What you can, and cannot achieve with logic alone

This topic is prompted by a an off-topic discussion in the "Why Christians Might Resist Belief in Evolution" thread, that sprang up between myself and Objectivitees.

First, an off-topic rant:

I wrote a summary of the argument but lost that post, TWISE now, due first to me accidentally pressing the back button on my browser, and the second time because when I pressed submit, the system had apparently logged me out, and asked me to log back in. When I did, I was staring at an empty "post a new topic" field.

I REALLY hope this site would do what many others do - mainly preserve the text entered in the text box, so that if you accidentally or due to some website technical inanity navigate out of the page before submitting, you could simply press the "back" button and be returned to the previous page ALONG WITH the text you spent twenty minutes writing.

*Sigh*

Last try.

The basic argument from Objectivitees that I'm disputing is that logic is sufficient to gain knowledge of the outside world. I'm saying that no, it is not - logic may be necessary in evaluating claims, but it is not sufficient, without empirical evidence of the outside world, to establish the truth of any claim (that is not self-contradictory) about the outside world.

To show why this is so, I made a hypothetical, imagining a man that had been raised in a box, isolated from the outside world (and creatively named "man-in-a-box").

Now if you were to give this man the following sylogism:
1.All rocks are made of cheese
2.The moon is made of rock
3.Therefore the moon is made of cheese

The man-in-a-box, having had no experience of either rocks, cheese, or indeed the moon, using only pure logic, could only say that the argument is valid, that if the premises are true, then the conclusion inevitably follows. What he could not deduce is whether or not the premises are true or false; and thus he could not say whether the moon was made of cheese or not.

Objectivitees, suggested that the man could use the law of identity to note that since "rocks and cheese" are different words, the premise is false.

This is clearly a gross misapplication of the law of identity, for a few reasons; the law of identity does not prevent things being composed of other things, or words being synonymous with each other - and, indeed, I must also point out that the law of identity states specifically that claims of the form "x is x and not-x" cannot be true, and there's nothing in the syllogism that would both affirm and deny the same thing.

For the law of identity to apply, the sylogism would have to look something like this:
1.All rocks are made of not-rocks
2.The moon is made of rocks
3 Therefore the moon is made of not-rocks

Now here one could validly say, without empirical evidence of any kind, that premise 1 is false, because it violates the law of identity.

In the original sylogism, there's nothing even remotely like this - the law of identity does simply not apply.

To demonstrate the point, that applying the law of identity in this strange way would lead to false conclusions when applied consistently, I proposed another syllogism to give to the man-in-the-box:

1.All rocks are made of atoms
2.The moon is made of rocks
3.Therefore the moon is made of atoms

If the man-in-the-box could validly use the law of identity to deduce that the moon is not made of cheese, the very same method would lead him to deduce that the moon is not made of atoms either.

Clearly there is a gross missaplication of the law of identity here - as I've shown above.

Objectivitees then says:

Objectivitees wrote:
Steg, let's have this discussion another time, because every time you go there, it seems you are suggesting Logic is not a means to derive knowledge, thereby denying the first premise of my argument, which then means you cannot claim to have any kind of knowledge.

There's a subtle distinction here that I simply do not think you are getting. Indeed, you cannot use logic alone to derive knowledge, with the exception of ruling out certain claims.

So, for example, if someone says that they've drawn a square circle, I don't need any empirical evidence to know that they are either mistaken or lying - this is because there is an inherent logical contradiction in claiming that something is simultaneously a square and a circle; the definitions of these words rule each other out.

However, if someone says to me that there is a ring of invisible pixie dust circling the Earth at a distance of a billion light years, I cannot use logic alone to either deny or affirm this claim. All I can say is that there is nothing logically self-contradictory in the claim, and thus I cannot deny the possibility of the claim on the basis of logic alone.

I must resort to empirical knowledge of the world - the absence of evidence for pixies, or pixie dust, and the impossibility of detecting such a ring, even if it were to exist. Based on these empirical considerations, I can say that it is probably not reasonable to accept the claim.

Surely, in that, I used logic - at least my reasoning does not contradict any laws of logic - but that is a case of applying logic to empirical evidence to produce a reasonable conclusion.

So notice the a-symmery - logic alone can be used to deny the possibility of some claims, but it cannot be used again - alone, independent of empirical evidence - to positively establish the existence or nature of something in reality.

So no, you can't use logic alone to derive knowledge, other than placing boundaries on what might be true.

Hi Steph,
Interesting read. I buy your logic, and confess I have not read the other thread.

However, I have a question for you. Do you think logic alone can derive knowledge in the mathematical realm?
Being a maths teacher, I notice that logic alone is used in pure maths to derive knowledge.

