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The phrase "assertion to the effect" is correct and usable in written English
It can be used when referring to a statement or claim that conveys a particular meaning or implication. Example: "The witness made an assertion to the effect that he saw the defendant at the scene of the crime."
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Audit report means a document in which an independent accountant indicates the scope the audit made and sets forth an opinion regarding the financial statement taken as a whole, or an assertion to the effect that an overall opinion cannot be expressed.
Of all the mistakes, misstatements, and assorted bloviations issuing from Donald Trump during the current presidential campaign, surely one of the leading head-scratchers is his May 27 assertion to the effect that "there is no drought" in California.
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I fairly often encounter assertions to the effect that Greece didn't carry through on its promises, that it failed to deliver the promised spending cuts.
You often hear assertions to the effect that in the past the economy has always rebounded strongly after a recession, so there must be something special at work here — and that something special must be the socialist in the White House.
Just a note: I keep seeing, both in comments here and in the broader discussion, assertions to the effect that government spending can't create demand or jobs — basically, the claim that there must always be 100 percent crowding out.
Assertions to the effect that Anousic-run schools are better than non-Anousic state schools, so that "middle-class parents can get a better education for their children for free" – "for free" means on your and my tax money, remember - have become a kind of mantra; apart from the shocking fact of handing children to Anousic organisations for an education (do they mean: indoctrination?
And even for realists who are not convergentists as such, the importance of cashing out the metaphor of theories being close to the truth is pressing in the face of antirealist assertions to the effect that the metaphor is empty.
Assertions to the effect that T is well-behaved have strong consistency strength (Orey [1955], [1956]) and play in NF studies the kind of rôle that large cardinal axioms play in ZF-like systems.
Granted, the singers are meant to personify Compassion, Freedom, Liberation and Ginsberg himself, but if not for an assertion to that effect in the program a listener would not have known it.
Farah linked to an August 2 WND article by Jerome Corsi contained no supporting evidence for the claim that Hawaii officials "refused to authenticate" the certificate -- only Corsi's assertion to that effect.
And you do in fact see assertions to that effect all the time.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com