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November 3rd, 2011, 06:26 AM  #1 
Joined: Nov 2011 Posts: 12 Thanks: 0  new preprint about an interesting new approach to logic
I’d like to inform you of a preprint (named 'A different approach to logic') I have published in the PhilSci archive at http://philsciarchive.pitt.edu/8823/ . The paper is also archived in the arxiv site at http://arxiv.org/abs/1107.4696 . The paper is about an approach to logic that differs from the standard firstorder logic and other known approaches. It should be a new approach I’ve created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. Further details in the abstract and introduction (and in the paper, of course). The paper is written in Microsoft Word, a version in LaTeX (with other improvements, possibly) is somehow planned but (due mainly to my current forced lack of time) i cannot make a prevision when it will be available. Mauro Avon 
November 3rd, 2011, 09:19 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,826 Thanks: 84  Re: new preprint about an interesting new approach to logic
The paper is written quite informally. On one hand that makes it easy to read, but on the other it's often hard to decide the exact meaning you intend. For example, I had trouble determining the precise meaning of your setbuilder notation, the interpretation of the # operator, and the relationship between the generic strings (free monoid?) and the logic itself. Some things are mysterious to me. At first I thought an univocal soop was just a function with a finite domain, but your definition of restriction means that for a given univocal soop f (with more than, say, 2 elements in its domain), most subsets S of its domain are such that the restriction f/S does not exist. How odd! What prompted this choice? Can you explain what the purpose/intent of this "new approach" is? Can you comment on its expressiveness compared to standard logics (as you say, Enderton's)? Some parts of it are so weak that I thought it was supposed to be similar to Robinson's Q, but as I go further I see features that seem to give it more power. Is anything known about its consistency strength? (I read section 8 but I'm not convinced by its arguments.) 
November 3rd, 2011, 01:13 PM  #3 
Joined: Nov 2011 Posts: 12 Thanks: 0  Re: new preprint about an interesting new approach to logic
thanks for your comments i begin to reply to some of your remarks, in the next days i'll try to respond to the others. about the setbuilder notation: stands for the expression used in mathematics. as regards the # operator, its meaning is 'the meaning of'. page 5: For a constant c we denote by #(c) the meaning of c. page 2: If t is an expression with respect to context k and ? is a state on k, we’ll be able to define the meaning of t with respect to k and ?, which we’ll denote by #(k,t,?) . as regard the univocal soop, f/S is not defined for some subsets since i suppose i didn't need to define it for those subsets. If you find in the paper one place where i use f/S for one of those subsets please let me know, because clearly this would be a mistake (hopefully a correctable one since f/S should be definable for all the subsets). The choice of the 'univocal soop' is probably not the best one, and this is one of the improvements i plan. But you cannot use just one function because if you look at the definition of on page 10, you see there you have to build , and this requires to have something like a sequence. The domain of a function is not ordered, the domain of a univocal soop is ordered. 
November 3rd, 2011, 03:24 PM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,826 Thanks: 84  Re: new preprint about an interesting new approach to logic Quote:
Quote:
Quote:
Also, I don't understand what value K(n) has  why you even defined it.  
November 4th, 2011, 01:39 AM  #5 
Joined: Nov 2011 Posts: 12 Thanks: 0  Re: new preprint about an interesting new approach to logic
again about the setbuilder notation: i just tried to explain that, in my system, the meaning of the expression will be the same meaning that expressions like are intended to have when used in most mathematics books. again about the # operator: basically it is simply a function which associates a meaning to every constant (page 7). Then definition 2.1 (which is an inductive definition process) defines #(k,t,?) for each context k, t expression with respect to k and ? state on k. For more details you have to read definition 2.1. about K(n): this is defined inductively in definition 2.1. In fact you can see that K(1) is defined to be equal to {?} (page 10), and also on page 10 i define K(n+1) for an arbitrary positive integer n, assumed to have defined K(n). 
November 4th, 2011, 01:57 AM  #6  
Joined: Nov 2011 Posts: 12 Thanks: 0  Re: new preprint about an interesting new approach to logic Quote:
I’ve created this approach proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. And you find further details about that purpose/intent in the abstract and introduction. Quote:
1) In firstorder logic we have terms and formulas and we cannot apply a predicate to one or more formulas, this seems a clear limitation. With our system we can apply predicates to formulas. 2) In our system the setbuilder notation is enclosed as an expressionbuilding pattern, and this is an advantage over standard logic. 3) In our system we can easily express secondorder and allorder conditions. Quote:
 
November 4th, 2011, 06:29 AM  #7  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,826 Thanks: 84  Re: new preprint about an interesting new approach to logic Quote:
 
November 4th, 2011, 06:33 AM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,826 Thanks: 84  Re: new preprint about an interesting new approach to logic Quote:
 
November 4th, 2011, 06:59 AM  #9  
Joined: Nov 2011 Posts: 12 Thanks: 0  Re: new preprint about an interesting new approach to logic Quote:
Quote:
 
November 4th, 2011, 10:26 AM  #10 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,826 Thanks: 84  Re: new preprint about an interesting new approach to logic
The second incompleteness theorem. If your system can prove itself consistent then it isn't.

November 4th, 2011, 11:39 AM  #11 
Joined: Nov 2011 Posts: 12 Thanks: 0  Re: new preprint about an interesting new approach to logic
well, of course there are many instances of my system. i am not proving the consistency of one instance within the instance itself, i am proving the consistency of a generic instance, and though mine is an informal proof it can probably be formalized within another instance of my system. I don't think it can be formalized within the system itself, how will you do this?

November 4th, 2011, 11:40 AM  #12  
Joined: Nov 2011 Posts: 12 Thanks: 0  Re: new preprint about an interesting new approach to logic Quote:
 
November 4th, 2011, 11:47 AM  #13 
Joined: Nov 2011 Posts: 12 Thanks: 0  Re: new preprint about an interesting new approach to logic
trying to be more accurate, you (probably) can prove the consistency of one instance of my sytem S1 using another instance S2, but this should not violate the second incompleteness theorem

November 4th, 2011, 12:26 PM  #14 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,826 Thanks: 84  Re: new preprint about an interesting new approach to logic
Can you point out what you're using in S2 that isn't in S1?

November 5th, 2011, 11:19 AM  #15 
Joined: Nov 2011 Posts: 12 Thanks: 0  Re: new preprint about an interesting new approach to logic
Suppose we can prove the consistency of S1 within S2. We want to show that we can’t use the same proof within S1. The proof in S2 uses the symbol #. We indicate with #2 the function that maps each 3tuple (k,t,?) to its meaning within S2. We indicate with #1 the function that maps each 3tuple (k,t,?) to its meaning within S1. # is a constant symbol in S2 and #2(?,#,?) = #1. Now assume we can use # as a constant within S1, and that # has the same meaning it had in S2. Therefore #1(?,#,?) = #2(?,#,?) = #1 . Therefore #1 would be a member of its image, and as far as I know, no function is a member of its image. 

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