
Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Posted:
Oct 13, 2012 4:30 PM


On Oct 14, 1:48 am, George Greene <gree...@email.unc.edu> wrote: > > > 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
These are just INFERENCES mind you.
You still owe me some BASE THEOREMS, otherwise all your "proofs" are Oracular.
I'll print the table here, nice one!
http://en.wikipedia.org/wiki/List_of_rules_of_inference

RESOLUTION IS STATED AS:
((p v q) ^ (!p v r)) > (q v r)
So you have to get
xRr <> !xRr p = xRr !p = xRr
???
Mind you, only 1 inference rule is needed, modus ponens and it can pattern match the rest as axioms, since modus ponens will CREATE A NEW FORMULA as well as derive it, so there only has to be one inference rule that instantiates new formula.
But anything in the form LHS > RHS can be used to instantiate a new line, fine with me. ;)
Herc

