The Instigator
shlh1514
Con (against)
Losing
33 Points
The Contender
Tatarize
Pro (for)
Winning
37 Points

One equals to two, 1=2

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Voting Style: Open Point System: 7 Point
Started: 1/15/2008 Category: Science
Updated: 9 years ago Status: Voting Period
Viewed: 2,007 times Debate No: 1808
Debate Rounds (5)
Comments (9)
Votes (18)

 

shlh1514

Con

Hello people, this debate will be good, hopefully. 1=2 No way. Let me show my evidence by doing A & B'S.

1=a a*10=10 10-a=9
2=b b*10=20 20-a=19

No way they are the same.

That's why 1 is lower then 2.

1*100=100
2*100=200 no way they can equal unless you round up.

1.99999999 and 2*2=4
* 2
3.99999998
Tatarize

Pro

Resolved: One equals to two.

* Math debates are retarded. The voters know this. I know this. Regardless of the arguments involved for simply instigating the debate you should be voted down. I hold that this should be the general policy of all the voters on this site for the purpose of ending this nonsense. If you see a math debate up for review, I recommend all people vote against the instigator.

* Arguing in favor of one equals two places me in the unique position of causing a clear contradiction. If 1 = 2 and 1 = 1, then we can use this flaw to come to any conclusion. Bertrand Russell once gave a speech noting that with a false statement and valid argument (no logical flaws in the argument) you can conclude anything. An audience member challenged him on this point saying, "certainly you can't take an statement like 2 + 2 = 5 to conclude that you are the pope!" - On the spot, Russell assembled the following proof:

2 + 2 = 5,
4 = 5 (subtracting four from both sides)
1 = 2
The pope and I are two, but we see that two equals one: The pope and I are one.
Therefore, I am the pope.

I bring up this story to show that, it is true that with 1=2 there is a contradiction. However, this doesn't make anything false. The problem with contradictions is that they make everything true. We avoid them out of utility not necessity. "One equals one and only one", is a requirement for coherent useful mathematics. Absolutely nothing in your opening statement or topic requires that I abide by a useful form of mathematics.

There is no requirement to prove that 1=2, just as there's no requirement that 1=1. I assign one to equal one if I want useful mathematics. For the sake of this debate, I do not.

I assign, 1=2. This allows anything to be true.

Godel's incompleteness theorem shows us that a system cannot be both complete and consistent. A system cannot prove that it is valid, unless that system is not valid. With an inconsistent system we can prove anything, even the validity of this system (within itself). For incomplete systems, we are missing parts, like the ability to conclude that 1=2 or the system is valid. The utility of the resulting system is non-existent as utility of a system is in those things it cannot show (false things).

* Taking this understanding into account. Your disproof of my position is clearly flawed. I can quite easily prove those things you claimed are not equal.

1) You concluded that 9 = 19, since this is false 1=2 is absurd.
2) 1 is lower then 2 (therefore they cannot be equal)
3) 100 = 200 (impossible without rounding up)

However, as previously established. I can easily conclude all of these things. I can conclude that I am the pope. I have assigned 1 to be equal to 2, therefore I am no longer bound by an incomplete system of mathematics.

A) 1=2, multiply both sides by 10.
10=20, subtract 1 from each side.
9 = 19. Reflexive property.
19 = 9.
Previous conclusion: 9 = 19
Substitute: 9 for 19.
19 = 19. Divide by 19.
1 = 1.

B) If X is less than Y, X cannot equal Y.
Let X = 1.
Let Y = 2.
1 is less than 2, therefore 1 != 2
However, we know that, 1 = 2.
- Contradiction. The original statement is false. A number can be less and equal at the same time.

3) 100 = 200,
1 = 2, Multiply by 100
100 = 200.
Previous conclusion: 100 = 200.
Divide equivalence:
100/100 = 200/200
1 = 1

* We see now that your previous conclusions simply assumed that 1 could not equal 2 because you mistakingly believed that 9 could not equal 19 or that 100 could not equal 200. However, if one accepts that 1 = 2 then these assumptions are inaccurate. Further, you would need to prove that 9 could not equal 19 and to do this you would need to justify the system of mathematics from within the system of mathematics. However, by Godel we can tell that this could not be accomplished by any coherent system.

