Scott Mayers wrote: ↑Sat May 18, 2019 4:51 pm
PeteOlcott wrote: ↑Wed May 15, 2019 9:44 pm
Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015)
1.4 An Axiom System for the Propositional Calculus page 27-28
A wf C is said to be a consequence in S of a set Γ of wfs if and only if there is a
sequence B1, …, Bk of wfs such that C is Bk and, for each i, either Bi is an axiom
or Bi is in Γ, or Bi is a direct consequence by some rule of inference of some of
the preceding wfs in the sequence. Such a sequence is called a proof (or deduction)
of C from Γ. The members of Γ are called the hypotheses or premisses of the proof.
We use Γ ⊢ C as an abbreviation for “C is a consequence of Γ”...
I only have to use these links to avoid violating copyright law.
If you really do understand formal proofs of mathematical logic
then you don't need to go to these linked textbook pages.
It you can't understand what I mean by formal proof by the above paragraph alone
then you simply lack too much of the required prerequisite knowledge.
The textbook pages take someone all the way through the whole process
of learning everything about formal proofs of mathematical logic.
What you are showing above is ONLY about the proof of completeness and consistency of Propositional Calculus. It has no means of proving you know what you've read. What I complained about linking of you is that you are NOT arguing here but referencing YOUR work elsewhere rather than attempting to reconstruct it step by step here. Because every textbook on logic differs on how they express things, you require representing how YOU understand things in an appeal to the reader here, not by expecting us to do the work for you.
So you NEED to:
State a 'thesis' statement about your position when being 'formal'.
Define your terms and symbols here UNIQUELY.
Preface your proof's argument in colloquial language and express how you are going to go about proving/disproving your thesis.
...and I'm sure many others. But you have yet to clarify precisely what you are wanting to prove, what motivated your challenge, and why you think your perception of the prior theorems is appropriate.
This single stipulated set of constraints transforms any formal system of any
level of logic into a complete and consistent formal system without any
If the notion of True(x) is defined as provable from axioms and axioms are
stipulated to be finite strings having the semantic property of Boolean true
then every expression of language that not a theorem or an axiom is not true.
Here is what one reviewer on another forum said last night:
I agree with you that if your definitions are accepted then we can eliminate incompleteness.
I had to goad him into carefully studying and testing my specification before he would say
this. Prior to this he kept dismissing my words out-of-hand as ridiculous without even
bothering to read them.