x=x is a fact, rather than just an assumption or a factoid.
Debate Rounds (4)
In this debate, I'll be defending the contention that x=x is a fact, whilst Con will be arguing against it.
x - A letter that is in this case representing a number
Fact - A true statement
Assumption - A baseless statement
Factoid - A statement widely believed to be true but isn't true.
1. No abuse of semantics
2. No trolling
Round 1 is for acceptance.
Anyways, I accept your challenge although I would like to add two definitions:
mathematical fact - a provable fact (by proof is understood a mathematical proof)
axiom or postulate - a statement that is held to be self-evidently true - it cannot be proven although it can be used as a "fact" in a mathematical proof.
I am going to assume that by "x=x is a fact" you mean a mathematical fact and argue against that standpoint.
Thank you to Steve Jobs (if that is his real name) for accepting this debate. About the assumption Con makes at the end of his round one, I just want to make it clear that x=x is the fact that I'm arguing for, however the evidence will not necessarily have to be mathematical. Remember, abuse of semantics is against the rules of this debate.
• The Argument
P1) If T=X, X=T
P3) If T=X and X=T, then X=T=X
C2) (Following P2, C1 and P3) X=T=X
P4) If X=T=X, X=X
C3) (Following C2 and P4) X=X
P1 - This is pretty self-explanatory. If something equals something else, that something else equals that something.
P2 - Let's say in this case that T also represents the number X is representing
C1 - Follows P1 and P2 logically
P3 - Pretty self-explanatory also. X=T, which equals X.
C2 - Follows P2, C1 and P3 logically
P4 - If something equals something that equals something, it follows logically that the first something equals the last something. In this case, X has equal value to T, which has equal value to X. This means that if X=T=X, then X=X.
C3 - Follows C2 and P4 logically
• Argument from the law of non-contradiction
P1) If the law of non-contradiction is true, X=X
P2) The law of non-contradiction is true.
C) ∴, X=X
P1 - If the law of non-contradiction is true, nothing that's a contradiction in terms can exist or be true in the case of a statement. An X that is not an X is a contradiction in terms, as one statement is positive and one is negative, and they're both talking about the same thing
P2 - No two mutually exclusive propositions can be true. For example, "X is Y" and "X is not Y" are mutually exclusive propositions and they can't both be true when they're talking about the same thing.
C - Follows logically
Pro's first "proof" assumes that X=X in P3, when he states that If X=T and T=X, then X=T=X. By making this statement, he is implicitly assuming that T=T because if T != T (!= means not equal to) then that step could not be made. To help, think of T1 and T2 as two different variables. Without knowing that T1 = T2 or X=X you cannot simplify T1 = X and T2=X. Therefore, Pro is using a circular argument in his first "proof"
In his second "proof" Pro uses the Law of Non-Contradiction to prove that X must equal X. The problem here is that The law of noncontradiction is derived from the law of identity, which states that X must equal X. (Law of noncontradiction: http://en.m.wikipedia.org...; Law of identity: http://en.m.wikipedia.org...) Therefore, once again, Pro is using circular arguments.
There are two points I want to make here.
1. It is possible for X != to X if, for example, we consider x to vary with time, and then consider two x's at different points in time.
2. The most basic of Laws is the Law of identity: which states that X must equal X assuming a static X. This is also known as the Postulate of identity, which highlights the fact that X=X cannot be proved , therefore me must take it for granted as an axiom - still true, but not a mathematical fact.
• Con's "Rebuttals"
My First Argument
Con's problem with the third premise is unjustified, as I made it clear that T has a value and thatvalue is the same as X's. It doesn't necessarily have to be 'T', just another thing that is or represents the number X is representing. As long as T has a value and it is the sames as x (it is), the third premise is sound. What Con is basically objecting to is that T has a value and it is the same as X's. Remember, it doesn't necessarily need to be "T", it could be an actual number such as 5. Either way, it still works:
X= The value of X = X
My argument from the law of non-contradiction
I'd just like to point out that my argument here is not circular,as the law of non-contradiction doesn't derive from the law of identity, it just means that if the law of non-contradiction is true, then the law of identity is true. So, my argument that if the law of non-contradiction is true then x=x and the law of non-contradiction is true so x=x is sound.
• Con's "arguments"
1. But if the two X's are different, then we're not talking about the same thing and you're point is irrelevant. X=X when both X's are representing the same thing.
2. Note in this debate how I did not agree to Con's assumption that he meant proving X=X as a mathematical fact. I said: "I just want to make it clear that x=x is the fact that I'm arguing for". I didn't say "I just want to make it clear that x=x is the mathematical fact I'm arguing for". This means that in his second point, Con conceded by saying that x=x is true.
-The fault in his argument against my rebuttal is that once again he uses the assumption that X=X to show his statements to be true. In this case, he uses 5 instead of T, but his logic remains based on 5 equaling 5. Also, I'd like to challenge Pro to show that X=5=X => X=X WITHOUT assuming X=X. It's impossible
Again, there is the problem that the law of non contradiction is based on the law of identity, which pro refuses to admit, even though I posted links.... Also, X=X AND X!=X is a contradiction, but X!=X by itself does not violate the law of noncontradiction
Argument 1: Here I was not trying to show that x != x could be true if you took x to vary with time.
Argument 2: To explain, I am using axiom/postulate to mean basically the same thing asan assumption. Therefore x=X is an assumption - something we have to assume to be true in order for math to work. Therefore it is theoretically possible for x=x to be false for certain values of x. Therefore I did NOT concede by saying x=x is true
Argument 3: (new argument) Another way to show we have to assume x=x is true is to consider words or strings as they are called in programming. In a sentence, the strings "Abracadabra" and "abracadabra" mean the same thing, but they are not comprised of the same bite code ('A' is considered different from 'a' in computer science. Therefore, we can have x != x and x=x to be true at the same time if we consider two different equality statements.
I'm going to have to quit in this case because I forgot to put "a certain number" in my definition of x instead of just "a number".
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