You could only prove my position wrong if and only if, my position is correct.

1) To justify mathematics in such a way to disallow the 1=2, requires that mathematics is incoherent, by Godel's incompleteness.
2) If mathematics is incoherent then it contains a contradiction.
3) With a contradiction you can prove anything.
4) 1=2 is something.
5) With a contradiction you can prove 1=2.

Conclusion: You can prove my position wrong, if and only if you can prove my position right

I am not asked to prove that 1 = 2 in a coherent mathematics. Therefore, assigning 1 = 2 allows for the proof that 1 = 2 by permitting everything to be proved.

Either one equals two or one does not equal two. Only one result may be accepted. Therefore, only two results may be accepted. We have two options: "equals two" or "does not equal two" -- Therefore, two of the two options are correct. One equals two and one does not equal two. Therefore, one equals two.

QED.

----

I have proved that 1 = 2 mathematically, as well as proven by Godel's incompleteness theorem that you cannot prove that I am wrong, without proving that I am right.

http://en.wikipedia.org...

Your argument cannot succeed.
Debate Round No. 1
shlh1514

Con

Hello and thanks for accepting this debate.

Now onto the debate. You said that "The pope and I are one." Really? I in definition is 1. the nominative singular pronoun, used by a speaker in referring to himself or herself. http://dictionary.reference.com...
And the pope is one person. Meaning that 1+1=2, thus excluding the fact you need another 1 to get to two. You also said that "1=2, multiply both sides by 10.
10=20, subtract 1 from each side.
9 = 19. Reflexive property.
19 = 9.
Previous conclusion: 9 = 19
Substitute: 9 for 19.
19 = 19. Divide by 19.
1 = 1."

That meaning that 1=1 can only be the one number equaling one.

You said "1 is less than 2, therefore 1 != 2
However, we know that, 1 = 2.
- Contradiction. The original statement is false. A number can be less and equal at the same time."

1! is 1*1 which equals to 1 not 2.

You said that
We see now that your previous conclusions simply assumed that 1 could not equal 2 because you mistakingly believed that 9 could not equal 19 or that 100 could not equal 200. However, if one accepts that 1 = 2 then these assumptions are inaccurate. Further, you would need to prove that 9 could not equal 19 and to do this you would need to justify the system of mathematics from within the system of mathematics. However, by Godel we can tell that this could not be accomplished by any coherent system.

Really I am trying to prove that 1=2 impossible by showing basic multiplication. Not put division into it. But then that meaning that 1*2= 2 or 4.

1!=1 2!=2

The space of numbers between 1 and 2 are infinite and saying 1=2 contradicts that.
Tatarize

Pro

"That meaning that 1=1 can only be the one number equaling one."
"The space of numbers between 1 and 2 are infinite and saying 1=2 contradicts that."

You misapprehend obviously. If one equals two then *every* statement is logically true. 1=1 is true, as is 1=50 as is 1=2 and 1!=2 and 1 != 50. Assigning 1=2 makes mathematics incoherent. There are an infinite number of numbers between 1 and 2 and there are also 0 numbers between the two. With incoherent mathematics you can prove everything.

The topic didn't say that I needed to use a useful form of mathematics which correlates to the real world, rather I need to show that 1=2 and it does. And by Godel's incompleteness. I am no longer bounded by by being incomplete. I can actually PROVE that 1 = 2, whereas you're mathematics is incomplete and cannot actually prove that 1 = 1 without invoking the correlation between the real world and that number system or just assuming it.

"Either one equals two or one does not equal two. Only one result may be accepted. Therefore, only two results may be accepted. We have two options: "equals two" or "does not equal two" -- Therefore, two of the two options are correct. One equals two and one does not equal two. Therefore, one equals two."

