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The phrase "used the same logic" is correct and usable in written English.
You can use it when comparing reasoning or thought processes between different situations or arguments. Example: "In both cases, the researchers used the same logic to arrive at their conclusions."
Exact(11)
And they used the same logic to arrive at their ultimate decision.
Another couple, fresh from a nearby bar and looking a little impatient as a drizzle began to fall, used the same logic but with a twist.
RYAN LIZZA: Not sure about that, but what is so striking when you report out how the two men came to support the individual mandate, is that that they used the same logic and were convinced by the data rather than their ideological prejudices.
Here, we used the same logic, a hypothetical observer who cannot detect a mass without also knowing where that mass is, to work out the proportion of trials on which correct detection combined with incorrect localisation could be due to lucky guesses.
And then I started noticing that sometimes I used the same logic on my body.
(Donald Trump used the same logic for his pick for Education Secretary, Betsy DeVos also a longtime ALEC supporter).
Similar(49)
Back on the street he uses the same logic.
We use the same logic in criminal law.
The Colts will use the same logic to draft Luck.
There, India could use the same logic for pursuing extremists into Pakistani territory.
We can use the same logic for what is happening above the eye-line.
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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