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The phrase "a common intelligence" is correct and usable in written English.
It can be used to refer to a shared or collective understanding or cognitive ability among a group of people or entities.
Example: "In collaborative projects, a common intelligence among team members can lead to more innovative solutions."
Alternatives: "shared intelligence" or "collective intelligence".
Exact(3)
Yet a common intelligence runs through it all.
Looking at the techie crowd in a "not cynical way," she saw a "common intelligence" around her.
Typical examples include variables defined as various test scores of individuals, as such scores are thought to be related to a common "intelligence" factor.
Similar(57)
There are some in Brussels who want the EU to develop a common intelligence-gathering service, as well as a "European Army" (see below).
It was to establish a "joint common intelligence picture" of the global threat of improvised explosive devices, or IEDs.
"The third reason is the agencies"—a common term for Pakistan's intelligence services.
Seeing their interaction up close, you come to realize how much Stanley and Blanche actually have in common: intelligence and wit and a healthy appetite for sex and for drink.
In 1926, the Supreme Court in Connally v. General Construction Company expressed the principle as a matter of constitutional due process, saying, "A statute which either forbids or requires the doing of an act in terms so vague that men of common intelligence must necessarily guess at its meaning and differ as to its application violates the first essential of due process of law".
And, finally, an oath may not be so vague that "men of common intelligence must necessarily guess at its meaning and differ as to its application, (because such an oath) violates the first essential of due process of law". Cramp v. Board of Public Instruction, 368 U.S., at 287, 82 S.Ct., at 280.
The common intelligence tests include the various Wechsler Intelligence tests; Peabody Picture Vocabulary Test; Bender Visual Motor Gestalt Test; Thematic Apperception Test; Draw-A-Person; Stanford Binet; Slossen; Otis Self-Administering Test of Mental Ability.
It is vague because 'men of common intelligence (must) speculate at their peril on its meaning.' Whitehill v. Elkins, 389 U.S. 54, 59, 88 S.Ct.
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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