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November 3rd, 2011, 06:26 AM   #1
 
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new preprint about an interesting new approach to logic

Id like to inform you of a preprint (named 'A different approach to logic') I have published in the PhilSci archive at

http://philsci-archive.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 first-order logic and other known approaches. It should be a new approach Ive 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
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November 3rd, 2011, 09:19 AM   #2
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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 set-builder 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.)
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November 3rd, 2011, 01:13 PM   #3
 
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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 set-builder 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, well be able to define the meaning of t with respect to k and ?, which well 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.
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November 3rd, 2011, 03:24 PM   #4
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Re: new preprint about an interesting new approach to logic

Quote:
Originally Posted by avonm
about the set-builder notation: stands for the expression used in mathematics.
Yes, that's my problem. What exactly is allowed in the statement, "used in mathematics"? A formal system needs to define this. It's not clear what your system allows. (Unless it's not a new system but just a notation, in which case this is fine.)

Quote:
Originally Posted by avonm
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, well be able to define the meaning of t with respect to k and ?, which well denote by #(k,t,?) .
I read that, I just didn't know what it meant. I still don't.

Quote:
Originally Posted by avonm
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.
That's another good example of something that isn't defined. What, precisely, is K(n)? You describe it as a set of "'contexts'" but don't say what that is or which set it comprises.

Also, I don't understand what value K(n) has -- why you even defined it.
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November 4th, 2011, 01:39 AM   #5
 
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Re: new preprint about an interesting new approach to logic

again about the set-builder 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).
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November 4th, 2011, 01:57 AM   #6
 
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Re: new preprint about an interesting new approach to logic

Quote:
Originally Posted by CRGreathouse
Can you explain what the purpose/intent of this "new approach" is?
Basically that's what i said in the first message:

Ive 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:
Originally Posted by CRGreathouse
Can you comment on its expressiveness compared to standard logics (as you say, Enderton's)?
So far I haven't theorems of comparison but many remarks about this are in the abstract and introduction. Just some examples:

1) In first-order 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 set-builder notation is enclosed as an expression-building pattern, and this is an advantage over standard logic.

3) In our system we can easily express second-order and all-order conditions.

Quote:
Originally Posted by CRGreathouse
Is anything known about its consistency strength?
Consistency follows from the soundness as proved in the first lines of section 8.
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November 4th, 2011, 06:29 AM   #7
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Re: new preprint about an interesting new approach to logic

Quote:
Originally Posted by avonm
Consistency follows from the soundness as proved in the first lines of section 8.
Section 8 (as I previously intimated) is incorrect. It should be obvious from Goedel that this statement cannot hold (since you can embed Peano arithmetic into this system).
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November 4th, 2011, 06:33 AM   #8
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Re: new preprint about an interesting new approach to logic

Quote:
Originally Posted by avonm
In first-order 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.
So what advantage does it have over standard second-order (or higher-order) logic?
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November 4th, 2011, 06:59 AM   #9
 
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Re: new preprint about an interesting new approach to logic

Quote:
Originally Posted by CRGreathouse
Section 8 (as I previously intimated) is incorrect. It should be obvious from Goedel that this statement cannot hold (since you can embed Peano arithmetic into this system).
what formulation of goedel theorem do you refer to? you may be confusing incompleteness and inconsistency

Quote:
Originally Posted by CRGreathouse
So what advantage does it have over standard second-order (or higher-order) logic?
this might be an interesting topic to explore, but as far as i can remember higher order logic systems have a certain level of complexity and are based on different types of variable, so they are a bit unnatural/counterintuitive
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November 4th, 2011, 10:26 AM   #10
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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.
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