Is it possible to form adjective "morally" by deriving it from the noun "moral" (as in "of the story")?

Recently I used the word morally as an adjective formed from the noun moral. The concept I wanted to describe was that some statement is morally correct if you are able to agree with it intuitively but you don't necessarily have evidence or proof for it. So the statement would be a moral of some precise story (meaning evidence or proof).

Then it was pointed out to me that this might not be correct because morally is usually understood as referring to morality. So I looked it up in the dictionary and, indeed, I weren't able to find the meaning I wanted. So my questions are:

1. Is it correct to use word morally in the meaning I referred to above?
2. More generally, how do I know that words I form (e.g. by deriving adjectives from nouns) are correct? Is it enough to look in the dictionary and by not finding the word there, could I conclude that the word (or some special meaning of the word) indeed does not exist?

Solution 1:

You are likely to be misunderstood if you use it in that way.

I think part of the problem is that you are using (or at least defining) the noun moral in an odd way. To me it is not the 'proof' or 'evidence' of a story: it is the lesson that is embodied in the story. (Perhaps you mean that the story is 'proof' or 'evidence' of the moral - that makes more sense, but I would not think of it as either 'proof' or 'evidence', just as 'exemplification').

So I think for you the link to 'moral' is "I don't have a story to back this belief up". I am fairly confident that you will not be understood in that sense unless you give a lot of context.

Solution 2:

Interestingly, your definition

you are able to agree with it intuitively but you don't necessarily have evidence or proof for it

pretty closely describes a jargon usage of morally in mathematics. It’s a comparatively recent usage, I think, and somewhat subfield-specific. It typically describes a statement which isn’t itself true, but succinctly captures the gist of a more complicated statement that is actually true. E.g.:

Giving a grading on A corresponds at least morally to giving a C-action on spec(A).

This is morally correct but technically wrong, as pointed out in Jose's answer and the comments to the question (including yours).

These and many more examples can be found at mathoverflow; Eugenia Cheng has an essay on the topic.