Originally posted by royalchickenHmmm, I think you may be courting some Lewis Carroll type regresses here, but I'll have to wait and see. Anyway, you claim that the assumption x on one line and ax on another will not entail that both y and ay are writable, but merely that iiaxcxigiycayii is writable. But application of Rule 5 on the assumptions, combined with application of Rule 7, allows us to write the consenquent of the symbol string you give above. This consequent is iycayi, so iycayi is writable. Hence, by application of Rule 6, this conjunction can be broke apart to yield both y and ay. Hence, both y and ay are writable. Hence, your system of rules is inconsistent.
No, you don't show that y is writable and ay is writable by assuming ax and x. You show that iiaxcxigiycayii is writable. This violates no rules, because it is not of the form izcazi that is prohibited.
It's a question of proof and subproof. I really should have followed Douglas Hofstadter's example and used something like his ''fantasy rul ...[text shortened]... orrow.
''Subproof'' is not my word--it belongs to one of the long messages bbarr sent me.
Originally posted by bbarrDamn, now I think you're right! This stuff
Hmmm, I think you may be courting some Lewis Carroll type regresses here, but I'll have to wait and see. Anyway, you claim that the assumption x on one line and ax on another will not entail that both y and ay are writable, but merely that iiaxcxigiycayii is writable. But application of Rule 5 on the assumptions, combined with application of Rule 7, allows ...[text shortened]... both y and ay. Hence, both y and ay are writable. Hence, your system of rules is inconsistent.
is too hard for me. Maybe I just need to
stick to being polite to people.
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How interesting-seeming. I'm going to have to read this when I have a couple hours to spare.
However, just to cause a little trouble... how does this system (which I have at least skimmed) account for the imperfection in any symbolic communication (including speech) by virtue of the fact that it is separate and different from the thing it supposedly signifies? See generally Derrida, "Limited, Inc." See also Plato, forms, etc.
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Originally posted by bbarrThe rules are just a shorthand for a list of every possible statement of the form iixcyiciwczi....igui.
Hmmm, I think you may be courting some Lewis Carroll type regresses here, but I'll have to wait and see. Anyway, you claim that the assumption x on one line and ax on another will not entail that both y and ay are writable, but merely that ...[text shortened]... and ay are writable. Hence, your system of rules is inconsistent.
NG doesn't differ materially from any other presentation of a propositional calculus I have seen, although you are correct. Come to think of it, I haven't seen the 'Lewis Carroll regress' dealt with anywhere, so it's possible that PCs have such problems inherently.
I'll say more, but I have to go now.
Actually, come to think of it, drop rule 13 entirely. Given the other rules, the notion that x and ax taken together imply everything can e deduced, and since there exist unwritable strings, proofs by contradiction are possible.
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Originally posted by royalchickenOK, show me.
The rules are just a shorthand for a list of every possible statement of the form iixcyiciwczi....igui.
NG doesn't differ materially from any other presentation of a propositional calculus I have seen, although you are correct. Co ...[text shortened]... re exist unwritable strings, proofs by contradiction are possible.
Premises:
1) x
2) ax
Now, please construct a derivation, applying only those rules mentioned, concluding with the following symbol string:
C) y
A derivation of this sort will show that from a contradiction you can derive any arbitrary symbol string. Also, please include in your derivation line justifications (i.e., state the Rule you are applying and the lines to which you are applying that Rule).
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Originally posted by bbarrA quick note concerning my post immediately above:
OK, show me.
Premises:
1) x
2) ax
Now, please construct a derivation, applying only those rules mentioned, concluding with the following symbol string:
C) y
A derivation of this sort will show that from a contradiction you ...[text shortened]... u are applying and the lines to which you are applying that Rule).
I've gone over your list of rules again, and, as I mentioned to you via PM, you need a rule that corresponds to normal propositional calculus' rule of disjunction introduction. That is, you need a rule such that if x is writable, then ixeyi is writable. With this rule, you could construct the following derivation:
1) x
2) ax
_____________
3) ixeyi from Disjunction Introduction
4) iaxgyi from your Rule 11
5) iaxciaxgyii from your Rule 5
6) y from your Rule 7
Hence, given a contradiction, any arbitrarily selected symbol string can be introduced with as a disjunct, and then derived as above. Now, other than with a rule such as disjunction introduction, I see no way your current set of rules allow for the introduction of new basic symbols into NG derivations, and hence no way to derive arbitrary propositions from contradictions.
Originally posted by bbarrYou are right. We have two new rules now, namely disjunction introduction as you say and an explicit rule for reductio ad absurdum, namely that if we introduce x and derive y and ay then we may eliminate our introduction of x and conclude ax.
A quick note concerning my post immediately above:
I've gone over your list of rules again, and, as I mentioned to you via PM, you need a rule that corresponds to normal propositional calculus' rule of disjunction introduction. That is, you need a rule such that if x is writable, then ixeyi is writable. With this rule, you could construct the following ...[text shortened]... bols into NG derivations, and hence no way to derive arbitrary propositions from contradictions.
This cool now?
Another viewpoint:
''There's a very subtle thing I don't like about the new rule that let's you conclude ax when x leads to a contradiction. It's the fact that you can't know for certain that a given set of assumed statements is consistent, and thus you can't single out one as the culprit, or add a new one and be sure that that new one led to the contradiction.''
Originally posted by royalchickenHoorah! Yes, this is now super-cool.
You are right. We have two new rules now, namely disjunction introduction as you say and an explicit rule for reductio ad absurdum, namely that if we introduce x and derive y and ay then we may eliminate our introduction of x and conclude ax.
This cool now?
Then, Russ-like, I have made a new thread to talk about NG proper (I can identify one small problem still with its suitablility as a propositional calculus, and one problem with propositional calculi in general). This thread is now for the discussioni of the application of NG in arguments and such. I will address your other difficulty in our game; it is mostly due to a silly inference by me.