- These alleged experts are no help.
- These so-called experts are no help.
Can anyone explain the difference?
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Edited (because OP changed question):
Alleged, as an adjective, means that something was said to have taken place, but it has not been proven. It is often used when reporting about a person or incident that occurred, but the person has not yet been tried and convicted of the crime or the incident has not been verified by authorities. Unfortunately it is frequently used incorrectly. In your first example sentence, alleged means "asserted to be true, often without or before proof."
A usage note from the American Heritage Dictionary entry on using alleged as an adjective says:
An alleged burglar is someone who has been accused of being a burglar but against whom no charges have been proved. An alleged incident is an event that is said to have taken place but has not yet been verified. In their zeal to protect the rights of the accused, newspapers and law enforcement officials sometimes misuse alleged. Someone arrested for murder may be only an alleged murderer, for example, but is a real, not an alleged, suspect in that his or her status as a suspect is not in doubt. Similarly, if the money from a safe is known to have been stolen and not merely mislaid, then we may safely speak of a theft without having to qualify our description with alleged.
So-called doesn't have the legal weight the term alleged has. In the context of your example, it means that what follows is "incorrectly or falsely termed" or that the speaker or writer doubts the truth of the following term. For example: The so-called experts did not know anything that could help me. So-called also has a definition of "commonly called", but I believe it is more often used to cast doubt on the term that follows.
A usage note from the American Heritage Dictionary entry for so-called says:
Quotation marks are not used to set off descriptions that follow expressions such as so-called and self-styled, which themselves relieve the writer of responsibility for the attribution: his so-called foolproof method (not "foolproof method").
Alleged can always be used in place of so-called, though the latter is less formal.
So-called cannot however, always be substituted for alleged for 3 reasons:
1) As noted in the comments, alleged can also be used as a verb. So-called is always an adjective.
2) The aforementioned question of formality.
3) in some specific, formal, legal contexts, alleged carries special meaning with regard to accusations of criminal intent this is true because of the presumption of innocence under which many courts operate.
The way the terms are used in the question, it's not so much the literal definitions of the words. The terms are used in a general derogatory way and in that sense, they have slightly different meanings.
"Alleged" plays on the meaning that the claim of being an "expert" has yet to be proven. In this context, it's meant to cast doubt that the "expert" has any credentials. It doesn't so much cast doubt on whether the person has knowledge but whether you can formally rely it. Alleged is commonly used when the "expert" can "talk the talk", but the detractor lacks the knowledge to refute it. Instead of attacking the argument, they attack the credentials.
"So called" is meant to imply that the "expert" doesn't really know what they're talking about, but unknowledgeable people have just accepted that they are an expert. For example, Hollywood personalities often take up causes for which they have no education, but they eloquently repeat incorrect information they have been fed, as if they personally know something about the subject. A detractor would call them a "so called expert" because people accept them as having expertise for reasons unrelated to merit.