2

Though the box and diamond are the more common representations of the strong and weak modal operators in modal logic, “L” and “M” are also used. I suspect that those letters were chosen because they each refer to English or German words, but I can't find any reference that mentions which ones.

4

From the Polish logician and philosopher Jan Łukasiewicz who invented the Polish notation for logic (named after his nationality) :

M ϕ for możliwość : possibility

L ϕ for konieczność : necessity [unfortunately: K was alredy used for koniunkcja : conjunction].

For M, see Jan Lukasiewicz, Selected works, North-Holland (1970), Philosophical remarks on many-valued systems of propositional logic (Polish ed., 1920), page 154:

it is possible that p, in symbols Mp, [where] "M" corresponds to the words "it is possible that".

Łukasiewicz used "L" for "it is necessary that" in: Aristotle Syllogistic From the Standpoint of Modern Formal Logic (1st ed, 1951).

But an earlier use is in Robert Feys, Les systèmes formalisés des modalités aristotéliciennes, Revue Philosophique de Louvain, Volume 48 (1950).

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