Proof of logic

[cyborg]

Scarcely.

What does it fail to capture?

What is incompleteness if not asking meaningless symbols to be meaningful and then asking them what their meaning is?

 

[Complexity]

Proof of logic....well, here's my take on it:

In the end, logic and math are pseudosciences. Very good ones, but still pseudosciences, because they base all of their claims on intuition. Now, they don't have to. They could (and probably will in future) derive the rules, etc. from human psychology or some similar source. Math and logic are needed to do this, however, so for now we have to stick with intuitive rules (which happen to be quite good and quite complex).

I don't regard math or logic as sciences at all, let alone 'pseudosciences'.

I've addressed this in earlier threads and won't rehash it here.

Math is the study of consequences. The axioms that are posited do not need to have any consistency with reality as we know it - the only requirement is that they be consistent as a set of axioms.

Drop the term 'pseudosciences' - it adds nothing to the discussion, and is likely to be misunderstood as woo.

 

[TuftedPuffin]

I don't regard math or logic as sciences at all, let alone 'pseudosciences'.

I've addressed this in earlier threads and won't rehash it here.

Math is the study of consequences. The axioms that are posited do not need to have any consistency with reality as we know it - the only requirement is that they be consistent as a set of axioms.

Drop the term 'pseudosciences' - it adds nothing to the discussion, and is likely to be misunderstood as woo.

In this case, woo was kinda the comparison I was making, though I agree that there are huge differences.

Math is the study of consequences, but it also makes claims. For example, the real number system is presented as equivalent to our intuitive concepts of number. The consequences are the rigorous stuff, but the choice of axioms is often based on achieving some "real world result", despite the fact that math alone can't judge whether the result accords with the real world. That's why I compare math to pseudoscience.

 

[Complexity]

What does it fail to capture?

What is incompleteness if not asking meaningless symbols to be meaningful and then asking them what their meaning is?

I'll use 'iff' as an abbreviation for 'if and only if'

We're working with statements - strings of symbols that are considered to be 'well-formed' if they conform to a specified grammar. One such grammar is that of the first-order predicate calculus; another is that of the second-order predicate calculus. More on these later.

An interpretation of a statement or a set of statements provides meaning to the symbols in the statement(s). For every interpretation of a set of statements, each statement in the system is either true or false under that interpretation.

A statement that is true under every interpretation of the system is called a tautology.

A statement that is false under every interpretation of the system is called a contradiction.

A set of statements is consistent if it they don't imply a contradiction.

Pick a consistent set of well-formed statements and assume that they are true. This set is our set of axioms.

Some statements may exist that are true iff each of the axioms is true (under interpretation). These statements are called consequences of the set of axioms. These statements are also called theorems.

A set of axioms and the set of its theorems are called a theory.

Another important idea, besides consequence, is that of derivability.

A logical calulus is a set of procedures, a logical machine, for taking one or more statements and constructing new statements. It also provides the grammar that statements must conform to to be considered well-formed.

One such logical calculus is called the first-order predicate calculus (FOPC). Another logical calculus, the second-order predicate calculus (SOPC), enhances the first-order predicate calculus by adding in the ideas of 'for every' and 'there exists'.

Whenever we use the procedures of a logical calculus to construct a new statement from one or more existing 'input' statements, we say that we are deriving the new statement from the 'input' statements, and that the new statement is derived from the 'input' statements.

Which statements can we feed into the logical calculus machine? Any of the axioms and any statements which were derived from the axioms, directly or indirectly.

We will say that a statement is derivable iff there is a sequence of derivations that starts with the axioms and results in the construction of that statement.

A logical calculus is sound iff, for every set of axioms that conform to the calculus, every derivable statement is a consequence of that set of axioms.

In other words, given the input of axioms or theorems, a sound logical calculus can only construct theorems.

A logical calculus is complete iff, for every set of axioms that conform to the calculus, every theorem is derivable.

Goedel proved that the first-order predicate calculus is complete; that is, every true statement of a FOPC theory, every theorem, can be derived from its set of axioms. That impressed mathematicians.

Goedel then proved that the second-order predicate calculus is incomplete; that is, there exist SOPC theorems that can not be derived, statements that are true that can not be proved to be true.

