Is all logic/reasoning circular?

[yougene]

No it doesn't. It only applies to

consistent formal, computably enumerable theories that prove basic integer arithmetical truths.​

Those are the hypotheses of the theorem, and thus are the only things that this theorem can prove incomplete.

True enough, however the nature of axioms are universal regardless of what construct they are used in.

The elementary theory of Euclidean geometry and the elementary theory of real number arithmetic are notable examples of consistent formal, computably enumerable, complete theories. (In fact, they are essentially the same theory)

I'm just an armchair thinker here so you'll have to forgive my lack of in depth experience with either one of those. My understanding is it's impossible to have a construct be both simultaneously complete AND consistent due to the fact that all constructs rest on axioms. It's just the nature of the relationship we have with reality. We can clearly see that 1 + 1 = 2, and use that as an axiom to build up the construct consistently however you can't use that construct to prove that 1 + 1 = 2, since the construct assumes that it's true already. In the end it would be like saying 1 + 1 = 2 therefore 1 + 1 = 2.

Furthermore, I'm fairly certain that there exist consistent formal theories that can prove their own consistency. (which, of course, requires that the theory fails to include integer arithmetic, or that it fails to be computably enumerable)

A theory can prove it's own consistency, it just can't be complete at the same time.

No, they are said to be incomplete in the sense that there exists a statement in its language that it can neither prove nor disprove.

True enough

 

[yougene]

IMHO, the main contribution of this theorem is that such a thing can be done, so it's worth seeing what else might be incomplete, not that it says anything about anything else.

True, the thing is it applies to all axiomic systems. Basically everything.

At some point, you have to reach axioms. I don't know of any such infinite regression. If you try to force it, you'll just end up restating stuff.

In a consistent system you end up where you started. What can be done is the axioms can be pawned off to another construct to test it's provability(out of the box so to speak). The problem with this is the other construct itself rests on axioms(whose own axioms can be pawned off to another construct that rests on axioms to infinity).

I don't have any experience with this however an earlier conversations noted to me that the infinite nature of number set theory illustrates this as well.

 

[Hurkyl]

My understanding is it's impossible to have a construct be both simultaneously complete AND consistent due to the fact that all constructs rest on axioms.

And I'm telling you that your understanding is incorrect.

The content of Tarski's theorem¹ is that the elementary theory of real arithmetic², or equivalently the elementary theory of Euclidean geometry, is simultaneously consistent complete, consistent, and computably enumerable.


An easier theorem is to prove that any consistent theory can be extended to become a complete, consistent theory. From this it follows that there does exist a consistent and complete theory of integer arithmetic. (so, by Gödel's theorem, the completion must not be computably enumerable)

In the end it would be like saying 1 + 1 = 2 therefore 1 + 1 = 2.

And that's a perfectly valid proof.

A theory can prove it's own consistency, it just can't be complete at the same time.

Do you have a reference for that? I can't seem to find one.


1: alas, there are a few that go by this name, and Wikipedia doesn't have an entry for the one I'm talking about

2: if you want a buzzword to start investigating, try real closed field.

 

[yougene]

And I'm telling you that your understanding is incorrect.

The content of Tarski's theorem¹ is that the elementary theory of real arithmetic², or equivalently the elementary theory of Euclidean geometry, is simultaneously consistent complete, consistent, and computably enumerable.

I fail to see how this doesn't rest on unproveable axioms(e.g 1 + 1 = 2).

Maybe I'm not interpreting the Godel's incompleteness theory correctly but all math is essentially sitting on unproveable givens(as apparent as they can be).

And that's a perfectly valid proof.

It's a perfectly valid observation, a perfectly valid observation but you can't "prove" its validity by using mathematics(since mathematics simply assumes it's true). You can only observe it. As such it's proof lays outside of the formal system in direct observation.

Do you have a reference for that? I can't seem to find one.

It's just Kantian philosophy. It's not specific to mathematics.

1: alas, there are a few that go by this name, and Wikipedia doesn't have an entry for the one I'm talking about

2: if you want a buzzword to start investigating, try real closed field.

Interesting stuff, I just may have to try and wrap my head around it.

 

[honestrosewater]

It seems like "proof" is being used in more than one way. To some people, an axiom is a proof of itself, so there is no such thing as an "unproveable axiom".

What do you take "proof" to mean? Logic and mathematics have defined the term for themselves already. You seem to be using a different meaning.

Also, a deductive system doesn't even need to have any axioms. You can have rules that allow you to deduce sentences from "nothing" (or the empty set).

 

[yougene]

It seems like "proof" is being used in more than one way. To some people, an axiom is a proof of itself, so there is no such thing as an "unproveable axiom".