By the limitations of the mathematics you are using, you cannot prove that 1=1 or any statement within that mathematics has a foundation. I can however prove everything because I have ASSIGNED 1 to equal 2.

Because one equals two, every statement is true. And by restrictions established by Godel's you actually cannot provide a foundation for your assertions.

"That meaning that 1=1 can only be the one number equaling one."

No. There is no restriction on what numbers can equal 1 anymore. All numbers equal 1 and none of them do. The coherency of mathematics is gone. True = false.

Your only recourse is to suggest that your intro required coherency and it didn't.

"1! is 1*1 which equals to 1 not 2." -- No. It equals both!

-----

You didn't require coherency. So I assigned the numbers to be equal. I proved that they are within the incoherent system and by Godels you cannot do the same. My case is proven and your case is unable to be proven.
Debate Round No. 2
shlh1514

Con

Really hard to say anything, but 1=2 is impossible theories by definition is
1. a coherent group of general propositions used as principles of explanation for a class of phenomena: Einstein's theory of relativity.
2. a proposed explanation whose status is still conjectural, in contrast to well-established propositions that are regarded as reporting matters of actual fact.
3. Mathematics. a body of principles, theorems, or the like, belonging to one subject: number theory.
4. the branch of a science or art that deals with its principles or methods, as distinguished from its practice: music theory.
5. a particular conception or view of something to be done or of the method of doing it; a system of rules or principles.
6. contemplation or speculation.
7. guess or conjecture.
http://dictionary.reference.com...

If you spot number 2,6,and 7 theories are just a guess or a way to put things to a prospective to what could of happened.
Thus, 1=2 is impossible unless you by a property into the equation, but me trying to exclude that thus and mathematics without using a property will have 1=2 impossible.
Tatarize

Pro

When I say "1=2", it is not part of any theory. Theories are models for explaining the real world. And, as noted assigning 1 to equal 2 creates an incoherent system which would be incompatible with any theory because it's inconsistent with the real world. When dealing with math, we are, by definition, dealing with the abstract. We produce a coherent system because coherent systems are useful in the non-abstract world. Adding one thing to a box which already has one things gives us two things in the box.

However, that relationship with reality is not a requirement for these numbers and they could very well mean whatever I say they mean. 1 does equal 2. As a consequence everything can equal anything and every statement is true (even false ones).

You could argue that in *your* system one cannot equal two, however there are one problems with this:

1) I am in favor of this position and proving something in your system doesn't negate the assignment I've made.
2) Due to limitations of coherent systems, you can't prove your system. And due to an utter lack of limitations to incoherent systems, I can prove my system.

I assign one equal to two, the resulting mathematical system is useless but cannot be proven false and can be logically proven true. You can have useful systems or limitless systems. This debate didn't require the former so I'm taking the latter. Other than practical matters such as the relationship the system has to reality there's nothing suggest your system is true while my system is false. And again, Godel's uncertaintly says that you cannot prove your system true while I actually can and have proven that my system is valid:

-- Either one equals two or one does not equal two. Only one result may be accepted. Therefore, only two results may be accepted. We have two options: "equals two" or "does not equal two" -- Therefore, two of the two options are correct. One equals two and one does not equal two. Therefore, one equals two.

Try to assemble a similar valid proof for your system; it simply cannot be done.

I have proven my case.
You have not proven your case.
Further, I have proven that you cannot prove your case.
I have refuted your arguments against my case.

1=2 isn't a theory nor is 1=1. Theories refer to science not math. In math we deal with abstracts. If 1=2, so be it. If the system produced by this assignment is incoherent, so be it.

There really isn't much more to say, you can't win.
Debate Round No. 3
shlh1514

Con

shlh1514 forfeited this round.
Tatarize

Pro

* I have proven my case by assignment of values between 1 and 2.
* I have shown that incoherent mathematics are permitted by the topic.
* I have proven that my mathematics are coherent.
* I have proven that my opponent cannot (by Godel's) prove his mathematics are coherent.
* I have argued that math debates are stupid and whoever starts them should be voted down as a general rule.
* My opponent has forfeited.