Like the proof that the square root of 2 is irrational.
For example,
http://books.google.com.au/books?id=YJCkBOyrP4gC&pg=PA22&lpg=PA22&dq=euclid+irrational+proof&source=bl&ots=GLlVgCssgr&sig=2O6f8sNDCvSQyzEvIGdLBHh4dQ8&hl=en&ei=Uc3uSrDMDoTWsgO1huX1Aw&sa=X&oi=book_result&ct=result&resnum=3&ved=0CBAQ6AEwAg#v=onepage&q=euclid%20irrational%20proof&f=false

I think your proposition from the final line in your thread is not true for everything, but may indeed be true for some things. Mathematical proofs are one exception.

Got a notification for a reply in this thread, and thought that Objectivitees might have surpriced me and actually replied to this - no such luck, but I'm not sorry; fascinating question indeed Snafu!

I think that logic in pure math can tell you about the equivalence or non-equvalence of mathematical statements - what it does not tell you is whether that mathematical structure has any correspondence in the physical universe we inhabit. For that you'd need the actual empirical evidence.

It seems to me that mathematical statements are almost like tautologies - often very complex tautologies.

2 + 2 = 4

Is this not just another way of saying that 4=4? Mathematical analysis of real world events nevertheless relies on observation of that which is being modeled - it isn't enough to make a mathematical model, and expect our universe to oblidge with something to which that model applies.

For example, Johannes Kepler, when he made his astronomical investigations, after getting the idea that positioning the sun in the center, instead of the Earth, simplified the orbits of the planets immensely, noticed that the fact that there are only 5 planets (as was thought at the time), and there being just 5 platonic solids, related the two "facts" together, to produce his first model which involved some sort of geometric perfection. Alas, his observations forced him to reject his beatiful mathematical model, as he realized that the paths of the planets around the Sun are not described by perfect circles, but ellipses.

His first idea MIGHT have been true - it was mathematically consistent. Yet observation relegated that model to the status of an intellectual curiosity, rather than an actual correspondence to reality.

Hi again
I fear you may have misunderstood what I was saying.
I did not mean to say that logic in pure maths was about equivalence, eg) 2 + 2 = 4
What I hoped to say was that logic alone proves some things in the mathematical realm. I gave one example, ie) the square root of 2 is irrational. This is quite different to your 2 + 2 = 4 line of thinking.

I stand by my position, logic alone can derive knowledge in parts of our universe, maths being an example,and the square root of 2 being irrational gave an explicit example.
So steph, with respect, your position was falsifiable, and has been falsified in my mind at least. You may wish to qualify your statement to exclude mathematical knowledge, & then I'm stuck.

Another example would be the area of a circle. The well known equation of area=pi*r*r can be logically derived quite easily. It does not need to be compared to any real circles for verification.

Mathematicians delve into some pretty wacky stuff, some of which have found practical real world applications, some have not. I remember reading Brian Greene's "The fabric of the cosmos" (I'm trusting my memory here), and he recanted going to some mathematicians about Calabi Yau Manifolds in their relation to string theory, and was surprised that mathematicans had been researching that area of maths without any known physical application.

Snafu - the position I was defending in my initial post, and the position I was arguing against Objectivitees was that pure logic alone cannot be used to derive knowledge of the outside world.

Mathematical proofs can surely be used to derive new knowledge about mathematics, but that's a self-contained area of thought, and whether or not it has a connection to the outside world, has to be determined empirically.

For example, you mention the equation for the area of a circle, and indeed, you can derive, with pure logic, the knowledge that in any context where Euclidean geometry applies, this equation will hold true. However, what you cannot establish by mathematics alone is whether Euclidean geometry applies to the observable world - for that, you need empirical observation. Once you have that observation that yes, at least to the best approximation we can achieve with our methods, the universe does appear to be flat, and euclidean geometry does seem to apply, THEN you can use your equations to derive knowledge of the empirical world.

But this is entirely what I was saying - that empirical observation needs to play a role when determining how things are in the real world; logic surely plays a vital part in deducing new information, but it has to be grounded, at some point, on observation.

No wucken furries. I understand now where you are coming from.

I find it interesting however that what we call knowledge gained from the empirical observations you describe may prove false in the future. One example would be Newtons laws of motion which were tested emprically and thought to be correct for a time. Was this knowledge? or Belief? Einstein came along and overthrew the newtonian view with the relativistic view. His theory (logic) made predictions, one of which was the bending of starlight. This was empirically tested and found correct during an eclipse. So again, does the empirical testing of relativity give us true knowledge? I think probably not. It only gives us a theory which hasn't been proved false yet, and cannot be proven true beyond all doubt.

This leads to the question of what is knowledge anyway? But I'm getting out of my depth. Perhaps a philosopher can chime in.