Is there a name for the relation between a proposition and the proposition formed after applying the diamond modal operator?

I don't know if there is a name for it but, since a negative proposition is the negation of another proposition for e.g. the proposition that "it is not the case that it is sunny" is the negation of "it is sunny". Does that mean that the proposition "It is possible that it is sunny" stands to the proposition "It is sunny" in some sort of relationship, if so does it have a name?

• Some authors occasionally call it "possibilification", see e.g. von Wright, A New system of modal logic. But generally the word is used loosely for introducing possibility somehow somewhere. Apr 14 at 1:06
• This does not exactly answer your question, but adding a box or diamond to a proposition is sometimes described as giving it a modal force. Apr 14 at 4:46

Is there a relation? Well, diamond and square certainly are unitary operators that transform a proposition. Is it named and does the name have currency? That is, do they have a grammatical presence as nouns and verbs like 'negation' and 'to negate', what one would call a lexical definition, a description of current usage? I don't think so. In logical modality

the modal formula ◊P can be read as "possibly P" while ◻P can be read as "necessarily P"

If you're looking for a parallel grammatical construction to "X is the negation of Y, then you seem to be out of luck. Of course, no one can stop your from coining language and persuading others to use it, so you might go with:

For necessity, X is the necessitation of Y (since one speaks of the Necessitation Rule)

It's a little akward, but I think it's understandable. I guess like "~ negates a proposition" you might say "◻ necessitates a proposition".

For possibility, X is the possibilization of Y

That's more awkward and there doesn't seem to be any use of the term at all even if it comports with being parallel gramatically. Could you say "◊ possibilizes a proposition"? Again, you could. There's actually an entry for it in the urban dictionary which purports itself to be a catalog of neologisms. Google Ngram viewer also shows a climb in usage. If you Google "possibilize a proposition", it returns at least one paper that uses the language.

The moral of the story here is that this would, as far as I can tell with my limited resources, be idiosyncratic, non-canonical language. As such, anyone using these constructions would be best off to initially use scare quotes and perhaps have an explicit mention of the effort to move such language into convention.

• Thanks for your help JD! I found this book about propositions in the section logic cambridge.org/core/books/… which you might like to read Apr 13 at 21:18