Logic, can it be for everything?

[Prague]

Hey, I found PF and have been posting in the physics forums. I learned some stuff there and figured why not try out something else, I may learn something. So what is this Logic stuff? Can this-- which looks like math-- actually solve everyday things? Or longterm things for that matter?

 

[C0mmie]

In short, the answer to your question is: No.

Have you looked at Godel's theorem? It shows that, from what I understand, that any axiomatic system has to be incomplete, in the sense that there will aways be true statements that you cannot derive within the system. There are probably people on this forum that have a better understanding of this than I do, so correct me if I'm wrong.

Oh, and also, an application of logic requires certainty as far as the premises go, so you can prove that IF you accept the axioms THEN you have to believe something else is true, but what necessitates us to accept one set of premises or axioms over another?

 

[Pengwuino]

@Prague

Get a girlfriend and then you won't even have to ask this question :P

 

[honestrosewater]

In short, the answer to your question is: No.

Have you looked at Godel's theorem? It shows that, from what I understand, that any axiomatic system has to be incomplete, in the sense that there will aways be true statements that you cannot derive within the system. There are probably people on this forum that have a better understanding of this than I do, so correct me if I'm wrong.

Google Gödel's Completeness Theorem. Do you know what an axiomatic system is? If not, that would be a good place to start.

 

[C0mmie]

Google Gödel's Completeness Theorem. Do you know what an axiomatic system is? If not, that would be a good place to start.

A set of rules for deriving true statements.

 

[honestrosewater]

A set of rules for deriving true statements.

And how do you know if a statement is true? You need some other things too. Here's a quick outline.
You start with a language that contains a set of symbols. You string the symbols together to get a set of strings. You select some of the strings to get a set of formulas.
You define a valuation that tells you whether each formula is true or false. If a formula is true under every valuation (i.e. if it is always true), that formula is called a tautology.
You then define a calculus which consists of a set of axioms and a set of inference rules. If a formula can be derived from the calculus, that formula is called a theorem. Now, soundness and completeness are properties of calculi. A calculus is sound iff, for any formula F, if F is a theorem, then F is a tautology. A calculus is complete iff, for any formula F, if F is a tautology, then F is a theorem.
Everything above, minus the valuation and tautologies, is called a theory (or system). If every axiom or a set of rules to determine which formulas are axioms is given, then the theory is called an axiomatic theory. If a theory has finitely many axioms or there can be given a finite set of rules to determine which formulas are axioms (i.e. an algorithm), then the theory is called axiomatizable or finitely axiomatizable.
Make sense? Do you have a statement of Gödel's Completeness or Incompleteness Theorems around? Edit: If not, you can search PF; They've been discussed many times here. Hurkyl and matt grime are reliable sources.

 

[C0mmie]

@honestrosewater

I wasn't looking for an explanation of what logic is. I've studied it for quite a bit and I'm taking a symbolic logic course right now. I just wasn't sure if my understanding of Godel's Theorem was correct. Yes, the other posts on this subject did turn out to be helpful.

 

[honestrosewater]

@honestrosewater

I wasn't looking for an explanation of what logic is. I've studied it for quite a bit and I'm taking a symbolic logic course right now. I just wasn't sure if my understanding of Godel's Theorem was correct. Yes, the other posts on this subject did turn out to be helpful.

Okay, great- you'll probably see a completeness proof soon then. I was just going on your earlier statements and considering the other people who might be reading the thread. Glad others were helpful.