Negation Paradox

TheMadFool

Argument A

1. All statements can be negated [assume for reductio ad absurdum]

2. If all statements can be negated then this statement can be negated [premise]

3. If this statement can be negated then this statement can't be negated [premise]

4. If this statement can't be negated then not all statements can be negated [premiseMP]

5. This statement can be negated [1, 2 MP]

6. This statement can't be negated [3, 5 MP]

7. Not all statements can be negated [4, 7 MP]

8. All statements can be negated AND not all statements can be negated [1, 2 Conj]

9. Not all statements can be negated [1 - 8 reductio ad absurdum]

QED

Now, consider the statement, This statement can't be negated

Argument B

1. Either this statement can't be negated can be negated or this statement can't be negated can't be negated [premise]

2. If this statement can't be negated can't be negated then this statement can't be negated can't be negated [premise]

3.. If this statement can't be negated can be negated then this statement can be negated [premise]

4. If this statement can be negated then this statement can't be negated [premise]

5. This statement can't be negated can be negated [assume for conditional proof]

6. This statement can be negated [3, 5 MP]

7. This statement can't be negated [4, 6 MP]

8. If this statement can't be negated can be negated then this statement can't be negated [5 - 7 conditional proof]

9. This statement can't be negated can't be negated or this statement can't be negated [1, 3, 8 CD]

QED

TonesInDeepFreeze

↪TheMadFool

Every sentence can be negated, simply by putting a negation sign in front of the sentence. Doing that is purely a syntactical operation. It does not mean that we are asserting the negation.

So "This sentence can be negated" is true. That is, N is true since N can be negated by writing

~N

/

N = This sentence can be negated.

~N = This sentence can't be negated — TheMadFool

There's slippage there in what 'this sentence' refers to.

In "This sentence can be negated", "this sentence" refers to "This sentence can be negated".

But in "This sentence can't be negated", "this sentence" refers to "This sentence can't be negated".

So "this sentence" refers to two different things in your writeup.

That is just a foible of English that "this" changes meaning by context.

So, to avoid ambiguousness, you probably have to reformulate "This sentence can't be negated" without "this sentence".

Then we can see how the rest of your argument fares.

TheMadFool

@TonesInDeepFreeze Please take a look at the OP.

TonesInDeepFreeze

↪TheMadFool

That's a major re-edit of the OP after several edits. Before I reply, is that your final edit?

TheMadFool

That's a major re-edit of the OP after several edits. Before I reply, is that your final edit? — TonesInDeepFreeze

That's the best I can do.

TonesInDeepFreeze

If you wish to revise further, then please state any further amended arguments in new posts, so that my replies are still pertinent relative to the posts I replied to.

Argument A

1. All statements can be negated [assume for reductio ad absurdum]

2. If all statements can be negated then this statement can be negated [premise]

3. If this statement can be negated then this statement can't be negated [premise]

4. If this statement can't be negated then not all statements can be negated [premiseMP]

5. This statement can be negated [1, 2 MP]

6. This statement can't be negated [3, 5 MP]

7. Not all statements can be negated [4, 7 MP]

8. All statements can be negated AND not all statements can be negated [1, 2 Conj]

9. Not all statements can be negated [1 - 8 reductio ad absurdum]

QED

Now, consider the statement, This statement can't be negated

Argument B

1. Either this statement can't be negated can be negated or this statement can't be negated can't be negated [premise]

2. If this statement can't be negated can't be negated then this statement can't be negated can't be negated [premise]

3.. If this statement can't be negated can be negated then this statement can be negated [premise]

4. If this statement can be negated then this statement can't be negated [premise]

5. This statement can't be negated can be negated [assume for conditional proof]

6. This statement can be negated [3, 5 MP]

7. This statement can't be negated [4, 6 MP]

8. If this statement can't be negated can be negated then this statement can't be negated [5 - 7 conditional proof]

9. This statement can't be negated can't be negated or this statement can't be negated [1, 3, 8 CD]

QED — TheMadFool

I'll mention the first problem I find in each Argument, before going on to the rest of it:

Argument A:

3. If this statement can be negated then this statement can't be negated [premise] — TheMadFool

That is a false premise.

Also, "this statement" in that premise denotes "If this statement can be negated then this statement can't be negated", but "this statement" in line 2 denotes "If all statements can be negated then this statement can be negated". So "this statement" is used ambiguously in the argument.

And if "this statement" weren't ambiguous, then 3. is just the negation of 2, while 2. comes from 1. So 1. and 3. are inconsistent. So, of course, we can derive a contradiction is we assume both 1. and 3. There would be no point in your excercise.

Argument B:

1. Either this statement can't be negated can be negated or this statement can't be negated can't be negated [premise] — TheMadFool

"this statement can't be negated can be negated" is not grammatical, so I don't know that it is supposed to mean.

Maybe you mean:

"this statement can't be negated" can be negated.

