I think the positivists came up with the idea that a statement is meaningless if it cannot be proven. But obviously this isn't true: there are a lot of scientific theories that cannot be proven yet, and some theories that may never be able to be proven, so this cannot be true. So is there a better criteria someone came up with or not? Because the claim that a statement is meaningless if it cannot be proven cannot be proven either.
First of all logical positivists criterion for meaningfulness of a propositional statement or hypothesis is called verification principle, thus it's all about empirical verifiability not any form of provability or derivability wished in a complete strong deductive system since provability can be achieved without empirical verification like in logic and math and is regarded as mere semantic language convention (a tool) at best if not meaningless tautology at worst per positivists.
Verificationism, also known as the verification principle or the verifiability criterion of meaning, is the philosophical doctrine which maintains that only statements that are empirically verifiable (i.e. verifiable through the senses) are cognitively meaningful, or else they are truths of logic (tautologies)
As you intuited, this principle itself cannot be empirically verified since many scientific theories and hypotheses cannot be conclusively verified exhaustively, so later Carnap advocated confirmation instead of verification.
Logical positivists within the Vienna Circle recognized quickly that the verifiability criterion was too stringent. Notably, all universal generalizations are empirically unverifiable, such that, under verificationism, vast domains of science and reason, such as scientific hypothesis, would be rendered meaningless... Hahn argued that the verifiability criterion should accede to less-than-conclusive verifiability...
In 1936, Carnap sought a switch from verification to confirmation. Carnap's confirmability criterion (confirmationism) would not require conclusive verification (thus accommodating for universal generalizations) but allow for partial testability to establish "degrees of confirmation" on a probabilistic basis. Carnap never succeeded in formalizing his thesis despite employing abundant logical and mathematical tools for this purpose.
Later Popper's falsificationsim inherited the logical empiricism nature while changed verification or confirmation to his famous falsification.
Karl Popper's The Logic of Scientific Discovery proposed falsificationism as a criterion under which scientific hypothesis would be tenable. Falsificationism would allow hypotheses expressed as universal generalizations, such as "all swans are white", to be provisionally true until falsified by evidence, in contrast to verificationism under which they would be disqualified immediately as meaningless.
Below is the the summary judgement of falsification within philosophy and science circles from the same reference.
Although Karl Popper's falsificationism has been widely criticized by philosophers, Popper is often praised by many scientists. Verificationists, in contrast, have been likened to economists of the 19th century who took circuitous, protracted measures to refuse refutation of their preconceived principles.
If you want to know whether a statement is likely to be meaningful, look at the process that generated the statement. Look at the selection pressure that this process imposes on statements to filter out the "wrong" ones and keep the "right" ones. Is it strong selection pressure, or weak? Is it systematic and impersonal, or informal and social? Is it based in fact or opinion? Do influential parties in the process have a strong reason to lie or exaggerate, and how strongly does the process punish them for doing so?
If the process involves systematic, impersonal, formal, fact-based strong selection of ideas, then statements coming out of that process are more trustworthy. Mathematics, hard sciences, engineering, chess, and sports are areas where if you fail, you fail objectively. Therefore, statements in these fields tend to be more grounded.
If the process involves informal, social, opinion-based, weak selection of ideas, then statements coming out of that process are less trustworthy, and may not be meaningful. Statements made in advertising, political lobbying, entertainment, fashion, and alternative medicine tend to succeed or fail based on whether people subjectively like them, while being fairly detached from hard evidence.
Ultimately, a statement is meaningful if it makes some significant difference to us whether it's true or false, beyond just how we feel about the statement.
- Does it propose some state of affairs in the physical universe? Then it is meaningful, although sometimes it may be hard to verify, and just because it is meaningful does not make it true.
- Does it describe some property of a formal, rigorous system, such as mathematics? Then it is meaningful.
- Can we see how we might translate it into one of the above? Then maybe it is meaningful.
- Is it so vague that we can't even sketch out how it relates to a physical state of affairs or a formal system? Then perhaps it is not meaningful, or needs more development to become meaningful.
We can ask, of a statement P, "What other propositions would hold as a result of P being true, and what would be the impact of those other propositions being true?"
However, in philosophy we may deal with "precursors" to meaning - statements whose meaning we are still uncertain of. And just because we might not have pinned down the precise meaning yet, does not mean there is none. "X is moral" or "X is conscious" come to mind in this category.
I start from the assumption that it is statements and words that can be said to be meaningful or meaningless.
A statement or a word will be meaningful for anyone who has a meaning in mind to associate to it. Thus, meaningfulness is a personal and subjective fact. Different persons will inevitably disagree as to whether particular statements are meaningful or not. Most conspicuously, the Chinese tend to be able to make sense of statements in Chinese only while the Germans tend to be able to make sense of statements in German only.
There is an interesting distinction between statements and words in this respect. We learn the words with our mother tongue and even when learning a foreign language. Mostly, people have zero problem associating a meaning to most words in the language they know.
While meaning often involves some interaction of the subject with the particular concrete objects referred to by words, this is apparently not a necessary condition at least for words best described as abstract. The concept of God for example shows that we are able to arrive at a meaning through abstraction. Essentially, the concept of God is arrived at as a logical conjunction of predicates whose meaning originally came from concrete objects. In this instance, God is essentially abstracted from humans themselves and understood as a sort of super-human.
The concept of God is also a social construct, but it is clear that individuals can work out their own private menagerie of quirky concepts they may or may not then publicise for the benefit of others, and we can think of philosophers like Hegel and Kant in this respect.
Statements are clearly different from words in terms of their meaning. A meaningful statement first requires that the words it is made of be themselves meaningful as per the above. But, this is not sufficient. It is also necessary that there be no contradiction between the different words involved. Thus, statements need to be logical.
This leads to distinguish between two levels of meaningfulness. First, what I would call "literal meaningfulness", which is obtained once all the words in the statement are meaningful. And then what I would call "semantic meaningfulness" when there is no contradiction between the words involved in the statement. Both levels of meaningfulness are necessary to achieve statement meaningfulness.
Thus, there is no hard connection between meaningfulness and the reality of the world, but there is nonetheless a loose connection with our experience of the world, both as individuals and as a collective. Individuals can make sense of their own personal experience, but the connection to a linguistic community will dramatically increase the number and range of the abstraction we can understand as individuals.
Logical positivism should be seen as essentially an ideological moment in philosophy, a logical absurdity, historically dated, at a time where the conflict between idealism and materialism climaxed.
Common sense is good enough to specify what meaningfulness means to most people. And only common sense is good enough, by logical necessity. And it works. Mankind has survived thousands of years on this basis. Logical positivism should have been debunked as illogical from the start.
This also shows that people can make statements which are illogical, including intellectuals. The reason is that our philosophical views can be quite sprawling and complex, to the point where the absurdity of them is lost to us. We continue to make claims which are absurd until it is pointed out to us that they are absurd. We can contradict ourselves. This implies that our theories are collections of statements, and that at any one time we can only entertain some of these statements, which leaves room for theories or views involving statements that will be individually meaningful but collectively meaningless.