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The phrase "Following this logic" is a perfectly correct and usable phrase in written English
You can use it to introduce an idea that follows logically from a previously established idea. For example: "The government's proposed tax plan will disadvantage low-income earners and benefit higher-income earners. Following this logic, it will likely exacerbate economic inequality in the country."
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Following this logic, we would all feel better if we were insulted, even the police.
Following this logic, Broyard embarked upon an impersonation: "I never act sick.
Following this logic, I have a 55 percent chance of getting the Nobel".
Following this logic to its conclusion, Derbyshire has declared that babies cannot feel pain until they are 1 year old.
Following this logic, Bernanke and two colleagues proposed that the Fed become more transparent and publicize an explicit inflation target.
Following this logic to its conclusion would require a circuit court to order partial manual recounts upon the mere filing of a contest.
Following this logic, if I saw someone in a wheelchair who I thought lacked self-esteem, I could insult them, to make them stronger.
Following this logic, Bill O'Reilly could secretly be a Manchurian Candidate for Hamas because of his prolific knowledge of Arabic, as gleaned from his usage of "loofah" and "falafel" when allegedly attempted to sexually harass a female producer.
Following this logic, Banerji ponders if fashion magazines, designers, models and fashion shows that encourage fast fashion should also be added to the list of those responsible for improving wages.
Following this logic, what's needed is a state that bankrolls scientific research at midcentury cold war levels – without the comparatively high tax rates and social spending that accompanied it.
Following this logic, Antarctica immediately comes to mind as a promising candidate.
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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