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<<On the definition that P is true if P>>

This is a definition by intuition: We say that a statement P is true if and only if P.

This definition seems quite reasonable, but if we discuss it deeper, then some disquiet would emerge.

(1)
We may check some statements for examples. We have, by definition, that

"1+1=2" is true if and only if 1+1=2.
"2+1=3" is true if and only if 2+1=3.
"2+2=4" is true if and only if 2+2=4.
"4+3=7" is true if and only if 4+3=7.
.....

As the statement P changes, the judging condition would immediately changes, but a question arises: How do they change? If it is true that the right hand sides change as left hand sides change, they must change in some given rule. This means, there is a relation between the statement P (left hand sides) and the condition P (right hand sides).

(2)
We might find that this definition links that statement P and an event P. To be in detail, when we want to check whether "1+1=2" is true, we have to, according to definition, check whether the event 1+1=2 occurrs.

In the part of the event 1+1=2, we care about the properties about 1, +, =, 2, by which we can judge the occurrence abouot this event (1+1=2); however, in the statement form, if we view "1+1=2" as a statement, then it is merely a string of symbols. This means, when saying " "1+1=2" is true", we are saying that "something" is true, with the "something" replaced by the ordered symbols ------ '1', '+', '1', '=', '2'.

If we agree that the property of a statement is like that, then the statement "1+1=2" seems to have nothing to do with the event 1+1=2. If it does, then we will answer the problem from (1), which is:

As the given statement P changes, how to get the condition Q for this statement P to be true?

To be more clear, we have to rewrite the definition first, with refreshed concept about "a statement", as follow:

We say that a statement P is true if and only if Q(P), where Q(P) is what we will check to verify whether "P is true". For example, as P chosen as "1+1=2", then Q(P), that is, Q("1+1=2") would be 1+1=2.


(3)
We can rather hardly construct an assignment Q to assign each given string of symbol (statement) P to the proper condition that fixes our intuition for ourselves to check if "P is true" holds. In fact, they are ultimately different. We can hardly operate a symbol string and finally form a condition. Indeed, if we want to do that, that is , if we want to form the condition 1+1=2 from the statement "1+1=2", we might say, we can choose the first symbol "1" of the statement, and then choose the second one, "+", and then, "1", "=", "2". Combining them together, we claim that we get the condition 1+1=2. No! It is merely the same statement, what we've done is only forming the same string '1', '+', '1', '=', '2'.

A thought then comes to mind that forming the statement "1+1=2" and forming the condition 1+1=2 seems to follow different systems of rules, where the former is the system concerning strings (sequences) operating, while the later seems to be a more "intrinsic" system, which involves the rules about "sentences", "grammar", "language", and "inference".

Hence we conclude that Q(P) can not be expressed formulatedly by us. We can neither assign each statement P to its Q(P) as we expect one by one since they can't be terminably listed, nor fix the condition Q unchangable, which means, Q does not depend on the choice of P------ Q is a constant condition. But the thought is even worse.

For example, if we let Q be 1+1=2, then for the statement "1+1=2", since 1+1 is indeed 2, we conclude that "1+1=2" is true. But we can perform similar argument: for the statement "3+1=2", since 1+1 is indeed 2, we also conclude that "3+1=2" is true (don't forget our assumption that Q is constant, and set to be 1+1=2 by us), which is strange!!

Since there is no proper way to explain the meaning of this definition, therefore, we can say that the definition that "a statement P is true if and only if P" makes no sense.


(F)
If P is replaced by 1+1=2, then we are trying to express "1+1=2" is true if and only if 1+1=2. However, Programming linguistically speaking, "1+1=2" is a string whose elements are '1', '+', '1', '=', '2', or more precisely we can translate them into ASCII codes; while the right hand side is purely in Peano system. They are essentially different, and then there is nothing else to do.

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