Array-focused programming language
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No complicated loop processing to apply a function to a array of arrays. Functions are defined in a way that they will typically operate the same way on any number of array dimensions. This, along with the clear syntax, leads to very compact code that can be comprehended in a single line, rather than spread out over many pages.
Pro Clear syntax
There is no operator precedence to memorize, as everything is evaluated right-to-left. E.g., in APL 3*10+3 = 39. You do have to type in some otherwise unusual characters, such as ↓ and ∊, but those are easy enough to pick up -- and they have the advantage of being easily remembered once understood, as they often have some connection to common mathematical symbols.
You can seriously implement Conway's Game of Life in one line. There's a reason we do algebra with symbols instead of story problems. APL is good as a language of thought, since you can hold entire algorithms in your head at once.
APL symbols are only used by APL. You have to learn how to type them and how to read them. It doesn't work well with standard text editors , version control systems, search engines, or web forums. This makes it difficult for a beginner to find help.
Con Does not prepare you for most of the practical programming languages of today
While APL does have a strong use in certain areas (mostly mathematically intensive applications), it is a Domain-Specific language. That along with the fact that its syntax is not similar to C-like or other common syntax forms means that learning APL and expecting it to help you with learning other languages is like learning Calculus and expecting it to make English easier.
Con Write-only language
Maybe you can learn to read it with experience. And an interpreter. Reading APL is like reading a college math book. You might have to study a single line for fifteen minutes to understand what it's doing. And that's if you're an expert at APL. This also applies if you wrote it yourself more than a month ago. hopefully you have comments.
Con Flawless diamond
You can't extend the language itself. (J does this better.) Of course, what's built in is quite powerful.