The Liar Paradox and Curry's Paradox are solvable

Solvable in this case means a logically valid true or false answer to each paradox without denying either the Law of Non-Contradiction (like paraconsistent logic) or the Law of the Excluded Middle (like intuitionistic logic).

I argue that Arthur Prior's solution (Inspired by Buridan and Pierce) to the Liar's Paradox is sufficient to call it a non-paradox. Using the similar logic, we can also solve Curry's Paradox. The paradoxes are as follows:

Liar's Paradox
This sentence is false.

Curry's Paradox
If this sentence is true, then Santa Claus exists.

Rules:
Round 1: Intro/Acceptance
Round 2: Arguments
Round 3: Rebuttals/Conclusions

I accept, and look forward to a good debate!

I thank my opponent for accepting this challenge.

===Liar's Paradox===

The Liar's Paradox is a self-referential statement, originally formulated by Eubulides of Miletus in the 4th Century BC. The Liar's Paradox is as follows this sentence.

"This sentence is false."

Now at face value, it seems that if we say that the sentence is true, then is false. If we say it is false, then it is true. We see that either way, we find a contradiction. I will show how the Liar's Paradox is not a paradox that leads to contradictory conclusions but is in itself, a contradiction. This make the sentence false.

The solution I believe that works is the one laid down by Arthur Prior (1914-1969), who claimed that he found the method in the writings of both Jean Buridan (1300-1358) and Charles Sanders Peirce (1839-1914). The solution states that every proposition has a hidden part, namely "It is true that". So when we examine any proposition, we can add those four words and see that it is true.

It is true that Socrates was a man.

It is true that the Earth is round.

It is true that SocialPinko changes his profile picture way too often.

Etc.

Using this standard, let us examine what the Liar's Paradox is really saying.

It is true that this sentence is false. The previous statement is equivalent to saying "It is true that this sentence is true and it is true that this sentence is false" which is a contradiction and therefore, false.

It is true that this sentence is true if and only if it is true that this sentence is true and it is true that this sentence is false.

~S<---->(~S&S)
If we say that ~S is false, then we see that, without paradox, the paradox is false.

http://www.iep.utm.edu...
http://plato.stanford.edu...

===Curry's Paradox===
Curry's Paradox is similar to the Liar's Paradox but rather uses the negation of the Liar's Paradox in a modus ponens argument. Originally formulated by Haskell Curry (1900-1982) and is as follows this sentence.

If this sentence is true, then Santa Claus exists
This sentence is true
Therefore, Santa Claus exists

Normally, we would formulate this argument as follows:

(P-->Q)
P
Q

But in this case, we must examine what is really being said in the argument. The "This sentence is true" in the first premise is referring not to the antecedent but to the entire conditional "If this sentence is true, then Santa Claus exists". The second premise, though identical in words to the antecedent of the first premise, is referring to itself which is a difference sentence. Therefore, the conclusion does not follow from the premises, it is invalid.

R=If this sentence is true, then Santa Claus exists.
Q=Santa Claus exists
S=This sentence is true

(R-->Q)
S
Q

We can clearly see that Curry's Paradox is an invalid argument.

http://plato.stanford.edu...
http://en.wikipedia.org...'s_paradox

Thank you, AnalyticArizonan, for posting your argument.

My opponent's analysis looks decent until he claims the following two statements to be equivalent:
"It is true that this sentence is false."
"It is true that this sentence is true and it is true that this sentence is false."

Two statements are logically equivalent if they they have the same truth value at all times [1]. My opponent claims that the second statement is false by the law of non-contradiction, which is true. However, if we assume equivalency and thus that the first sentence is false, it instead resolves to true, leading to the very paradox that we are debating. A paradoxical statement cannot be equivalent to a false statement. Therefore, my opponent's analysis of the Liar's Paradox is illogical and invalid, and the Liar's Paradox remains a paradox.

http://www.iep.utm.edu...
http://plato.stanford.edu...
http://en.wikipedia.org...

http://en.wikipedia.org...
http://kleene.ss.uci.edu...

