george
Posts:
800
Registered:
8/5/08


Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Posted:
Oct 13, 2012 11:48 AM


> > NO, a deductive system IS NOT a list of formulas. > > > JEEzus.
On Oct 12, 8:37 pm, camgi...@hush.com wrote: > That's a quote from WIKI
So FUCKING what?? That's a quote in the context OF THAT treatment. And you should have NAMED the treatment or article. Nothing IS EVER "a quote from WIKI". What it IS is "a quote from the Wikipedia article on <whatever>." DIFFERENT wikipedia articles are going to treat the SAME topic DIFFERENTLY!!
Any time anybody is trying to explain anything to you, they may have to tell you what THEIR terms mean in the context of THEIR presentation. This does NOT obligate OTHER people to use those terms in the same way. FOR EXAMPLE: IF YOU GOOGLE (jointly) the pair of terms "Deductive System" and "Inference Rule", you will get, FROM THE WIKIPEDIA ARTICLE ON FirstOrder Logic, (all that follows is quoted from the article) >> Deductive systems A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for firstorder logic, including Hilbert style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. They are also often called proofs, but are completely formalized unlike natural language mathematical proofs.
A deductive system is sound if any formula that can be derived in the system is logically valid. Conversely, a deductive system is complete if every logically valid formula is derivable. All of the systems discussed in this article are both sound and complete. They also share the property that it is possible to effectively verify that a purportedly valid deduction is actually a deduction; such deduction systems are called effective.
A key property of deductive systems is that they are purely syntactic, so that derivations can be verified without considering any interpretation. Thus a sound argument is correct in every possible interpretation of the language, regardless whether that interpretation is about mathematics, economics, or some other area.
In general, logical consequence in firstorder logic is only semidecidable: if a sentence A logically implies a sentence B then this can be discovered (for example, by searching for a proof until one is found, using some effective, sound, complete proof system). However, if A does not logically imply B, this does not mean that A logically implies the negation of B. There is no effective procedure that, given formulas A and B, always correctly decides whether A logically implies B. Rules of inference Further information: List of rules of inference
A rule of inference states that, given a particular formula (or set of formulas) with a certain property as a hypothesis, another specific formula (or set of formulas) can be derived as a conclusion. The rule is sound (or truthpreserving) if it preserves validity in the sense that whenever any interpretation satisfies the hypothesis, that interpretation also satisfies the conclusion.
For example, one common rule of inference is the rule of substitution.

