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This narrower sense of logic is related to the influential idea of logical form.
What Peirce meant by "logical critic" is pretty much logic in the ordinary, accepted sense of "logic" from Aristotle's logic to present-day mathematical logic.
"This defies our sense of logic and common sense.
I began to lose sense of logic and proportion.
He would have strong opinions without any sense of logic".
Many philosophers are distinctly uneasy about the wider sense of logic.
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Hence, the conjunction has a model in the sense of logic-without-identity if and only if F has a model in the sense of logic-with-identity; and the completeness of elementary logic (with identity) can thus be inferred.
In this respect, logicians' prospects have been enhanced by the development of a semantical theory of modal logic, both in the narrower sense of modal logic, which is restricted to logical necessity and logical possibility, and in the wider sense, in which all concepts that exhibit similar logical behaviour are included.
The term 'logic' was understood in Bolzano's time, as also by Bolzano himself, not in the narrow sense of formal logic, as it is commonly used nowadays, but rather in the broad sense which includes — besides formal logic — also both epistemology and the philosophy of science.
Salonen conducts it with cool lucidity and a sense of remorseless logic.
In his presentation of United States policy, especially in the area of nuclear strategy, there was a sense of irresistible logic.
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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