Math is typically abstract, however, when forced to correlate to things in reality we must disallow contradiction (because reality lacks them). Nothing about the original topic asked for coherency, so my argument succeeds.

It is a narrow line, but if we are just dealing with numbers, assigning the numbers to equal each other just allows for contradictions. Contradictions aren't wrong, they just make the system you are using completely absurd, pointless, incoherent, useless and limitless. It doesn't negate my position in favor of creating such a system.
Debate Round No. 4
shlh1514

Con

2 + 2 = 5,
4 = 5 (subtracting four from both sides)
1 = 2
You said that and if subtracting 4 from each side then the answer would equal 0=1 not 1=2. "The pope and I are two, but we see that two equals one: The pope and I are one.
Therefore, I am the pope." How are you the pope? Really how can one person be equal to another. If what you say is correct then the whole world is equal to the pope. If one number equals another, then what is the use of numbers overall? If one equals two. "One equals two and one does not equal two. Therefore, one equals two."
How does that statement make any sense. You are contradicting yourself. If one doesn't equal two then how can it equal two?
Tatarize

Pro

* I have argued that math debates are stupid and whoever starts them should be voted down as a general rule.
* I have proven my case by assignment of values between 1 and 2.
* I have shown that incoherent mathematics are permitted by the topic.
* I have proven that my mathematics are coherent.
* I have proven that my opponent cannot (by Godel's) prove his mathematics are coherent.
* My opponent has forfeited (round 4).

------------------------------

* For the pope equation I meant three. But, good catch.

"How are you the pope?"
- I would be the pope on the grounds that the pope and I would be one.

One person can't be equal to another, it's a contradiction in the real world. Incoherent mathematics does not have any relevance in the real world. This is why we usually prohibit contradiction, not because there's any real restriction to the system. My opponent doesn't understand the purpose of my point there. I am showing that within an incoherent system proving anything is possible.

The statement makes perfect sense, and of course it's contradictory! Have you even paid any attention at all? Contradiction is allowed in incoherent mathematics! Everything is allowed! One can equal two as well as not equal two if when we're talking about "one" and "two" they are abstract concepts not describing anything in reality. You don't seem to be following the argument:

Why can't things be contradictory?

I can and have proved that with contradiction I can prove my system to be valid. When things are allowed to be contradictory, correct things can be proven correct (incorrect things too) and the system itself can be shown to be valid. However, by Godel's incompleteness theorem I know that your answer (or any readers answers) will be unjustified. The reason for this is coherency. A coherent system cannot prove it's own coherency. Think about it this way, if the system could prove it's own coherency, what if it were actually incoherent? Then the system and the proof of the system are both incoherent.

This is Godel's incompleteness theorem:
http://en.wikipedia.org...
----------------------------
T1 can prove that if T2 proves the consistency of T1, then T1 is in fact consistent. For the claim that T1 is consistent has form "for all numbers n, n has the decidable property of not being a code for a proof of contradiction in T1". If T1 were in fact inconsistent, then T2 would prove for some n that n is the code of a contradiction in T1. But if T2 also proved that T1 is consistent, i.e. there is no such n, it would itself be inconsistent. We can carry out this reasoning in T1 and conclude that if T2 is consistent, then T1 is consistent. Since by second incompleteness theorem, T1 does not prove its consistency, it can't prove the consistency of T2 either.

This easy corollary of the second incompleteness theorem shows that there is no hope of proving e.g. the consistency of first order arithmetic using finitistic means provided we accept that finitistic means are correctly formalized in a theory the consistency of which is provable in PA. It's generally accepted that the theory of primitive recursive arithmetic (PRA) is an accurate formalization of finitistic mathematics, and PRA is provably consistent in PA. Thus PRA can't prove the consistency of PA. This is generally seen to show that Hilbert's program, which is to use "ideal" mathematical principles to prove "real" (finitistic) mathematical statements by showing that the "ideal" principles are consistent by finitistically acceptable principles, can't be carried out.

