioned above), then it is still valid to discuss it in scientific terms because it is then the most basic of sciences, and thus the starting point. Again, is there an alternative method that performs better than logic?

[Ariphaos]

His argument is obveiously flawed, however I am having a hard time poking holes in it without reverting to the "logic" he is criticizing me about. By supporting a scientific worldview using logic, he claims my beliefs are incestuous. Is this circular reasoning on my part or his?

His belief is not directly solipsist, it seems to be more of 'we are mortal therefore we cannot think a perfect thing.'

There are two problems with this view.

The first is that certain statements of logic can be made on a very fundamental level, and that every digital device in existence relies on several to several billion such fundamental statements. When they don't work, it's a problem either with our interpretation of said logic (programming bugs) or a failure of the hardware in question. The idea that the concept of OR, AND, and NOT statements could be fundamentally 'wrong' can't even enter into the equation.

The second problem has been mentioned - logic works. You have two apples, someone gives you two more apples, the realization that you only have four to eat, give away or plant is pretty fundamental.

[Patrick Degan]

Hasn't David Hume gotten his ass kicked-in enough lately?

[Darth Wong]

His argument is obveiously flawed, however I am having a hard time poking holes in it without reverting to the "logic" he is criticizing me about. By supporting a scientific worldview using logic, he claims my beliefs are incestuous. Is this circular reasoning on my part or his?

His belief is not directly solipsist, it seems to be more of 'we are mortal therefore we cannot think a perfect thing.'

That is a particular case of the ad-hominem fallacy: judging an idea by its source.

There are two problems with this view.

The first is that certain statements of logic can be made on a very fundamental level, and that every digital device in existence relies on several to several billion such fundamental statements. When they don't work, it's a problem either with our interpretation of said logic (programming bugs) or a failure of the hardware in question. The idea that the concept of OR, AND, and NOT statements could be fundamentally 'wrong' can't even enter into the equation.

The second problem has been mentioned - logic works. You have two apples, someone gives you two more apples, the realization that you only have four to eat, give away or plant is pretty fundamental.

Perhaps more to the point, science relies upon outside data for its premises. Whether you accept that the universe is real or not (and one must reiterate that anyone who seriously denies the existence of objective reality is just arguing for the sake of arguing), the fact is that the universe was obviously not created by science. Therefore, observations of the universe are external to science, ergo science is a logical construct which uses data external to itself as its premises.

Religion, on the other hand, creates its own premises. Its Scriptures, its testimonies, its revelations ... 100% of it is created by members of the religion. The entire enterprise is wholly circular; you cannot find a piece of religious "evidence" which was not created by a person belonging to that religion.

Even if we accept his argument that you can judge an ideology by the imperfection of its human authors, religion is still inferior to science.

[drachefly]

I had an argument along these lines not long ago.

The main strategy is to make him clarify his objection. There are many possible objections he could be making.

1) We cannot assume the laws of logic actually are what we say they are. In particular, I was dealing with the law of noncontradiction. How do we know that, given a proposition A, that A is not both true and not true?

Solution: Start before we know what 'true' and 'false' and 'proposition' mean. Define proposition such that each proposition is in one of two sets -- true, or false. Also let the logical contrary of each proposition fall in the other one. Then if you find a statement which is both true and false, you know it's not a proposition.

2) We cannot assume that the universe actually has rules that do not change.

Solution: if the rules change, then the fundamental rules would include the change.

2a) Now, how do know that the fundamental rules include no such changes?

Solution: We don't. However, this has no impact on our actions, since if they do change, any action we took is equally liable to completely screw us up. The only wise reaction might be a shift not to trust delicate things; but that was already wise. It doesn't take the rules of the universe changing to break delicate things.

3) Due to the above statement, why do you say your models are true?

Solution: every model that people actually use has limits on its applicability. These limits mean that the model is self-describing when it can be used. It is up to you to figure out when a model can be applied. If it isn't useful in a given situation, that doesn't make the model wrong.

[Talanth]

He goes on to claim that science is based on logic which makes only assumptions.

I'm sure Kuroneko can corect me if I'm wrong, but this seems almost identical to the problems they had with Mathematics a few decades ago. Someone (Goldel someone, I think) wanted to know if the logical structure of maths could stand up on its own and never, ever contradict itself (IE the math is indipendent of its human discoverors). He ended up managing to prouve that for any systen of (arythmetical) logic you will always be able to find a statement that cannot be prouved or disprouved. He also managed to prouve that in order to increace the number of statements that can be prouved/disprouved you need to increace the number of anxioms used in the system (anxioms are self-evident truths of the most fundimental type), but that even if you had an infinite list of anxioms you would still have at least one statement that could not be prouved or disprouved.

So not only do we not know absolutly that our logical system is without flaw, but we have prouved that we can never know if there is a flaw in the ststem. Prouved using the logical sytem itself.


Isn't maths fun!

[Talanth]

Found it!

Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency [if and only if] it is inconsistent.

So if we ever prouve that science is logicaly consistant then we automaticly know that it is not.

Mathworld is a good scource for all mathsie bits. Also from Wikipedia for a more wordie version.

