Negation Paradox
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
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
Comments (50)
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
/
Quoting 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.
That's a major re-edit of the OP after several edits. Before I reply, is that your final edit?
That's the best I can do.
Quoting TheMadFool
I'll mention the first problem I find in each Argument, before going on to the rest of it:
Argument A:
Quoting 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:
Quoting 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.
P.S.
In Argument A, 2. does not need to be taken as a premise. 2. follows from 1. by UI.
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"
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'.
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'.
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).
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.
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.
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.
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.
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.
Quoting TheMadFool
That's good. Though, it's logically true anyway.
Next is to fix line 3 of Argument A.
So,
Quoting TheMadFool
If you like, think of it as, if "this statement can be negated" then (its negation is) "this statement can't be negated"
There is no apparent meaning in placing '(1)' between the adjective 'this' and the noun 'statement'.
Now I've said that three times.
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.
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.
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.
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.
You've missed the point of the numbering.
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.
I've tried explaining to you that "this" is, as you said, is ambiguous but the point is precisely that.
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.
I'm done here! You raised some good objections and I responded to them adequately.
Yes, contextual but that's exactly the point!
Show me a reference that claims that "not all statements can be negated" and that "either this statement can't be negated or this statement can't be negated can't be negated".
Then I don't need your exercise.
Are you a genius then? :roll:
Why do you object then?
There's no reason for me to do that.
Back up your claim!
Not in logic. And probably not in anything. So what?
Then why did you say "You're utterly obtuse" as if it mattered to you? :chin:
Yes.
Quoting TheMadFool
Quoting TonesInDeepFreeze
:chin:
Because it's worth pointing out the reason you can't understand the most basic things.
Now you are reaching your true level: emoticons.
Just what the doctor ordered for people like you I guess!
Give me a basic thing you can understand and we'll see how well you fare.
As if 'obtuse' is seriously "mudslinging".
But my remark does bear amendment. It's not that you're obtuse, it's that you are willfully so.
Here's a logical puzzle for you.
IF I am obtuse THEN you're facing an acute shortage of intelligence
I am obtuse!
You were doing better with emoticons.
You resumed the "discussion", not me.
Thanks for the compliment Unfortunately I can't seem to be able to say the same about you. You seem to be bad at everything! :smile: