Start with a true sentence. Replace a sub-formula with its contrapositive. Is result true?

[Boojatta]

Is the following statement true, false, or meaningless?

"If 0/0 isn't equal to 25, then 0 is equal to 0."

Consider the following:

#1. A non-zero factor that appears in both the numerator and denominator of a fraction can be canceled.

In other words...

#2 For every b, and every c, if b isn't equal to zero, then (cb)/b = c.

Replace "if b isn't equal to zero, then (cb)/b = c" with its contrapositive to obtain:

#3 For every b, and every c, if (cb)/b isn't equal to c, then b equals zero

Now, in particular plug in b = 0 and c = 25, to obtain:

#4 If 0/0 isn't equal to 25, then 0 = 0.

[babylonsister]

1. You go first. nt

[Boojatta]

2. I did go first. I started the thread.

[babylonsister]

5. And I bookmarked it, in the event someone understands where you're coming from. nt

[Lint Head]

3. And a partridge in a pear tree.

[Tansy_Gold]

4. Sometimes. n/t

[Speck Tater]

6. 0 + 0 = 1 for very large values of 0. But seriously...

Since x/0 for all X is undefined, then 0/0 is undefined. However, 25 IS defined, so a defined quantity cannot be equal to an undefined quantity, therefore "If 0/0 != 25" must be true.

On the other hand, "0 = 0" is true under all conditions. Therefore ANYTHING can be used to imply "true". I can say "If the moon is made of green cheese then 0 = 0, and that is a valid logical statement. Therefore, the fact that the conditional is true is irrelevant to the truth of the implied clause.

[FiveGoodMen]

10. "If the moon is made of green cheese then 0 = 0" seems a little misleading

"If the moon is made of green cheese then 0 = 0"

would seem to state that a Green Cheese Moon implies that 0 = 0 (i.e. "GCM ==> (0=0)")

In that case (0!=0) ==> !GCM

But really, they are unrelated and implication doesn't come into the picture at all.

[Boojatta]

17. "But really, they are unrelated and implication doesn't come into the picture at all."

There's a subtle distinction between different concepts of implication. On the one hand, there's the ordinary concept of truth-functional "if ... then." On the other hand, there are various other concepts referred to by such terms as "indicative conditional", "causal conditional", "counter-factual conditional", "logical consequence", etc. Provided that you are told that the implication concept that is being used is truth-functional, there shouldn't be anything misleading about a statement such as "if the moon is made of green cheese then 0 = 0."

[WheelWalker]

27. +1

[NYC_SKP]

7. The statement is true.

Next question?

:P

[Boojatta]

24. When you wrote that, did you intend to assert that the following statement is true?

"If 0/0 isn't equal to 25, then 0 = 0."

Note that the above statement is labeled as "#4" in the Original Post of this thread.

I ask because I can't guess whether the smilie of a tongue sticking out is to be understood specifically in connection with "next question?" or is to be understood as an emotional reaction to the Original Post itself and to me for posting it.

[WheelWalker]

28. Unless your statement is false.

[Orrex]

8. If the Repubs have their way, no one will have access to contrapositives

[qazplm]

9. well played

.

[Thor_MN]

11. I thought the GOP WAS the Party of Countrapositives.

[Dr. Strange]

21. If some one has access to contrapositives, then the Repubs have not had their way.

:D

[Duer 157099]

12. You lost me at "start with a true sentence."

"What is truth?"

[WheelWalker]

29. A "true sentence" is a term of art....

It frequently follows a "true bill".

[Boojatta]

13. Kick

[Ready4Change]

14. I reject your initial statement.

X/0 is not a valid operation for any value of X, including X=0.

[Boojatta]

15. If you intend to reject as meaningless every statement that refers to an expression of the form X/0

then shouldn't you reject as meaningless the following statement?

"X/0 is not a valid operation for any value of X."

[Ready4Change]

16. Who wrote your code?

Because that post could have been the output of an Eliza like program.

[hunter]

18. Occasionally I've wondered that myself about our friend Boojatta

B-)

[Taitertots]

19. Only if you limit yourself to the real number system

I don't really understand how, but my advanced calculus professor said it is possible to divide by zero. You have to use quaternions, hyperreal numbers, or some other thing I don't yet understand.

[bananas]

20. Yes, for example hyperreals are used in model theory

and Boojatta mentions "a model-theoretic interpretation" in another thread: http://www.democraticunderground.com/discuss/duboard.php?az=view_all&address=228x66192

[Dr. Strange]

22. Okay.

"I've read many of Boojatta's threads and if I ever start to understand them, I will fear for my sanity."

I shall now replace "if I ever start to understand them, I will fear for my sanity" with its contrapositive.

"I've read many of Boojatta's threads and if I am not fearing for my sanity, then I must not have begun to understand them."

I believe this sentence is also true, much like the statement "If 0/0 isn't equal to 25, then 0 = 0."

[Name removed]

23. Deleted message

Message removed by moderator. Click here to review the message board rules.

[Boojatta]

25. Kick

[DetlefK]

26. You are wrong. Here is my counter-proof:

You did not say what b and c are! If you consider them operators or tensors more complicated than a scalar (e.g. a matrix), then your proof is incorrect.

You are going from
b unequal 0 -> (cb)/b=c
to
b = 0 <- (cb)/b unequal c

Written different, it looks like this::
b unequal 0 -> (cb-bc)*f(x)=0
should result in
b=0 <- (cb-bc)*f(x) unequal 0

For example, let's take the Pauli-matrizes b=sigmax and c=sigmaz. Et voilà: You are wrong, because they don't commutate and neither of them is zero. (Well, their traces are 0, but I have no idea what influence they should wield. The counter-proof also works if you take b as the operator d/dc .)

[dimbear]

30. In most logical systems, the universe implies a tautology.

Just as a contradiction implies the universe. Obviously you can select any other set of axioms you like, but it's likely to be lonely. #4 is a case of the subject line.

[BadgerKid]

31. Original statement could be T or not well formed depending....

The statement "0/0 isn't equal to 25" could be straight-up false. On other hand, one wonders whether it makes sense to do the comparison in the first place since 0/0 is undefined whereas 25 is a defined and definite concept.

The statement "0 == 0" is true.

So either you have F -> T (which is True in first-order logic), or the if-part is not well formed and in that case I don't know what the value of "not well formed implies T" should be.

[Boojatta]

32. If you say that "0/0 isn't equal to 25" isn't well-formed, then you're not finished.

You should explain the following.

First:
If "0/0 isn't equal to 25" isn't well-formed, then does this failure to be well-formed arise purely as a matter of formal reasoning, or does it depend upon your interpretation of the symbols representing zero and division? If it's a matter of interpretation, then how can you support the allegation that the formal reasoning in the Original Post of this thread isn't valid? If it's a matter of formal reasoning, then are you actually prepared to abandon the principle that when a variable is universally quantified upon, we are entitled to substitute any constant for the variable? After all, that principle takes us from #3 to #4 in the Original Post of this thread.

Second:
Is an ordinary proof merely the first section of a complete proof? Should a complete proof also include a second section that demonstrates that all expressions that appear in the first section of the proof are well-formed expressions? That would certainly be very inconvenient, and would also be a change from standard practices.