## Statements are true?

• 255
What does it mean to say that a statement is true?

Is it, as some assert, that a statement has the property of truth or is it that a statement is merely labelled as true?

The trouble with asserting that truth is a property of a statement is in finding a logical process by which the property of truth can be identified.(Tarski's artificial meta-system fails to answer this question.)

On the other side, if statements can only be labelled as 'true' by someone who considers that the statement conforms with the facts of the world as they see them or that it conforms with the axioms of a formal abstract system then the above-mentioned problem evaporates.

Then statements like' this statement is true' make no more sense than 'this statement is blue'. and statements like ' this statement is false' would no longer be problematic.
• 352
I am for the second one: conformity with the facts of the world.
• 1.3k
What does it mean to say that a statement is true?
That if people believe that statement and use it to inform their actions, they will be more likely to make useful decisions related to what the statement refers to.
• 1.2k
Is it, as some assert, that a statement has the property of truth or is it that a statement is merely labelled as true?

Most generally, it is labelled as true.

When you feed a logic sentence to a theory, you need to provide the two-tuple ($\ulcorner s\urcorner$,s) to the formal system. Example:

("Socrates is mortal",true)


There are exceptions. Some but not all sentences happen to be provable in the theory T from existing tuples ($T \vdash s$). According to Gödel's semantic completeness, a statement provable from true two-tuples, is also true (in all models for T) ($T \vDash s$):

$T \vdash s \Rightarrow T \vDash s$

Meaning: if sentence s is provable from T then sentence S is also true in all models for T.

The trouble with asserting that truth is a property of a statement is in finding a logical process by which the property of truth can be identified.

This is exactly what is NOT possible.

Tarski's theorem concerning the undefinability of the truth predicate makes this impossible. A theory is not allowed to evaluate the truth of a sentence. So, this does not exist:

$s \leftrightarrow isTrue(\ulcorner s \urcorner)$

Such isTrue predicate cannot possibly exist. In terms of computability, the following cannot be done:

if(T.eval("Socrates is mortal")) then
doThis()
else
doThat()
end


Such $T.eval(\ulcorner s \urcorner)$ function that can evaluate the truth of any logic sentence s in a first-order (number) theory T cannot exist, and is therefore uncomputable. The fact that some -- just some -- logic sentences will be provable and can therefore be computed to be true, is the exception. It is not the rule.

Tarski's artificial meta-system fails to answer this question.

Well, given the theorem about the undefinability of the truth in the object theory, Tarski proposes a work around. He injects the truth via an encompassing meta-theory. The meta-theory is an extension of the object theory that can evaluate the truth of sentences in the object theory but not in itself.

If T1 is the object theory and T2 is the meta theory, then T2 feeds two-tuples to T1:

($T1.accept(\ulcorner s\urcorner, T2.eval(\ulcorner s\urcorner))$)

This convoluted protocol seeks to bypass the limitations imposed be the fact that T1.eval(($\ulcorner s\urcorner$) is undefinable.

Then statements like' this statement is true' make no more sense than 'this statement is blue'. and statements like ' this statement is false' would no longer be problematic.

The truth predicate is part of the meta theory, T2.eval($\ulcorner s\urcorner$), but the provability predicate is part of the object theory, T1.isProvable($\ulcorner s\urcorner$).

So, if you do not need to prove anything in the object theory T1, then yes, the truth of its statements does not matter. If you want to prove sentences in T1, then the truth of its statements is an essential extra piece of mostly external information.

The T1.isProvable($\ulcorner s\urcorner$) predicate is rather weak. It cannot handle all true sentences s. The following is Gödel's incompleteness theorem, expressed in Tarski's convention T:

$\exists \ulcorner s\urcorner (T2.eval(\ulcorner s\urcorner) \leftrightarrow \neg T1.isProvable(\ulcorner s\urcorner) )$

Which means: T1 is inconsistent (It will prove false statements) or there are true statements that can be expressed in the language of T1 that are not provable from T1. You can use the expression above to derive Gödel's second incompleteness theorem: If T1 is capable of proving its own consistency, then T1 is provably inconsistent.

All of this kicks in, for a first-order theory that embodies enough arithmetic to reach the level of complexity of Q (=Robinson's arithmetic).
• 255
What does it mean to say that a statement is true? — A SeagullThat if people believe that statement and use it to inform their actions, they will be more likely to make useful decisions related to what the statement refers to.

Yes, this is exactly right. Without this link words, statements and even philosophy are meaningless.
• 255
When you feed a logic sentence to a theory, you need to provide the two-tuple (┌s┐ ⌜s⌝\ulcorner s\urcorner,s) to the formal system. Example:

("Socrates is mortal",true)

Who is 'feeding a logic sentence to a theory'?

What is a 'logic sentence'? - as opposed to a 'non-logic sentence'?

What theory are you referring to?

And why does it 'need' a two-tuple?

