What's the difference between a statement and a boolean expression?

atran · 2013-10-15 07:38:28

Thank you. So, is it safe to say that a boolean expression is a mathematical statement?

Bob · 2013-10-15 18:42:36

hi atran,

Yes.

Bob

Bob · 2013-10-15 18:49:12

hi Stefy,

I think the Prior argument is clever and interesting but I think it's a fiddle. The argument appears to confuse "TRUE" and "This is TRUE" as the same and so leads to an erroneous result. (Notice that the Wiki writer expresses the point as if it is not generally accepted.)

It is a requirement that the truth state of mathematical statements is never in doubt. I don't think "This is FALSE" fits this requirement, therefore it isn't a mathematical statement.

Bob

anonimnystefy · 2013-10-15 20:57:27

As I said, I do not think it confuses true and "this us true". Can you explain to me why you think your case (iv) lead to a paradix? (Or as the Daleks woukd say simply: "EXPLAIN! EXPLAIN! EXPLAAAIN!" )

Bob · 2013-10-15 22:17:14

hi Stefy,

I thought I had done that already in an earlier post but I'm happy to do it again. I'm assuming that Wiki have correctly stated the argument.

.... every statement includes an implicit assertion of its own truth. Thus, for example, the statement, "It is true that two plus two equals four", contains no more information than the statement "two plus two equals four", because the phrase "it is true that..." is always implicitly there.

It is unclear whether that means

TRUE AND A = A

OR

"A is TRUE AND A = A

so I explored both.

The first diagram below shows the truth table for the first possibility and the third diagram shows the other.

I accept the first as the truth tables are the same.

I do not accept the second as the truth tables are different.

The argument then uses the second interpretation to generate an expression that is shown in my second (and fourth) truth table.

I accept that this statement is FALSE. But I do not accept that the statement is equivalent to the statement "this statement is false".

So I do not think this argument clears up the problem.

Furthermore, let's pretend it does and see where it leads.

We are being asked to accept that

"This statement is true and this statement is false" is FALSE and also that

"This statement is false" is equivalent to "This statement is true and this statement is false".

using substitution that means that

"This statement is false" is FALSE.

This means that "This statement is TRUE" and we are back to the paradox.

Bob

Agnishom · 2013-10-18 02:29:35

Hi Bob,

Do you agree with Arthur's resolution? Yes / No?

Bob · 2013-10-18 07:48:10

hi Agnishom,

YES/NO is not possible.

Firstly, some assumptions:

(i) Wiki has correctly given the 'proof'.

(ii) By Arthur's resolution you mean Prior's proof that there is no paradox, but rather the statement is FALSE.

On that basis here's what I think:

(iii) A = "TRUE" AND A ... tick

(iv) A = "This is TRUE" AND A .... X

(iii) and (iv) are not the same thing.

(v) "This is TRUE" AND "This is FALSE" is a FALSE statement .....tick

(vi) There is no paradox ......X

Summary answer = NO

Bob

anonimnystefy · 2013-10-18 10:02:12

bob bundy wrote:

hi Agnishom,
YES/NO is not possible.
Firstly, some assumptions:
(i) Wiki has correctly given the 'proof'.
(ii) By Arthur's resolution you mean Prior's proof that there is no paradox, but rather the statement is FALSE.
On that basis here's what I think:
(iii) A = "TRUE" AND A ... tick
(iv) A = "This is TRUE" AND A .... X
(iii) and (iv) are not the same thing.
(v) "This is TRUE" AND "This is FALSE" is a FALSE statement .....tick
(vi) There is no paradox ......X
Summary answer = NO
Bob

I disagree on iv)

Agnishom · 2013-10-18 17:55:37

Hmm, I see

Bob · 2013-10-18 22:30:29

Stefy wrote:

I disagree on iv)

Explain! Explain! Explaaain!

Bob

Agnishom · 2013-10-18 23:49:42

Maybe, stefy will tell that This is True is always true

anonimnystefy · 2013-10-19 00:00:27

No, I will say that "This is true" always has the value of A.

To Bob:

I can't wait for November 23rd!

Agnishom · 2013-10-19 18:34:26

To anonimnystefy:

Of which year?

Math Is Fun Forum

#26 2013-10-15 07:38:28

Re: What's the difference between a statement and a boolean expression?

#27 2013-10-15 18:42:36

Re: What's the difference between a statement and a boolean expression?

#28 2013-10-15 18:49:12

Re: What's the difference between a statement and a boolean expression?

#29 2013-10-15 20:57:27

Re: What's the difference between a statement and a boolean expression?

#30 2013-10-15 22:17:14

Re: What's the difference between a statement and a boolean expression?

#31 2013-10-18 02:29:35

Re: What's the difference between a statement and a boolean expression?

#32 2013-10-18 07:48:10

Re: What's the difference between a statement and a boolean expression?

#33 2013-10-18 10:02:12

Re: What's the difference between a statement and a boolean expression?

#34 2013-10-18 17:55:37

Re: What's the difference between a statement and a boolean expression?

#35 2013-10-18 22:30:29

Re: What's the difference between a statement and a boolean expression?

#36 2013-10-18 23:49:42

Re: What's the difference between a statement and a boolean expression?

#37 2013-10-19 00:00:27

Re: What's the difference between a statement and a boolean expression?

#38 2013-10-19 18:34:26

Re: What's the difference between a statement and a boolean expression?

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