berry's paradox

Four valued logic Q&A 1998.5

9. What is a form of the 4^th valued proposition?

10. How does the 4-valued logic treat a self- reference proposition?

11. Berry's paradox is a third-valued proposition?

12. Liar paradox is a 3^rd-valued proposition?

Q&A1234 | Q&A5678

Q&A9 : What is a form of the 4th valued proposition ?

Usually the proposition can exist itself with an object which it's proposition signify. However, the 4^th valued proposition has not this form.

Example of 4^th valued proposition : Nothing can exist in no-time-space.

We image usually something from this sentence in the topological space in spite of no time-space(p).

So we have to deny something itself(p=>-p), because it's something has time and space. Then we try again to image same other thing in no-time-pace(-p=>p). If we image another thing, it is same result(p=>-p). Then it invents a paradoxical, reflexive repetition(p<=>-p). This type of proposition has a hidden dual implication in it's proposition. Then we have to deny again this real appearance(p<=>-p) of the 4^th valued proposition itself ((p<=>p)=>-(p<=>-p),because this real appearance of the 4^th valued proposition is something which can only exist in time-space.

But, What is this -(p<=>-p)? ,and so on.

So the real appearance of 4^th valued proposition is always 3^rd valued proposition which deny itself. It can be nothing but catch a temporary appearance.Therefore, the 4^thvalued proposition has always the self-denial form which has the hidden implication to deny the signification or the proposition itself. The 4^th valued proposition wishes always to imply nothing or to mean nothing. In our thinking, there are such a lot of proposition which imply the logical ultimate existence without time and space.

The 4^thvalued proposition takes the denial form of the 3^rdvalued proposition by which the 4^th valued proposition may invent such ideal and logical and no-real existence without "time and space".

The form of 3^rd valued proposition (p<=> -p) is

p=>-p=>p=>-p=>p=>-p.......or

p=-p(p(-p(p....... .

The form of 4^th valued proposition -(p<=>-p) which deny the 3^rd valued proposition is next.

p=>-p=>-(p=>-p)=>-(p=>-p=>-(p=>-p))=>-(p=>-p=>-(p=>-p)=>-(p=>-p=>-(p=>-p)))......or

p= -p(-p(-p(-p(-p......

I think that this form express adequately the endless self-denial proposition.

This form of the 4^th valued proposition does not leave perfectly all of affirmatives in the 3^rd valued proposition. Therefore, the 4^th valued proposition have only content which cannot stop denying itself. That is, this fourth valued propositions deny either time or the space ultimately. The comprehensible example of this type of proposition is “Self-denial proposition".

Thus, this type of proposition differs definitely from the true or false proposition and the third valued proposition in the respect. Moreover, it is possible to exist logically only in exclusive with true or false or third propositions.

So, this proposition cannot exist with the third valued proposition, or, with a true or false proposition.

Q&A10 : How 4-valued logic treat a self- reference proposition?

A：4- valued logic thinks about the self-reference proposition as an implication which the antecedent "protasis" imply the consequent "apodosis"

This means that it is not necessary to analyze the self-reference proposition beforehand at which level it is like 2-valued logic. And then this means that the truth tables of this logic treats the self-denial proposition ordinarily in our world even if the truth table of implication in 2-valued logic assign false(F) to self-denial proposition(p=>-p).

First of all,4-valued logic depends on the most basic fact that the self-reference proposition generates oneself after the antecedent which describes something. So 4-valued logic recognizes a self-reference proposition as 3-valued proposition which has a hidden dual implication.

Why 4-valued logic think so? Because talking about the proposition about which was talked now is very ordinary in a usual idea, it is necessary for 4 valued logic to treat a self-reference proposition as a logical proposition. The 4-valued logic does not treat a self-reference proposition as a special proposition. The 4-valued logic recognizes all propositions in the truth table. This is an important characteristic of 4-value logic.

Q&A11: The berry's paradox is a third valued proposition?

Ａ：4 valued logic defines that the berry's paradox is the third valued proposition.

Berry's paradox : The phrases of a language that refer to numbers can be ordered, alphabetically and according to length. There will be a definite set of integers named by those phrases of less than any given length. In particular there will be some integer which is the least integer not namable in fewer than nineteen syllables. But this phrase "the least integer not namable in fewer than nineteen syllables" then names this number, yet itself contains fewer than nineteen syllables.(From the oxford dictionary of philosophy by Simon Blackburn)

Now, Proposition p = the least integer not namable in fewer than nineteen syllables exists

As the integer which fill this proposition p exists, the proposition p is true. But this proposition p which refer to numbers does not fill proposition p itself. But, this proposition p approves correctly in first step.

Therefore, berry's paradoxical proposition has the form implication p=>-p=>p, but this 3 step implication has a simple form or a first stage of a endless paradoxical proposition.Because this proposition has the endless implication or dual proposition, this berry's proposition is the 3^rd valued proposition.

Q&A12: Liar paradox is the 3^rd valued proposition?

A: Liar paradox is the 3^rd valued proposition

Liar paradox owing to Epimenides. The simplest example is the sentence "This sentence is false" which must be false if it is true, and true if it is false. Other versions of the Liar introduced pairs sentences, as in a slogan on the front of a T-shirt saying "The sentence on the back of this T-shirt is false" and one on the back saying "The sentence on the front of this T-shirt is true". ( From the oxford dictionary of philosophy by Simon Blackburn)

Now, Proposition p =this sentence is false

The proposition p "this sentence is false" must be false if it is true, and true if it is false. This is arranged p=>-p=>p which is a simple step or a first stage of a endless paradoxical proposition. So this sentence is the 3^rd valued proposition which has a dual endless implication.