Furthermore, Goedel proved that if you try to patch up the calculus so that this true statement becomes derivable, you inevitably result in a calculus that has other true statements that are not derivable. You can't ever, ever fix it. Any logical system of sufficient complexity has theorems that can't be proven.

This shocked the hell out of mathematicians.

I hope that this made some sense. I left out a lot and probably screwed something up.

 

[cyborg]

This shocked the hell out of mathematicians.

I hope that this made some sense. I left out a lot and probably screwed something up.

Complexity - you think I don't understand this. I do. What I am trying to do is give the fundamental 'why' as to 'why we should not be shocked by this result' because to me it is not just correct: if you frame the problem in the correct way things being the other way would be just plain weird.

Mathematics is powerful enough to ask about its own 'meaning'. It does, however, remain totally meaningless.

1 does not have a meaning. 0 does not have a meaning. X does not have a meaning. Y does not have a meaning.

When you build a mathematics from these meaningless symbols and then assign meaning to them (from the human perspective) you can then ask questions which equate to shifting these symbols about until there is a result.

Building atop that you can build questions about questions. Once you do that you can ask: "Is the meaning of this statement correct?"

Whilst this is a well-formed question in our formal system it is as meaningless as the symbols that formed the system. The question cannot be answered - the meaning is arbitrary because the output symbols are just as meaningless as the input symbols. The only way you can recognise a 'meaning' is to attach a computation between these symbols. But there are an arbitrary number of ways of linking these symbols together. You are left then again being forced to recognise a 'meaning' in this and so on ad infinitum ad infinitum.

This is why Godel numbering is so clever - by expressing the computation in the same way he expressed the result he could produce the fundamental fallacy of thinking that one set of symbols was fundamentally 'meaningful'. By making it all numbers we are reduced to asking the absurd question:

"Is this set of numbers more meaningful then this set when they are the same set?"

The question is well-formed but meaning can only occur in a system with bias - the axioms. A formal system where symbols come together in purely arbitrary ways would be, I'm sure you'd agree, totally chaotic. But we are forced to make such arbitrary decisions in order to get 'meaning'.

I blame our brains for having this confusion about their own nature.

BTW what meaning do you think this set of symbols might have?

0Y1XY0X1XY0X1XY1XYXY1XYXYXYXY

And what about this set?

01XYXY11010X01X010XY1Y0XY1Y0Y1

In a way one might also say that the notion that such a thing as a formal system exists is the greatest fallacy of all...

 

[Complexity]

Sigh.

Meaning is given to a theory through an interpretation. Many interpretations are usually possible.

At most levels of mathematics, mathematicians work with some interpretation (usually a common interpretation) in mind.

I have no idea what, if any, interpretation might be valid for your strings of symbols. You have provided no grammar.

I disagree with you about formal systems, of course.

 

[Ichneumonwasp]

I don't think so.

Godel was about the impossibility of a system of logic that is inclusive of all truth.
Jetlag is asking about the actual rules of logic, or how you get from premises to conclusions.
Not that I understand why he asks such a question.

BJ

Um, no he's not. You may not be aware of his earlier posts in other threads. I was trying to help him out. Godel is what he is really asking about. He is trying to find room for God. Godel thought he had (or, at least room for the Platonic Forms).

He is really asking about the foundations and limitations of logic.

 

[cyborg]

Meaning is given to a theory through an interpretation. Many interpretations are usually possible.

Yes.

So you're going to have to tell me: how does an interpreter prove its own validity in this case?

At most levels of mathematics, mathematicians work with some interpretation (usually a common interpretation) in mind.

Of course. There's no problem until you start interpreting interpreters.

I mean, I presume that you believe that the mind is isomorphic to some mathematical system, some style of computation?

I have no idea what, if any, interpretation might be valid for your strings of symbols. You have provided no grammar.

Exactly. But could you create one and would you find one grammar more meaningful than another? Why would you choose that grammar? Can the grammar absolutely justify this choice?

I disagree with you about formal systems, of course.

That formal systems exist?

Time to weigh the memes.

 

[BillyJoe]

Um, no he's not. You may not be aware of his earlier posts in other threads. I was trying to help him out. Godel is what he is really asking about. He is trying to find room for God. Godel thought he had (or, at least room for the Platonic Forms).

He is really asking about the foundations and limitations of logic.

No, my first exposure to jetlag.
Now I understand why I didn't understand why he asked that question.
He was actually asking a different one.