And that is true. And "this statement can't be negated" is false.

"this statement can't be negated can't be negated" is not grammatical.

Maybe you mean:

"this statement can't be negated" can't be negated.

And that is false. And ""this statement can't be negated" can be negated" is true.

TonesInDeepFreeze

↪TheMadFool

P.S.

In Argument A, 2. does not need to be taken as a premise. 2. follows from 1. by UI.

TheMadFool

Also, "this statement" in that premise denotes "If this statement can be negated then this statement can't be negated", but "this statement" in line 2 denotes "If all statements can be negated then this statement can be negated". So "this statement" is used ambiguously in the argument. — TonesInDeepFreeze

Not necessarily. Do you accept that, "this statement can be negated" refers to itself? Yes, of course.

If so, if I say "IF this (1) statement can be negated THEN this (2) statement can't be negated", this (1) refers to "this (1) statement can be negated and this (2) refers to "this statement can't be negated"

TonesInDeepFreeze

"IF this (1) statement can be negated THEN this (2) statement can't be negated" — TheMadFool

Putting '(1)' between 'this' and 'statement' is not coherent. And putting '(2)' between 'this' and 'statement' is not coherent.

Maybe you mean:

If "this statement can be negated" is true, then "this statement can't be negated" is true.

Or with the 'N' you used in a previous edit:

'N' stands for 'this statement can be negated'.

Then your premise 3. is:

If N then ~N.

And, since N is also 'N can be negated', 'If N then ~N' is 'If N can be negated then N cannot be negated'.

And then there is no ambiguity with 'this statement'.

Or, using your '(1)' and '(2)':

'(1)' stands for 'this statement can be negated'.

'(2)' stands for 'this statement can't be negated',

Then your premise 3. is:

If (1) then (2).

And 'this statement' in (1) denotes 'this statement can be negated'. And 'this statement' in (2) denotes 'this statement can't be negated'.

And then there is ambiguity with 'this statement'.

TonesInDeepFreeze

↪TheMadFool

Your exercise is not in a mathematical context, which is okay, but it's worth noting comparison with mathematics (I'm simplifying here).

Consider:

This sentence is not provable.

There is only one 'this sentence' in Godel's argument.

The "self reference" of 'this sentence' is okay, because the actual formal sentence doesn't use 'this sentence'. It is paraphrased more fully:

The sentence with Godel-number n is not provable in theory T.

And the above sentence has Godel-number n.

Consider:

This sentence is false.

There is only one 'this sentence' in Tarski's argument.

But the actual formal sentence doesn't use 'this sentence'. It is paraphrased more fully:

The sentence with Godel-number n is false in a model of theory T.

And the above sentence has Godel-number n.

And Tarski proves that if 'is false in a model of theory T' can be defined within the theory T, then the theory is inconsistent. So it's not even a matter whether the "self-reference" of 'this sentence is false' is okay; rather, in a consistent theory, we are not even capable of saying 'this sentence is false'.

TonesInDeepFreeze

↪TheMadFool

Bottom line for your exercise:

'This sentence can be negated' is true and not paradoxical.

'This sentence can't be negated' is false and not paradoxical.

My guess is it is not easy, even if possible, to get a paradox from merely syntactical considerations ('sentence', 'negation', et. al). Paradoxes usually arise from semantical considerations ('true', 'false', 'definable', et. al).

TonesInDeepFreeze

↪TheMadFool

There's an interesting angle on this.

In the language of PA we can express.

Sentence S can't be negated.

It's false, but it can be stated in the language.

And, I'm not sure, but I suspect we can have:

The sentence with Godel-number n can't be negated.

And also have the above sentence have Godel-number n.

And "the sentence with Godel-number n can't be negated" would be false, as seen by the fact that, contrary to what the sentence claims, we would simply show that the negation of "the sentence with Godel-number n can't be negated" is also a sentence in the language of PA.

But if a contradiction in PA could be derived from this, then that would prove the inconsistency of PA.

Some brilliant mathematicians have spent a large part of their lives trying to prove (contrary to mathematical consensus) that PA is inconsistent.

If anyone proves that PA is inconsistent then it would be huge headline news, not just in mathematics but generally. It would be "earth shaking". Now, that is not itself an argument that the TheMadFool's exercise is not correct, but it puts this in perspective that one would be extremely doubtful that his argument to his conclusion could be made correct even with needed redaction.

TheMadFool

Putting '(1)' between 'this' and 'statement' is not coherent. And putting '(2)' between 'this' and 'statement' is not coherent. — TonesInDeepFreeze

Why?

"This statement can be negated" is a statement in itself. So is the statement, "this statement can't be negated". Why should that fact suddenly and without reason cease to be in the statement; "if this statement can be negated then this statement can't be negated"?

How would you express the fact that IF "this statement can be negated" THEN (the negation is) "this statement can't be negated? Exactly the way I did of course.

