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Topic: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Replies: 39   Last Post: Oct 14, 2012 11:56 PM

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 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 First-Order 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 first-order 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 first-order 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 truth-preserving) 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.

Date Subject Author
10/5/12 Graham Cooper
10/5/12 Frederick Williams
10/7/12 Charlie-Boo
10/5/12 Graham Cooper
10/5/12 Frederick Williams
10/5/12 Graham Cooper
10/7/12 Graham Cooper
10/8/12 Graham Cooper
10/9/12 Graham Cooper
10/11/12 Graham Cooper
10/12/12 Graham Cooper
10/12/12 Graham Cooper
10/12/12 camgirls@hush.com
10/12/12 Richard Tobin
10/12/12 camgirls@hush.com
10/13/12 george
10/13/12 Graham Cooper
10/14/12 george
10/13/12 Graham Cooper
10/13/12 george
10/13/12 george
10/13/12 Graham Cooper
10/14/12 Graham Cooper
10/14/12 Graham Cooper
10/14/12 Graham Cooper
10/5/12 Scott Berg
10/5/12 Curt Welch
10/6/12 Mike Terry
10/6/12 Graham Cooper