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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://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 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 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.)
 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 set-builder notation: $\lbrace \rbrace ( x_1 : \varphi_1 , \dots , x_m : \varphi_m , \varphi )$ stands for the expression $\lbrace \varphi | x_1 \in \varphi_1 , \dots , x_m \in \varphi_m \rbrace$ 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 $K(n)^+$ on page 10, you see there you have to build $h || ( y, \varphi)$ , 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
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Re: new preprint about an interesting new approach to logic

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 Originally Posted by avonm about the set-builder notation: $\lbrace \rbrace ( x_1 : \varphi_1 , \dots , x_m : \varphi_m , \varphi )$ stands for the expression $\lbrace \varphi | x_1 \in \varphi_1 , \dots , x_m \in \varphi_m \rbrace$ 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, we’ll be able to define the meaning of t with respect to k and ?, which we’ll 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 $K(n)^+$ on page 10, you see there you have to build $h || ( y, \varphi)$ , 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.

 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 set-builder notation: i just tried to explain that, in my system, the meaning of the expression $\lbrace \rbrace ( x_1 : \varphi_1 , \dots , x_m : \varphi_m , \varphi )$ will be the same meaning that expressions like $\lbrace \varphi | x_1 \in \varphi_1 , \dots , x_m \in \varphi_m \rbrace$ 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

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Re: new preprint about an interesting new approach to logic

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 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:

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:
 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.

November 4th, 2011, 06:29 AM   #7
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Re: new preprint about an interesting new approach to logic

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 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).

November 4th, 2011, 06:33 AM   #8
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Re: new preprint about an interesting new approach to logic

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 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?

November 4th, 2011, 06:59 AM   #9

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Re: new preprint about an interesting new approach to logic

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 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

 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

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Re: new preprint about an interesting new approach to logic

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 Originally Posted by avonm I don't think it can be formalized within the system itself, how will you do this?
correct this sentence: I don't think it can be formalized within the instance itself, how will you do this?

 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 3-tuple (k,t,?) to its meaning within S2. We indicate with #1 the function that maps each 3-tuple (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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