sodium

To Lazcatluc,

You make the point that GB (or your smart person) doesn't need to be omniscient in order to understand a simple deduction. But that isn't the question. With God, we aren't expecting that he is deducing these things. He just knows them. So, if a statement is true, it follows that God believes it. With GB, this follows only as the result of a bit of intellectual effort, which he may not engage in. In fact, it is reasonable to believe he won't, if he realizes that he cannot develop a true belief with regard to the proposition.

To bd-from-kg,

You claim that this situation would be absurd or ridiculous. It is perhaps somewhat amusing. It's kind of frustrating to feel that you can logically figure out the truth value, yet know you can't. The conclusion is strange, but it does follow, and it fits with the actual real life experience when you're faced with one of these statements. I find it harder to imagine that a statement is meaningless, when we clearly know its meaning, and can even guess at the truth value.

If you want to render such statements meaningless, I wonder what general rule you would adopt for doing so. Would any statement that is true, yet cannot be consistently asserted by any particular person (or perhaps machine) be meaningless?

You mention the idea of levels of statements. But keep in mind, my proposition is not a statement about the truth value of statements. Statements about the truth value of statements can result in logical contradictions. But a statement about belief cannot in itself, do this. It is only when you start assuming that belief always follows from truth and vice versa that you get a contradiction.

bd-from-kg

sodium:

If you can believe that it possible to prove from pure logic that it's impossible for someone to believe a meaningful proposition that he knows to be true, there's no hope for you. I'm outta here.

sodium

I encourage people who have read my posts to consider whether that is in fact what I'm saying. But here's an analogy I hope will shed some light on the subject.

Let's say I have a book full of true statements that I call the Book of Truth. I claim that my book contains any true statement you can think of. You try guessing a few statements, and I show you where they are written in the book. Then, you get an idea for a stumper. You suggest the statement, "This statement is not written in the Book of Truth".

You think you've got me stumped, but I try a surprising new tactic. I claim that the question of whether the statement is written in the book is meaningless. You point out that even for a meaningless sequence of letters we should be able to find whether it is in the book or not. I claim that for some statements, we simply cannot ask whether or not they're in the book. It is meaningless to do so. Something about self-reference.

Of course you wouldn't believe me. But is it so different to claim that it is meaningless to ask whether a particular person believes a particular statement? Not if we believe that those beliefs are actually part of reality. Of course, we could define "belief" so that it is not part of reality, for example, as lazcatluc has done, by claiming that people should be taken to "believe" certain statements if they logically follow in certain ways from other statements that they know to be true. Then, you aren't talking about something out there in the real world. You're talking about a hypothetical concept. One that can be proven to contradict itself.

But if you think that the beliefs of God are out there, in the real world, then we are in the same position as with the book. To say that some beliefs are in the mind of God, some beliefs are not, and some it is meaningless to ask, doesn't make any sense.

Tenpudo

This is the same sort of argument that Saint Augustine (I think) used -- "God, by definition, exists."
Of course, personally I prefer the Jelly Donut rejoinder, myself...
"Applied theology is delicious!"

bd-from-kg

sodium:

OK, I’ll give it one more try, since this is as good an example as any to illustrate what’s wrong with this whole style of argument.

The statement “This statement is not written in the Book of Truth” is indeed meaningless – that is, it does not express a proposition. What you’re trying to do, in case after case, is to go from “X is meaningless” to “X is true” or “X is false”. But “X is meaningless” can never entail either “X is true” or “X is false”. If it seems to entail either of these things, it can only be because X is indeed meaningless.

Thus in your fantasy, the statement “This statement is not written in the Book of Truth" is not written in the Book of Truth, but this fact doesn’t make it true. It’s not in the Book of Truth because it’s meaningless. Being meaningless cannot make a statement true. This is pure, unadulterated nonsense.

sodium

I didn't "go from 'X is meaningless' to 'X is true' or 'X is false'". Assuming the statement is not written in the book, then it is true because we know what it means, and if we check whether what it is saying is true or false, we find that it is true. If it were written in the book, when we did the check, we would find that the statement was false. In neither case is there any trouble assigning the proposition a truth value. And in neither case do I try to derive the truth value by claiming that the statement is meaningless. That wouldn't make any sense.

bd-from-kg

sodium:

Please. You ask us to consider the statement:

(#)This statement is not written in the Book of Truth

You say:

And in your latest post you make the point even more unambiguously:

To understand the logic here we need to recall that you’re assuming for the sake of argument that (#) is meaningless. What do you have to say about this case?

(i) We can still find out whether (#) is in the Book of Truth.
[And in this case it obviously won’t be in the Book of Truth since it isn’t true, being meaningless.
(iii) But this is precisely what (#) asserts: that it’s not in the Book of Truth. (We know what it means, and this is it.)
(iv) So (#) is true.

In other words, “(#) is meaningless” entails “(#) is true”.

But this is absurd, as I pointed out. It’s impossible in principle that for any statement X, “X is meaningless” entails “X is true” (or “X is false”).

The fallacy is in the claim that (#) asserts that it’s not in the Book of Truth. Of course it seems at first sight to assert this, but on analysis we find that it doesn’t.

There are two possibilities:

(A) The “Book of Truth” is logically impossible – i.e., it’s a meaningless concept, like “square circle”.
(B) The “Book of Truth” is a meaningful concept. In that case it’s meaningful to talk about whether a given statement is in it ( or more precisely, whether a given statement would be in it if it existed). Then we have three subpossibilities:

If (A), it’s meaningless to talk about whether a statement is in it or not. (Actually I think it can be proved rigorously that this is the case.) In that case (#) is immediately meaningless.

