There are two common misconceptions about logical positivism. 1. the positivists, like New Atheists, set out to prove that non scientific theories like religion and metaphysics, are false and only science can be true, and 2. Logical positivism fails at its own criterion of meaningfulness.
(1) has got LP backwards. LP allows religions and metaphysical systems as not just true, but necessarily true, while it's the scientific theories that are possibly false, not necessarily true at all.
That's the strange beauty of LP. The difference LP draws between theories is not between the true or the false, but "meaningful" and "not meaningful", using a peculiar definition of "meaning". LP doesn't touch on any other aspect or virtue of non verifiable theories, their aesthetic value, their mystery or charm or inspirational insight, their moral or social value. Just their meaning, where "meaning" in LP is used as a theoretical jargon for "phenomenal informational impact -- how the world of phenomena and events are and are not." The challenge of unpacking their use of "meaning" such that it isn't circular is the reason for (2).
(2) is flatly false. Apply LP to LP and it verifies. (2) also assumes that LP is a theory and not either a definition or description or an axiomatic system or merely a kind of practical advice like Popper's demarcation.
The popular misconception has it that if LP is a theory it should apply to itself, but LP can't itself be verified. I think people who say this must not have tried to apply LP to LP, maybe because "it doesn't apply to itself" is self-reflexive and so clever-sounding that they don't bother to experiment to verify whether the clever is also true. Whatever their reason for why they don't apply it to itself, we can apply it here and now:
LP says that theories are meaningful (in the sense of "tells us how the world is or is not", "what's in the phenomenal world and what isn't") if their statements and predictions about the world are verifiable. Is this assertion verifiable? Sure. God is not a directly verifiable object. It's not meaningful in the LP sense of telling us how the phenomenal world is. Religion is meaningless in that sense. Are the bones of dinosaurs verifiable? Yes. Archeology is meaningful in the LP sense of telling us about the phenomenal world, in this case where to find dinosaur bones. LP is verified by both these cases. LP is meaningful in the LP sense of meaningful, telling us how the world is and is not.
This is all crude and simplistic, but it shows how to apply LP to LP. Let's try again with something more substantial.
Creationism cannot predict the fossil record. There's no book of trilobites and dinosaurs in scriptures, and scriptures don't need them. It doesn't tell us how the world is, phenomenally. That's an unverifiable theory, and notice, it's a necessarily true theory -- no empirical evidence can prove it false. (The New Atheist will complain about it's internal contradictions, but those are logical disproofs, not evidential disconfirmations, and LP is concerned only with evidence. That's a huge difference.) Archeology does predict the fossil record. Treatises on trilobites and dinosaurs belong to science. Biology and archeology tell us what we will and will not find when we dig into the earth and find bones. Are they true? Well, not necessarily. They are the most likely theories of the topic given the evidence currently available. True? Who knows what we'll discover tomorrow? And that's one difference between the religious or metaphysical theories and the scientific theories -- according to LP. What we discover tomorrow could trash our current science. It will never trash the religious or metaphysical theories. Are these differences between creationism and archeology verifiable? Yes, the difference seems to be verified. That difference is the LP criterion, the LP "theory".
You may have already noticed that creationism is strictly ambiguous over verificationism, since it doesn't predict, so you could say verificationism can't apply to creationism. IOW, the problem is not that verificationism doesn't apply to verificationism -- it does -- it's that verificationism doesn't apply to the necessarily true and meaningless (in the LP sense) theories.
Right at the outset, it's important to know that Karl Popper identified an essential flaw in LP. Verifiability runs into the inductive fallacy. Verifying a theory supports the theory but can't prove an explanatory theory, that is, it can't prove a theory that predicts the possible (as compared with post hoc descriptions of a closed set of observations). Popper replaced verificationism with falsificationism -- that a "meaningful" explanation must identify the conditions under which it would be false. The consequence is that scientific theories are never provably true, they are just the ones that haven't yet been proven false. There are weaknesses in falsifiability too, but it was an important advance over LP's primitive verificationism. LP was using its confirmation bias to confirm its theory of confirmationism. It's a typically human failure to use clear Bayesian reasoning.
The weakness of LP is not alone its confirmationism. It also defined "meaningfulness" in terms of verificationism and vice versa. Their criterion of science was circular. That's because LP's use of "meaning" is not a theory at all. It's a definition or axiom or maybe a kind of practical advice. Definitions generally don't apply to themselves. "Blue" is not blue, and it would be irrelevant even if it were blue. "Blue" can be used as a kind of practical advice: you can view these objects as having this common property of being blue. "Look at these -- they all have a kind of similar hue. For convenience let's call them 'blue' so they're 'blue-ish'." That's all there is to a definition. And if you can provide for all or many possible additional individuals ("that thing there should be included in the set") all the better. Most definitions don't apply to themselves, though some do: the set of definitions that define themselves, for example. Not very practical.
Another weakness of LP, most obvious in Wittgenstein's Tractatus, was the belief that there could be atomic facts, indivisible facts independent of any other facts or ideas or theories. But facts are partly theoretical -- they are conjectural, dependent on the likelihood of the theory -- and like theory, their value is their falsifiability. There's a frequency theory of hues that classes navy blue with sky blue although Russians distinguish them as distinct hues. When I was a child I refused to wear anything navy blue so repulsive to me was this color. My favorite color was sky blue. Which fact is relevant -- that navy blue and sky blue are opposite ends of a single color or that they are two colors? Depends on the theory and its purpose. (See "true but wrong" on this blog.) "Whales are giant fish" belongs to a biological taxonomy that sufficed for the deity in the Book of Job, and that book makes effective, memorable use of it. I have no problem with the "whales are giant fish" theory. It's just not useful for science, a predictive theory of what's out there in the phenomenal world and how it got there. People can wear different hats, you know.
The lack of conjectural theory in LP led to Wittgenstein's private language argument, a kind of reductio ad absurdum of his verificationism, applying verificationism to the mind. Can experiential states be verified, he asks. Well, on a verificationsit model of truth, no. Rather than seeing this as a disproof of the verificationist premise, and rather than seek a better conjecture, he oddly, and perversely, embraced the absurd result that the experiential is meaningless, and advocated for a kind of behaviorism that prevailed in philosophy and the sciences until Chomsky in 1956 demonstrated that such a behavioral program couldn't account for the productivity and inventiveness -- the creativeness -- of speech, that the mind played a necessary role in behavior. Chomsky's program was a better conjecture that led to a better understanding of the mind.
The common view holds that Wittgenstein's later views is a rejection of his earlier logical positivism. I think that's another misconception. His later views are, I think, best understood as pushing his earlier views to their extreme and often counterintuitive and even absurd logical consequences, an insistence on biting the philosophical bullets one after another. It's a wonder he had any teeth left.
So much for the strange beauty and the misconceptions.