An important theorem

[jason_t]

Theorem: I am always right.

Proof: Let S be the statement "Statement S is false or I am always right."

There are two possibilities. Either statement S is false or statement S is true.

If statement S is false then both of the clauses in the disjunction must be false. That is "statement S is false" is false and "I am always right" is false. But in this case "statement S is false" is false means that "statement S is false" is true and therefore we have "statement S is false" is both false and true, a logical impossibility.

Therefore statement S is true. But if statement S is true, one of the clauses in the disjunction must be true. That is "statement S is false" is true or "I am always right" is true. If the former is true then "statement S is false" is true. But in this case "statement S is false" is true means that "statement S is false" is false and therefore we have "statement S is false" is both false and true, a logical impossibility.

We are left with one possible conclusion: "I am always right" is true.

QED.

[Philo]

Statement S is neither true nor false, since the first disjunct is neither true nor false, therefore we cannot conclude that statement S is true.

[J. Stew]

True and false, can there be true without false? Really just 'something' that is relatively true and relatively false at the same time. False is relative to true, and true is relative to false, two sides of the same coin.

By 'right' do you mean you are always true? Well if you are always true then you are always false too, but what about when you are true and false, where are you then . . .

[atrifix]

I came up with this some years ago when I was a sophomore in philosophy. This commits the fallacy in a much more subtle way. At the time I was convinced that there was probably something wrong with it, but didn't know exactly what it was.

Theorem: There is no God.

(1) Let God be an omniscient being, where omniscience is defined as the ability to know anything that is knowable. (premise)
(2) A deduction is sufficient for knowledge. (premise)
(3) God must know everything that I know. (from 1)
(4) Let S = "God does not know that this sentence is true." (definition)
(5) S is then logically equivalent to "God does not know that S is true." (substitution)
(6) If God were to know S, then S would have to be true, but then God would not know S. Thus we have a contradiction. So God does not know S.
(7) I know S. (from 2,5,6)
(8) Hence, I know something that God does not. (from 6,7)

(C) There cannot be any omniscient being. (=> There is no God; from 3,8)

[guesswest]

Surely it's just a simple case of (4) not being possible if (1) (or (3)) are true?

[guesswest]

And why do the green people hardly ever post on here? Are you angels? /images/graemlins/ooo.gif

[atrifix]

[ QUOTE ]
Surely it's just a simple case of (4) not being possible if (1) (or (3)) are true?

[/ QUOTE ]
Well, of course, but the argument is a reductio that seeks to establish (1) (and hence (3)) is false. Since (4) is just a definition, it makes much more sense to reject (1) than claim that (4) is somehow impossible.

[gumpzilla]

[ QUOTE ]

There are two possibilities. Either statement S is false or statement S is true.

[/ QUOTE ]

Clearly you're not a constructivist.

[sweetjazz]

Corollary: Jason's proof is fallacious.

Proof: Let S be the statement "Statement S is false or Jason's proof is fallacious."

There are two possibilities. Either statement S is false or statement S is true.

If statement S is false then both of the clauses in the disjunction must be false. That is "statement S is false" is false and "Jason's proof is fallacious" is false. But in this case "statement S is false" is false means that "statement S is false" is true and therefore we have "statement S is false" is both false and true, a logical impossibility.

Therefore statement S is true. But if statement S is true, one of the clauses in the disjunction must be true. That is "statement S is false" is true or "Jason's proof is fallacious" is true. If the former is true then "statement S is false" is true. But in this case "statement S is false" is true means that "statement S is false" is false and therefore we have "statement S is false" is both false and true, a logical impossibility.

We are left with one possible conclusion: "Jason's proof is fallacious" is true.

QED.

[/ QUOTE ]

[Silent A]

[ QUOTE ]
There are two possibilities. Either statement S is false or statement S is true.

[/ QUOTE ]

Statement S is gibberish (neither false nor true).

[jason_t]

[ QUOTE ]
Corollary: Jason's proof is fallacious.

Proof: Let S be the statement "Statement S is false or Jason's proof is fallacious."

There are two possibilities. Either statement S is false or statement S is true.

If statement S is false then both of the clauses in the disjunction must be false. That is "statement S is false" is false and "Jason's proof is fallacious" is false. But in this case "statement S is false" is false means that "statement S is false" is true and therefore we have "statement S is false" is both false and true, a logical impossibility.

Therefore statement S is true. But if statement S is true, one of the clauses in the disjunction must be true. That is "statement S is false" is true or "Jason's proof is fallacious" is true. If the former is true then "statement S is false" is true. But in this case "statement S is false" is true means that "statement S is false" is false and therefore we have "statement S is false" is both false and true, a logical impossibility.

We are left with one possible conclusion: "Jason's proof is fallacious" is true.

QED.

[/ QUOTE ]

Your proof is incorrect. And since I am always right, my statement that your proof is incorrect is correct and my proof is not fallacious.

[CallMeIshmael]

[ QUOTE ]
[ QUOTE ]
Corollary: Jason's proof is fallacious.

Proof: Let S be the statement "Statement S is false or Jason's proof is fallacious."