 

[JetLeg]

Yes.



If you knew anything about logic, you'd not need to frame such a question.

It's True!

Ok, please explain.

 

[JetLeg]

Um, no he's not. You may not be aware of his earlier posts in other threads. I was trying to help him out. Godel is what he is really asking about. He is trying to find room for God. Godel thought he had (or, at least room for the Platonic Forms).

He is really asking about the foundations and limitations of logic.

Actually, I am much less sophisticated than that.

If you cannot prove logic, then how can you be so sure of it that you disprove god with it?

 

[streamlet]

I'll take a stab at that.

Skeptics aren't saying, "We proved there is no god!" What people are saying is, "At this time, we have found no evidence of god that doesn't have another, simpler, explanation. This could change when new evidence appears."

 

[BillyJoe]

Skeptics aren't saying, "We proved there is no god!" What people are saying is, "At this time, we have found no evidence of god that doesn't have another, simpler, explanation. This could change when new evidence appears."

Actually, some sceptics do say: "There is no god".
But, you are correct, what they actually mean is something like: "At this time, we have found no evidence of god that doesn't have another, simpler, explanation. This could change when new evidence appears."

 

[cyborg]

If you cannot prove logic, then how can you be so sure of it

You're asking about two different types of certainty.

1) How can we be sure logic 'works' on a mechanical level? Because it's very simply constructable.
2) How can we be sure logic 'works' on an explicative level? Because we can make it reflect the world effectively.

that you disprove god with it?

You have to give 'god' meaning first, not ask us to remove meaning from it.

 

[Ichneumonwasp]

Actually, I am much less sophisticated than that.

If you cannot prove logic, then how can you be so sure of it that you disprove god with it?

We cannot disprove God with or without logic. We can use logic and evidence to disprove certain views of God, but not views of God.

 

[JetLeg]

You're asking about two different types of certainty.

1) How can we be sure logic 'works' on a mechanical level? Because it's very simply constructable.
2) How can we be sure logic 'works' on an explicative level? Because we can make it reflect the world effectively.



You have to give 'god' meaning first, not ask us to remove meaning from it.

'god' is usually said to be mysterious, unknowable, beyond human abilities to understand, so one should not really talk about the meaning I think, but -

'god' is more about attaching emotions to it, transcendent feelings, contemplations of the sacred, mystery and deep feelings...

 

[cyborg]

'god' is usually said to be mysterious, unknowable, beyond human abilities to understand, so one cannot really talk about his meaning I think.

You use of language betrays you. You say it is mysterious, unknowable and so on then personify the thing you are talking about by calling it a 'him'.

'god' is more about attaching emotions to it, transcendent feelings, contemplations of the sacred, mystery and meaning of life...

Yes. Concepts attach themselves to other concepts.

Linking the concepts of 'god' and 'real' isn't meaningful if it is not knowable.

 

[JetLeg]

I could actually call it a her, since it is more of an 'it'

 

[cyborg]

You could call it anything you want - but the meaning you have arising from doing so is purely synthetic unless you can link it to something knowable.

 

[slingblade]

Ok, please explain.

Not my job. You are able to educate yourself, as I educated myself. The fact that you asked a question out of ignorance, and that I chose to give a terse answer, in no way obligates me to you.

I chose to take an intro course in logic when I attended college a couple of years ago. Because of that course, I've done independent study in the subject, on my own. My life experience (48 years' worth), combined with my personal thought processes, have made it clear to me that my life is better with a grounding in logic than it was without it.

But I did not come to that conclusion after I convinced someone to lead me by the nose to it. In fact, I feel that if you are asking other people to convince you logic is worthwhile, you aren't ready for logic yet.

I see that "proof of God" has come up in the thread. God can't be proven or disproven. Logic has helped me form my own conclusion that there probably isn't a god or gods, and that my belief in it/them has allowed me to harm myself for many years. I chose to eliminate that harm. I choose to not believe in god(s) any longer. I choose to believe in nothing that I can't prove to myself, and if that proof is available, then it's no longer a belief, but a knowledge, a gnosis.

However, all of this is quite personal. Your mileage will most definitely vary. If you want to know about logic, study it. Form your own opinions, after an honest and sincere exploration--of this, or any other subject that interests you. But the spoon is broken. You can't be fed with it.

Why would you want to be?