When I use "-->" I intend it to mean "implies". What are the different meanings you contend I'm using it to mean "at different points in the same expression"?


PairTheBoard

Post #66:

Is S a tautology considered as a statement within the axiomatic system ZFC?

Post #72:

In words I believe I'm asking whether the following statement is a tautology.

S': "If we can show that ZFC+ implies the Continuum Hypothesis then we have the Continuum Hypothesis in ZFC+."

----

So we have two things that it could mean beyond "implies"

1) "Consider the following statement within the axiomatic system"
2) "If we can show that..."

Neither of these carry the same meaning as "implies" in a formal mathematical sense.

And you still haven't addressed the fact that I've never claimed anything about such a statement being valid or invalid. I merely said that it's not tautological.

What you don't really seem to follow is that the statement of the given form as pure mathematical symbols has absolutely no meaning in terms of trying to interpret statements.

Both of the following are true statements:

P --> [(P-->True) --> True]
P --> [(P-->False) --> False]

So the truth value of CH (under whatever hypotheses) is absolutely irrelevant, so trying to argue something related to their truth values is absolutely irrelevant. It just doesn't matter what you shove into those slots, you're going to get a true statement. But it's true because there's no way it can be false. No because of anything related to ZFC or ZFC+ or assuming you're under such-and-such an axiomatic system. It's a tautology. It's true by virtue of the structure of the sentence, not because the P or the Q (or the CH) is true or false.

I still don't see why people think these are true. If it means "P proves that if P proves X is true then X is true" that is equivalent to P---->Con(P).

But PTB was talking about statements in which P is a set of axioms and not a statement that is either true or false. Do you think "ZFC+ proves that if ZFC+ proves the CH, the CH is true" is true or false. I agree that if P is a proposition with value T or F then the statements are true, but I didn't think thats what PTB was trying to say.

I cringe a little when I see the terms "True" and "False" thrown around. If what's being added to ZFC is CH, then CH is "True in ZFC+CH". If what's being added to ZFC is ~CH then CH is "False in ZFC+(~CH)". Whether or not a proposition like CH is "implied" by the axioms of an axiomatic system (along with the rules of logic) seems like the more natural way to look at it to me.


PairTheBoard

Right....typically people talk about the formal notion of provability in a system like ZFC and the semantic notion of truth, that cannot be formalized in something like ZFC.

It seems to me that the statement

S: ZFC --> [(ZFC-->CH) --> CH]

is true because it has the logical form of a tautology. But if you fill out a truth table for it, what does it mean to identify CH as true or false in the table? True or False in ZFC? How does that make sense when we know CH is independent of ZFC? Is there some true or false fact of the matter for CH independent of ZFC? I'm doubtful that notion makes any sense. For that matter, I'm doubtful it makes sense to identify true or false values to the axioms of ZFC itself.

However, I do think it makes sense to talk about the rules of inference. And in the case of the statement S above I think it makes sense to apply those rules to recognize that S is a tautology, whether we state it formally as above or less formally with words. Of course, as soon as you bring words into it someone will quibble over them.


PairTheBoard

It depends on what exactly is meant by the sentence, which is why I asked for it to just be written out without using symbols like ---> which might mean different things to different people.

Right they can be consistent or not but true/false doesn't make a ton of sense.

Stating something with symbols isn't any more formal than stating it with words. Symbols convey things quicker if everybody agrees on what they mean but just cause confusion is they don't.

This doesn't really make that much sense. You can't pretend that the logical formality is without meaning until we put words to them.

There's a fairly standard language that is used:

* "If P then Q" (This is my preferred phrasing)
* "P is sufficient for Q"
* "Q is necessary for P"

There's not too much debate as long as the language you use is fully consistent with the truth tables that are created.

* If ZFC then ( if (if ZFC then CH) then CH ).
* If ZFC then ( (if ZFC then CH) is sufficient for CH).
* If ZFC then ( CH is necessary for ZFC ) is sufficient for CH.

And in the end, it's just a boring statement, like all good tautologies are.

As I said to Aaron, I intend "-->" to mean "implies". I believe "If ... Then ..." is also correct. As in "P-->Q" means "P implies Q" or equivalently
"If P Then Q". All of those are used throughout mathematics so they should be ok. I've not seen "P proves Q" used so not sure what to make of it.

Then there's also P<-->Q meaning "P if and only if Q" sometimes written
"P iff Q". Meaning "(P-->Q) and (Q-->P)".

So what do we really mean when we say these things? If, say, we're working in ZFC and we say "Lemma1 --> Theorem1" what do we mean? I think we mean that the rules of inference provided by the axioms of ZFC (along with the rules of logic) allow us to display a sequence of allowable primitive inferences where,

lemma1-->P1-->P2-->...Pn-->Theorem1

ie. there is a "proof" made up of primitive inferences by which Theorem1 follows from lemma1. If this is not what it technically means it seems to come down to that in practice from all the math I've ever done.

So maybe lemma1 "proves" theorem1 is not bad.


PairTheBoard

If you want to include a category like "logically independent" then you're no longer functioning in a binary logic. Not that there's anything wrong with this, but it would require more language and delicacy so that you agree on the distinction between consistent and logically independent.

I somewhat reject that true/false doesn't make sense, but rather that true/false for the statement P depends on assumptions outside of the given axiomatic system. It's still one or the other and not both, but you don't get to know which one it is. This is not that different to me as saying that x is a solution to x^2 = 1. This means that x is one of two values, but you don't know which one.

That would invoke a different set of underlying logical concepts. This language is creates a different logic for for "true but unprovable" statements.

I'm not saying anything here about "true" statements that are unprovable.


PairTheBoard