This corollary is actually what makes the second incompleteness theorem epistemically relevant. As Georg Kreisel remarked, it would actually provide no interesting information if a theory T proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of T in T would give us no clue as to whether T really is consistent; no doubts about T's consistency would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a theory T in some theory T' which is in some sense less doubtful than T itself, e.g. weaker than T. For most naturally occurring T and T', such as T = Zermelo-Fraenkel set theory and T' = primitive recursive arithmetic, the consistency of T' is provable in T, and thus T' can't prove the consistency of T by the above corollary of the second incompleteness theorem.
---------------

Coherent systems can't prove their own coherency. I advocate, by assigning 1 to equal 2, that my system is incoherent. As a consequence, I can prove the validity of my system and my opponent cannot.

-- Either one equals two or one does not equal two. Only one result may be accepted. Therefore, only two results may be accepted. We have two options: "equals two" or "does not equal two" -- Therefore, two of the two options are correct. One equals two and one does not equal two. Therefore, one equals two.

By Godel's theorem, I have fulfilled my burden of showing 1 = 2 and he CANNOT. It's actually impossible for him to construct a similar proof unless he is using incoherent mathematics. In incoherent mathematics, 1 = 2, always (one contradiction allows for all contradiction).

I have not only proven my statement valid, I have proven that he cannot prove his statement valid. The fact that he seems confused on this point and doesn't so much as try shows the weakness in his case.

Further, I'm sick of these stupid math debates. You people should just vote them down when ever you see them. Whoever is first in the debate, started the stupid debate, they should lose just for that. This is not to say I didn't win this debate, I crushed my opponent's argument as well as perhaps informed readers a bit about the foundations of mathematics. Nothing in the topic or introduction required coherency (a relationship to the physical world would have implied such a requirement, none was made). I won hands down. However, math debates are stupid, let people get their win ratio up the old fashion way... by winning.

Thank you for the debate.
Debate Round No. 5
9 comments have been posted on this debate. Showing 1 through 9 records.
Posted by Tatarize 9 years ago
Tatarize
I wouldn't teach it to anybody. For all the good it does to win the debate it is admittedly completely worthless. If you accept 1=2 then anything equals anything.
Posted by keni08 9 years ago
keni08
who knew that 1 could equal 2. I say stick with what we teach the kids in kindergarten and let these imaginary numbers equal the number that society has chosen that they should equal. Overall, job well done, for something so simple, this was well over my head.
Posted by Tatarize 9 years ago
Tatarize
Sorry, I've made that mistake since in was a kid; it's not a grammar mistake.

Hopefully they read the bulleted stuff. I tried to divide it up a bit. Explain Godel's in one argument, explain the assignment of variables in another, and the use of incoherent math in another.

Padfoot, Mr. Black, you are simply moving the math into the realm of the useful. I specifically contended that this was not a requirement as given. I contended that with 1=2 the math becomes useless and limitless. Limitless math can prove itself valid whereas due to a restriction of Godel's coherent math cannot. I am okay with that. You can't use math which allows 10 pounds = 2 ounces in the real world. But you can allow math which allows 10 = 2, it just isn't useful for anything (unless that 10 is binary).

I categorically did not use a theory; I used a theorem. The difference is a theory is a model to describe things in the real world (theories would require coherent math). A theorem is actually proven with a rigorous logical mathematical proof like the one given to show that .999... = 1. Where it actually is absolutely true that incoherent systems can prove their validity and absolutely true that coherent systems cannot. There is no opinion with a theorem or relationship to the real world, it is a mathematical proof that that statement is absolutely true.
Posted by padfo0t 9 years ago
padfo0t
You guys are silly, and bad at grammar:

"That's why 1 is lower then 2."

Correct: That's why 1 is lower THAN 2

I have a prediction... No one, with the exception of myself, has read the entire debate. Your arguements are too long so the people reading these debates vote for what they think is right. I see where the proposition is getting at, but the opposition is correct.

