Re: [math-fun] true self-referential sentences with false contrapositives
I understand why the second sentence is called the literal contrapositive, but the "this" in that sentence refers of course to something other than the "this" in the first sentence. Which is at least one reason it's only a "literal" but not actual contrapositive. --Dan Jim wrote: << The sentence "If two plus two equals four, then the premise of this sentence contains numbers" is true, while its literal contrapositive "If the premise of this sentence does not contain numbers, then two plus two does not equal four" is false. Can anyone find a more elegant example of a true self-referential sentence of the form "If ..., then ..." whose literal contrapositive is false?
Sometimes the brain has a mind of its own.
"If 3 is a prime, then the contrapositive of this sentence is false." The contrapositive is: "If the contrapositive of this sentence is true, then 3 is not prime." which is false. (The contrapositive of the contrapositive is the original statement, which is true.) Sincerely, Adam P. Goucher
On Wed, May 11, 2011 at 2:13 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
"If 3 is a prime, then the contrapositive of this sentence is false."
The contrapositive is:
"If the contrapositive of this sentence is true, then 3 is not prime."
which is false. (The contrapositive of the contrapositive is the original statement, which is true.)
You claim that the first sentence is true and the second is false. Why is this any more correct than the claim that the first statement is false and the second statement is true? It's similar to statements like 1. Statement 2 is false 2. Statement 1 is false You can assign the truth values (T, F) or (F, T) without contradiction. But I'm not sure it's really correct to say that these are meaningful statements that are true or false at all. Is the statement "This statement is true" true or false? I don't find this any more paradoxical than the trio 1. The statement directly below this one is false. 2. The statement directly below this one is false. 3. 2 + 3 + 5 Statements 1 and 2 are the same statement, yet they have different truth values. A more troublesome paradox: What about the statements: 1. 2 + 2 = 4 2. You will mail me a check for a thousand dollars tomorrow. 3. An odd number of these three statements are true. Which of these 3 statements are true, and which are false? Andy
You claim that the first sentence is true and the second is false. Why is this any more correct than the claim that the first statement is false and the second statement is true? It's similar to statements like
1. Statement 2 is false 2. Statement 1 is false
You can assign the truth values (T, F) or (F, T) without contradiction.
In either case, the contrapositive of the true statement is false. So, we can assume wlog that the first statement is true, and its contra- positive is false. Sincerely, Adam P. Goucher
On Wed, May 11, 2011 at 11:40 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
You claim that the first sentence is true and the second is false. Why is this any more correct than the claim that the first statement is false and the second statement is true? It's similar to statements like
1. Statement 2 is false 2. Statement 1 is false
You can assign the truth values (T, F) or (F, T) without contradiction.
In either case, the contrapositive of the true statement is false. So, we can assume wlog that the first statement is true, and its contra- positive is false.
There's a difference between "Statement 1 is true" and "The assumption that Statement 1 is true does not lead to a contradiction". For example "Statement 1 is true" and "Statement 2 is false" can't both be correct, and in this case, both ""The assumption that Statement 1 is true does not lead to a contradiction" and "The assumption that Statement 1 is false does not lead to a contradiction" are true. Andy
participants (3)
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Adam P. Goucher -
Andy Latto -
Dan Asimov