The thing about formal systems, which are necessarily abstract, is that any proof or truth they generate is only applicable within that particular formal system.

In order to apply the theorems of a formal system to some other system, a mapping is required between the elements or symbols of one system with those of the other. This mapping process is not a logical deductive process and so when the mapping is complete the deductive proof and certainty provided by the formal system cannot be carried over onto the other system.

All too often formal systems and semantic systems are conflated, this is unjustifiable.
• 234
I'm not an epistemologist here... just throwing out my 2 cents here so someone can crush it and I can go back to the drawing board.

The boring wittgensteinian in me would just relate truth to language games: that thing outside is a "tree" because of its use in our language, just like it would be "derevo" if we were in russia (which I believe means wood as well as tree). it would be absurd to point to it and ask which one it "really" is - tree or derevo?

going outside of language though - if we're able to do this - it does seem that certain languages are superior to others. if i were to go down this route and explain this it would be a different type of truth than the simple one explained above.

even the word "truth" has different meanings across culture and languages. russians distinguish between "pravda" and "instina": that which is apparently true and that which is an unshakeable fact of the universe.
• 3
Statements are relatives to what you assume for real. If you assume nothing, no statement can be true.
• 1.2k
Who is 'feeding a logic sentence to a theory'?

The user of the theory.

What is a 'logic sentence'? - as opposed to a 'non-logic sentence'?

A logic sentence evaluates to true or false. A non-logic sentence may evaluate to something else or to nothing at all. Example:

5+3 --> non-logic sentence, because it evaluates to a number.
It is raining now --> logic sentence, because it evaluates to a boolean value.

And why does it 'need' a two-tuple?

The truth of a sentence can generally not be determined by the theory. The general case is that it must be externally supplied. Other properties of such sentence could be decidable by the theory. For example, the number of characters in a logic sentence is decidable in a sufficiently strong theory. In the general case, the truth value of the sentence is not.

All too often formal systems and semantic systems are conflated, this is unjustifiable.

The term "semantic" itself is already a quite confusing and annoying one in model theory. The term "semantic" is used to indicate truth value while the term "syntactic" is used to indicate provability. This does not mean such semantics have a real-world interpretation or other such meaning. In fact, everything is synctactic in mathematics. Therefore, the meaning of the term "semantic" in mathematics is quite different from what you could otherwise expect. I do not really like that situation but there is nothing that I can do about it! ;-)
• 255
Who is 'feeding a logic sentence to a theory'? — A Seagull
The user of the theory.

What is a 'logic sentence'? - as opposed to a 'non-logic sentence'? — A Seagull
A logic sentence evaluates to true or false. A non-logic sentence may evaluate to something else or to nothing at all. Example:

5+3 --> non-logic sentence, because it evaluates to a number.
It is raining now --> logic sentence, because it evaluates to a boolean value.

Since we have agreed that the truth of a statement is a label and not a property, how can a statement be 'evaluated' to a truth or falsity?

Just because one can apply a statement to a formal system, this does not mean that it is expedient or useful to do so. It seems to me to be a pointless exercise.
• 1.2k
Since we have agreed that the truth of a statement is a label and not a property, how can a statement be 'evaluated' to a truth or falsity?

Actually, you are right. The term 'evaluated' is a bit ambiguous here.

It is a syntactic thing. For example, you can legitimately write:

s1 $\leftrightarrow$ s2

That is legitimate. It means that (s1,s2) is (true,true) or (false,false). The tuple of variables (s1,s2) could, for example, "evaluate" to (true,true). You could also say that it then "resolves" to (true,true).

This is ambiguous because s1 looks like "true" or "false" while $\ulcorner s1 \urcorner$ could look like "Socrates is mortal". The term "evaluate" does not mean in that case that the theory can determine its truth. It just means that it can replace the symbol "s1" by "true".

Properties, on the other hand, are implemented through predicates:

Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common. So, for example, when P is a predicate on X, one might sometimes say P is a property of X.

Even though it would mean exactly the same as the sentence above, the following is not allowed:

truth($\ulcorner s1 \urcorner$) $\leftrightarrow$ truth($\ulcorner s2 \urcorner$)

because truth cannot be defined as a predicate.

In another case, sometimes a sentence is actually even provable in the theory, and then through the computational steps of some proof procedure, the sentence "evaluates" to true.

The term to "evaluate" can be ambiguous, actually; especially in this context ...
• 255
Even though it would mean exactly the same as the sentence above, the following is not allowed:

truth(┌s1┐ ⌜s1⌝\ulcorner s1 \urcorner) ↔ ↔\leftrightarrow truth(┌s2┐ ⌜s2⌝\ulcorner s2 \urcorner)

You are referring to a specific formal system. And you can have many theorems and analyses within that formal system. You can even specify an interpretation of 'truth' within that system. But you cannot apply that system to statements about the real world without going outside of that system to a domain where the rules, axioms and 'truths' of the system do not apply.
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal