## Thursday, October 29, 2015

### The truth value of unprovable statements

Many statements which are not provable, at least not by us, are either true or false, and we agree that they are either true or false even if we have no way of determining whether they are true or false. A good example would be
1)______ committed the Jack the Ripper murders. For any name you put in, they either did or did not commit those crimes. Yet we have no doubt that there is someone (or more than one person) for which that statement is true, just as it is false of everyone else.

2) There is life of comparable intelligence to our own on other planets.

Ilíon said...

*Every* proposition is either true or false. A "statement" that is neither true or false isn't a statement, it's not nonsensical noise.

Unknown said...

I wonder if (2) is really an example of unprovable statement or is merely an unproved statement.

A proof is a formal demonstration, beginning with axioms. Thus the notion of proof within a formal axiomatic system.

But our minds are far more than manipulators of axioms. Thus, we can perceive truth of statements that can not be proved within a formal system. It is precisely these statements that are unprovable. Eg Goedel's statement-given axioms of some mathematical systems, a proposition can be build that is unprovable within the system but we can see it to be true.

World of Facts said...

(2) is certainly provable but almost just as certainly never going to be proven during our lifetime. Does that elevate it to a status of quasi non-provable?

So, isn't the general point here that some propositions, even if technically provable, are so hard to prove that they are indistinguishable from non-provable ones?

Beside the existence of intelligent life elsewhere in the universe, questions regarding the universe being infinite or whether a universe-creating god exists are along those lines too.

Well, except if you believe in that kind of god.

World of Facts said...

Ilíon, you're mostly right here I think, but what you call nonsensical noise is still some form of proposition. Basically you define 'proposition' and 'statement' to implicitly include a condition of validity. Statements are always either true or false; otherwise, they are not valid and thus not statements.

However, what usually matters most, and I am wondring whether that's actually why you made that comment, is that when a statement is not proven false, it doesn't imply that it's true*; even if we know it must be either true or false.

*Fixed typos!

Unknown said...

I think "provable" is a misleading word for the kind of propositions listed in OP.
It is more suited to mathematical or metaphysical propositions but not really for empirical propositions. As Illon says, they are immediately perceived to be true or false.

Steve Lovell said...

I think Ilion is talking more about things like "This sentence is false" which can't be said to be either true or false. If that is what Ilion means, then I agree with him.

planks length said...

Steve,

I would imagine that Ilion was thinking about statements like "The color purple weighs 18 pounds."

Nonsensical noise.

Ilíon said...

Vishal Mehra: "I think "provable" is a misleading word for the kind of propositions listed in OP."

It's not that "provable" is the wrong word to use, it's that people just don't understand (and frequently refuse to understand) what the word 'to prove' means. -- It's due to that misunderstanding of the word that people quote the old saying, "The exception proves the rule", and get its meaning entirely backward. -- It's due to that misunderstanding of the word that people hear that some place is a "proving ground" and their response is, "Huh?"

'To prove' means to test a thing (an object or a proposition) by an appropriate metric, to see whether it does the job for which it's intended. So, of course, you have to understand what the appropriate metric is to test your thing.

When people say things like, "Nothing is ever proven in science", what they demonstrate is that they don't understand what 'to prove' means. Well, that and that scientists frequently don't even attempt to test their hypotheses before asserting them as facts.

Vishal Mehra: "It is more suited to mathematical or metaphysical propositions but not really for empirical propositions."

No, it's that proving a mathematical (or metaphysical) proposition is a special sub-set of proving. Mathematics is entirely logical/non-empitical, and so a mathematical proposition can be proven entirely by logical relationships. Metaphysical propositions generally have at least some empirical content or component, and so proving them generally requires at least some empirical reference. An automobile is a very physical thing, and so proving a new automobile design involves very little logic and quite a bit of empirical content.

Vishal Mehra: "As Illon says, they are immediately perceived to be true or false."

That's not at all what I meant (and I certainly wouldn't say it).

VR had written, "Many statements which are not provable, at least not by us, are either true or false, and we agree that they are either true or false even if we have no way of determining whether they are true or false."

And my response was, "[Well, no, that's not correct --] *Every* proposition is either true or false [-- regardless of whether we know how to determine which it is]", and "[some series of words strung together to look like a sentence] that is neither true nor false is not a statement [of a proposition], it's nonsensical noise." (I now see that a stray "not" found its way into my written statement, making the literal statement I posted, in contrast to what I meant, nonsensical)

Ilíon said...

Steve Lovell "I think Ilion is talking more about things like "This sentence is false" which can't be said to be either true or false. If that is what Ilion means, then I agree with him."

planks length: "I would imagine that Ilion was thinking about statements like "The color purple weighs 18 pounds.""

You two are in violent agreement with one another (and with me).

Ilíon said...

Hugo Pelland: "Ilíon, you're mostly right here I think ..."

*sigh* I'm always right. Even that one time when I thought I had been wrong, it turns out that I was mistaken [about having been wrong].

Hugo Pelland: "... but what you call nonsensical noise is still some form of proposition. Basically you define 'proposition' and 'statement' to implicitly include a condition of validity. Statements are always either true or false; otherwise, they are not valid and thus not statements."

That's simply what the terms 'statement' and 'proposition' mean

A [series of words strung together in a sentence-like manner] is not a 'statement' by virtue of looking like a sentence, but rather by virtue of the logical correspondence of its meaning to reality (*); this correspondence is either 'yes' or 'no'. A "statement" with no meaning, or with no logical correspondence to reality (*), is not a statement.

A 'proposition' is a proposed statement; it is a statement that one proposes as corresponding to reality (*). A "proposition" that isn't a statement isn't a proposition.

(*) And, no, someone who refuses even to think about what I have said is not going to falsify it by reference to Frodo.

Ilíon said...

Hugo Pelland: "However, what usually matters most, and I am wondring whether that's actually why you made that comment, is that when a statement is not proven false, it doesn't imply that it's true*; even if we know it must be either true or false."

Good night! I'm the last person in the world who would make/assert *that* mistake. As it turns out, one of the things so many people get so pissed-off at me about is that I *won't* allow their (generally) implicit assumption of that, or sometimes explicit assertion, to go unchallenged.

A statement is always either true or false; we know that about *every* statement, even if we don't know how to test some particular statement to determine whether it is true or false.

Since we know that *every* statement is either true or false, we *also* know that the truth-value of the negation of any statement is the negation of the truth-value of the original statement. Thus, we *also* know that to prove the negation of a statement is logically equivalent to having proven the statement itself. Thus, even if we can't figure out a way to directly test some proposition, if we can figure out a way to test its negation, then we can thereby test the original proposition.

Ilíon said...

By the way, PL, have you ever smelled the color nine?

B. Prokop said...

"*sigh* I'm always right. Even that one time when I thought I had been wrong, it turns out that I was mistaken [about having been wrong]."

Is this an example of one of those statements that is "nonsensical noise"?

Ilíon said...

"Is this an example of one of those statements that is "nonsensical noise"?"

No, this statement is one of those demonstrating the truth of Vishal Mehra's statement that "But our minds are far more than manipulators of axioms."

World of Facts said...

Well said Ilíon, makes perfect sense regarding what statements are. That being clear, what do you think the truth value of a statement such as 'The universe has always existed' is? And why do you think Atheists are always lying?

World of Facts said...

Well said Ilíon, makes perfect sense regarding what statements are. That being clear, what do you think the truth value of a statement such as 'The universe has always existed' is? And why do you think Atheists are always lying?

B. Prokop said...

Hugo,

The universe cannot have always existed. For that to be the case, there would have of necessity been a point of time infinitely in the past, from which our present moment ("now") would have been an infinite amount of time into the future. For that hypothetical point in the infinite past to have existed (being at some point "now"), our present time could never have been arrived at. But we are demonstrably here, in the "now". Ergo, there must have been some initial point in time, from which there are no preceding points. Otherwise, we could not be here. We would forever be in the infinite future.

So the universe has not always existed.

Jezu ufam tobie!

World of Facts said...

That makes no sense in the context of our current understanding of spacetime. Because you're partially right in that 'time' as we currently experience it has not always existed; the furthest back we can think of is the Big Bang. That doesn't the universe is dependent on the existence of that time or spacetime.

Gyan said...

To test is only one meaning of the word To Prove. Among other meanings there is to demonstrate the truth of something.
And I do not think unprovable in this context means untestable but it means undemonstrable.

Ilíon said...

"To test is only one meaning of the word To Prove. Among other meanings there is to demonstrate the truth of something."

That's not a separate class of meaning of the word, it's a special case.