[Surlethe]

I'm fairly certain Gödel's proof relies on self-referential statements, such as "This statement is false", or "This statement cannot be proven"; he went on to show that any system of axioms complex enough to encode the natural numbers -- i.e., complex enough to do simple arithmetic -- is naturally incomplete. I was discussing this with a professor, though, and he said that logicians can look at mathematics from the "outside", and see that it's not likely that the system is self-contradictory. That is, mathematics is inherently incomplete, but no mathematician thinks the work is all in vain, because it's unlikely that the incompleteness will lead to fundamental contradictions.

[Darth Wong]

Correct me if I'm wrong, but isn't the demand for proof of consistency similar to the demand for proof of non-existence, ie- a burden of proof fallacy? Shouldn't you have to show the inconsistency in order to prove that a system is inconsistent, rather than having to prove that it is consistent?

Consider the ancient Socratic method of interrogation: the idea is interrogate the subject until an inconsistency can be found. If you assumed that every idea was inconsistent until proven consistent, then there would be no need. You could simply fold your arms and say "prove to me that you are not contradicting yourself." It seems absurd.

[Ariphaos]

Correct me if I'm wrong, but isn't the demand for proof of consistency similar to the demand for proof of non-existence, ie- a burden of proof fallacy? Shouldn't you have to show the inconsistency in order to prove that a system is inconsistent, rather than having to prove that it is consistent?

We tend to call consistant things laws. They happen again and again, and are assumed to be true not for 'absolute proof' but simply because assuming otherwise is not useful. One form denial is solipsism, another is "Last Thursdayism".

Another is saying that physics as we know it is complete bunk, God moves every last particle according to his whim.

You can make such statements, and they aren't outright wrong in a 'well it could be', sense, but if you believe say, the Aristotilean concept of Inertia over Newtonian you can always cut your break lines and see how far it gets you.

[Darth Wong]

Correct me if I'm wrong, but isn't the demand for proof of consistency similar to the demand for proof of non-existence, ie- a burden of proof fallacy? Shouldn't you have to show the inconsistency in order to prove that a system is inconsistent, rather than having to prove that it is consistent?

We tend to call consistant things laws. They happen again and again, and are assumed to be true not for 'absolute proof' but simply because assuming otherwise is not useful. One form denial is solipsism, another is "Last Thursdayism".

Another is saying that physics as we know it is complete bunk, God moves every last particle according to his whim.

You can make such statements, and they aren't outright wrong in a 'well it could be', sense, but if you believe say, the Aristotilean concept of Inertia over Newtonian you can always cut your break lines and see how far it gets you.

What does that have to do with the proposition, raised earlier, that mathematics and science are flawed because you can't prove they're consistent? I'm just pointing out that it's absurd to demand proof of consistency rather than presenting proof of inconsistency.

[Ariphaos]

What does that have to do with the proposition, raised earlier, that mathematics and science are flawed because you can't prove they're consistent? I'm just pointing out that it's absurd to demand proof of consistency rather than presenting proof of inconsistency.

Just a (to me) more solid reason than 'it seems absurd'. That being, you can't make use of such claims ("But it will save your soul! ... crap notwithstanding). It's a dead-end philosophy, and given their infinite number, not worth the mental power to bother with.

[Surlethe]

Correct me if I'm wrong, but isn't the demand for proof of consistency similar to the demand for proof of non-existence, ie- a burden of proof fallacy? Shouldn't you have to show the inconsistency in order to prove that a system is inconsistent, rather than having to prove that it is consistent?

If the demand for "proof" of consistency or inconsistency is mathematical, then either claim requires proof. Even in mathematics, we can have evidence without actual proof -- for example, mathematicians have come up with arguments for the infinitude of twin prime pairs, but haven't actually proven it -- but the burden of mathematical proof lies on both sides of the claim.

However, when you're talking about logic itself, then mathematical proof doesn't apply, as far as I know; the demand of proof for inconsistency is, essentially, an existence claim, so it boils down to the burden of proof anyway. Besides, since the use of logic is necessary for a proof, if your claim is that logic is flawed, then don't you contradict yourself in proving that you're correct?

Consider the ancient Socratic method of interrogation: the idea is interrogate the subject until an inconsistency can be found. If you assumed that every idea was inconsistent until proven consistent, then there would be no need. You could simply fold your arms and say "prove to me that you are not contradicting yourself." It seems absurd.

Gödel's proof ensures that, mathematically, at least, a mathematician can never answer that question. Generally, mathematicians operate on the assumption that the system is consistent. It's important to distinguish between the colloquial definition of proof, as in the demand "prove there are no contradictions", and the mathematical definition, which implies that you'll deduct the fact that there are no contradictions from axioms. In some sense, the man standing there with his arms folded and demanding proof of consistency is equivocating on "proof".

EDIT: And, of course, I don't see how "inconsistent" implies "flawed", as if calculus is somehow wrong because mathematics contains Gödel statements.

[drachefly]

Gödel's proof gives a recipe for generating pairs of contradictory axioms, either of which is consistent with the parent framework (obviously, you can't take both. You can take neither, though).