Numbers are assigned/have a numerical value that makes them larger or smaller than the next. It would be rediculous to say that 10 pounds is equal to 2 ounces. Everyone knows this is not true. I agree, math debates are stupid but please do not critisize me in your responce when you see that I have had a debate named '.9999999(etc.) can be proven to have the same value as 1', in which I encourage you both to vote in.

I will not be voting in this debate because you both contradict eachother so well. I agree that 1 does not equal 2, but I also agree that it can be said that '1=2' with the proper wording.

I encourage, for lack of a better term, both of you fine debaters to leave the arguement alone because no one can actually prove that 1=2. If it were so, math would be false, and we don't want little kids to think they are being taught something fake, do we?

Also, don't respond by saying that I have bad grammar because I don't have enough time, not to mention pacience, to deal with either of you. As you can see, I have never misused the words than and then interchangedly. I enjoyed reading your debate.

One suggestion to Tatarize, do not use theories to prove your point. That is why they are called theories. Theories are opinionated, not fact. You can't prove anything in the world unless you know everything and us humans are not at all close to that and I think we will never be.

I would like to comment that Tatarize has a very unique and slightly scary image representing him/herself.

More few words of wisdom: nobody on debate.org is forced to tell the truth. /:
Posted by Tatarize 9 years ago
Tatarize
Trying to hide the fact that you divided by zero isn't worth my time. I'll just assign the two and deal with the consequence. In this case it's the ability to prove that I am right and to point out that my opponent cannot prove that he is right.

Godel and incoherent mathematics. My argument doesn't rely and trying to pull a fast one on the voters. It just assigns the two and says "so what".
Posted by GaryBacon 9 years ago
GaryBacon
There actually is an algebraic proof that 1=2. Obviously there is a flaw in the proof, but it is hard to detect. Start with the given equation: a=b
Now multiply both sides by "a" which gives a^2=ab
Subtract b squared from both sides to give a^2-b^2=ab-b^2
The first side of the equation can now be factored using the difference of two perfect squares. The second side can also be factored by removing b and leaving it outside of the parentheses. This gives the following:

(a+b)(a-b)=b(a-b)

Now dividing both sides of the equation by (a-b) we are left with:

a+b=b

Since the starting equation was a=b, we can substitute on letter for another by virtue of the substitution postulate. In this case, we will substitute the letter b in place of the letter a. This gives:

b+b=b

Which becomes 2b=b

Now divide both sides of the equation by b and we are left with 2=1

Note: I did mention that there is a flaw
Posted by Rousseau 9 years ago
Rousseau
I can't really vote off of arguments, because neither side was really arguing a fair point. However, I will vote on educational purposes. Tatarize wins because of the educational content, and the complete waste of time the debate was.
Posted by Tatarize 9 years ago
Tatarize
In base 1, no numbers can be expressed. My argument is strictly that because his intro has no relationship to reality I can go ahead and use incoherent forms of mathematics. In all of which, 1 does in fact equal 2.

Being freed from the shackles of coherency, I can prove my position logically as valid (because incoherent systems can prove anything even their own validity). Whereas to argue that 1 does not equal 2 requires a coherent mathematics which despite being useful and correlating to reality cannot actually be proved accurate.

In the end, the only differences are... my mathematics allows every statement to be true and false, whereas his mathematics only allows true statements to be proven true. This, as a consequence of Godel's incompleteness, limits his ability to prove the validity of his system whereas I have no such limitation.

I assign one equal to two, the resulting mathematical system is useless but cannot be proven false and can be proven true. You need limits to be useful. If science could prove everything it would be worthless, it needs to be limited to only proving true things. Faith can prove everything, and faith is thusly worthless and fundamentally incoherent. You can have useful systems or limitless systems. This debate didn't require the former so I'm taking the later.
Posted by Thoreau 9 years ago
Thoreau
Helpful hint: 1 and 2 have no meaning (and therefore can have any meaning) when working in Base 1 (which has no 1's or 2's).

Therefore, in base 1 1=2 because neither means anything and can mean anything, so we can say this and be entirely correct, making it true.
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Vote Placed by keni08 9 years ago
keni08
shlh1514TatarizeTied
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