There are two possibilities. Either statement S is false or statement S is true.

If statement S is false then both of the clauses in the disjunction must be false. That is "statement S is false" is false and "Jason's proof is fallacious" is false. But in this case "statement S is false" is false means that "statement S is false" is true and therefore we have "statement S is false" is both false and true, a logical impossibility.

Therefore statement S is true. But if statement S is true, one of the clauses in the disjunction must be true. That is "statement S is false" is true or "Jason's proof is fallacious" is true. If the former is true then "statement S is false" is true. But in this case "statement S is false" is true means that "statement S is false" is false and therefore we have "statement S is false" is both false and true, a logical impossibility.

We are left with one possible conclusion: "Jason's proof is fallacious" is true.

QED.

[/ QUOTE ]

Your proof is incorrect. And since I am always right, my statement that your proof is incorrect is correct and my proof is not fallacious.

[/ QUOTE ]


Check and mate

[lightw1thoutheat]

godel anyone?

[guesswest]

[ QUOTE ]
[ QUOTE ]
Surely it's just a simple case of (4) not being possible if (1) (or (3)) are true?

[/ QUOTE ]
Well, of course, but the argument is a reductio that seeks to establish (1) (and hence (3)) is false. Since (4) is just a definition, it makes much more sense to reject (1) than claim that (4) is somehow impossible.

[/ QUOTE ]

You have to reject (4) because it's internally inconsistent. It can be rephrased 'Let S = God knows that this sentence is true and does not know that this sentence is true', since you've just defined god as omniscient. If the argument proves anything it just proves that (4) is non-sensical. It amounts to 'if god is not omniscient', 'then god is not omniscient'.

[atrifix]

[ QUOTE ]
You have to reject (4) because it's internally inconsistent. It can be rephrased 'Let S = God knows that this sentence is true and does not know that this sentence is true', since you've just defined god as omniscient.

[/ QUOTE ]
I don't understand this. If it's internally inconsistent, we shouldn't have to also invoke the reductio premise. There's no way you can possibly rephrase S in that way without also invoking (1).

[ QUOTE ]
If the argument proves anything it just proves that (4) is non-sensical. It amounts to 'if god is not omniscient', 'then god is not omniscient'.

[/ QUOTE ]
The argument doesn't prove anything, because it's fallacious, but I don't think that this is what it amounts to.

[atrifix]

[ QUOTE ]
You have to reject (4) because it's internally inconsistent. It can be rephrased 'Let S = God knows that this sentence is true and does not know that this sentence is true', since you've just defined god as omniscient.

[/ QUOTE ]
One other thing I should note is that even invoking (1)--which you can't do if you want to show that (4) is internally inconsistent--doesn't allow for this rephrasing of S, because it has not been demonstrated that S is knowable.

[guesswest]

S isn't knowable because it says nothing, it cancels itself out. And by rephrasing I meant paraphrasing, ie that's what (4) already says.

[atrifix]

I don't understand--how does S already say that? S says just what I defined it to say, and nothing else. I'm confused, at any rate.

[guesswest]

Probably just because I'm making my point badly. (4) is a non-statement, it'd have to be 'god does not know that this sentence is true: (sentence)'. So yes it's unknowable, in the sense that there is no content (and that also makes (7) false).

[atrifix]

[ QUOTE ]
(4) is a non-statement, it'd have to be 'god does not know that this sentence is true: (sentence)'.

[/ QUOTE ] I don't agree with this; self-referential statements are fine. (edit: as an example, "This sentence is true" isn't nonsense, it's a tautology.) I don't know what I think about the rest of it.

I don't think it can be demonstrated that S is internally inconsistent--at least, not to my knowledge. The key point is this, though: it also cannot be demonstrated that S is a (bivalent) proposition. When formalized, this can be demonstrated for finite sets of propositions (perhaps even a stronger conclusion can be proven for enumerable sets of propositions. Maybe someone more well-versed in logic can answer this.). However, it cannot be shown for infinite (or uncountably infinite) sets. If we assume that God is infinite, the argument is not valid.

[The DaveR]

[ QUOTE ]
Theorem: I am always right.

The remainder of the proof is simple alegbra.

[/ QUOTE ]

FYP.

[ZenMasterFlex]

[ QUOTE ]
Theorem: I am always right.

Proof: Let S be the statement "Statement S is false or I am always right."

There are two possibilities. Either statement S is false or statement S is true.

If statement S is false then both of the clauses in the disjunction must be false. That is "statement S is false" is false and "I am always right" is false. But in this case "statement S is false" is false means that "statement S is false" is true and therefore we have "statement S is false" is both false and true, a logical impossibility.

Therefore statement S is true. But if statement S is true, one of the clauses in the disjunction must be true. That is "statement S is false" is true or "I am always right" is true. If the former is true then "statement S is false" is true. But in this case "statement S is false" is true means that "statement S is false" is false and therefore we have "statement S is false" is both false and true, a logical impossibility.

We are left with one possible conclusion: "I am always right" is true.

QED.

[/ QUOTE ]

Simply stunning.