That is, if you keep on applying that method to an axiomatic system, you'll never produce a contradiction.

[Darth Wong]

I still don't see how any of this can apply to science. In mathematics, you are expected to prove propositions. In science, you are expected to show that your theories are highly consistent with observed data, and then challenge others to disprove them. That's a huge difference.

Besides, the notion that you can't create an axiomatic system which is complete enough to allow direct logical deduction of every conceivable statement consistent with that system generates a "so what" reaction from me. I don't see why it means or even implies that there's anything wrong with these systems.

[Kuroneko]

Someone (Goldel someone, I think) wanted to know if the logical structure of maths could stand up on its own and never, ever contradict itself (IE the math is indipendent of its human discoverors). He ended up managing to prouve that for any systen of (arythmetical) logic you will always be able to find a statement that cannot be prouved or disprouved.

As a historical note, there seems to be no indication that Gödel set out to follow Hilbert's programme to reduce mathematics to formal logic system which is complete and consistent, but he was certainly directly responsible for its demise.

He ended up managing to prouve that for any systen of (arythmetical) logic you will always be able to find a statement that cannot be prouved or disprouved.

That depends. If one is is dealing with second-order logic, it is possible to construct an arithmetical system which is complete. The problem is that the statements of higher-order systems are not recursively enumerable, so that although this system would have all propositions as either true or false, in some sense one cannot know the full extent of the system in the first place.

I'm fairly certain Gödel's proof relies on self-referential statements, such as "This statement is false", or "This statement cannot be proven"; he went on to show that any system of axioms complex enough to encode the natural numbers -- i.e., complex enough to do simple arithmetic -- is naturally incomplete.

Indeed, although 'natural' is sometimes taken to be something less mathematically precise, to distinguish it from propositions explictly constructed to be independent, like those of Gödel. One of the more well-known examples is the convergence of the Goodstein sequences, which appear to grow roughly as n^n^n at first, but nevertheless converge to zero. The fact that they do so is unprovable in Peano arithmetic, but provable in ordinals.

That is, mathematics is inherently incomplete, but no mathematician thinks the work is all in vain, because it's unlikely that the incompleteness will lead to fundamental contradictions.

Someone has facetiously stated that Gödel's incompleteness theorem proves that mathematicians will always be employed. It might have been Erdős.

Besides, the notion that you can't create an axiomatic system which is complete enough to allow direct logical deduction of every conceivable statement consistent with that system generates a "so what" reaction from me. I don't see why it means or even implies that there's anything wrong with these systems.

It means that some questions may be well-formed and yet still be unanswerable in mathematics, which in principle carries over to physical theories as well. This is considered much less of a big deal than it was originally, partly because Hilbert's ambitions, which Gödel annihilated, did not stop with mathematics--he had hoped to axiomatize physics as well.

This ambition was somewhat justified at the time. Following the success of Lagrangian formalism (with its connection between conservation laws and symmetries), both gravity and electromagnetism were interpreted as geometry. For a while, it looked as if physics really was reducible to some kind of mathematical formalism. That view doesn't hold up in light of particle physics, however.

[Surlethe]

I still don't see how any of this can apply to science. In mathematics, you are expected to prove propositions. In science, you are expected to show that your theories are highly consistent with observed data, and then challenge others to disprove them. That's a huge difference.

I suppose that's why it's important to distinguish between mathematical and scientific "proof" when dealing with situations where the two are both involved. Standing back and saying, "Prove your system has no contradictions" is like the bastard child of an equivocation and a burden of proof fallacy.

Besides, the notion that you can't create an axiomatic system which is complete enough to allow direct logical deduction of every conceivable statement consistent with that system generates a "so what" reaction from me. I don't see why it means or even implies that there's anything wrong with these systems.

It seems to me like a black and white fallacy: either the system is perfect and can explain everything, or it's utterly worthless -- kind of like creationist protestations that science must be invalid because it isn't perfect, or trekkie arguments that Alderaan must not have been destroyed by a direct energy weapon because the description provided by that mechanism is not entirely accurate.

[Kuroneko]

It seems to me like a black and white fallacy: either the system is perfect and can explain everything, or it's utterly worthless -- ...

Oh, certainly not! The severity of the problem of logical/mathematical incompleteness depends on one's scientific philosophy. For the typical falsificationism commonly held today, it really is nothing more than a "so what" moment. For the logical positivism popular in the early 20th century, however, it is a horrifying result.

[Surlethe]

It seems to me like a black and white fallacy: either the system is perfect and can explain everything, or it's utterly worthless -- ...

Oh, certainly not! The severity of the problem of logical/mathematical incompleteness depends on one's scientific philosophy. For the typical falsificationism commonly held today, it really is nothing more than a "so what" moment. For the logical positivism popular in the early 20th century, however, it is a horrifying result.

That's my point: to dismiss a system out of hand because it can never be complete is simply bifurcation; the fact varying initial philosophies produce varying reactions is consistent with the argument against bifurcating between perfect and useless. Besides, any objective philosopher will agree that mathematics and science, though incomplete, are quite useful; to dismiss their usefulness because of incompleteness is therefore sophistry of the highest order.