TonesInDeepFreeze

"IF this (1) statement can be negated THEN this (2) statement can't be negated" — TheMadFool

I can't make sense of that with the numbers interposed as written. I don't know what is meant by interposing a number between an adjective and what it modifies.

TheMadFool

↪TonesInDeepFreeze

Tak the liar sentence, this sentence is false

The logic proceeds as follows:

1. IF this sentence is false is true THEN this sentence is false is false.

2. IF this sentence is false is false THEN this sentence is false is true

The "this" refers not to the entire statements in line 1 and 2 but to the liar sentence.

TonesInDeepFreeze

Whatever you mean to say, if you want me to understand it (especially to understand it exactly) then you need to rewrite it without a number between an adjective and what the adjective modifies.

TonesInDeepFreeze

Liar paradox:

If "This sentence is false" is true, then "This sentence is false" is false.

and

If "This sentence is false" is false, then "This sentence is false" is true.

TheMadFool

↪TonesInDeepFreeze

Yes!

1. Either "this statement can't be negated" can be negated or "this statement can't be negated" can't be negated [premise] — TheMadFool

TonesInDeepFreeze

↪TheMadFool

That's good. Though, it's logically true anyway.

Next is to fix line 3 of Argument A.

TheMadFool

↪TonesInDeepFreeze

Nothing to fix, the negation of "this statement can be negated" is "this statement can't be negates".

So,

If this statement can be negated then this statement can't be negated [premise] — TheMadFool

If you like, think of it as, if "this statement can be negated" then (its negation is) "this statement can't be negated"

TonesInDeepFreeze

Nothing to fix, — TheMadFool

There is no apparent meaning in placing '(1)' between the adjective 'this' and the noun 'statement'.

Now I've said that three times.

TonesInDeepFreeze

the negation of "this statement can be negated" is "this statement can't be negates". — TheMadFool

Again, you miss the point:

In "This statement can be negated", 'this statement' denotes "This statement can be negated".

but

In "This statement can't be negated", 'this statement' denotes "This statement can't be negated".

So 'this statement' is used to denote two different things.

As I mentioned, that ambiguity comes from the fact that 'this' is contextual.

You have to set up your presentation so that it stays clear of those kinds of natural language foibles.

TheMadFool

Again, you miss that: — TonesInDeepFreeze

Refer to my post about the liar statement.

The liar statement = This sentence is false

The logic, I'm told, proceeds as follows:

1. IF this statement is false is true THEN this statement is false is false.

2. IF this statement is false is false THEN this statement is flase is true.

The "this" refers to "this statement is false" and not the statements 1 and 2

Also,

Suppose we consider the statement "god exists". What is its negation? "god doesn't exist. In other words, the negation of "god exists" is "god doesn't exist".

Now, look at "this statement can be negated". If it can be negated, the negation is "this statement can't be negated". Put simply, "this statement can't be negated" is implied by "this statement can be negated" and this logical relationship is expressed as:

IF this statement can be negated THEN this statement can't be negated.

TonesInDeepFreeze

↪TheMadFool

I answered your post about the liar. Now you're just flat out ignoring that answer.

And, still you are not facing that putting '(1)' between 'this' and 'statement' makes no sense.

TonesInDeepFreeze

Suppose we consider the statement "god exists". What is its negation? "god doesn't exist. In other words, the negation of "god exists" is "god doesn't exist". — TheMadFool

Yes, but that fails with 'this statement' in the mix because 'this' is contextual.

Also, you misuse the concept of 'implies'.

Each of your posts express even more of your confusions. It's exponential.

TheMadFool

answered your post about the liar. Now you're just flat out ignoring that answer.

And, still you are not facing that putting '(1)' between 'this' and 'statement' makes no sense. — TonesInDeepFreeze

You've missed the point of the numbering.

TonesInDeepFreeze

You've missed the point of the numbering. — TheMadFool

You are utterly obtuse. I miss the point of the numbering because your use of it is not grammatical. Make it grammatical if you would like me to understand whatever point you have.

TheMadFool

Yes, but that fails with 'this statement' in the mix because 'this' is contextual. — TonesInDeepFreeze

I've tried explaining to you that "this" is, as you said, is ambiguous but the point is precisely that.

TonesInDeepFreeze

I've tried explaining to you that "this" is, as you said, is ambiguous but the point is precisely that. — TheMadFool

Then the moral of your exercise is trivial. It merely highlights what we already know: English pronouns and demonstrative pronouns are contextual and if used carelessly can cause ambiguity.

TheMadFool

You are utterly obtuse. I miss the point of the numbering because your use of it is not grammatical. Make it grammatical if you would like me to understand whatever point you have. — TonesInDeepFreeze

I'm done here! You raised some good objections and I responded to them adequately.

TheMadFool

Then all you've done is highlight what we already know: English pronouns and demonstrative pronounds are contextual. — TonesInDeepFreeze

Yes, contextual but that's exactly the point!

Negation Paradox

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