Truth table for incrementing and decrementing 4-bit numbers
The second F column are decremented until row six; it is a mistake in the table. And another one in the last row of the second F column, they forgot to represent a bit


  [I]f we decrement 0000, if it's signed, we get 1111, but why do we
  have a P = 1 there?


Because there is still a carry bit there, the bits are [C][3][2][1][0] where C is the carry and the rest are your four bits. A -1 is represented in signed binary as all 1's, including the carry bit, 11111. You can see P is high anytime the four bits rollover, so P is like the carry bit, a pseudo fifth bit. Just move the P column in front of its F column and it'll be obvious.