My opponent then claims that Curry's Paradox is "invalid." If by this he means that it remains a paradox, then I have won this debate. If he means that the statement is false, then he is incorrect. Clearly, if the sentence is true, then the conclusion that "if this sentence is true, then Santa Claus exists" is true by substitution, and therefore it follows from the original sentence that Santa Claus exists. To use my opponent's labels to summarize the paragraph logically:

S -> R
If this sentence is true, then if this sentence is true, then Santa Claus exists.
S & R -> Q
If this sentence is true and Santa Claus exists if this sentence is true, then Santa Claus exists.
S -> Q
If this sentence is true, then Santa Claus exists.

Therefore, the conclusion does follow the premises logically. The only issue, of course, is that Santa Claus does not exist, and even if he did, we could assert, "If this sentence is true, then Santa Claus does not exist," and come to the opposite conclusion. Hence, Curry's Paradox.

http://plato.stanford.edu...
http://en.wikipedia.org...

I have demonstrated both the Liar's Paradox and Curry's Paradox to be indeed paradoxes. Good luck with your response, AnalyticArizonan.

===Liar's Paradox===
"However, if we assume equivalency and thus that the first sentence is false, it instead resolves to true, leading to the very paradox that we are debating."

I concede that Con was able to refute the explanation offered as I reexamine my argument.

I appreciate his intelligent analysis of my explanation.

===Curry's Paradox===
"My opponent then claims that Curry's Paradox is "invalid." If by this he means that it remains a paradox, then I have won this debate. If he means that the statement is false, then he is incorrect."

A paradox is a logically valid argument or proposition that leads to contradiction. Of course, I disagree that I am incorrect that Curry's Paradox is not false.

"S -> R
If this sentence is true, then if this sentence is true, then Santa Claus exists."

If you say this sentence is true in that same sentence, you are referring to the entire sentence, not the antecedent. So even if we grant Con's first premise, we still cannot say: "This sentence is true" as the second premise for that would be a new proposition referring to itself, not the antecedent of premise 1.

I have second argument regarding both of these paradoxes but as I did not put them forward in round 2, I will not put them in now. I believe that Con showed that my explanation about the Liar's Paradox was inadequate to disprove it as a paradox but Con was not able to show how my disproving of Curry's Paradox as paradox was wrong.
So vote Pro if you think my arguments were better and/or I can win with half the resolution.
I thank my opponent for this insightful debate.

Thank you, AnalyticArizonan, for your response.

My opponent has conceded that he has not solved the Liar's Paradox, thus conceding the debate.

Regarding Curry's Paradox, every time I said "this sentence," I was actually referring to the original sentence. Perhaps this will clear things up:

R: If R, then Q.
Q: Santa Claus exists.
S: R is true.

1. If R, R. (Law of Identity)
2. If R, if R, Q. (Substitution of Definition of R)
3. If R, Q. (Combination of conditionals)
4. R (Substitution of Definition of R)
5. Q (Syllogism of 3 and 4)

Now, I'm not quite sure what my opponent means in his refuting paragraph. "This sentence is true" was never a premise, as my opponent claims. Every statement made in my logical proof in each round had the conditional "If this sentence is true" until the original sentence "If this sentence is true, then Santa Claus exists" was logically proven. It was a conclusion, not a premise.

Now, my opponent claims that he may still win this debate, but he cannot. He had the burden of solving both the Liar's Paradox and Curry's Paradox, and as he already conceded one, it does not matter that his attempt to solve Curry's Paradox is also unsuccessful. The resolution is negated.

AnalyticArizonan, if you'd like to try your new argument to solve the paradoxes at some point in the future, I'd be happy to debate you on it, as well, although perhaps with more rounds. Regardless, thank you for this interesting debate, and I look forward to debating you again sometime in the future!