When I first began writing papers I tried to describe using prose. I found that readers were confused and there was much more ambuigity than I anticipated. By using mathematical notation, it becomes a lot less likely that you are misinterpetted and often takes less space. It's kind of like the difference between providing code versus a written description of the algorithm.

The problem is assuming that the reader is already familiar with the notation, and consequently not explaining it.

If it is a mathematical paper it is targeted at other mathematicans eho should be able to understqnd the notation.

My math professors would teach maths under the presumption we knew mathematical notation. I felt I got thrown in the deep in during first-year calculus units but I also now feel strangely adept at reading maths so it had a positive effect.

the other problem is assuming the reader knows nothing, going too deep into the foundations supporting the conclusion.

mathematical knowledge is like scaffolding. a reader has to gradually build up and buttress the lower layers before attempting to move up. skipping steps does no one any favours.

Natural language can be extremely ambiguous though, and while the ambiguity may be resolved easily from context, someone in OP's situation doesn't have that context. The precision mathematical notation can afford (more readily than, say, English) actually helps. But of course, only of one knows the meaning...

I'm in a somewhat similar position to OP, it's extremely rare that I'd find a whole paper accessible, the best way I know is just to get gradually more specific, from a broad undergraduate textbook on 'logic' or whatever other similarly broad area, and hone in on the more specific area of interest. Notation will be introduced, and by the time the last 5% is, 80% will be second nature, and the remaining 15% will be familiar and 'let's see.. I think it was used in this book..'

Can you give an example?