Meaningless vs. Undecidable

[honestrosewater]

Are the following definitions correct?
A statement S is meaningless if its solution set is empty.
S is undecidable if its solution set contains contradictory solutions.
:confused:
I am having problems understanding those terms. If someone can give me a precise explanation, I would appreciate it.
___
Sorry, momentary lapse of judgment. This isn't really the appropriate forum.

[matt grime]

Statements do not have solution sets.

S is undecidable in a formal system X, with axioms A(X) if both A and S, and A and notS are not inconsistent.

Examples:

the continuum hypothesis, and the axiom of choice are undecidable in ZF. (proof mainly due to Cohen.)

A statement/proposition U=>V is vacuous if U is always false, which is as close to meaningless as I can get for you I think.

I hope Hurkly can clear this up, he seems to understand this best.

[honestrosewater]

S is undecidable in a formal system X, with axioms A(X) if both A and S, and A and notS are not inconsistent.
So if
(A \wedge S) \wedge (A \wedge \neg S)
is true, then S is said to be undecidable in X? Or if the above is "not false" or "possible"? I don't understand precisely what "not inconsistent" means.

Or is A considered to not be a statement?

Are undecidable statements only possible if A(X) does not include both the Laws of Non-Contradiction and the Excluded Middle?
I am thinking about this, but I could use a hint.

[matt grime]

I was carefully trying not to say consistent.

What it means is if I assume S is true I can contradict none of the axioms in A, and similarly if I assume S is false then I still can contradict none of the axioms in A. Or I can neither prove S is true, nor S is false from the axioms, A.


A isn't a statement, A is the collection of axioms of the underlying theory. (usually we talk of ZF).

The last thing you wrote is definitely not true. I gave you two examples of undecidable statements in ZF, where we are implicitly using predicate logic.

For instance the parallel postulate is 'independent' of the other axioms of geometry - neither it nor its negation can be deduced from the other axioms. (proof - we construct models satisfying the other axioms one of which has the parallel postulate true, the other has it false, eg the hyperbolic model and the euclidean model.)

[honestrosewater]

You rock. Thank you.

Some further clarifications:
The undecidability of S in X can only be determined if S is well-formed in X?

Can decidability be extended to all possible truth-values in X? That is, if T, F, and P are all possible truth-values in X, can you say:
S is undecidable in X if neither (S is T) nor (S is F) nor (S is P) can be proven from A? Or is it restricted to only true and false?

[loseyourname]

Are the following definitions correct?
A statement S is meaningless if its solution set is empty.
S is undecidable if its solution set contains contradictory solutions.
:confused:
I am having problems understanding those terms. If someone can give me a precise explanation, I would appreciate it.
___
Sorry, momentary lapse of judgment. This isn't really the appropriate forum.

I get the impression you were originally asking this question in an informal context. You'll need to specify what kind of "meaningless" you are referring to. A statement is said to be factually meaningless if the truth value of that statement cannot be determined in principle. Take, for instance, the statement "blok is greener than wolt." Because we don't know what the words "blok" and "wolt" refer to, there is no way to assign a truth value to this statement. Put into logical terms, this statement is not a proposition, and so no propositional calculus can be applied to any argument containing this statement. Such an argument would not be a truth-functional argument. Of course, that statement is also grammatically meaningless because it contains words that don't exist.

However, there is no need to use an example that contains made-up words. Take the statement "I am 24." This statement is also factually meaningless if taken out of context, because we do not know what the word "I" refers to. If it's referring to me, then it is true. But if you make the statement, it might not be. This statement can also not be part of a truth-functional argument, unless it is prefaced by another statement that made explicit who the word "I" is referring to. Any statement whose truth value is thus indeterminable is factually meaningless, even though clearly the latter statement has a grammatical meaning, whereas the former does not.

There is also another way in which a statement can be factually meaningless. Take the statement "Aristotle is more of a female than Socrates." Even though we know what every word refers to, and all of the words are real words, there is still no way to evaluate the truth value of this statement, simply because neither Aristotle nor Socrates is female, and neither can be more or less female than the other. You might simply contend that the statement is false, but according to the rules of logic, the negation of any false statement is true, and clearly the statement "Aristotle is less female than Socrates" is not true.

I don't know how to answer the questions you asked initially, because as Matt pointed out, statements don't have solution sets, so their solution sets can be neither empty nor can they contain contradictions.

[honestrosewater]

You'll need to specify what kind of "meaningless" you are referring to.
Wow, I meant to say "undefined", not meaningless. For instance, 1/0 is "undefined" in a field. (I am familiar with 1/0, but not with "undefined")
Sorry, don't ask how I made and repeatedly overlooked that mistake. Your explanations have not gone to waste, though. Thank you.

Put into logical terms, this statement is not a proposition, and so no propositional calculus can be applied to any argument containing this statement.
Can a statement be both "factually meaningless" and "undecidable"? I would assume not, but this would answer my question, "The undecidability of S in X can only be determined if S is well-formed in X?"

[loseyourname]

Actually, any given statement is factually meaningless if its truth value is undecidable. There is no way for a statement itself to be undecidable. How do you decide a statement?

[honestrosewater]

Actually, any given statement is factually meaningless if its truth value is undecidable.
Oh, right. I was only thinking of your first example. Wow, I'm developing quite a bad track record in this thread.
There is no way for a statement itself to be undecidable.
I know, just an oversight.