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This is valid in this framework.
This is valid for everywhere in North America.
This is valid here, too.
And this is valid.
On some level, this is valid.
Suppose this is not so.
Suppose this is so.
On the other-hand, supposing that (2.6) is valid, by (2.11) and (1.2) (for, it follows. If, then (2.7) is naturally valid; if, setting (2.14).
Yet for no reading of "if" which is stronger than the truth-functional reading is CP valid — at least this is so if we treat "&" and "~" in the classical way and accept the validity of the inference: (I) ~(A&~B); A; therefore B. Suppose CP is valid for some interpretation of "If A, B".
Open image in new window Suppose (22) is valid with j1 ≠ 0, then, in particular, - 2 p arg c ~ ′ ( ζ ~ 1 - ) - ( ψ 1 + 2 π j 1 ) < π max { p, q } Open image in new window.
Suppose that (3.1) is valid.
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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