What do you take "proof" to mean? Logic and mathematics have defined the term for themselves already. You seem to be using a different meaning.

I'm talking about proving somethings existence using a formal system without being circular.

Also, a deductive system doesn't even need to have any axioms. You can have rules that allow you to deduce sentences from "nothing" (or the empty set).

The process of deduction introduces givens that aren't neccasarily proven. Assumptions about our relationship to reality, logic, etc...

 

[honestrosewater]

I'm talking about proving somethings existence using a formal system without being circular.

So you are talking about a formal proof, then? How about the following definition?

Definition. If L is a formal language and F is a set of L-formulas, a proof of an L-formula f_n from F is a non-empty finite sequence of formulas f₁, f₂, ... f_n such that, for each f_i, at least one of the following is true:
1) f_i is an axiom
2) f_i is in F
3) f_i follows by your rules from some combination of L-formulas occurring earlier in the sequence.

There's no point in continuing to use words whose meanings aren't agreed on, yes?

The process of deduction introduces givens that aren't neccasarily proven. Assumptions about our relationship to reality, logic, etc...

What is "the process of deduction"? A proof? What is a given? An axiom? Are you again claiming, with different words, that there exist unprovable axioms?

What formal deductive system says anything about reality? Formal deductive systems don't actually "say" anything in the usual sense. If you want to interpret them as making promises that they cannot keep, that sounds like it's not their problem.

And how is that a response to my comment? You seemed to have problems with axioms, so I thought it might be helpful to point out that you don't necessarily need them.

I suppose this thread might be on the way back to "but how do you know that your deductive system(/your rules/your car/the key to a lock) works?".And by the bye, what exactly is wrong with being circular in the sense of having your conclusion be one of your premises, besides perhaps usually being a waste of time?

 

[yougene]

So you are talking about a formal proof, then? How about the following definition?

Definition. If L is a formal language and F is a set of L-formulas, a proof of an L-formula f_n from F is a non-empty finite sequence of formulas f₁, f₂, ... f_n such that, for each f_i, at least one of the following is true:
1) f_i is an axiom
2) f_i is in F
3) f_i follows by your rules from some combination of L-formulas occurring earlier in the sequence.

There's no point in continuing to use words whose meanings aren't agreed on, yes?

I'm not talking about a proof. I'm talking about proof as in being able to prove it.

I agree, it was probably a mistake to bring Godel's theorem into this. What I'm talking about here isn't particular to Math.

What is "the process of deduction"? A proof? What is a given? An axiom? Are you again claiming, with different words, that there exist unprovable axioms?

A given is something that seems to be readily apparent and is assumed to be true. An axiom is considered a given yes.

What formal deductive system says anything about reality? Formal deductive systems don't actually "say" anything in the usual sense. If you want to interpret them as making promises that they cannot keep, that sounds like it's not their problem.

And how is that a response to my comment? You seemed to have problems with axioms, so I thought it might be helpful to point out that you don't necessarily need them.

I assumed people would be ok with me using the term axioms outside of mathematics. Myth of the given seems to be the more appropriate phrase when talking generally. A deduction begins with a premise which is the given. We can split hairs here but it's essentiallly the same thing.

And by the bye, what exactly is wrong with being circular in the sense of having your conclusion be one of your premises, besides perhaps usually being a waste of time?

There's nothing wrong with it, I was just pointing out that it's not neccasarily circular because the givens can then be validated from another perspective(which also rest on givens)

 

[Hurkyl]

I'm not talking about a proof. I'm talking about proof as in being able to prove it.

What do you mean by "to prove" if you don't mean "to provide a proof"?

 

[yougene]

What do you mean?

 

[kmarinas86]

What do you mean?

=P

Look, all logic relies on definitions. A logical statement must have defined components. They are "defined", therefore they rest on axioms. You cannot have logic without conventional methods of reasoning. Nevertheless, reasoning doesn't have to be circular.

claim 1: A therefore B
claim 2: B therefore C
result: A therefore C

A, B, and C

But just because A, B, and C are true, does that really mean that A therefore B, and B therefore C?

Suppose we know these:

fact 1: A therefore B
fact 2: B therefore C
result: A therefore C

Then by knowing that A is true, then we can know that B and C are true. But knowing A, B, and C is not the clincher that establishes A therefore B, and B therefore C. Ways to make this estabilished (in the sciences) involve making it acceptable by the (scientific) parties involved with the subject. This is done groupwise.

An example of circular reasoning is this:

I don't have a job because I don't have any skills.
I don't have any skills because I didn't go to school.
I didn't go to school because I didn't have any money.
I didn't have money because I didn't have a job.