If (B), we have three possibilities:

(B1) (#) is true.
(B2) (#) is false
(B3) (#) is meaningless.

But if (B1), then (B2), and if (B2), then (B1). Both of these are contradictions, so these possibilities are eliminated. That leaves (B3).

Thus regardless of whether the Book of Truth is a meaningful concept, (#) is meaningless. But a meaningless statement does not assert anything, regardless of superficial appearances. So either the Book of Truth is a meaningless concept, or (#) is not in the Book of Truth. But in either case (#) isn’t true, because it doesn’t assert that it’s not in the Book of Truth. It doesn’t assert anything at all. It’s just a meaningless string of symbols.

[Note: For the record, I think that it can be rigorously proved that (A) is correct: the “Book of Truth” is a meaningless concept. But your argument fails to show this. Fallacious arguments sometimes have true conclusions.

Finally, a couple of comments regarding the “meta-hierarchy” of statements that I described earlier:

The criterion for determining the level of a given statement is not what statement(s) it makes truth claims about, but simply what statements it refers to. Thus the fact that some of the statements you discuss talk about whether a statement is believed rather than whether it’s true doesn’t save them from being meaningless by this criterion.

The reason that the levels can be infinite is that a statement can refer to an infinite class of statements. For example “Any statement of the form ‘x cubed plus y cubed equals z cubed’, where x, y, and z are nonzero integers, is false”, refers to an infinite number of statements. This doesn’t disqualify it from being meaningful, since all of the statements in question are “Level 0”. But more complicated statements can be devised where the statements referenced include statements of level N for every positive integer N. Such a statement is still meaningful, but its level is omega – the order of the integers. On the other hand, a statement that refers to all statements, such as “For any statement S, S is in the Book of Truth if and only if it expresses a true proposition” is out of court because it cannot be assigned a level – not even an infinite one. [This is the fundamental reason why the “Book of Truth” is (I believe) a meaningless concept, but it isn’t the “rigorous proof” I referred to earlier.]

sodium

First, as a slight digression, the word "meaningless" may be a bit misleading. Some statements are indeed meaningless in that we have no idea what they're saying, or at least don't know precisely enough. But a statement like "this statement is false" doesn't fall under that category. The problem with that statement is that it cannot be successfully assigned a value of true or false for every state of the world. Such a statement can't be considered a proposition, because of assumptions we implicitly make about propositions.

Now, back to the problem at hand, you declare that the statement is meaningless if the Book of Truth doesn't exist. That's one possible interpretation, but it leads to a lot of mischief. We aren't used to propositions suddenly becoming meaningless under certain assumptions.

I interpret the statement as being false if the Book doesn't exist. I suppose I could rewrite the statement to make this clearer, but I hope you won't make me bother.

Now as for assigning levels to statements, what would you do with this statement:

Russell wrote "This statement is false" in his essay.

Since you can't successfully assign a level to "This statement is false", it follows that you can't assign a level to the larger statement, and so it must too be meaningless. But it sure seems like it means something. I guess we've proven otherwise though, through the power of level assignment, as you interpret it.

bd-from-kg

sodium:

It can be, but since I clearly defined what I mean by “meaningless” in this context (namely, “does not express a proposition”) this isn’t a problem here. Certainly it would be odd in some contexts to say that statements like “Close the door”, “What’s your name?”, and “Hello!” are “meaningless”, but this context I think that the convenience of having a word for statements that don’t express propositions outweighs the importance of adhering to standard usage.

I don’t see how you can get that. If the Book doesn’t exist, nothing is written in it, so if you insist on assigning a truth value to (#) for this case, it should be “true”. Of course, this doesn’t lead to a paradox, which perhaps is why you don’t like it...

You’ve got me there. In fact, it gets even worse. For example, we would surely want to say that “The statement ‘This statement is false" is neither true nor false” is true, yet this statement not only refers to the meaningless statement “This statement is false” but makes an assertion about its truth and falsehood. Clearly the presence of a reference to a self-referencing (or otherwise meaningless) sentence does not automatically disqualify a sentence from being meaningful. I still think that the “level” idea can work, but at this point I haven’t the foggiest idea how to do it.

Witt

bd-from-kg:

OK, I’ll give it one more try, since this is as good an example as any to illustrate what’s wrong with this whole style of argument.

The statement “This statement is not written in the Book of Truth” is indeed meaningless – that is, it does not express a proposition. What you’re trying to do, in case after case, is to go from “X is meaningless” to “X is true” or “X is false”. But “X is meaningless” can never entail either “X is true” or “X is false”. If it seems to entail either of these things, it can only be because X is indeed meaningless.

Thus in your fantasy, the statement “This statement is not written in the Book of Truth" is not written in the Book of Truth, but this fact doesn’t make it true. It’s not in the Book of Truth because it’s meaningless. Being meaningless cannot make a statement true. This is pure, unadulterated nonsense.
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Hi bd-from-kg,

I believe that you are correct here.

When antinomies occur, non-existence is present, and a repair to language useage is required.

The Russell Class, the barber paradox, and apparent propositions such as 'This statement is false', do not have any meaning.

'This statement is false' is not a meaningful statement, because it is neither true nor false, i.e. it is not an existent proposition at all.

Logic can only deal with extensions of things, and there is no extension of the expressions: This proposition is false, the class of those classes that are not members of themselves, the barber who shaves all and only those who do not shave themselves, etc..

When all truths and all falsities are listed, 'this proposition is false' is not on either list. It does not exist as a proposition.

It is similar to the statement, the present king of France is bald.

When all people who are bald and all people who are not bald are listed, 'the present king of France' is not on either list.
Because, it does not exist.

Witt