Fundamentals of Logic

[pgrmdave]

What are the fundamental ideas of logic that cannot be proven but must be accepted as true for any logical system to be true?

[Qfwfq]

What are the fundamental ideas of logic that cannot be proven but must be accepted as true for any logical system to be true?

I couldn't list all inference rules offhand but they haven't changed much since Aristotle wrote his book entitled Logic.

[Qfwfq]

This appears to be reasonable:

 

http://encyclopedia.laborlawtalk.com/List_of_rules_of_inference

[WhitePhoenix]

There are many categories and subjects of logic and the study of epistemology...( I finally get to use that word) Is there a specific topic you are interested in?

[Buffy]

Since you're still in school, try to find a course on "Predicate Logic" which seems like math, but when I was at Berkeley, it was taught in the Philosophy department and its better that way. To get at the crux of your question, it really boils down to Kurt Goedel's Incompleteness Theorem which you can Wiki here: http://en.wikipedia.org/wiki/Incompleteness_theorem

 

Logically,

Buffy

[Qfwfq]

"Predicate Logic" which seems like math

It's somewhat the other way around, logic is what math is based on. Goedel comes much later and after number theory.

 

I took a very brief spin through the site I found yesterday, it seems like a good course, starting from the start. It even has Lewis Carrol's amusing spoof on modus ponens, which illustrates what you said, we don't prove the inference rules, they are obviously reasonable and that's that, so we use them. If you want to go deeper into this disquisition you might like the concept of more general formal systems, abstractions which might not use logic at all.

 

There is also the related matter of formal systems which do include the inference rules of logic but add one or more axioms as well. Euclid's geometry and other possible geometries are a notable historic example. Look up Euclid + Riemann for this.

[Guest loarevalo]

Well, of course, the most fundamental assumption of logic is the existence of its object:

 

That Truth exists.

 

Along with that assumption there must be some definition (as to establish the possibility of Falsehood). Then again, even if the existence of Truth is only assumed (and it makes no sense to talk about proving that it does exist), the definition provided for it is entirely open - depending on the logical system. Some Truth can be relative, other absolute under all circumstances, Truth could also be ambiguos (like fuzzy logic) and there may also be indeterminancy.

 

It's not so simple as to T and F - even if it were, that would be some BOLD assumption.

[Qfwfq]

Logic can be treated by simply considering true and false as two values, an expression may have a value of true or false, tertium non datur. As such, it is a very fundamental branch of Mathematics.

 

Change this, or change the inference rules, and you are talking about a different formal system, e. g. "fuzzy logic" is a different formal system. There is no sense in proving or disproving a formal system, there is only sense in talking about how well it applies to something.

 

Along with that assumption there must be some definition (as to establish the possibility of Falsehood).

To put it very simply, the existence of negation in formal logic. In a formal system without negation, there's no such thing as proving an assert false.

 

Strictly, this is also supposing that an assert can't be simultaneously true and also false. However, there is an amusing theorem showing that this would make no sense.

[Guest loarevalo]

Strictly, this is also supposing that an assert can't be simultaneously true and also false. However, there is an amusing theorem showing that this would make no sense.

I agree. I would like to see that theorem - not that I don't believe; I just would like to see a formal argument stating that that true and false are disjoint, which argument must rely on some logic as well. I just doubt that such a theorem could really prove it, as a system cannot prove itself - or prove that itself is consistent.

[Qfwfq]

I just would like to see a formal argument stating that that true and false are disjoint

The theorem doesn't prove it, but it shows that it wouldn't make sense not to suppose it. One can just as well take it as an axiom.

 

It's amusing though that one can prove that, if an assert A and its negation are both true, then any assert B and its negation are true. Obviously, logic would be pointless if we didn't rule this out.

[kamil]

What are the fundamental ideas of logic that cannot be proven but must be accepted as true for any logical system to be true?

 

 

The fundamental law of logic which builds other laws of logic is:

Something is right if it is not impossible. In other words:

Something is right if things that contradict them are impossible.

e.g: A triangles internal angles add up to 180 degrees because you can not imagine a triangle whos angles add up to something else.

 

This law of logic, as you see, is the foundation of mathematics.

[Qfwfq]

I must disagree with that. Neither logic nor mathematics are based on "you can not imagine a....".

 

Given Euclid's axioms, it can be proven that all triangles have internal angles totalling pi radians. Change the last axiom and you can prove that they will always add up to less than pi, or always more than pi, with pi being the limit value.

[infamous]

In this age of relativism, logic has lost some strength in it's attempt to define absolute truth. Where there are no absolutes, logic suffers.

[Qfwfq]

Logic isn't based on absolutes.

[Guest loarevalo]

It's amusing though that one can prove that, if an assert A and its negation are both true, then any assert B and its negation are true. Obviously, logic would be pointless if we didn't rule this out.

I guess what you meant was that, if A was both true and false, there would be no point of making the distinction between truth and falsehood. I see. That makes sense. Yet, I believe that for logic to better model our current relationship with reality, logic can't make things so simple as to Truth and False cut and dry - it just doesn't work that way in this life. I don't believe that we can have absolute certainty about anything, perhaps only about the fact that we exist and that we think and are free agents in that sense.

 

Sorry Kamil, your point is very weak and if we were to evaluate it seriously we would find it circular.

 

I don't know if Logic suffers from relativistic views of truth, but I think that absolute truth isn't necessary to it because we can't have absolute truth anyways. No one really knows or can know how things are, but we can have some general idea about it - an approximate understanding of reality. Even though a great number of us may believe in God, and thus assert "God exists," we all understand that truth in diferent levels - may I call that understanding, "awareness."

 

If logic and science suffer from anything, it is from the lack of regoction of faith in their epistemological foundation. By faith, I don't mean religion, but faith as what fills the void of uncertainty, because there is no certainty in life.

[Buffy]

I guess what you meant was that, if A was both true and false, there would be no point of making the distinction between truth and falsehood. I see. That makes sense. Yet, I believe that for logic to better model our current relationship with reality, logic can't make things so simple as to Truth and False cut and dry - it just doesn't work that way in this life.

Predicate logic does have its place in "reality": That red light is either red or green and you're not going to be successful in court trying to use Heisenberg to say it was red and green at the same time. On the other hand, there are situations when "Fuzzy Logic" is useful, especially when building self-driving cars or washing machines or deciding to execute an accused murderer: http://en.wikipedia.org/wiki/Fuzzy_logic

 

Cheers,

Buffy

[Guest loarevalo]

Predicate logic does have its place in "reality": That red light is either red or green and you're not going to be successful in court trying to use Heisenberg to say it was red and green at the same time. On the other hand, there are situations when "Fuzzy Logic" is useful, especially when building self-driving cars or washing machines or deciding to execute an accused murderer

 

I used that same example of street light being either (discreetly) red or green to point that precisely, there isn't such a thing in absolute terms as Red or Green - we don't even see Red the same way, but we all have a general idea of what Red is and that idea is suficiently broad so that most of us understand Red the same way, even though we all don't see the same Red. Because our view and understanding of the world is so fuzzy, and we can't help, is that street lights aren't Red, Orange, Pink - that would be terribly confusing. With that example I hope you recognize that discreetness doesn't exist as a matter of reality - that's just how we perceive reality.

 

Also, it's no coincidence we perceive reality sufficiently equal so that society can work - that means that reality itself isn't different for all of us, that Absolute truths do exist after all, but we just can't perceive exactly as it is and thus absolute truths have no place in our understanding.

 

In the ultimate sense, we are not guided by truth and light, but by faith.