Circular reasoning can be used to describe viscious and virtous circles. They are valid arguments and can describe our natural world, yet they are circular.

So don't think of circular reasoning so negatively. They can and do make logical descriptions of reality. On the other hand, is all reasoning circular?

Well if there is a strict uni-directional casuality between A and B, then the system can be described without circular reasoning; actually it must not involve circular reasoning. For example:

fact 1: There are only two one pound balls on a given weight scale.
result: There are two pounds on this weight scale.

However, you cannot determine that there are two one pound balls simply by knowing the weight on the scale. Therefore sound circular reasoning concerning this is impossible. Circular reasoning is avoided when one reasons from the specific material to the general material, since you cannot logically deduce something specific (e.g. the two balls each weighing a pound on the scale) from a generality of what exists (e.g. the two pounds on the scale).

I can assure you that not all logic is circular. Also, we do not necessarily arrive a definition of words through deductive reasoning but often through trial an error (e.g. discovering what is a "cow"). Begging the question can be certainly avoided by avoiding deductive reasoning and sticking with inductive reasoning.

 

[yougene]

Then by knowing that A is true, then we can know that B and C are true. But knowing A, B, and C is not the clincher that establishes A therefore B, and B therefore C. Ways to make this estabilished (in the sciences) involve making it acceptable by the (scientific) parties involved with the subject. This is done groupwise.

Turn A, B, and C into a real statement, digg down deep enough, and you'll end up where you started.

An example of circular reasoning is this:

I don't have a job because I don't have any skills.
I don't have any skills because I didn't go to school.
I didn't go to school because I didn't have any money.
I didn't have money because I didn't have a job.

Circular reasoning can be used to describe viscious and virtous circles. They are valid arguments and can describe our natural world, yet they are circular.

So don't think of circular reasoning so negatively. They can and do make logical descriptions of reality. On the other hand, is all reasoning circular?

I have nothing against circular reasoning I find it rather useful.

Well if there is a strict uni-directional casuality between A and B, then the system can be described without circular reasoning; actually it must not involve circular reasoning. For example:

fact 1: There are only two one pound balls on a given weight scale.
result: There are two pounds on this weight scale.

However, you cannot determine that there are two one pound balls simply by knowing the weight on the scale. Therefore sound circular reasoning concerning this is impossible. Circular reasoning is avoided when one reasons from the specific material to the general material, since you cannot logically deduce something specific (e.g. the two balls each weighing a pound on the scale) from a generality of what exists (e.g. the two pounds on the scale).

The problem with that is we never ever directly experience the two balls or the scale. Our experience of the other is always mediated through the mind(culture, constructs, etc..). As such all similar thinking is laid down on assumptions that we hold about our relationship to the outside world. Once you start digging deep enough in the foundations of your reasoning, you start running in circles.

 

[honestrosewater]

I'm still confused about why you consider the axioms to be any more or less "provable" than the other theorems that you derive from them. All that a proof of Q from P says is that Q follows from P by your rules. From outside, a proof of Q from P doesn't give you Q. It gives you a proof of Q from P.

And why this relation (you can view syntactic and semantic consequence as relations just like any other relations) should bother someone more than other relations (does anyone have a problem with equality?) also escapes me.

 

[yougene]

I'm still confused about why you consider the axioms to be any more or less "provable" than the other theorems that you derive from them. All that a proof of Q from P says is that Q follows from P by your rules. From outside, a proof of Q from P doesn't give you Q. It gives you a proof of Q from P.

I consider them on equal footing.

How do we get P? And how did we get that? and so on?

And why this relation (you can view syntactic and semantic consequence as relations just like any other relations) should bother someone more than other relations (does anyone have a problem with equality?) also escapes me.

I don't have a problem with it. I was just discussing the limitations of only being able to observe reality through mediated constructs. That's not saying the rational pov is a mediating construct and the pre-rational pov is not. All of our experience of the outside world is mediated.

 

[honestrosewater]

How do we get P? And how did we get that? and so on?

Well, outside of the system, you don't get P. You can get a proof of P from P (and so on).

I don't have a problem with it. I was just discussing the limitations of only being able to observe reality through mediated constructs. That's not saying the rational pov is a mediating construct and the pre-rational pov is not. All of our experience of the outside world is mediated.

Oh. Roger.

Well, that is perception for you, I guess.

 

[JoeDawg]

:rofl:
Maybe I am asking about truth. Can we really find truth using the system of logic we have created?

Logic depends on premises. Within the context of those premises you can determine if something is true or false. Similarly one could argue that one would have to know the initial state of a given system to be able to state any 'absolute' truth about it. As to how that applies to 'reality', this is why knowing how the universe began is important.