On a mandolin, a full, four-fingered, closed-form D or G chord has one finger on the second fret and one on the seventh. Point this out to a guitar player and he'll say, "Whoa!" The frets on a mandolin are close.
Beginning mandolin players, intimidated by this, will stick to two-fingered chords with open strings. It's the mandolin-player analogue of guitar players who avoid barre chords.
There's second another useful approach to use: two-fingered closed chords -- two fingered chords where you just play the two strings you're fingering.
I think this sounds best on the middle two strings, but each pair has its own characteristic sound. By playing two-fingered chords, you get a better sense of what the different strings sound like.
I don't have trouble playing four-finger chords, but I use two-string chords to vary the sound.
It also gets you pick control. If you turn your mandolin to face you, you'll see that it's going to be impossible to hit just two (pairs of) strings without hitting the others, especially if it's the middle two strings and you're not looking. It's a nice surprise to discover that you can do it pretty easily.
Plus, unlike chords with open strings, you can chunk a two-finger chord. An open string rings until the vibrations die off, but after you play a two-finger chord, if you let up the pressure on your left hand, but keep the fingers on the strings, the sound just stops.
It helps you tease apart the pieces of your chords, too. Melodies are combinations of scales and arpeggios (notes in a chord). Once you're used to putting down pairs of fingers for chords, you'll see melodies fall into place more easily.
And, if you're just starting to learn four-finger chords, once your left hand knows where two of the fingers go, they'll serve as anchors to help you place the other two, on the fly.
Finally, they are double stops. When you're playing a note in a melody, playing it as part of a two-fingered chord lets you play a harmony note just above or below it.
Two-fingered chords -- they're not just for amputees.
