I heard this piano can operate half pedal, but how do I tell that my piano can give a 'half pedal' sound?

I've tried to distinguish it, but my pedal has recently gone squeaky (is there a way to fix this?. I hope it hasn't destroyed the ability to 'half pedal', because I can't see any difference in sound depending on the depth of the pedal pressed. From what I've tried, I only get an 'on' and 'off' response from the pedal. Have I broken it?
Working on:

Prokofiev - Sonata No. 3
Kapustin - Variations Op. 41
Chopin - Ballade No. 4