Logical positivism is the following doctrine:

A statement of fact is saying that something is the case, like “It is raining now”.

We all make the mistake of thinking that a statement of fact is either true or false but it isn't that simple. A statement can also be meaningless. It can look meaningful and still be really nonsense. Meaningless means that the statement is not factually significant (page 344, OCR Philosophy of Religion for AS and A2, Matthew Taylor, Editor Jon Mayled, Routledge, Oxon, New York, 2007). The statement may be emotionally significant but not factually significant. If the statement cannot be shown to be correct/incorrect by evidence or it is obviously true/false then it is meaningless.
In other words, logical positivism means that to say a statement makes sense or has meaning is to say it can be proven or at least shown to be likely to be true. Meaning is about what experience and evidence teaches you. So a statement can look meaningful and not be.

The Verification Principle is that a statement is gibberish (even if it seems to make sense verbally) when you don't know how to verify it sufficiently or show it to be wrong.
The Logical Positivists recognised that philosophy is all about the meaning of ideas and concepts and making the meaning clear. They used the Verification Principle as the ultimate and foundational weapon against religious metaphysics and theology which was guilty of making doctrines and claims which could not be supported or verified in principle or falsified. Or in practice either! Thus the only conclusion possible was that the religious were pretending to believe on sensible grounds but were really just talking about ideas they wanted to believe and imagine were true. A theological doctrine speaks of what a person wants to believe not what they think is true.
To sum up, only a statement having a truth value - meaning you can know if it is true or false - has meaning or is about anything. Ayer is the philosopher most associated with propounding logical positivism. "Ayer decided that a proposition is meaningful if it is known how to prove it true or false" (page 195, (Philosophy of Religion for A Level, OCR Edition, Anne Jordan, Neil Lockyer and Edwin Tate, Nelson Thornes Ltd, 1999).

When I say that a statement is meaningless I mean that it is neither true nor false because it is meant to be neither. That is to say it is not intended to be literal. I must verify before I speak in order for what I say to have real meaning. When I say there is a bear at my back door without evidence one way or the other, I am talking nonsense for I cannot mean it literally. When I state something when I can neither verify or disprove it I am just talking about nothing. My statement is absurd for it is worded as if it is meant to be true and it isn’t.

A statement that is meaningful to me will not be meaningful to another who cannot verify or disprove it.

AJ Ayer who founded Logical Positivism taught that if a statement is not verifiable then it is either meaningless or a tautology (truism). A tautology does have meaning (page 345). Ayer never said that all unverified statements are necessarily meaningless (page 345). A tautology is a statement like bachelors are unmarried men which you know is true but which you cannot verify by sense experience. You know it is true for bachelors are unmarried men and vice versa or in other words because the definition of bachelors is unmarried men.
Ayer taught that something could be practically verifiable like when you go to see if there is a statue down the road.
Another way something can be verifiable is if it is verifiable in principle. For example, we can't get to Mars to see if there is life on it but it can be verified in principle if there is life on it or not.
There is also strong verification. This is conclusive. It is like when you find out that somebody you have been warned about really is a thief when you catch them in the act.
There is weak verification when everybody tells you that there is a country called Spain though you have never been there and you know they couldn't all be lying (page 345 ibid).

If you deny that reason and sense-experience verify you finish up denying that there are any meaningful statements at all. Even this denial would be meaningless. But what if you say there is no contradiction in saying that the only meaningful statement is that there are no meaningful statements when you don't mean to include the meaningful statement you are making. Yes but you are saying reason is rubbish and you are reasoning that there are no meaningful statements. That is where the fatal flaw is.

There are few logical positivists today but all agree that its ideas are not hugely important but they do help.  You could argue that most statements in some weak way are meaningless despite being useful.  God and magical statements being the least testable would be top of the list.

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