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]

Originally Posted by 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?

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...s_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]

Originally Posted by Buffy

"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.

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.

Originally Posted by loarevalo

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.

Originally Posted by Qfwfq

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]

Originally Posted by loarevalo

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.