This thread is for questions about NG, and formal reasoning to prove statements about Ng, because I think we may be turning people off of the point of that post, which is to introduce people with low regard for formal reasoning to how it actually works. So, for starters, the following has been asked:
''Typically in Proof by Contradiction, you take a set of axioms that you know (or assume) to be true and consistent, add one that you suspect is false, and then show a contradiction. You can then make the proper conclusion, because there is only one suspect assumption. In your system, I would agree with your new rule, provided either: 1) x is the only assumption made, or 2) for all assumptions besides x, they are required to be writable and themselves not able to produce any contradiction.''
Can someone comment on the validity of (2)?
The current rules, just updated, are:
Rule 1.
If n is the largest number such that B(n) appears in an instance of NG, and a new basic symbol is introduced, it must be B(n+1).
Rule 1 does not have the same charm that the rest of them do, and is not especially important, but NG is a very picky game indeed.
The remaining symbols are those referred to by the rest of the rules. They are:
Definition. Regulated Symbols.
'a', 'c', 'e', 'g', 'i' are regulated symbols.
Now, to play, we need the rules and the procedure. Here, by convention, we use the letters w, x, y, z to refer to sequences of symbols which are either basic symbols or which can be formed from the procedure and the introduction of basic symbols.
Definition. Procedure.
Given some x, follow the rules to form a new sequence.
This gives us our concept called 'writability'. A sequence is writable relative to another sequence or sequences if applying the procedure to them yields the sequence in question.
Rule 2. The i-rule.
Given x and y, xcy, xey, xgy must be replaced with ixcyi, ixeyi, ixgyi. The former sequences are not allowed; the latter ones are.
Rule 3. The even-i-rule.
For any x, simply writing ix or xi is not allowed; ixi is allowed unless other rules prohibit it.
Rule 4. The ai-rule.
ax is allowed for allowable sequences x; iaxi is not.
Rule 5. The c-rule.
If x and y are writable, then ixcyi is also writable.
Rule 6. Antics.
If ixcyi is writable, then x is writable and so is y.
Rule 7. The cg-rule.
If ixcixgyii is writable, then so is y.
Rule 8. a nonproliferation.
If x is writable, then aax is writable and vice versa.
Rule 9. a proliferation.
As in Rule 8, ixgyi and iaygaxi may be interchanged.
Rule 10. Political correctness.
iaxcayi and aixeyi are interchangeable.
Rule 11. The ea-rule.
ixeyi and iaxgyi are interchangeable.
Rule 12. The Introduction Rule.
If after writing x, by assumption or through previous derivation, one can apply other rules to write y, then ixgyi is writable.
Rule 13. The Drowned Pythagorean Rule..
If x is assumed writable and iycayi can be derived, then ax is writable and the assumption of writability of x is dropped.
Rule 14. Introducing Dis Junk
If x is writable, then ixeyi is writable.
Originally posted by royalchickenOf course!
This thread is for questions about NG, and formal reasoning to prove statements about Ng, because I think we may be turning people off of the point of that post, which is to introduce people with low regard for formal reasoning to how it actually works. So, for starters, the following has been asked:
''Typically in Proof by Contradiction, you take ...[text shortened]... selves not able to produce any contradiction.''
Can someone comment on the validity of (2)?
If you have a set of propositions: A, B and C and you want to disprove propostition D by contradiction, then any contradiction assertained by supposing A,B,C and D to be true at the same time must be due to the invalidity of D alone.
If you are not certain of A,B and C's validity before-hand, then any one (or more) of these could be causing the contradiction.
Originally posted by howardgeeThis is correct. The person who asked that question wanted to see a discussion of how to determine which statement is causing the contradiction.
Of course!
If you have a set of propositions: A, B and C and you want to disprove propostition D by contradiction, then any contradiction assertained by supposing A,B,C and D to be true at the same time must be due to the invalidity o ...[text shortened]... en any one (or more) of these could be causing the contradiction.
Thanks for joining the discussion 🙂.
Vaguely. Time limits me!
I studied Philosophical logic at University and it was my pet topic.
Think I understand all the concepts, just that the notation is different.
Your proof by contradiction for example, we studied as "Reductio ad absurdum". (literally reduced to absurdity).
If assuming proposition 'A' leads to absurdity (contradiction) then we know that 'not A' must be true.
Originally posted by howardgeeI intentionally introduced nonstandard notation so that people reading the orignial post would think of it first as an abstract game to play with symbols, to reinforce the point that logic does not 'mean' anything; it merely is a way of establishing formal relationships.
Vaguely. Time limits me!
I studied Philosophical logic at University and it was my pet topic.
Think I understand all the concepts, just that the notation is different.
Your proof by contradiction for example, we studied as "Reductio ad absurdum". (literally reduced to absurdity).
If assuming proposition 'A' leads to absurdity (contradiction) then we know that 'not A' must be true.
Your description of 'reductio ad absurdum' is correct (my name for that Rule, the drowned Pythagorean, refers to a very famous RAA argument that supposedly caused Pythagoras to kill the man who made it). If you have time, you may be interested in the comments made by bbarr and Cribs in the other thread.
Originally posted by bbarrThis was a big issue, actually.
I dig this notation, and think it will be fun to work with. Hopefully, those who have been presenting questions about formal logic and deduction will take a close look at this system. One question: Are you planning on introducing universal and existential quantifiers and the rules for their use?
I wanted to strike a balance between completeness (in a pedagogical, not a Goedel, sense*) and hiding the fact, at least at the beginning, that NG was anything other than a game to play with symbols. I found it hard to explain too much without giving away the fact that it is logic I'm talking about, although the modifications I made to the rules in view of your comments (appreciated) are phrased so as to make it pretty obvious anyway.
Actually, Basic Symbols are ''B''s because I was leaving ''A'' open for a universal quantifier.
*Actually, I wanted to introduce some formal set of rules which contained a formalization of arithmetic...😉